Teaching Preservice Secondary Teachers HOW TO Teach Elementary Mathematics ConceptsAuthor(s): Robin S. KalderSource: The Mathematics Teacher, Vol. 101, No. 2 (SEPTEMBER 2007), pp. 146-149Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20876062 .

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Teaching Preservice

Secondary Teachers

HOW TO

Teach Elementary

Mathematics Concepts Robin S. Kalder

Secondary mathematics preparation pro grams throughout the United States focus on higher-level mathematics topics such as calculus, discrete mathematics, and

abstract algebra. Mathematics educators

agree that secondary mathematics teachers should have the deep understanding of the structure of math ematics that is inherent in these higher-level courses.

Mathematics methods courses required in these pro grams generally focus on the mathematics traditionally taught in middle school and high school classrooms, as identified by the individual states' standards.

By focusing on higher-level mathematics, how

ever, we are not preparing prospective mathematics

teachers to cope with the reality that exists, especially in inclusion classrooms: namely, that some secondary students do not have the skills they were expected to develop in elementary school. One of the general recommendations made by the authors of The Math ematical Education of Teachers is that teacher prepa ration "courses on fundamental ideas of school math ematics should focus on a thorough development of mathematical ideas" (CBMS 2001). Research has found that teachers of mathematics are more effec tive when they have a profound understanding of the content that is being taught and the content to which students have previously been exposed (Ma 1999). It is clear that professional programs must enable pre service teachers to develop the knowledge and skills

they need to teach elementary mathematics skills and

concepts when necessary. As early as 1904, David

Eugene Smith identified this need in his program descriptions of secondary mathematics teacher prepa

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ration (Donoghue 2006). Still, many such programs do not have a course like this as a requirement.

It is important that preservice teachers have a working knowledge of how basic mathematics

knowledge is developed. The majority of people who decide to become secondary mathematics teachers have a history of being successful as students in mathematics classrooms. This success has generally

been determined by students' having been able to

produce correct answers to mathematics questions by following accepted algorithmic procedures. Many of these preservice teachers know the procedures but do not know the concepts that underlie the procedures. How will these future teachers fulfill the NCTM vision of "offering students opportunities to learn

important mathematical concepts and procedures with understanding" (NCTM 2000) when they them selves do not have that conceptual understanding?

In an effort to remedy this gap in a traditional sec

ondary mathematics education program, I developed a course that is now a requirement for preservice

teachers at our university. This course, "Number

Systems from an Advanced Viewpoint," is designed for college students who plan to teach mathematics in middle or high schools. Its goal is to investigate thoroughly the development of elementary school

mathematics concepts. Preservice teachers already know how to "do" the mathematical procedures. In this course, they reflect on "why" we do mathemat ics the way we do, and they explore "how" to teach

mathematics from a conceptual foundation. Since this is one of the first mathematics methods

courses that our preservice teachers take, many of

them have never seen manipulative materials com

monly available in elementary schools?materials such as pattern blocks, fraction circles, base-ten

blocks, and Cuisenaire rods. Secondary preservice

teachers in this class are given the opportunity to

explore with the manipulatives, just as elementary preservice teachers do. Activities are designed to have the preservice teachers understand the concep

tual development of familiar algorithms. Preservice teachers in this course often begin to see the teach

ing of mathematics in an entirely new light and to

develop a greater desire to understand why the more advanced procedures of mathematics work as well.

One concept that has regularly generated interest

ing class conversations has been division of fractions. The first task that secondary preservice teachers were

given was to divide 3 3/4 by 1/2. These preservice teachers easily found the correct value of 7 1/2 by using the algorithm that they referred to as "keep, change, flip": Keep the first fraction the same, change the division to multiplication, and flip the second frac

tion, i.e., use the reciprocal to multiply. When I asked these same preservice teachers to create a story prob

lem to illustrate this problem, only a few were able to

Fig. 1 Fraction circles represent the dividend in the problem 3?-*--.

Fig. 2 Using the fraction circles to obtain the answer for 3? 4 2

do so. One preservice mathematics teacher wrote this:

I have three and three-quarters yards of fabric. If I only need one-half of that fabric to make a vest, how much fabric will I need?

This preservice teacher obviously confused dividing by one-half with multiplying by one-half. Few of these preservice teachers were able to explain why the "keep, change, flip" algorithm works. Instead, they offered the usual "explanation": "Dividing by one-half is the same as multiplying by two." When asked to describe an alternate method of dividing fractions, no one was able to supply one.

Subsequent to this discussion I gave these same

preservice teachers the following story problem:

I have three and three-quarters pounds of sugar in a bag that I want to separate into containers that hold exactly a half pound of sugar. How

many of these containers that hold exactly one half pound will I be able to fill?

Using a concrete example, I used fraction circles to illustrate the solution to this problem by saying, "We are going to divide 3 3/4 by 1/2. To do this, build the fraction 3 3/4 with your fraction pieces." (See fig. 1.)

"Since we are dividing 3 3/4 by 1/2, we should make as many pieces of size 1/2 as we can from the 3 3/4 shown above" (in fig. 1).

Since we had already used fraction circles to

explore the concept of representing equivalent frac

tions, students were able to replace each whole circle with two half circles and two of the one-quarter pieces with one half circle. A one-quarter-circle piece remained. (See fig. 2.)

I asked the students, "How many one-half pieces are there?" Students answered, "Seven." Then I

asked, "Are there any pieces left?" The students

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responