Quick Takes: Calculators in the Classroom

Using calculators in the classroom, especially in elementary school, has become a divisive issue. While some people think that children should be taught to use calculators from the time they enter school, others fear that learning to use calculators will rob children of the ability to do mental calculations. Much of the contentious atmosphere surrounding this issue arises from seeing calculator use as an either-or situation. Though important, calculators are only one method of computing. Both sides of this debate have some merit; so let's look at some of the points raised.

The National Council of Teachers of Mathematics (NCTM) maintains that each child should learn to solve problems by using a hand-held calculator and mental and written calculations. The NCTM clearly asserts that children need to master all these methods if they are to understand and use mathematics. Why does the NCTM feel so strongly about supporting young children's use of calculators?

Why would children need to use calculators?In concluding that children of all ages should be proficient with calculators, the NCTM did not intend to rule out other ways of solving problems. Children do need to learn to figure in their heads and to do paper and pencil work, but these methods alone will not meet contemporary needs.

Aversion to using calculators in schools contrasts with their general acceptance in the work place and the daily life of adults. Even a part-time job at the corner fast-food chain requires the use of a calculator in some form. If schools do not teach students to use these devices from an early age, the rising generation will lack necessary work skills. Calculators are one tool almost every employer expects employees to use. Calculators are ubiquitous in the work world and as important for employees as voice mail and word processing.

Word processing is an analogous skill. We do not require that students check all spelling with a dictionary rather than by the computer's spell check program. Instead, we expect students to be able to gauge the reasonableness of a spell check message using their own experiences drawn from reading, writing, and dictionary use. In the same way, young students can learn to compare the calculator's messages to the reasonable answers they have learned to expect from their evolving understanding of arithmetic. The issue is not should students use calculators in the classroom but how calculators should be used.

How could calculator use benefit students?When students do not have to worry about computation mistakes, they can focus on reasoning and problem solving. Teachers can help students see patterns, check estimates against reality, and solve complex problems, like those encountered in daily life, through the structured use of calculators. Children introduced to the calculator when they are young will find it easy and effective to use. Calculators should be used in the classroom for many reasons:

Calculators help students at all levels learn mathematically complicated material.

Even young children can use calculators to focus on the ideas behind computation rather than on the act of calculating.

Rather than hampering mathematical ability, calculator use can actually improve student achievement in mathematics, according to research.

Both the SAT and ACT now allow students to use calculators during testing, as do many state-level exams. Students who have not been comfortable with calculators from a young age may be at a disadvantage on these tests. In the recent Third International Mathematics and Science Study (TIMSS) fourth and eight graders who used calculators almost every day performed at higher levels than did those who never used one or only used it once or twice a month.

What does research show about classroom calculator use?Researchers have studied classroom calculator use for several decades and in many countries. Research in this area began to take off in the late 1970s. In a 1986 study in the Journal for Research in Mathematics Education, Hembree and Dessart analyzed 79 studies of calculator use and found the following:

Children who use calculators on tests have higher scores in both basic computation skills and problem solving.

Students who use calculators within a mix of instructional styles do not lose their paper and pencil skills.

Calculator use in the classroom improves the paper and pencil skills of students regardless of their ability levels.

Those who use calculators in class have better attitudes toward mathematics than children who do not use them.

How are calculators used effectively in the classroom?Too often schools approach calculator use casually and uncritically. While people understand the need for a strategic plan to incorporate computers into the curriculum, often they do not see a similar need for a systematic calculator strategy. All concerned must work together to figure out where the calculator fits into the overall goals of the curriculum and students' needs. Some strategies for effective classroom use follow:

Have students decide on the reasonableness of calculator answers by estimating before they do the calculation.

Use questions and discussion to help students think actively about the processes used to arrive at answers.

Incorporate open-ended problems or projects with several possible solutions (or no solutions) into classroom instruction.

Mix in problems that are easier to solve by hand or that become unwieldy on the calculator so students will become discriminating in calculator use.

Teach mathematics as an integrated discipline rather than as disconnected processes.

A frequently overlooked skill is knowing when to use a calculator and when other methods are more efficient. Simple exercises can show students that mental and written computations are often more useful than working with a calculator. For example, in one fourth-grade contest students with paper and pencil were to solve 20 simple subtraction problems working against an adult with a calculator. Since each problem could be solved mentally, the students did better than the adult, who had to key in each number and operation.

Students also need to learn to think about what can cause unreasonable answers on a calculator. Did you put numbers in at the wrong place, fail to put in a decimal, hold a key down long enough for a number to be entered twice?

With guided calculator use even young children can begin to see differences and relationships among arithmetic procedures. The difference between memorizing 5 x 5 = 25 and hitting 5+ on the keypad five times is that the latter activity can lead to discussions about connections between addition and multiplication and make the underlying patterns of multiplication apparent. With teacher guidance young children can become aware of larger numbers or even negative numbers at an earlier age than they have in the past.

In secondary school, calculators can help students develop their understanding of algebra and other advanced mathematics. Students will have an easier time learning advanced mathematical procedures if the foundation for complex calculator use is laid in the elementary grades.

Time for exploration is needed for effective calculator use. If students use calculators to figure out the relationship between the circumferences and diameters of many different round objects, they can get beyond problems of correct division and watch the concept of pi emerge. Doing such work adequately, however, requires that the teacher make the time to allow students to work with their own material until the concept is discovered and internalized. Once such time is set aside the calculator will repay with more time available for searching, developing hypotheses, and testing them. Without anxiety over basic mathematical processes, children will be able to concentrate on the applications and meanings of the world of numbers.

I N   T H E   F A C U L T Y   L O U N G E

"Hey Chris, I heard some good stuff about you last night."

"Well, that's unusual Pat. Who from?"

"Justin Franks. Remember him? I think he graduated about three years ago. He was a good kid but could have made higher grades then he did. He was on the track team. About this high. Blonde."

"Oh yeah, Justin was more interested in video games and his garage band than in mathematics as I recall. What is he up to now?"

"He's an assistant manager at Electronic Mart, and he said that because of your class he made assistant manager faster than anyone else who was hired when he was."

"Well gosh, I always knew my teaching was powerful, but I had no idea it could reach that far! What did he learn from me that helped him become assistant manager?"

"It was the work you did with calculators. Because he learned to use the functions and to key properly he was not stymied when he was introduced to the inventorying cash registers. Then, when he had to go to the next step and help make out purchase plans, he had a good grasp of how to manipulate the figures on the machine and set up different buying plans. His boss was so impressed that he sent Justin to special training in Little Rock. Then two months later, he was made assistant manager. When he finishes at the community college, he can get a company scholarship and go to state and get a marketing degree."

"Well, that's great. I always work my students really hard with scientific and graphing calculators. I try to find interesting problems for them. Most of them were based on stuff from business. Maybe that's what stuck with Justin."

"From what he said, I think it was mainly just your approach to the calculator. He even quoted you, 'The calculator is a tool to help people with their tasks and their thinking but never a substitute for thinking.' So do you think you could share some of these problems and methods with me? I've never let students use a calculator in my classes; maybe I'll start."

For more information, try the TIMSS web page: http://nces.ed.gov/timss/

Won't schools have to make sure every student has a calculator and won't that be expensive?The calculator is cheaper than other technological innovations. A national drive to ensure that each student has his or her own calculator may make more sense, economically and educationally, than a national push for more classroom computers. In most schools computers will continue to be shared resources but calculators could become individual resources used in class and at home. Three types of calculators are generally used in schools:

Arithmetic calculators cost less than $10 and have a numeric keypad with the four basic arithmetic operations, although some may also have percentage and square root keys. A single line of characters displays up to eight digits.

Scientific calculators have a broader range of functions and cost around $20. Some statistical abilities, a more extensive keypad with more than one function for certain keys, and scientific and engineering notation are common.

Graphing calculators have an extensive range of operations, a larger screen, more characters on the line, and the ability to move between displays and use alphabetic characters. They can graph data and symbolic expressions, cost close to $100, and are generally only appropriate in the higher grades.

About Quicktakes

Quick Takes was a publication of the Eisenhower Southwest Consortium for the Improvement of Mathematics and Science Teaching (SCIMAST) project, sponsored by the U.S. Department of Education under grant number R168R50027-95. The content herein does not necessarily reflect the views of the department or any other agency of the U.S. government or any other source. Available in alternative formats.

The Eisenhower SCIMAST project was located in the Southwest Educational Development Laboratory (SEDL) at 211 East Seventh Street, Austin, Texas 78701; (512)476-6861/(800)201-7435. SEDL is an Equal Employment Opportunity/Affirmative Action Employer and is committed to affording equal employment opportunities to all individuals in all employment matters.

Calculators are now in use in mathematics classrooms everywhere. They are used often in elementary classes, regularly in Junior High mathematics courses, and in every Senior High math and Science class.This page will attempt to explain some of the reasons why calculator use has become such an integral part of the mathematics classroom, and what benefits and drawbacks exist for students as a result.

First, you need to be aware that calculator use is built into the courses which are offered to students. Teachers are expected to provide opportunities for students to use them, and students must become proficient in their use. High School mathematics courses are designed around the use of a scientific/graphing calculator. As a result, students in lower grades must know how to use a calculator properly. This is not to say, however, that calculators are to be used for all operations.

Junior High School:

Students in grades 7, 8, and 9 are exposed to many new ideas in mathematics, including the topics of integer and fraction operations, and measurement formulas. We'll use these topics to help explain where calculators are helpful, and when they should not be used.

One important benefit of using a calculator in the Jr. High mathematics classroom is that it lets students cover more material during the year. And there is more material to cover ... students in these grades learn far more mathematics than their parents or grandparents did at an equivalent level.For example, the notion that formulas can be used to evaluate an area or volume requires that students perform many calculations, some of which would be quite lengthy if done with pencil and paper.For instance, a typical calculation at the grade 8 or 9 level might look something like this:

3.1416x2.82x11.6when working out the volume of a cylinder. Multiplying decimals is a skill learned in grades 5 and 6, and is one that (hopefully) all Jr. High students can do, even a three-step problem like this one. However, the process is time-

consuming, especially when you remember that errors will be made occasionally, so every step must be checked.The rationale for using calculators to do these calculations is that the considerable amount of time saved using a calculator will enable the teacher and students to explore many more formulas, relationships between formulas, and real problems, ... that they wouldn't have had time for otherwise.In other words, if doing a cylinder volume problem calculation by hand takes five minutes, but using a calculator requires only one minute, students will have more time to make sure they understand how the formula works, and more time to explore formulas for spheres, cones, and pyramids! More mathematics can be learned, because the calculator is doing the time-consuming drudgery of the arithmetic!This premise holds true in all math courses, right through to the end of grade 12. In fact, the amount of material in those courses is such that it is assumed that students will be using calculators to do the multiplying and dividing ... the courses could never be completed if they weren't!

This immediately suggests to some that students may be losing skills. Surely if students are no longer doing multiplication or division with big numbers after grade 6, they will lose these skills.

In fact, this is true. Students are losing skills.

Here's how the argument goes. It's a trade-off. If you assume that calculators will always be available for our use, then let's use them a lot. Once you understand the principles of long division, for instance, in grades 5/6, and can do long division by hand, it's no longer necessary to actually do it. In Jr. High, all long division (eg: 875.4 ÷ 7.9) is done on a calculator. The extraction of square roots on paper isn't even taught any more ... the square root of 678.92 is done on a calculator.By freeing students from these time-consuming arithmetical calculations, more time in the classroom can be devoted to learning more mathematics. Lots more mathematics.Does a student need to practice long division after grade 6, in order to be ready for the 'real world'? Of course not. We all use a calculator. We'll always be able to use a calculator for those types of calculations, so it's a skill we don't need to be good at! And it doesn't matter! These are skills that aren't needed any more.

A parent of a Pure Math 20 student might be faster at multiplying 6.78 by 11.134 by hand than their son or daughter. That's because the parent spent a lot of their years in school practising arithmetic operations, and not so much time learning mathematics! Today's student knows far more mathematics than the parent did even after grade 12, and can solve far more complex problems! And that's what it's all about.

However, this is not to say that some skills are not still useful, and the distinction is very important. Both 'real world' problems and Sr. High school math courses require students to be able to be good at 'mental math'. You

can't expect to pull out your calculator every time you need to multiply 6 times 8, and if you have to use a calculator in grade 10 math to do problems like -6 + 8, you won't finish the tests!In Jr. High math, we still require students to be proficient at some basic skills; operations that they can do in their head without a calculator. These include, for example, the times tables, integer operations, simple powers, and the order of operations. Students are not allowed to use a calculator when practicing these skills.In addition, some paper and pencil skills are important because they carry over into algebra, where most calculators can't (yet) operate. So in grades 7 through 9, students are expected to be able to do fraction operations by hand, and may not use programmed formulas to solve problems.

Students in grade 9 who are not proficient in these mental calculations don't often succeed in the advanced level Sr. High mathematics courses.

Senior High School:

The past twenty years has seen huge changes to the Sr. High mathematics curriculum, most especially in the area of relations and functions. Where students in Math 10 and 20 previously studied just the line and the parabola, they now are exposed to a wide range of functions, including graphs of the exponential, sinusoidal, absolute value, reciprocal, and cubic functions. This broadening of the curriculum is a direct result of being able to use scientific and graphing calculators in place of the cumbersome calculations that were formerly necessary, involving paper and pencil graph plotting, as well as logarithmic and trigonometric tables. In fact, the graphing calculator makes possible the study of relationships between functions that could never be done in high school before, due to the immense amount of time required to plot relations by hand. As mentioned already, present day math students are exposed to much more information than their parents were. This is a direct result of using calculators in the classroom.

Nevertheless, it is important that students use their calculators properly. While using a calculator's graphing capability to study the similarities between functions can broaden a student's grasp of relationships in mathematics, relying on that calculator to to basic operations like times tables facts and integer operations has exactly the opposite effect. Students are slowed in their work, often making assignments take much longer to do than they should. As well, the concepts which should be evident from appropriate use of the calculator, become lost in a maze of unnecessary calculator operations.Students who are enrolled in advanced Sr. High math courses who are not able to perform basic calculations successfully without a calculator very seldom do well. Often they fail.

It is for this reason that the Jr. High mathematics teacher must carefully choose when to allow the use of a scientific calculator, in order to allow students to become familiar with its operation, ... and when to not allow its

use, so that students gain a set of skills that will free them to use the calculator to its best advantage.

Descriptons, definitons, and HistoryThe first four-function and scientific handheld calculators appeared in the early 1970s. Their appearance gave rise to simplistic yes-no articles in newspapers and magazines. Yet those opinions had little effect because cost, fragility, and short battery life limited calculator use.

By the early 1980s, those deterrents lost force because of the appearance of solar-powered, hard-case, four-function and scientific calculators costing less than $10 and $15, respectively. And so the new generation of calculators began to be used. In 1985, the first user-friendly calculators appeared that could graph functions. Like their simpler counterparts, these calculators were too expensive to be widely adopted when they first appeared, but today many high schools require them for all or virtually all their students. The use of these calculators in secondary school has not generated as much controversy as the use of simpler calculators in elementary school, and they are required on many college-entrance tests. More recently, user-friendly calculators have appeared that can solve literal algebraic equations, manipulate algebraic expressions, differentiate and integrate, and solve systems of equations. 4

Application in classroomsCalculating, is what calculators are good at, which is why much more time could be spent on the other parts of the problem-solving process. The other parts are unavoidable when solving problems. Since these parts often are neglected in school-mathematics, I would say that students, who don’t have problem solving as a part of their own family tradition, actually are discriminated against this in school.1

An article in "Education World" lists several advantages to calculator use in the classroom. Calculators allow students to spend more time using critical thinking to solve problems, since they are not spending as much time on the tedious calculations that the calculators can perform for them. Calculators allow students to study concepts that would have been beyond their scope of understanding if the calculations would have to be performed manually. The calculators bring about a spark of interest to otherwise uninterested or bored students. They also simplify the tasks at hand allowing for more time to explore different methods to solving problems. Calculators also bring about more confidence in students. 13

CALCULATORS CAN CHANGE THE WAY WE TEACHBert Waits reports ten fundamental activities done with "hand-held visualization technology” in the classroom work of students in the Calculator and Computer Pre-

calculus project (a project involving more than 1,000 schools in the USA). These activities are:

1) Approach problems numerically. 2) Use analytic algebraic manipulations to solve equations and inequalities and then support using visual methods. 3) Use visual methods to solve equations and inequalities and then confirm using analytic algebraic methods. 4) Model, simulate and solve problem situations. 5) Use computer generated visual scenarios to illustrate mathematical concepts. 6) Use visual methods to solve equations and inequalities which can not be solved or are impractical using analytic algebraic methods. 7) Conduct mathematical experiments, and make and test conjectures. 8) Study and classify the behavior of different classes of functions. 9) Foreshadow concepts of calculus. 10) Investigate and explore various connections among different representations of a problem situation.2

"According to the National Council of Teachers of Mathematics (NCTM), the use of calculators along with traditional paper-and-pencil instruction enhances the learning of basic skills." In fact, the NCTM supports "the integration of the calculator into the school mathematics program at all grade levels in class work, homework, and evaluation." (Roberts, 1991, p.51) In other words, the use of calculators should not eliminate the teaching of the basic algorithmic skills and processes of mathematics. It should be properly integrated to reinforce the basic concepts that are being taught and to aid in the application of these math processes to real-world situations. The key words here are "proper integration". This implies teacher-supervised activities relating to the mathematical concepts being learned.3 How could calculator use benefit students? When students do not have to worry about computation mistakes, they can focus on reasoning and problem solving. Teachers can help students see patterns, check estimates against reality, and solve complex problems, like those encountered in daily life, through the structured use of calculators. Children introduced to the calculator when they are young will find it easy and effective to use. Calculators should be used in the classroom for many reasons:

Calculators help students at all levels learn mathematically complicated material.

Even young children can use calculators to focus on the ideas behind computation rather than on the act of calculating.

Rather than hampering mathematical ability, calculator use can actually improve student achievement in mathematics, according to research.

Both the SAT and ACT now allow students to use calculators during testing, as do many state-level exams. Students who have not been comfortable with calculators from a young age may be at a disadvantage on these tests. In the recent Third International Mathematics and Science Study (TIMSS) fourth and

eight graders who used calculators almost every day performed at higher levels than did those who never used one or only used it once or twice a month. 9

With guided calculator use even young children can begin to see differences and relationships among arithmetic procedures. The difference between memorizing 5 x 5 = 25 and hitting 5+ on the keypad five times is that the latter activity can lead to discussions about connections between addition and multiplication and make the underlying patterns of multiplication apparent. With teacher guidance young children can become aware of larger numbers or even negative numbers at an earlier age than they have in the past.

In secondary school, calculators can help students develop their understanding of algebra and other advanced mathematics. Students will have an easier time learning advanced mathematical procedures if the foundation for complex calculator use is laid in the elementary grades.

Time for exploration is needed for effective calculator use. If students use calculators to figure out the relationship between the circumferences and diameters of many different round objects, they can get beyond problems of correct division and watch the concept of pi emerge. Doing such work adequately, however, requires that the teacher make the time to allow students to work with their own material until the concept is discovered and internalized. Once such time is set aside the calculator will repay with more time available for searching, developing hypotheses, and testing them. Without anxiety over basic mathematical processes, children will be able to concentrate on the applications and meanings of the world of numbers. 9

Evidence of EffectivenessWhat does research show about classroom calculator use?

Researchers have studied classroom calculator use for several decades and in many countries. Research in this area began to take off in the late 1970s. In a 1986 study in the Journal for Research in Mathematics Education, Hembree and Dessart analyzed 79 studies of calculator use and found the following:

Children who use calculators on tests have higher scores in both basic computation skills and problem solving.

Students who use calculators within a mix of instructional styles do not lose their paper and pencil skills.

Calculator use in the classroom improves the paper and pencil skills of students regardless of their ability levels.

Those who use calculators in class have better attitudes toward mathematics than children who do not use them. 9

Hembree and Dessart (1992) reported the findings of a meta-analysis of the effects of pre-college calculator use. This research analyzed results from eighty-eight studies focused on students’ achievement and attitude. Each study involved one group of students using calculators and another group having no access to calculators. From

their analysis, Hembree and Dessart concluded that the calculator did not hinder students’ acquisition of conceptual knowledge and that it significantly improved their attitude and self-concept concerning mathematics.

Smith (1997) conducted a meta-analysis that extended the results of Hembree and Dessart. Smith analyzed twenty-four research studies conducted from 1984 through 1995, asking questions about attitude and achievement as a result of student use of calculators. As in the Hembree and Dessart study, test results of students using calculators were compared to those of students not using calculators. Smith’s study showed that the calculator had a positive effect on increasing conceptual knowledge. This effect was evident through all grades and statistically significant for students in third grade, seventh through tenth grades, and twelfth grade. Smith also found that calculator usage had a positive effect on students in both problem solving and computation. Smith concluded that the calculator improved mathematical computation and did not hinder the development of pencil-and-paper skills.

A recent large study examined effects over a longer term. The purpose of the project, Calculators in Primary Mathematics, funded by the Australian Research Council, Deakin University, and the University of Melbourne, was to have primary and elementary school students explore and develop number sense using calculators before standard algorithms were taught (Groves and Stacey 1998). It involved one thousand students and eighty teachers over a three-year period. The performance of students in the project was compared with that of a control group for the same schools using a written test, a calculator test, and an interview. The results showed that the project students performed better overall on a wide range of items including place value, decimals, negative numbers, and mental computation. No detrimental effects of calculator use were observed. Until curricular innovations such as that tried in Australia are implemented, we believe that students and teachers should distinguish among three tools of computation: mental arithmetic, pencil and paper, and calculators. For example, we would chastise any student who reaches for the calculator to find 3 x 4; we would suggest pencil and paper for calculating 27 x 340; and we would insist on using the calculator for 2.7568 x 345.8972 after the student estimates mentally an answer of 900 (3 x 300).

In conclusion, we recommend that schools strongly encourage the use of calculators in all aspects of mathematical instruction including the development of mathematical concepts and the acquisition of computational skills. We believe that calculator education is an obligation of schools to our society where calculators are in common, daily use. 6

Emphasis is placed on the role of technology and the appropriate concepts and skills related to its use. Changes in technology and the broadening of the areas in which mathematics is applied have resulted in growth and changes in the discipline of mathematics itself. The new technology not only has made calculations and graphing easier but has also changed the very nature of the problems important to mathematics and the methods mathematicians use to investigate them.

Critics and their rationale

David Gelernter, professor of computer science at Yale University, believes calculators should be totally eliminated from the classroom. He feels that allowing children to use calculators produces adults who can't do basic arithmetic, doomed to wander through life in a numeric haze. In 1997, California legislation would prohibit the use of calculators in schools prior to the sixth grade.3

One of the problems experienced with children who have used calculators from an early age without proper integration is that they have not had the opportunity to develop good "number sense". This is sometimes referred to as a "feel for numbers". The importance of estimating an answer before or after calculating is not understood. Why estimate, when the calculator will give you the exact answer? After all, a calculator is never wrong. That is virtually true, but the operator of the calculator is capable of error. 3

Beyond the anecdotal, there is also the input from the Third International Mathematics and Science Study (TIMSS). For the 8th grade assessment, the majority (>50%) of the students from three of the five nations with top scores (Belgium, Korea, and Japan) never or rarely (once or twice a month) used calculators in mathematics classes. In contrast, the majority of students (>65%) from 10 of 11 nations, including the United States, with scores below the international mean, used calculators almost every day or several times a week in mathematics classes (Beaton, Mullis, Martin, Gonzalez, Kelly, and Smith 1996) While such data do not prove that calculator usage is damaging to the development of mathematical skills, it would be folly to ignore this. Taken from Mathematics Framework for California Public Schools K–12 (draft, 1998) 8

From the TIMSS results it is clear that mathematical competence at the grades K–6 level does not require calculators. Two of the highest-achieving countries at the fourth-grade and eighth-grade levels, Singapore and Japan, use calculators sparingly in elementary schools. 5

"Some of us who were very early to use technology to alleviate drudgery, to visualize graphs and surfaces, to conduct helpful experiments, etc., are now alarmed at its use as a substitute for thinking. It even seems to deter problem solvers from producing general mathematical proofs by holding their focus to computing a few numerical examples.” (John Duncan, mathematics professor, University of Arkansas, American Mathematical Monthly 102, p. 194)

The worst effects of calculator usage are the following:

1. The loss of experience in simplifying and the consequent loss of the student’s (and the teacher’s or examiner’s) expectation that expressions should have any meaning. 2. The destruction within half a generation of a hard-won, effective algebraic symbolism, developed and proved over centuries, capable of being manipulated as a "calculus” for exact numerical and symbolic calculations, and its replacement by slavish verbatim copies of what appears in calculator displays.

3. The collapse within ten years of arithmetical fluency within the very best students, with the resulting loss of meaning for symbolic generalizations of numerical expressions. 4. The loss of all attempts to teach pupils to present solutions in forms that others can make sense of, and the decline into mere personal jottings en route to an answer, which is related—I suspect—to the next effect. (Anthony Gardiner, mathematician University of Birmingham, personal e-mail)

Considerations due to diversity and genderLest you are about to jump up and toss calculators into the classroom, let me caution you that this is where things get tricky. What if calculator use does not promote a "girl friendly” learning environment? According to the AAUW again, "Girls have developed an appreciably different relationship to technology than boys … and technology may exacerbate rather than diminish inequities by gender as it becomes more integral to the K–12 classroom.” More boys have and use technological tools and toys. Boys more often perceive themselves as going into careers, such as engineering, that require technology like calculators and computers. They can envision a payoff for learning to use these tools. We need to be sensitive to past and present inequities when structuring opportunities for all children. 7

Signed "life experiences", testimonies and storiesThe calculator, especially the high end graphing calculators, can have a profound impact on learning and teaching. (to be continued Brad Frey - Johnsburg High School)

As an assistant math teacher for gifted children I learned first hand how a calculator can be a learning tool. At first I felt that the students should have to do their calculations on paper as many of us did when we were in school. What surprised me was how the students used the calculators. They didn't just enter the problem to be solved and hit enter - they chose to program the calculator. This is where I became convinced that using calculators in the classroom was a positive for these students. ~RSmall

Until recently I never allowed a calculator in my class. But now with the state testing situation, it has become a must. I guess we need to look at it realistically, in life most everyone uses a calculator. By not teaching the students to use a calculator, we would be denying teaching our students a life skill. Whether is balancing a checkbook, or using it on the job, calculators play an important role in our lives. We need to teach our students the right way to utilize the calculator. – Dale Donner

I think that calculators shouldn't really be used in the educaitonal setting because we are having our students rely on technology instead of the students using their brain. I have a student who currently counts on her finger when adding if she doesn't have a calcluator around to do basic arithmatic. She will use a calculator to do simple addition and subtraction. I just think that by allowing students to use technology we

are cheating them because they aren't challenging themselves at all. --Brian Bucciarelli

Although I do not teach math, I have heard from math teachers about the students' reliance on calculators. Keep in mind, these are middle school students that are being referred to. One of the most disturbing facts is that students are using them to do basic mathematical operations such as addition, subtraction, and multiplication. When I was in school, using a calculator for such purposes would've been unheard of. We only used the calculator for complex problems and graphing purposes. We were expected to have those basic operations memorized. If students are allowed to use calculators for everything I think it is doing a disservice to our youth. I feel that calculators should be used when performing complex operations and when they are necessary, not just for calculating the simplest of problems. ~K. Kleckauskas

I understand the cognitive need for students to understand the basics of computation and the historical drive to keep that as a mainstay of K-12 math instruction. However, given the ever-growing breadth of information that students need to be exposed to prior to graduation as well as the increased demands of mandates, not to mention the increasing reliance on technology society-wide, I cannot help but to endorse the use of calculators in the classroom. Not only the use of the calculator but instruction on how to program for and complete complex functions. -Steve Svendsen

While some students do become too reliant on the use of calculators, their use should not be prohibited. Calculators, if used correctly, can enable a student that struggles with basic facts to learn more advanced skill without the worry of making a simple math mistake. In an upper level mathematics class, calculators can improve the comprehension of a student by making what may have been a long, tedious problem more concise. (My stats students would attest to this last fact.) I am not advocating that students should be allowed to use them in all cases. For example, in my stats classes the students are taught to work the problem by hand one day. The next class day, they are taught how to repeat the process using a calculator. In this way, they learn to appreciate what the calculator is doing for them. We have also been practicing programming skills - which require logical mathematics steps - in an effort to further apply our basic math knowledge. - M Foshee

Today's calculators are a very powerful and effective tool if used correctly. Teachers need to focus their efforts on getting the calculators to be used as a learning tool, not a computational crutch. However, they can also be a huge distraction. Today's graphing calculators are able to function as portable gaming systems as well. It is frustrating as a teacher when as you walk around the classroom trying to make sure everyone is on task you constantly catch studens playing games on their calculators. (I even had a parent call the principal and complain because I erased the games off of her son's calculator becuase he was playing them in class...) J. Linnenburger

The Calculator in the Elementary Classroom: Making a Useful Tool out of an Ineffective

Crutch

Erin McCauliffDepartment of Education and Human Services

Villanova University

Edited by Klaus Volpert

In the early 1980’s the hand-held calculator began to appear in elementary classrooms, and with its introduction came controversy. Would the use of the calculator take away from students’ ability to think and reason through problems? The purpose of this paper is to review research that addresses both the positive and negative effects of calculator use in the primary grades. The author will specifically address research findings that both support and challenge the use of calculators in primary grades. It is important to note that most research that supports the use of calculators, but also cautions that responsibility must lie with the teacher. One study showed a direct correlation between teacher training and calculator use. “Teachers who had received no training in the use of calculators were evenly divided between whether their students used calculators or did not. Teachers who had more training were likely to have students use calculators in their classroom.” (Porter, 1990) This paper will also address teachers’ attitudes toward calculator use, and will conclude with a summary of how the existence of calculators in the primary grades demands curriculum modification, and consequently, a reformation in teacher education.

In 1966, a development team at Texas Instruments invented a miniature calculator that would change the lives of many. One could use the device to perform simple mathematical computations more quickly and more precisely than with paper and pencil. This tool expanded the mathematical capabilities of everyone from high school students to businesspersons. Public interest in calculator use in schools has grown over the past twenty-five years, as they have become more affordable.

Until the hand-held calculator appeared, elementary school mathematics curricula stressed paper-and-pencil calculations out of necessity—it was the fastest way to find the answer to the problem. Today, however, the device can quickly do computations that used to take students many hours of instruction and practice to master. This poses an important question: How will the incorporation of this technological advance influence the development of students’ basic reasoning skills, specifically in the elementary class room? The National Council of Teachers of Mathematics (NCTM) made the following statement in 2000: “Technology should not be used as a replacement for basic understandings and intuitions; rather, it can and should be used to foster those understandings and intuitions.” This belief has not wavered much since 1974, when the NCTM issued a sweeping statement urging that calculators appear in school at all grade levels. They expected that the tool “would aid algorithmic instruction, support concept development, reduce demand for memorization, enlarge the scope of problem solving, provide motivation, and encourage discovery, exploration and creativity.” Yet, twelve years later, the calculator had been unsuccessful in redirecting the curriculum and had failed to enter most classrooms. (Hembree and Dessart, 1986) Today, the National Council of Teachers of Mathematics takes the position that calculators can and should be used in all mathematics classrooms, as long as they are implemented properly. “Appropriate instruction that includes calculators can extend students’ understanding of mathematics and will allow all students access to rich problem-solving experiences.” (NCTM, 2000) This qualification, appropriate instruction, is the reason for concern. In order for this technology to have a positive impact on students’ learning of mathematics, teachers must be educated as to how to put the calculator into practice. The calculator should be used as a supplement to learning, not as a replacement for learning computational algorithms.

Professional MandatesBefore addressing the research findings on the positive and

negative effects of calculator use in the elementary classroom, it is necessary to state that professional mandates exist. The National Council of Teachers of Mathematics published a position statement that speaks to the use of calculators in the education of the nation’s children. “The NCTM recommends the integration of calculators into the school mathematics program at all grade levels.” The committee goes on to explain the rationale behind their position:

“Research and experience support the potential for appropriate use to enhance the learning and teaching of mathematics. Calculator use has been shown to enhance cognitive gains in areas that include number sense, conceptual development, and visualization.”

The committee recommends that all students should have access to calculators. They state that all mathematics teachers should promote the use of this technology and that they should keep up with new skills by participating in professional development activities that are encouraged by the school district.

In her doctoral dissertation in 1990, Porter quoted a Reys and Reys study from 1987 that concluded “every school should have a clear calculator policy; otherwise, teachers in the same school at the same grade level may employ different rules for calculator use.”

Some states also have mandates that support the use of calculators in at all grade levels. With such directives come a responsibility for all school districts, administrators, and teachers. The next two sections will address research that has highlighted both the positive and negative effects of calculator use in the elementary grades.

Positive EffectsResearch highlights both advantages and disadvantages of utilizing

the calculator in elementary classrooms. However, most studies show no definite harmful effects from recommending a calculator for computation at an early age. It seems clear that if the calculator is used properly to enhance a curriculum, the students will reap many benefits. First, students can spend more time solving problems conceptually. “For example, a simple four-function calculator will allow students to use whatever operation is appropriate in a problem, regardless of whether they are confident of their own skill at carrying out that operation.” (Hembree & Dessart, 1986) Here, the students experience a computational advantage and become more secure in their abilities. Computation is important specifically because it is necessary to solve many mathematical problems. The particular method used, however, whether it involves mental math, paper and pencil, or a calculator, is just one part of the computation process. Students must also know what kind of computation to perform and be able to identify the appropriate numbers to use in computations. Hembree and Dessart (1986) assert “real mathematics means knowing a variety of strategies for solving problems and having the ability to apply them appropriately.” Using a calculator enables students to think more abstractly: It allows children to solve problems whose solutions are within theoretical, but not computational, grasp. Furthermore, “The use

of realistic data is motivational and helps children see connections between school mathematics and the mathematics used in the real world.” (Charles, 1999)

Hembree and Dessart’s research in 1986 reported the findings of a meta-analysis of the effects of pre-college calculator use. They analyzed the results of seventy-nine research reports that focused on students’ achievement and attitude. Each study involved one group of students using calculators and another group having no access to calculators. From their analysis, Hembree and Dessart concluded that the calculator “did not delay students’ acquisition of conceptual knowledge and that it significantly improved their attitude and self-concept concerning mathematics.” In this study, results show that “for problem solving with the calculator, the effects for low- and high-ability students were higher than the effect for average students. The calculator created not only a computational advantage but also a benefit in the selection of proper approaches to a solution.” It was also found that “in grades K-12 (except grade 4), students who use calculators in concert with traditional instruction maintain their paper-and-pencil skills without apparent harm.” Hembree and Dessart found that the use of calculators in testing produces much higher achievement scores than paper-and-pencil efforts, both in basic operations and in problem solving. This was true across all grades and abilities.

In general, these researchers found that students using calculators possessed a better attitude toward mathematics and more confidence than non-calculator students did. (1986) In fact, “The role of the calculator as a positive motivator for students has been documented in many studies. Several studies have reported increased confidence and improved attitudes toward mathematics as well as a greater persistence in problem solving when calculators are used.” (Porter, 1990; Driscoll, 1981) So not only will students be able to develop conceptual thinking skills with the use of a calculator, but they will also gain confidence in their mathematical abilities.

In 1997, Smith conducted a meta-analysis that extended the results of Hembree and Dessart. Smith analyzed twenty-four research studies conducted from 1984 through 1995, asking questions about attitude and achievement due to student use of calculators. As in the Hembree and Dessart study, test results of students using calculators were compared to those of students not using calculators. Smith’s study showed that the calculator had a positive effect on increasing conceptual knowledge. This effect was evident through all grades and statistically significant for students in third grade. Smith also found that calculator usage had a positive effect on students in both problem solving and computation and

did not hinder the development of pencil-and-paper skills. (DeRidder, Dessart, Ellington, 1999; Smith 1997)

Dockweiler & Shielack found that conceptual development “was fostered by the calculator’s quick capability to display numbers. This is directly to students’ concrete experiences with the numbers by using the calculator to reinforce the patterns generated in base ten materials. A calculator provides support for recording the connections between the concrete material and their symbolic representation. For example, many young students have difficulty counting with the combination of hundreds, tens, and ones represented by the pieces in the base ten materials. With the use of the calculator, students can explore the relationships between these place values.” (1992)

The proper use of calculators will also enhance number sense, conceptual development, and visualization. Number sense is a foundation for early success with mathematics. Calculators can help to develop the conceptual understandings and abilities that underlie strong number sense. Calculators are particularly powerful in enabling children to make and test conjectures and generalizations related to numbers and operations.

“Making and testing conjectures about counting patterns helps children understand number relationships, develops flexibility with numbers, and promotes the development of mental and paper-and-pencil computational strategies. For example, students can use a calculator to skip count by 5’s (press 0 + 5 =, and so on). Students can try the same process with other numbers and try to figure out what patterns emerge, and make predictions. The counting capability of the calculator allows students to focus on patterns that result from adding the same number repeatedly.” (Charles, 1999)

This type of activity can aid students in future studying of multiplication and division. “In upper elementary grades, students can use the calculator to explore the relationships among various representations of rational numbers.” (Reys &Arbaugh, 2001)

Negative EffectsUnfortunately, most teachers do not know how to implement the

calculator properly and hence, students are often at a disadvantage.First, if students do not understand the basic skills necessary to

move on, they may not have success in future classes. If the students are taught to rely on the calculator, even to only check answers, their confidence will suffer when the calculator is taken away. If one provides

calculators at an early age, students may not learn computational algorithms.

Secondly, calculators also provide an illusion of progress; students may experience a false sense of confidence and consequently, their motivation decreases.

As mentioned earlier, Hembree and Dessart found positive results for calculator usage in all grades except grade four, “where paper-and-pencil skills were hampered by the calculator treatment. Throughout the analysis, it had appeared that the calculator usage served the low or high ability student less well than the average student. Sustained calculator use by average students in Grade 4 appears counterproductive with regard to basic skills.” (1986)

Danielle McNamara, at the University of Colorado, examined a specific laboratory finding called the generation effect, and applied it to the elementary classroom.

“The generation effect refers to the finding that having students generate to-be-learned information themselves, rather than simply copying or reading the information enhances both short-term (e.g., Slamecka & Graf, 1978) and long-term (e.g., Crutcher & Healy, 1989) retention of information in various situations. Elementary school children learned simple multiplication by generating (i.e., computing the answers) or reading (i.e., reading the answers from a calculator display). The children were given a pretest, read or generate training, posttest, and a retention test after 2 weeks. (The children did not use calculators on these tests). Read training involved approximately half as much training time compared with generate training and was moderately effective. In terms of test time, read children showed a loss of efficiency after the 2-week delay compared with the generate children who showed no loss.” (1995)

Earlier in 1995, McNamara and Alice Healy conducted a similar study of adults. While the findings of this study implied that the use calculators would be ineffective for children at this specific skill level, the second study did not positively support one learning method over the other. However, the study did imply that allowing elementary school children to use calculators to solve addition and multiplication problems before basic skills were acquired would be detrimental to the learning process. This means that children should not use calculators, but should perform the operations mentally when learning new types of problems.

“One goal of the experiment was to examine how elementary-school-age children learn new multiplication facts best, reading the answer from a calculator display versus generating the answer and thus solving the problem mentally. Children in both conditions used a calculator; however, the principal difference between the read and

generate conditions was the point at which the answer was displayed on the calculator. In the read condition, it was before writing down an answer to a problem, and in the generate condition, it was afterward.” (1995)

The following is a table showing overall results.Figure I (McNamara, 1995)

McNamara concluded that “calculators were neither good nor bad for elementary aged schoolchildren, but that their value depended on the use to which they are applied, the stage of development of the basic arithmetic skills, and the cost of using them relative to the possible benefits in a realistic classroom setting.” (1995)

International ApproachesAustralia

“The Australian Association of Mathematical Teachers has a policy on school students’ use of calculators: It suggests that scientific calculators should be used by students in their early secondary schooling.

“The National Statement of Mathematics for Australian Schools (Australian Education Council, 1990) recommends that all students use calculators at all levels (K-12) and that calculators be used both as instructional aids and as learning tools. However, research has shown that, despite overwhelming support for the early introduction of calculators, a majority of infant teachers rarely or never use calculators in their classrooms.” The Calculators in Primary Mathematics project was based on the premise that the calculator, as well as acting as a computational device, is a highly adaptable teaching aid that has the potential to radically transform mathematics teaching by allowing children to experiment with numbers and construct their own meanings” (Groves, 1997).

A four-year research project investigated the effects of the introduction of calculators on the learning and teaching of primary mathematics in six Melbourne schools. Classroom observations confirmed that the use of calculators provided a rich mathematical environment for children to explore and promoted the development of number sense. “Despite fears expressed by some parents, there was no evidence that children became reliant on calculators at the expense of their ability to use other forms of computation. Extensive written testing and interview showed that children with long-term experience of calculators performed better overall on a wide range of items, with no detrimental effects observed.” (Groves, 1997)

Japan

The current Course of Study in Japan does not permit the use of calculators until after grade 4. “Moreover, Japanese primary teachers generally agree that the calculator is not appropriate in grades 1-3 (Reys, 1996; Senuma, 1994). Although a calculator might be visible on a teacher’s desk, it would be for the teacher’s personal use rather than for instruction. Japanese teachers are currently debating whether students should continue to learn about and use the abacus for calculation.” (Reys, Reys, & Koyama, 1996)

In 2000, James Tarr and others produced a study that determined trends in calculator use among 13-year-olds in Japan, the United States, and Portugal. “Data from both student and teacher surveys confirm that calculator use in eighth-grade classrooms varies substantially across nations. Perhaps most intriguing is the virtual absence of calculator use in Japanese eighth-grade mathematics classrooms, particularly given Japan’s technologically advanced society and its tradition of excellence in mathematics education.” The study shows that only 0.37 percent of students in Japan used calculators during mathematics lessons, while 43.03 percent of the US students used calculators.

Figure II (Tarr et. al., 2000)

Curriculum Change“It no longer seems a question of whether calculators should be

used along with basic skills instruction, but how.” (Hembree & Dessart, 1986) In 1990, Porter reminded us, “not only have calculators failed to enter most mathematics classrooms, but they have also failed to redirect the curriculum.” (Porter, 1990) “The goal is not to produce a calculator-driven curriculum, but one that integrates calculators in a meaningful way while promoting mental computation, estimation, problem solving and critical thinking skills.” (Reys & Reys, 1998)

In order for calculators to be a presence in the classroom, certain changes must take place in the curriculum. Teachers can encourage the use of calculator in elementary classrooms while promoting a positive attitude towards their use among parents and students. This may involve the use of calculators in estimation activities, problem solving experiences, and composition of word problems. However, programs must be in place to educate parents on the role of calculators in elementary mathematics teaching.

“Contrary to the fears of many, the availability of calculators and computers has expanded students’ capability of performing calculations. However, there is no evidence to suggest that the availability of calculators makes students dependent on them for simple calculations. Students should be able to decide when they need to calculate and whether they require an exact or approximate answer.” (Dresdeck, 1995)

Dresdeck asserts, “It is important to keep classroom calculators readily accessible to children. Their physical proximity and availability help to promote their use.” (1995)

“The NCTM encourages teachers to provide experiences that build the underlying concepts and argue that only after these ideas are carefully linked to paper-and-pencil procedures is it appropriate to devote time to developing proficiency.” (1989) Meanwhile, local school district and state curriculum guidelines may be sending a different message to teachers, requiring them to introduce and develop a mastery of standard computation algorithms by a certain grade level. Unlike many industrialized countries that have a clearly defined national curriculum specifying the content and placement of various mathematics topics, the US educational policy of control has contributed to uncertainty and bewilderment among some teachers about the relative emphasis of computation. Ultimately, without clear direction, teachers make their own decisions based on the mixed messages received from the collective forces of parents, fellow teachers, standardized assessments, curriculum materials, backgrounds that their students bring to the learning

environment, and their own beliefs about how children learn. This situation creates significantly different approaches to computation within and across schools, districts, and states. Reys and Reys proposed a sequence of computation and curricular emphasis:

Figure III (Reys & Reys, 1998)Comput

ational ToolPrimary

(Gr. K-2)Intermed

iate (Gr. 3-5)Middle

Grades (Gr. 6-8)Mental

Computation (invented thinking strategies)

Students are encouraged to develop and use invented computational strategies and to record work, as needed.

Students are encouraged to use mental computation when efficient for whole number, fraction, and decimal computation.

Written Computation (efficient paper-and-pencil strategies, invented and standard)

 

Students develop efficient algorithms for whole number computation. Standard algorithms are introduced as one method.

Students develop efficient algorithms for fraction and decimal computation. Standard algorithms are introduced as one method.

Estimation

Students are encouraged to make sense of data and answers and to develop strategies for estimating in measurement settings.

Students develop and share a variety of strategies to produce computational estimates and to judge the reasonableness of answers.

Calculator

Students use a calculator to explore patterns and relationships with numbers and operations

   

and as an efficient tool to do complex computations associated with solving problems.

Note: Examples of computational problems that are typically encountered and that need to be the focus at this level are included for illustrative purposes.  

The Role of the Teacher Though seldom heard, the views of primary teachers are important.

These educators influence calculator use in the classroom. Rousham and Rowland (1996) suggest that the potential of the calculator in primary schools is largely unrealized. They report variable enthusiasm from teachers about calculator use and say the quality of calculator use is generally disappointing. In an early evaluation of the National Numeracy Strategy, many teachers were said to lack confidence in using calculators as a teaching aid. (Houssart, 1997; OFSTED, 2000)

In 1997, Jenny Houssart carried out interviews with twenty-six teachers from a wide range of primary schools in England. “The main purpose of the interviews was to see which issues teachers chose to raise and in how much detail. The teachers were shown separate classroom tasks, and then asked to respond. One such task included a fairly prominent picture of a calculator.” In response, one teacher stated outright that she did not allow calculator use and another expressed clear reservations, which he linked to his view of the importance of mental arithmetic. Only one teacher was openly positive about calculator use; others were apparently low users by default. One interesting reason that arose for the absence of calculators in the classroom was the lack of awareness of the teaching and learning potential of calculators. Only one teacher believed the calculator was a tool for exploring number operations. For the others, checking seemed to be the main role for calculators, with some attention also paid to calculator use for its own sake in order that children knew how to use them. This small-scale study, therefore, suggests that for a majority of teachers interviewed, low use of calculators exists alongside a limited view of the potential of calculators. (Houssart, 2000) Although this particular study was conducted abroad, it

is quite possible that many teachers in the United States are in the same position.

In her 1990 dissertation, Priscilla porter reported on teacher attitude in the Irvine Unified School District in California.

“Teachers are mainly concerned about how calculators will affect students’ computational skills (Reys et al., 1980). The teacher factor remains the most important aspect of effective instruction (Vannatta & Hutton, 1980) A successful calculator program must include effective teaching materials correlated with the ongoing mathematical program. Teachers mention a need for workshops to develop and improve competence in the use of calculators. In 1987, Williams suggested that an extensive training program was needed for elementary teachers to be calculator-literate and to be able to teach students to use calculators effectively to learn mathematics.” (Porter, 1990)

Without teacher commitment to the use of calculators, the policies of professional organizations and state curriculum departments toward calculator use will go unheard and the advantages for using calculators suggested by research will never occur. It is imperative that teachers be educated in the use of this technology so that it may be used in the most appropriate and effective manner.

References

Charles, Randall I. (1999 May/June). Calculators at the Elementary School Level? Yes,

It Just Makes Sense. Mathematics Education Dialogues. 8.

Colman. (2003 March). Calculators. Youth Studies Australia, 22, 7.

Dessart, D., DeRidder, C., & Ellington, A. (1999 May/June). The Research Backs Calculators. Mathematics Education Dialogues, 2, 8.

Dresdeck, C. (1995 January). Promoting Calculator Use in Elementary Classrooms. Teaching Children Mathematics, 1, 300(6).

Groves, S. (1997) The Effect of Long-Term Calculator Use on Children’s Understanding of Number: Results from the “Calculators in Primary Mathematics Project.” Proceedings of the 16th Biennial Conference of the Australian Association of Mathematics Teachers, 158.

Hembree, R. & Dessart, D. (1986). Effects of Hand-Held Calculators in Pre-College Mathematics Education: A Meta-Analysis. Journal for Research in Mathematics Education, 17(2), 83-89.

Houssart, J. (1997). I Haven’t Used Them Yet: Primary Teachers Talk About Calculators. Micromath, 14-17.

Lehman, J. (1994 April). Technology Use in the Teaching of Mathematics and Science in Elementary Schools. School Science and Mathematics, 94, 194-201.

McNamara, Danielle S. (1995). Effects of Prior Knowledge on the Generation Advantage: Calculators Versus Calculation to Learn Simple Multiplication. Journal of Educational Psychology, 87, 307-318.

National Council of Teachers of Mathematics, NCTM Standards 2000: Principles and Standards for School Mathematics, April 2000.

Porter, Priscilla H. (1990). Perceptions of Elementary School Teachers toward the Status of Calculator Use in the Irvine Unified School District.

Reys, B. & Arbaugh, F. (2001 October). Clearing up the Confusion over Calculator Use in Grades K-5. Teaching Children Mathematics, 8(2), 90-94.

Reys, B., Reys, R., Koyama, M. (1996). The Development of Computation in Three Japanese Primary-Grade Textbooks. The Elementary School Journal, 96(4).

Reys, B. & Reys, R. (1998 December). Computation in the Elementary Curriculum: Shifting the Emphasis. Teaching Children Mathematics, 236-241.

Schielack, J. & Dockweiler, C. (1992 November). Elementary Mathematics and Calculators: Let’s Think About It. School Science and Mathematics, 92, 392-394.

Tarr, J., Mittag, K, Uekawa, K., Lennex, L. (2000 March). A Comparison of Calculator Use in Eighth-Grade Mathematics Classrooms in the United States, Japan, and Portugal: Results from the Third International Mathematics and Science Study. School Science and Mathematics, 100(3), 139-150.

http://www.brighthub.com/education/k-12/articles/30105.aspx

If the technology exists - just use it. This may be the thought of most of us but this approach may not be right at all times. Just learn about the pros and cons of using calculators in the classroom especially at the primary level of education

Introduction

Technology is advancing at an enormous rate and the mention of the calculator hardly arouses much interest since it is a sort of primitive device in these modern times when PDA, handhelds and programmable mobile devices rule the roost. Yet calculators are invariably used in the classroom or performing mathematical functions. Most of us consider this as an important weapon in the armory of our kids but do we really know all about calculators as we should know.

Technology in the ClassroomObviously a calculator is a very reasonably priced and useful

machine for carrying out the arithmetic functions. It helps kids to develop more interest in mathematics, which may not be the case otherwise. Yet there is another school of though which puts the responsibility of under development of mental skills if the brain is not exercised and the children are relying heavily on the use of calculators. This debate is going on for a long time perhaps ever since these devices were invented and came into popular use in the early 1980s.

What Research Has to Say?

People are free to form their opinions from their own experience or hearsay, but has there been any scientific research on this topic? Well there have been quite a few studies on this topic though I must say that the proof has not been conclusive on either side of the spectrum. For example a project carried out in the UK in the 1980s found that the use of calculators had a positive impact on the performance and efficiency of students. At the same time several other studies such as those carried out by Duffin in 1994 stressed that the use of calculators could lead to serious problems with students in that they deteriorate their mathematical skills to a great degree. These two are not the only studies and their have been several such studies in various parts of the world.

What Should a Teacher Do?

As with all other things in life, it is important to maintain a balance in the use of calculators in the classroom. The use of calculators should be prohibited at the primary level since that is the stage when kids absorb the basic mathematical understanding and skills, hence mental maths and/or manual calculations should be used at this stage. They might be allowed to use the calculators to learn how they operate but during the study of mathematics they should be own their own.

Of course as kids progress and their grade and level, and exhibit sufficient mental capabilities; the use of calculator and other devices should be

gradually allowed. This is equally necessary since at this stage the students need to learn other skills and would certainly like to speed up the process and not waste much time in doing manual calculations alone.

Another thing I would like to add is that though this seems very good as a theoretical advice, it may not be possible for an individual teacher or even a group of teachers to implement it at their workplace when the trend is usually the reverse. Hence this requires sustained efforts and an mixed approach where kids are taught mental mathematics and skills alongside the calculators. This can be done by

Devoting time to train children in manual maths Organizing competition at classroom level on manual maths Giving recognition to such skills by way or prize or honour

These activities can be carried out at smaller levels and would certainly go a long way to ensure that the future generations do not become a puppet of calculating machines but only use them wisely for their own good.

http://www.vanderbilt.edu/exploration/text/index.php?action=view_section&id=1422&story_id=347&images=

Published: August 20, 2008

Calculators are useful tools in elementary mathematics classes, if students already have some basic skills, new research has found. The findings shed light on the debate about whether and when calculators should be used in the classroom.

"These findings suggest that it is important children first learn how to calculate answers on their own, but after that initial phase, using calculators is a fine thing to do, even for basic multiplication facts," Bethany Rittle-Johnson, assistant professor of psychology in Vanderbilt's Peabody College of education and human development and co-author of the study, said.

The research is currently in press at the Journal of Experimental Child Psychology and is available on the journal's web site.

Rittle-Johnson and co-author Alexander Kmicikewycz, who completed the work as his undergraduate honors thesis at Peabody, found that the level of a student's knowledge of mathematics facts was the determining factor in whether a calculator hindered his or her learning.

"The study indicates technology such as calculators can help kids who already have a strong foundation in basic skills," Kmicikewycz, now a teacher in New York City public schools, said.

"For students who did not know many multiplication facts, generating the answers on their own, without a calculator, was important and helped their performance on subsequent tests," Rittle-Johnson added. "But for students who already knew some multiplication facts, it didn't matter -- using a calculator to practice neither helped nor harmed them."

The researchers compared third graders' performance on multiplication problems after they had spent a class period working on other multiplication problems. Some of the students spent that class period generating answers on their own, while others simply read the answers from a calculator. All students used a calculator to check their answers.

The researchers found that the calculator's effect on subsequent performance depended on how much the students knew to begin with. For those students who already had some multiplication skills, using

the calculator before taking the test had no impact. But for those who were not good at multiplying, use of the calculator had a negative impact on their performance.

The researchers also found that the students using calculators were able to practice more problems and had fewer errors.

"Teachers struggle with how to give kids immediate feedback, which we know speeds the learning process. So, another use for calculators is allowing students to use them to check the answers they have come up with by themselves, giving them immediate feedback and more time for practice," Rittle-Johnson said.

And, for many of the students, using calculators was simply fun.

"Kids enjoyed them. It's one way to make memorizing your multiplication facts a more interesting thing to do," Rittle-Johnson said.

"So much of how you teach depends on how you market the material — presentation is very important to kids," Kmicikewycz added. "Many of these students had never used a calculator before, so it added a fun aspect to math class for them."

"It's a good tool that some teachers shy away from, because they are worried it's going to have negative consequences," Rittle-Johnson said. "I think that the evidence suggests there are good uses of calculators, even in elementary school."

Rittle-Johnson is an investigator in the Learning Sciences Institute and the Vanderbilt Kennedy Center for Research on Human Development.

http://www.pbs.org/wgbh/misunderstoodminds/mathbasics.htmlMath and the jobs of the future

It is tempting for a parent to dismiss a child's math disability, especially when the parent has a history with a similar learning problem. For many people, mathematics is the most difficult and intimidating school subject they will ever face. It is commonly thought of as a subject that either comes naturally to a person or will never be easy.

Not long ago in the United States, math was a subject that could be fairly easily avoided in the professional world. In 1970, only nine percent of all jobs were considered technical. Opportunities abounded, even for those who struggled in math. If you disliked the subject or felt you were incapable of grasping mathematical concepts, you simply settled into a career that allowed you to avoid working with numbers.

As recently as a few decades ago, math held a position in our culture similar to the one that music, for example, holds today. Although most people recognize that mastery of a musical instrument can enrich the life of a child in the U.S., few consider musical ability to be a requirement for success. Failing to develop musically is unlikely to bring shame upon a person. In fact, it is not uncommon to hear people joking about their own tone deafness.

In this way, it is clear that learning disabilities of all types can be rendered more or less disabling depending on their context. In a nontechnological society, a child's math problem will not limit his success, just as in an illiterate society, a child's inability to read or write will not restrict her development. And while it is true that people can succeed without achieving advanced competency in math, a deficiency in certain basic math skills is more limiting now than it once was. Today, nearly a third of all jobs are classified as technical; most require far more computing skills than many jobs of the past. In response to the demands of an increasingly competitive technological world, mathematics requirements have been strengthened in the schools.

Although there is nothing that can eliminate a math disability, our society's demand for highly educated people makes it all the more important for parents and teachers to identify a child's strengths and weaknesses early, and follow strategies recommended by experts to help students overcome their difficulties.

Where Do I Begin?Home and School CollaborationLiving with or teaching a child who has difficulty thinking with numbers can be an emotionally charged experience. Frustration and confusion can complicate the conversation between

parents and teachers about what to do. Respect for each other and open communication can reduce tension and enable parents and teachers to benefit from each other's expertise and knowledge of the child from different perspectives. Working together, parents, teachers, and the children themselves can inform one another about how best to address the child's needs.

Back to Top

Parents and Teachers Communicating about MathematicsWhen you suspect a learning problem with mathematics, schedule a parent-teacher meeting to share information about the child. The following "talking points" can help structure the discussion.

Share observations of the child's mathematics profile and discuss where the breakdown is occurring. What are the worries or concerns? Does the child have problems with a particular subskill, such as multiplication facts or division procedures. Do difficulties in memory, language, attention, sequencing, spatial ordering, or higher-order cognition seem to affect the child's math skills?

Identify and discuss the child's strengths and interests. How can they be used to enhance his or her math skills and motivation to complete math assignments?

Clarify the instructional program. What mathematics program or text does the class use? Discuss how that approach is working for the child. Examine and evaluate accommodations, such as extra time or a smaller number of test or homework problems.

Acknowledge emotional reactions to the situation. Discuss how children who experience frustration or failure may become so fearful that they develop math anxiety. Some children may then turn their energy to acting out, or may withdraw from math tasks. Share strategies that have worked in the classroom and at home to help the child cope.

Discuss appropriate next steps. Establish a plan for ongoing discussion and problem solving. Should specialists be consulted? How can you best advocate for the child?

When a problem with math has been specified: o Learn more about the process of thinking with numbers from other experts, reference

books, and Web sites. See Resources.o Seek assistance from colleagues and experienced parents, professional organizations,

and support groups.o Request that the school's special education teacher or learning specialist observe the

child, then consult with you on strategies to use both in the classroom and at home.o Investigate the availability of professional help from math tutors or other math

specialists.

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Talking with Children about their Strengths and WeaknessesMoments of frustration as well as pride are common for children with math problems and for the adults who work with them. Some children give up and see themselves as failures; others exhibit behavior complications that relate to their difficulties with math.

Dr. Mel Levine suggests using a process called demystification, which, through open discussion with supportive adults, helps children learn to clarify and specify their differences and understand that, like everyone else, they have strengths and weaknesses. This process creates a shared sense of optimism that the child and adult are working toward a common goal, and that learning problems can be successfully managed. The following suggestions can help parents, teachers, and learning specialists work together to demystify children's difficulties with math.

Eliminate any stigma. Empathy can reduce children's discouragement and anxiety about their difficulties with math. Emphasize that no one is to blame, and that you know they often need to work harder than others to think with numbers successfully. Explain that everyone has differences in the way they learn. Reassure children that you will help them find ways that work for them. Share an anecdote about how you handled a learning problem or an embarrassing mistake.

Discuss strengths and interests. Help children find their strengths. Use concrete examples, but avoid false praise. You might tell a child who seems to make friends quickly, "You're a real people person." Value children's interests. To a child who enjoys drawing, you might say, "Try drawing pictures of math problems as you solve them." Identify books, videos, Web sites, or

places in the community that can help children build on their strengths and interests.

Discuss areas of weakness. Use plain language to explain what aspect of math learning is difficult for the child. For example, you might say, "You may have difficulty completing a multi-step math problem not because you don't know your math facts, but because it is hard for you to remember the procedures for completing the problem."

Emphasize optimism. Help children realize that they can improve -- they can work on their weaknesses and make their strengths stronger. Point out future possibilities for success given their current strengths. Help children build a sense of control over their learning by encouraging them to be accountable for their own progress. A child who has difficulty remembering multiple steps in solving a math problem, for example, can learn to use subvocalization strategies to organize and guide his or her effort.

Teach explicit meta-cognitive strategies when needed. For some students, a teacher will need to provide direct instruction to help children think about their approach (including previewing), pursue facts, and self-monitor. Other students may need strategies to help check the precision or the reasonableness of their answers. Remember that explaining meta-cognitive approaches only once won't be sufficient for some students. They may need repeated instruction and practice in how to apply these strategies.

Identify an ally. Help children locate a mentor -- a favorite teacher, a teacher's aide, or a neighbor -- who will work with and support them. Explain to children that they can help themselves by sharing with others how they learn best. Older children can explain the strategies that work for them, while younger ones may need adult support. Encourage children to be active partners with their allies.

Protect from humiliation. Help children strengthen self-esteem and maintain pride by protecting them from public humiliation related to their learning differences. Always avoid criticizing children in public, and protect them from embarrassment in front of siblings and classmates. For example, do not ask children to solve math problems in front of their classmates at the chalkboard. Downplay confrontational or competitive aspects of mathematics, particularly those that create anxiety such as speed drills. Explore alternate ways of covering and assessing these skills.

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What Can I Do?Suggestions and StrategiesYou may use the following suggestions and strategies to help children who are experiencing problems with mathematics. Many of those listed are accommodations -- they work around a child's differences by offering alternative approaches to learning material. Checking work is one example of a suggestion that might help. Strategies -- more research-based methods -- are designed to specifically strengthen a weakness. From the suggestions and strategies described below, select those that you and the child think might work best.

Maintain consistency and communication across school and home settings is vital. For example, if a tutor explains math concepts in one way, the classroom teacher takes another approach, and parents yet a third, this may compound problems rather than solve them.

General SuggestionsTeach basic concepts using concrete objects. Let children explore number concepts by adding and subtracting objects in the room (for example, add the legs of a chair to find the number four or subtract crayons from a box). Move from concrete materials to pictorial representations to numbers (abstract representations).

Provide specialized materials. To help children organize their calculations, have them use graph paper (or lined paper turned sideways) to keep numbers in columns. Encourage the use of scrap paper to keep work neat, highlighters for underlining key words and numbers, and manipulatives such as Cuisenaire rods, base-ten blocks, or fraction bars.

Make your expectations explicit. Tell children the procedures you would like them to use when solving a problem, and model each procedure for them. Have a child then tell you what he is expected to do. Some students benefit by having a math notebook filled with examples of completed problems to which they can refer if they become overwhelmed or confused.

Use cooperative math-problem-solving activities. Provide opportunities for children to

work in groups when solving math problems. Encourage them to share their thinking aloud as they solve problems. Reinforce efficient strategies using multiple pathways.

Provide time for checking work. Emphasize that completing math assignments is a process. Encourage children to become comfortable reviewing their work, making changes, or asking questions when they are unsure of their answers.

Give children opportunities to connect mathematical concepts to familiar situations. For example, when introducing measurement concepts, have children measure the height of classmates and family members, or the weight of their book bags when empty and when full. Ask children to estimate the measurements (guessing how much taller the refrigerator is than the stove) before solving the problem. Point out how math is used in everyday life, such as when examining bus schedules or filling out catalogue order forms.

Help children apply math concepts to new situations. Show children how to use percentages to understand the price of a jacket on sale at the mall or the amount of their allowance spent on snacks.

Provide tutors. Tutors can assist children with weak math subskills (such as multiplication and division). Arrange for tutors during summer months or after school to boost performance and ensure that the child retains his skills.

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Specific StrategiesStrategies for > Memory> Language> Attention> Production

Strategy Tips: Decide which strategies to try by observing the child and identifying the ways in which he or she learns best.

o It may take several attempts to see positive results from one strategy. Don't give up too soon.

o If the first few strategies you try do not improve the child's skills, try others.o Most of these strategies can be adapted for use with different age groups.

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MemoryProvide the technology tools needed for problem solving. Encourage children to think mathematically, even if they have not mastered basic skills. For example, let them use computer spreadsheet programs and calculators when the goal of the math activity is to develop problem-solving skills as opposed to calculation skills.

Teach basic math facts. Use explicit instruction to promote student mastery. Put a few selected unknown facts on index cards. Put strategies for remembering on the back of the cards. Cards can be put on notebook rings. Add new facts as previous ones are learned. Build practice into lessons. Also, routinely conduct cumulative reviews of skills and knowledge to help children develop automaticity with math facts.

Use rule books. Ask children to keep a notebook in which they write math rules in their own words. Encourage children to use rule books with classroom or home assignments by looking up the rule in the book and talking about it. Rule books could have a math vocabulary section and a strategy section for recording "tricks" that help with the operations.

Teach subvocalization as a strategy. Show children how to quietly repeat sequences (such as numbers and procedures) under their breath while working. Practice the strategy by giving them a sequence of numbers or directions and having them quietly repeat them back to you.

Practice subskills. Help children recall math subskills (like multiplication) more automatically with the use of flashcards and drills. Play a game in which you quiz a child about math facts and record how many he answers correctly. To build motivation, have the child record her own progress each day. Together, review progress periodically.

Teach math in more than one mode. Children respond well when math is taught in a variety of ways -- visually (such as demonstration), verbally (such as using oral explanations), and experientially (such as setting up a mock store) -- so that children have an opportunity to

process and use math information in multiple ways.

Use games. To enhance active working memory, play mental math games. For example, "What two numbers can be multiplied to get 24? How many different combinations can you find?" Gradually build up a child's ability to hold a long problem (How much is 4 + 2 - 1 x 3?) in memory. Make sure the child understands the reason for playing the game.

Review patterns. Use flash cards to review patterns, such as key words that provide clues to the operation of a word problem, or geometric patterns or shapes within complex visual designs.

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LanguageFocus on the information provided in word problems. Have children separate the necessary information for solving the problem from unnecessary details.

Teach mnemonic strategies for solving word problems. Choose strategies that suit the child's learning style. One strategy is TIPS: Think (read and paraphrase), Information (what numbers and information do you need in order to solve the problem), Problem (write equation), Solve.

Encourage children to put problems into their own words. Teach children to read for meaning when trying to identify the operation to use for solving a math problem. Have them verbalize the problem before trying to solve it.

Teach math vocabulary. Review the meaning of key words and phrases commonly used in mathematics problems, such as "all" or "total" in addition problems ("How much money did they spend in all?" "What was the total amount of the grocery bill?"). To help children identify key terms in problems, ask them whether a problem requires a particular procedure, and have them underline the word or term that gave the answer away. Include new vocabulary in their rule books (see above).

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AttentionTeach children how to preview an assignment. Help them to see the importance of thinking ahead before beginning the task. For example, cue them to ask, "Which math operations will I need next?"

Teach children how to self-monitor. During a task, show children how to stop and assess how well they are progressing. For example, tell them, "Every 10 minutes you will need to stop and check your answers." Teach children to ask themselves questions such as "How is it going?" and, "Do I need to make changes?" "Does my answer make sense?" and "Does my answer match my estimate?"

Help children maintain mental energy. Allow them to take frequent breaks while completing math assignments. Suggest that they get up and walk around during these breaks.

Teach self-checking strategies. Have students change to a different color pen when they have finished their work, becoming a "test checker" instead of a "test taker." This will help them notice their errors. For students who continue to make attentional errors in calculation, despite instruction and practice with self-checking, permit the use of a calculator for checking.

Help children stay focused. Let them choose the best place to do assignments, or allow them to listen to music if that helps their concentration.

Provide a model. Work through the mathematical problem with the child, verbalizing or demonstrating each step. Especially with homework, assist the child by doing the first problem together.

Identify topics of interest to children. Explore mathematical concepts in relation to motivating topics, such as building a skateboard ramp, tracking a satellite's orbit around the earth, discovering how the pyramids were built, or saving money in an interest-bearing account. Ask children to help you identify topics for mathematical problems.

Build a foundation for multi-step problems. Be sure the child understands basic one-step problems (problems requiring only one math operation) before advancing to those that require multiple operations.

Isolate steps. Have children focus on one step at a time. For example, provide mathematical activities in which children identify only (1) what the question is asking them to find, (2) which

information is necessary to answer the question, and (3) which operations should be used in solving the problem.

Complete each step. Explain to children that even good problem solvers rarely skip steps when solving problems, though they may appear to.

Reduce the amount of data on a page. Children with spatial problems often become overwhelmed by large amounts of visual data on a page. Reduce the number of math problems or the number of diagrams to interpret per page. Remove unessential visual features.

Have children draw pictures to represent what is going on in a math problem. Suggest they draw representations of objects from the problem (for example, three shirts, a 6-by-12 foot garden plot).

Make auxiliary tools available. Provide calculators, graph paper for aligning numbers, or templates for tracing geometric shapes.

ProductionBecause math difficulties can affect a child's performance and ability to get work done, the following strategies are designed to help children improve their organization skills, work habits, and overall production.

Use assignment books. Teach children to use assignment books and "To Do" lists to keep track of their short- and long-term assignments, tests, and quizzes. Use peers to help monitor other children's assignment books. Most schools have a "homework hotline" on voicemail or homework posted on the school Web site. These resources provided by the school can help you support a student who does not yet record assignments consistently without reminders.

Provide models of assignments and criteria for success. Give children a clear sense of how a final product might look by showing examples and sharing exemplary products (such as providing a workbook of sample problems completed correctly). You might make work from last year available and draw the children's attention to specific qualities of the work (for example, "Notice how lining up the columns makes the problem easier to understand."). Do not, however, compare children's work with that of peers or siblings.

Build in planning time. Give children five minutes of planning time before beginning an assignment. Provide guidance in effective planning when necessary.

Use stepwise approaches. Require children to break down tasks into parts and write down the steps or stages. Compile steps of frequent tasks into a notebook for easy reference during work assignments. For long-term assignments, provide a due date for each step of the assignment.

Teach proven strategies. Provide children with specific age-appropriate strategies to use in checking work. For example, use TIPS: Think (read and paraphrase), Information (what numbers and information do you need in order to solve the problem?), Problem (write equation), Solve. Children can create a reminder card to keep on their desk or in their assignment book for quick reference to the strategy.

Stress the importance of organization. Have children preview an assignment and collect the materials they will need before starting it. Guide children in keeping their materials and notebooks organized and easily accessible. In middle and high school, conduct intermittent "notebook checks" and grade organization and completion. At the beginning of the school year and a week before each check, give a list of requirements. Emphasize the positive impact that organization and preplanning will have on the completed project or assignment. By grading organization, you will emphasize its value in the learning process.

Let children wait to turn in work. The day before an assignment is due, have children review their work and check it with a parent. This will give the children enough perspective to catch errors or add more details and produce better results in the end.

Encourage self-evaluation. Set a standard of work quality or criteria for success for children to follow, and allow them to self-assess the quality of their work before turning it in. If the grade matches the child's appraisal, give extra points for good self-assessment. Rubrics are one way for students to assess their own work.

Set goals and record progress. Have children set a short-term goal, such as completing all homework for the week. Record their daily progress toward the goal for children to observe. Graphic recording, such as plotting their own line graphs, may be particularly reinforcing for some children. Reward improvement at home.

Practice estimating. Children may benefit from estimating answers to math problems and science experiments. Stress the real-life benefits of estimating and understanding what the correct answer might look like.

Eliminate incentives for frenetic pacing. Remove any positive reinforcement for finishing first. State the amount of time a task should take. This will slow down children who work too quickly and will speed up children who work too slowly.

Provide consistent feedback. Create a feedback system so children understand which behaviors, actions, or work products are acceptable and which are not. Use specifics to praise good work and recognize when children use strategies effectively. Say, for example, "I like the way you drew a table to help explain the problem," or "Asking to take a break really seemed to help you come back and focus."

Try a mentor. Some children may benefit from a mentor who will work with them to analyze their academic progress, brainstorm alternative strategies, and provide recognition of progress. The mentor must be seen as credible, and may be an individual from either inside or outside the school.

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MATHEMATICS:  Basics | Difficulties | Responses

Give your self 1 minute to do the questions below.

Sequence Activity: Multistep ProblemsFollow all four instructions below to solve each of the three problems. Enter your answer into the space provided.

A. Multiply the third number in the first row by the seventh number in the third row. B. Add this result to the fifth number in the second row. C. Add to this total ten times the fourth number in the third row. D. Subtract the eighth number in the first row from the result.

Problem 1: 6 5 8 7 4 5 6 8 4 3 2 1 9 5 6 4 2 1 6 5 1 5 1 3 2 3 5

Answer:

Problem 2: 7 5 4 9 9 5 4 4 1 2 5 1 4 8 9 6 6 8 5 7 5 7 5 7 6 8 2

Answer:

Problem 3: 1 2 3 7 6 5 4 3 2 8 4 3 2 1 6 5 4 8 6 5 5 8 1 7 5 12 6

Answer:

A snapshot of mathematics problems and implications

Math disabilities, like other learning disorders, have the power to keep children from performing up to their potential in school and beyond. At no time in our history has this notion been truer. As the world's reliance on technology has grown, so too has the demand for people who can think in the abstract terms of math and science. The disparity between those who learn math with relative ease and those who struggle with math disabilities is widening at an alarming rate. Here are some statistics that suggest why and underscore the importance of early intervention.

Struggling Kids o Nine-year-olds with math

disabilities have, on average, a first-grade level of math knowledge.

o Seventeen-year-olds with math disabilities have, on average, a fifth-grade level of math knowledge.

o Experts estimate that for every two years of school, children with math disabilities acquire about one year of mathematical proficiency.

o Children with math disabilities often reach a learning plateau in seventh grade, and acquire only one year's worth of mathematical proficiency in grades seven through twelve.

o Thirty-five percent of children with learning disabilities drop out of high school.

o Fourteen percent of students with learning disabilities (compared to 53 percent of students in general population) attend post-secondary school within two years of leaving high school.

Changing Emphasis o In the 1950s and 1960s, the

United States, pushed by the space race with the

Soviets, introduced "new math," a movement away from everyday problem solving toward a focus on abstract structures, patterns, and relationships.

o Schools stiffened math-course requirements for graduation and introduced minimum competency testing in response to a 1983 government report titled "A Nation at Risk."

o In 1983, the typical college graduate's income was 38 percent higher, on average, than the typical high school graduate's income.

o In the late 1980s, the National Council of Teachers of Mathematics revised content and methods standards for the teaching of mathematics. At the same time, standards-based tests with rigorous math sections were included as part of the graduation requirements in many schools.

o In 1997, the typical college graduate's income was 73 percent higher, on average, than the typical high school graduate's income.

Difficulties with Mathematics

What Can Stand in the Way of a Student's Mathematical Development?Math disabilities can arise at nearly any stage of a child's scholastic development. While very little is known about the neurobiological or environmental causes of these problems, many experts attribute them to deficits in one or more of five different skill types. These deficits can exist independently of one another or can occur in combination. All can impact a child's ability to progress in mathematics.

Incomplete Mastery of Number FactsNumber facts are the basic computations (9 + 3 = 12 or 2 x 4 = 8) students are required to memorize in the earliest grades of elementary school. Recalling these facts efficiently is critical because it allows a student to approach more advanced mathematical thinking without being bogged down by simple calculations.

Try it yourself. Experience a problem with basic facts.

Computational WeaknessMany students, despite a good understanding of mathematical concepts, are inconsistent at computing. They make errors because they misread signs or carry numbers incorrectly, or may not write numerals clearly enough or in the correct column. These students often struggle, especially in primary school, where basic computation and "right answers" are stressed. Often they end up in remedial classes, even though they might have a high level of potential for higher-level mathematical thinking.

Difficulty Transferring KnowledgeOne fairly common difficulty experienced by people with math problems is the inability to easily connect the abstract or conceptual aspects of math with reality. Understanding what symbols

represent in the physical world is important to how well and how easily a child will remember a concept. Holding and inspecting an equilateral triangle, for example, will be much more meaningful to a child than simply being told that the triangle is equilateral because it has three equal sides. And yet children with this problem find connections such as these painstaking at best.

Making ConnectionsSome students have difficulty making meaningful connections within and across mathematical experiences. For instance, a student may not readily comprehend the relation between numbers and the quantities they represent. If this kind of connection is not made, math skills may be not anchored in any meaningful or relevant manner. This makes them harder to recall and apply in new situations.

Incomplete Understanding of the Language of MathFor some students, a math disability is driven by problems with language. These children may also experience difficulty with reading, writing, and speaking. In math, however, their language problem is confounded by the inherently difficult terminology, some of which they hear nowhere outside of the math classroom. These students have difficulty understanding written or verbal directions or explanations, and find word problems especially difficult to translate.

Difficulty Comprehending the Visual and Spatial Aspects and Perceptual Difficulties.A far less common problem -- and probably the most severe -- is the inability to effectively visualize math concepts. Students who have this problem may be unable to judge the relative size among three dissimilar objects. This disorder has obvious disadvantages, as it requires that a student rely almost entirely on rote memorization of verbal or written descriptions of math concepts that most people take for granted. Some mathematical problems also require students to combine higher-order cognition with perceptual skills, for instance, to determine what shape will result when a complex 3-D figure is rotated.

Try it yourself. Experience a visualization challenge.

Signs of Math Difficulties

Output DifficultiesA student with problems in output may

o be unable to recall basic math facts, procedures, rules, or formulas o be very slow to retrieve facts or pursue procedureso have difficulties maintaining precision during mathematical worko have difficulties with handwriting that slow down written work or make it hard to read

latero have difficulty remembering previously encountered patternso forget what he or she is doing in the middle of a math problem

Organizational DifficultiesA student with problems in organization may

o have difficulties sequencing multiple stepso become entangled in multiple steps or elements of a problemo lose appreciation of the final goal and over emphasize individual elements of a problemo not be able to identify salient aspects of a mathematical situation, particularly in word

problems or other problem solving situations where some information is not relevanto be unable to appreciate the appropriateness or reasonableness of solutions generated

Language DifficultiesA student with language problems in math may

o have difficulty with the vocabulary of matho be confused by language in word problemso not know when irrelevant information is included or when information is given out of

sequenceo have trouble learning or recalling abstract termso have difficulty understanding directionso have difficulty explaining and communicating about math, including asking and

answering questions

o have difficulty reading texts to direct their own learningo have difficulty remembering assigned values or definitions in specific problems

Attention DifficultiesA student with attention problems in math may

o be distracted or fidgety during math taskso lose his or her place while working on a math problemo appear mentally fatigued or overly tired when doing math

Visual Spatial or Ordering DifficultiesA student with problems in visual, spatial, or sequential aspects of mathematics may

o be confused when learning multi-step procedureso have trouble ordering the steps used to solve a problemo feel overloaded when faced with a worksheet full of math exerciseso not be able to copy problems correctlyo may have difficulties reading the hands on an analog clocko may have difficulties interpreting and manipulating geometric configurationso may have difficulties appreciating changes in objects as they are moved in space

Difficulties with multiple tasksA student with problems managing and/or merging different tasks in math may

o find it difficult to switch between multiple demands in a complex math problemo find it difficult to tell when tasks can be grouped or merged and when they must be

separated in a multi-step math problemo cannot manage all the demands of a complex problem, such as a word problem, even

thought he or she may know component facts and procedures

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Math Learning Disabilities By: Kate Garnett (1998)

While children with disorders in mathematics are specifically included under the definition of Learning Disabilities, seldom do math learning difficulties cause children to be referred for evaluation. In many school systems, special education services are provided almost exclusively on the basis of children's reading disabilities. Even after being identified as learning disabled (LD), few children are provided substantive assessment and remediation of their arithmetic difficulties.

This relative neglect might lead parents and teachers to believe that arithmetic learning problems are not very common, or perhaps not very serious. However, approximately 6% of school-age children have significant math deficits and among students classified as learning disabled, arithmetic difficulties are as pervasive as reading problems. This does not mean that all reading disabilities are accompanied by arithmetic learning problems, but it does mean that math deficits are widespread and in need of equivalent attention and concern.

Evidence from learning disabled adults belies the social myth that it is okay to be rotten at math. The effects of math failure throughout years of schooling, coupled with math illiteracy in adult life, can seriously handicap both daily living and vocational prospects. In today's world, mathematical knowledge, reasoning, and skills are no less important than reading ability .

Different types of math learning problemsAs with students' reading disabilities, when math difficulties are present, they range from mild to severe. There is also evidence that children manifest different types of disabilities in math. Unfortunately, research attempting to classify these has yet to be validated or widely accepted, so caution is required when considering descriptions of differing degrees of math disability. Still, it seems evident that students do experience not only differing intensities of math dilemmas, but also different types, which require diverse classroom emphases, adaptations and sometimes even divergent methods.

Mastering basic number factsMany learning disabled students have persistent trouble "memorizing" basic number facts in all four operations, despite adequate understanding and great effort expended trying to do so. Instead of readily knowing that 5+7=12, or that 4x6=24, these children continue laboriously over years to count fingers, pencil marks or scribbled circles and seem unable to develop efficient memory strategies on their own.

For some, this represents their only notable math learning difficulty and, in such cases, it is crucial not to hold them back "until they know their facts." Rather, they should be allowed to use a pocket-size facts chart in order to proceed to more complex computation, applications, and problem-solving. As the students demonstrate speed and reliability in knowing a number fact, it can be removed from a personal chart. Addition and multiplication charts also can be used for subtraction and division respectively. For specific use as a basic fact reference, a portable chart (back-pocket-size, for older students) is preferable to an electronic calculator. Having the full set of answers in view is valuable, as is finding the same answer in the same location each time since where something is can help in recalling what it is. Also, by blackening over each fact that has been mastered, overreliance on the chart is discouraged and motivation to learn another one is increased. For those students who have difficulty locating answers at the vertical/horizontal intersections, it helps to use cutout cardboard in a backward L-shape.

Several curriculum materials offer specific methods to help teach mastering of basic arithmetic facts. The important assumption behind these materials is that the concepts of quantities and operations are already firmly established in the student's understanding. This means that the student can readily show and explain what a problem means using objects, pencil marks, etc. Suggestions from these teaching approaches include:

Interactive and intensive practice with motivational materials such as games…attentiveness during practice is as crucial as time spent

Distributed practice, meaning much practice in small doses…for example, two 15-minute sessions per day, rather than an hour session every other day

Small numbers of facts per group to be mastered at one time…and then, frequent practice with mixed groups

Emphasis is on "reverses," or "turnarounds" (e.g., 4 + 5/5 + 4, 6x7/7x6)…In vertical. horizontal, and oral formats

Student self-charting of progress…having students keep track of how many and which facts are mastered and how many more there are to go

Instruction, not just practice…Teaching thinking strategies from one fact to another (e.g., doubles facts, 5 + 5, 6 + 6, etc. and then double-plus-one facts, 5 + 6, 6 + 7, etc.).

(For details of these thinking strategies, see Garnett, Frank & Fleischner, 1983, Thornton.1978; or Stern, 1987).

Arithmetic weakness/math talentSome learning disabled students have an excellent grasp of math concepts, but are inconsistent in calculating. They are reliably unreliable at paying attention to the operational sign, at borrowing or carrying appropriately, and at sequencing the steps in complex operations. These same students also may experience difficulty mastering basic number facts.

Interestingly, some of the students with these difficulties may be remedial math students during the elementary years when computational accuracy is heavily stressed, but can go on to join honors classes in higher math where their conceptual prowess is called for. Clearly, these students should not be tracked into low level secondary math classes where they will only continue to

demonstrate these careless errors and inconsistent computational skills while being denied access to higher-level math of which they are capable. Because there is much more to mathematics than right-answer reliable calculating, it is important to access the broad scope of math abilities and not judge intelligence or understanding by observing only weak lower level skills. Often a delicate balance must be struck in working with learning disabled math students which include:

a. Acknowledging their computational weaknessesb. Maintaining persistent effort at strengthening inconsistent skills;c. Sharing a partnership with the student to develop self-monitoring systems

and ingenious compensations; and at the same time, providing the full, enriched scope of math teaching.

The written symbol system and concrete materialsMany younger children who have difficulty with elementary math actually bring to school a strong foundation of informal math understanding. They encounter trouble in connecting this knowledge base to the more formal procedures, language, and symbolic notation system of school math. The collision of their informal skills with school math is like a tuneful, rhythmic child experiencing written music as something different from what he/she already can do. In fact, it is quite a complex feat to map the new world of written -math symbols onto the known world of quantities, actions and, at the same time to learn the peculiar language we use to talk about arithmetic. Students need many repeated experiences and many varieties of concrete materials to make these connections strong and stable. Teachers often compound difficulties at this stage of learning by asking students to match pictured groups with number sentences before they have had sufficient experience relating varieties of physical representations with the various ways we string together math symbols, and the different ways we refer to these things in words. The fact that concrete materials can be moved, held, and physically grouped and separated makes them much more vivid teaching tools than pictorial representations. Because pictures are semiabstract symbols, if introduced too early, they easily confuse the delicate connections being formed between existing concepts, the new language of math, and the formal world of written number problems.

In this same regard, it is important to remember that structured concrete materials are beneficial at the concept development stage for math topics at all grade levels. There is research evidence that students who use concrete materials actually develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, may better understand mathematical ideas, and may better apply these to life situations. Structured, concrete materials have been profitably used to develop concepts and to clarify early number relations, place value, computation, fractions, decimals, measurement, geometry, money, percentage, number bases story problems, probability and statistics), and even algebra.

Of course, different kinds of concrete materials are suited to different teaching purposes (see appendix for selected listing of materials and distributors). Materials do not teach by themselves; they work together with teacher guidance and student interactions, as well as with repeated demonstrations and explanations by both teachers and students.

Often students' confusion about the conventions of written math notation are sustained by the practice of using workbooks and ditto pages filled with problems to be solved. In these formats, students learn to act as problem answerers rather than demonstrators of math ideas. Students who show particular difficulty ordering math symbols in the conventional vertical, horizontal, and multi-step algorithms need much experience translating from one form to another. For example, teachers can provide answered addition problems with a double box next to each for translating these into the two related subtraction problems. Teachers can also dictate problems (with or without answers) for students to translate into pictorial form, then vertical notation, then horizontal notation. It can be helpful to structure pages with boxes for each of these different forms.

Students also can work in pairs translating answered problems into two or more different ways to read them (e.g., 20 x 56 - 1120 can be read twenty times fifty- six equals one thousand, one hundred and twenty or twenty multiplied by fifty-six is one thousand, one hundred, twenty). Or, again in pairs, students can be provided with answered problems each on an individual card; they alternate in their demonstration, or proof, of each example using materials (e.g., bundled sticks for carrying problems). To add zest, some of the problems can be answered incorrectly and a goal can be to find the "bad eggs."

Each of these suggestions is intended to move youngsters out of the rut of thinking of math as getting right answers or giving up. They help create a frame of mind that connects understanding with symbolic representation, while attaching the appropriate language variations.

The language of mathSome LD students are particularly hampered by the language aspects of math, resulting in confusion about terminology, difficulty following verbal explanations, and/or weak verbal skills for monitoring the steps of complex calculations. Teachers can help by slowing down the pace of their delivery, maintaining normal timing of phrases, and giving information in discrete segments. Such slowed down "chunking" of verbal information is important when asking questions, giving directions, presenting concepts, and offering explanations.

Equally important is frequently asking students to verbalize what they are doing. Too often, math time is filled either with teacher explanation or with silent written practice. Students with language confusions need to demonstrate with concrete materials and explain what they are doing at all ages and all levels of math work, not just in the earliest grades. Having students regularly "play teacher" can be not only enjoyable but also necessary for learning the complexities of the language of math. Also, understanding for all children tends to be more complete when they are required to explain, elaborate, or defend their position to others; the burden of having to explain often acts as the extra push needed to connect and integrate their knowledge in crucial ways.Typically, children with language deficits react to math problems on the page as signals to do something, rather than as meaningful sentences that need to be read for understanding. It is almost as though they specifically avoid verbalizing. Both younger and older students need to develop the habit of reading or saying problems before and/or after computing them. By attending to the simple steps of

self-verbalizing, they can monitor more of their attentional slips and careless errors. Therefore, teachers should encourage these students to:

Stop after each answer, Read aloud the problem and the answer, and Listen to myself and ask, "Does that make sense?"

For youngsters with language weakness, this may take repeated teacher modeling, patient reminding and much practice using a cue card as a visual reminder.

Visual-spatial aspects of mathA small number of LD students have disturbances in visual-spatial-motor organization, which may result in weak or lacking understanding of concepts, very poor "number sense," specific difficulty with pictorial representations and/or poorly controlled handwriting and confused arrangements of numerals and signs on the page. Students with profoundly impaired conceptual understanding often have substantial perceptual-motor deficits and are presumed to have right hemisphere dysfunction.

This small subgroup may well require a very heavy emphasis on precise and clear verbal descriptions. They seem to benefit from substituting verbal constructions for the intuitive/spatial/relational understanding they lack. Pictorial examples or diagrammatic explanations can thoroughly confuse them, so these should not be used when trying to teach or clarify concepts. In fact, this subgroup is specifically in need of remediation in the area of picture interpretation, diagram and graph reading, and nonverbal social cues. To develop an understanding of math concepts, it may be useful to make repeated use of concrete teaching materials (e.g., Stern blocks, Cuisenaire rods), with conscientious attention to developing stable verbal renditions of each quantity (e.g., 5), relationship (e.g., 5 is less than 7), and action (e.g., 5+2=7). Since understanding visual relationships and organization is difficult for these students, it is important to anchor verbal constructions in repeated experiences with structured materials that can be felt, seen, and moved around as they are talked about. For example, they may be better able to learn to identify triangles by holding a triangular block and saying to themselves, "A triangle has three sides. When we draw it, it has three connected lines." For example, a college freshman who had this deficit could not "see" what a triangle was without saying this to herself when she looked at different figures or attempted to draw a triangle.

The goal for these students is to construct a strong verbal model for quantities and their relationships in place of the visual-spatial mental representation that most people develop. Consistent descriptive verbalizations also need to become firmly established in regard to when to apply math procedures and how to carry out the steps of written computation. Great patience and verbal repetition are required to make small incremental steps.

It is important to recognize that average, bright, and even very bright youngsters can have the severe visual-spatial organization deficits that make developing simple math concepts extremely difficult. When such deficits are accompanied by strong verbal skills, there is a tendency to disbelieve the impaired area of functioning. Thus, parents and teachers can spend years growling, "She's just not trying…She doesn't play attention…She must have a math phobia…It's probably an emotional problem."

Because other accompanying weaknesses usually include a poor sense of body in space, difficulty reading the nonverbal social signals of gesture and face, and often nightmarish disorganization in the world of "things," it can be easy to mistake the problem for a constellation of emotional symptoms. Misreading the problems in this way delays the appropriate work that is needed both in mathematics and the other areas.

In summaryMath learning difficulties are common, significant, and worthy of serious instructional attention in both regular and special education classes. Students may respond to repeated failure with withdrawal of effort, lowered self-esteem, and avoidance behaviors. In addition, significant math deficits can have serious consequences on the management of everyday life as well as on job prospects and promotion.

Math learning problems range from mild to severe and manifest themselves in a variety of ways. Most common are difficulties with efficient recall of basic arithmetic facts and reliability in written computation. When these problems are accompanied by a strong conceptual grasp of mathematical and spatial relations, it is important not to bog the student down by focusing only on remediating computation. While important to work on, such efforts should not deny a full math education to otherwise capable students.

Language disabilities, even subtle ones, can interfere with math learning. In particular, many LD students have a tendency to avoid verbalizing in math activities, a tendency often exacerbated by the way math is typically taught in America. Developing their habits of verbalizing math examples and procedures can greatly help in removing obstacles to success in mainstream math settings.

Many children experience difficulty bridging informal math knowledge to formal school math. To build these connections takes time, experiences, and carefully guided instruction. The use of structured, concrete materials is important to securing these links, not only in the early elementary grades, but also during concept development stages of higher-level math. Some students need particular emphasis on the translating between different written forms, different ways of reading these, and various representations (with objects or drawings) of what they mean.

An extremely handicapping, though less common math disability, derives from significant visual-spatial-motor disorganization. The formation of foundation math concepts is impaired in this small subgroup of students. Methods to compensate include avoiding the use of pictures or graphics for conveying concepts, constructing verbal versions of math ideas, and using concrete materials as anchors. The organizational and social problems that accompany this math disability are also in need of long-term appropriate remedial attention in order to support successful life adjustment in adulthood.

In sum, as special educators, there is much we can and need to do in this area that calls for so much greater attention than we have typically provided.

About the authorDr. Garnett received her doctorate from Teachers College, Columbia University. Over the last 18 years Dr. Garnett has been on the faculty of the Department of Special Education, Hunter College, CUNY where she directs the masters program in Learning Disorders. She is currently with The Edison Project, where she is the architect of their Responsible Inclusion/Special Edison Support.

References

ReferencesClick the "References" link above to hide these references.

Allardice, B.S., & Ginsburg, H.P. (1983). Children's psychological difficulties in mathematics. In Ginsburg, H.P. (Ed.) The development of mathematical thinking. New York: Academic Press.

Badian N.A. (1983). Arithmetic and nonverbal learning. In Myklebust, H.R., (Ed.), Progress in Learning Disabilities, Volume V. New York: Grunt & Stratton.

Brown, A.L., & Campione, J.C. (1986). Psychological theory and the study of learning disabilities. American Psychologist, 14, 1059-1068.

Bruni, J.V., & Silverman, H.J. (1986). Developing concepts in probability and statistics - and much more. Arithmetic Teacher 33,34-37.

Cohn, R. (1971). Arithmetic and learning disabilities. In Myklebust, M.R., (Ed.) Progress In learning disabilities, VOL 11, New York: Grunt & Stratton.

Fleischner, J.E., Garnett, K., & Shepherd, M.J. (1982). Proficiency in arithmetic basic fact computation of learning disabled and nondisabled children. Focus an Learning Problems In Mathematics 4,47-55.

Garnett, K., Frank, B., & Fleischner, JX (1983). A strategies generalization approach to basic fact learning (addition and subtraction lessons, manual #3 multiplication lessons, manual #5). Research Institute for the Study of Learning Disabilities. New York: Teachers College, Columbia University.

Goodstein, M.A., & Kahn, H. (1974). Pattern of achievement among children with learning difficulties. Exceptional Children. 50.47-49.

Harrison, M., & Harrison, B. (1986). Developing numeration concepts and skills. Arithmetic Teacher, 33,18-21, SO.

Herbert, E. (1985). One point of view: Manipulatives are good mathematics. Arithmetic Teacher, 32.4.

Johnson, .0. (1987). Nonverbal learning disabilities. Pediatric Annuals, 18, 133-141.

Johnson, D., & Blalock, J. (1967). Adults with learning disabilities: clinical studies. Orlando, FL Grunt & Stratton

Kosc, L. (1974). Developmental dyscalculia. Journal of Learning Disabilities, 7,164-177.

Lovitt, T.C., & Curtiss, K.A. (1968). Effects of manipulating an antecedent event on mathematics response rate. Journal of Applied Behavior Analysis, 1, 329-333.

McKinney, J.D., & Feagans, L. 1980 Learning Disabilities In the Classroom (Final project report). Chapel Hill: University of North Carolina, Frank Porter Graham Child. Development Center.

Paulos, J.A. , The odds are you're Innumerate. Now York Times, Book Review, January 1, 1989.

Steen, LA.. (1987). Mathematics education: A predictor of scientific competitiveness. Science, 237,251-252,302.

Stem, M. (1967). Experimenting with numbers (rev. ed.) Cambridge MA: Educators Publishing Service.

Strang, JD, & Rourke, B.P. (1985). Adaptive behavior of children who exhibit specific arithmetic disabilities and associated neuropsychological abilities and deficits in BP Rourke (Et) Neuropsychology of learning disabilities. New York: The Guilford Press

Suydam, M.N. (1984). Research Report: Manipulative Materials. Arithmetic Teacher, 31, 27.

Garnett, Ph.D., Kate. "Math Learning Disabilities." Division for Learning Disabilities Journal of CEC (1998).

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Teaching students with learning difficulties in mathematicsWhat does the research tell us?Anne Bayetto, Lecturer in Education, Flinders University

Introduction

Classroom educators already know there is a wide range of student abilities within a year level and with this come significant planning and programming issues. Commenting on mathematics teaching, Elkins (2005) notes there has been a move away from the transmission model of content delivery for all, or what has been referred to as

However, there is a dilemma for educators. On the one hand, they are told not to teach to the whole class and, on the other hand, that a completely individualized mathematics program, or even multiple groups, will likely pose organizational challenges. While there will always be situation specific factors to consider, research has highlighted fundamental principles that maximize mathematics learning for all students. It may be the intensity and amount of learning that needs to be different for students with learning difficulties. As the National Council of Teachers of Mathematics (NCTM) (2006) states, learn it well

Why might students have mathematical learning difficulties?

A small number of students is identified as having a discussed.

It is timely to preface these points by reiterating that learning occurs in settings that are supportive and caring. Students with learning difficulties may already have an external locus of control i.e. they believe they can’t improve their mathematical capacities. It is when they feel confident to have a go, make mistakes, discuss and question, that engagement and achievement will occur.

Instruction

Sherman, Richardson and Yard (2005, p 3) remind us that

In order to work within curriculum guidelines while accommodating the diversity of students in their classrooms, educators need to be realistic and systematic in the way they structure their mathematics programme. The benefits of cross curricular teaching cannot be overemphasized. It could well be that use of an engaging, and age appropriate, theme is the way into developing conceptual knowledge and skills. For example, a topic such as patterns could have students exploring patterns not only in mathematics but also in Health and Physical Education (team games), Society and Environment (climate, history), Arts (dance), and Design and Technology (measurement processes used when designing and constructing). However, Tucker, Singleton and Weaver (2002, p 3) suggest that

Westwood (2000) and Carnellor (2004) highlight the importance of educators using a judicious blend of constructivist and explicit teaching with ample guided practice/scaffolding toward independence.

Where does this leave one-to-one instruction and drill activities that have long been the mainstay of many mathematics remediation programs? Before any practice is undertaken, a secure understanding of underpinning concepts, where new learning is linked to previous learning, must be assured. If not, it may become a cycle of practise and forget, practise and forget. How often has one heard said, content not be taught year after year, in almost the same manner of delivery. Students who did not

So what might an educator do?

So what might an educator do to acknowledge the wide diversity in a group and honour what students can do and need to do next? Following is a collection of key teaching issues in no particular order but all worthy of reflection.

1. Use

2. Provide opportunities to work us that

3. Use problem must be warmly acknowledged rather than discounted as not being

4. Use educators are able to model and think aloud as they tackle algorithms and problems. Additionally, there is a wide variety of materials that can be used on the overhead projector e.g. transparent counters, clocks, and calculators.

5. Confirm language, check for understanding and not assume that nods and smiles are indicating comprehension.

6. Have students keep

7. Play games

8. Use technologyresources.

Computer

It’s interesting that some educators believe the use of calculators is making students lazy and yet employers expect their staff to have an effortless capacity to use them. (What about the use of graphics calculators in year 12 exams?) Carnellor (2004, p 54) takes a more proactive view in relation to students with mathematical difficulties when she says

9. To optimize learning for students who already have mathematical difficulties it is essential that educators have a robust Professional learning through collegial sharing of practices, participation in workshops and self-selected reading must be part of every educator’s repertoire.

10. Choose published materials carefully

11. NCTM (2006) reminds us that

Summary

Hastening students through a curriculum at the expense of understanding is short sighted and inefficient. (Recent research supported by the Department of Education, Science and Training (2004) suggested that Year Four was the time to think about a first introduction of algorithms!) Rather, educators need to work in a cycle of assess, plan, program, assess etcetera and meet the student at the point of their knowing and what they need to know next. As educators consider what they might do tomorrow, next week or next year, it is challenging to reflect on what effective practices should be maintained or taken up, and what practices have had their time and are best left behind. Students with learning difficulties are like all other students: they must be taught mathematics in a way that engages and dignifies them as learners.

References

Booker, G. (2000).

Booker, G., Bond, D., Sparrow, L. & Swan, P. (2004).

Booker, G. (2004).

Carnellor, Y. (2004).

Department of Education, Science and Training (DEST). (2004).

Department of Education, Training and Employment (DETE). (2001).

Elkins, J. (2005). Numeracy. In A. Ashman & J. Elkins. (Eds.).

Hannell, G. (2005).

National Council of Teachers of Mathematics (2006).

Peterson, M., Hittie, M. & Tamor, L. (2002). http://www.coe.wayne.edu/wholeschooling/WS/WSPress/Authentic%20MultiLvl%206-25-02.pdf

Pincott, R. (2004).

Sherman, H.J., Richardson, L.I. & Yard, G.J. (2005).

Swan, P. (1996). Tucker, B.F., Singleton, A.H. & Weaver, T.L. (2002). Teaching mathematics to all children. Upper Saddle River, NJ: Pearson Education.

Westwood, P. (2000).

 

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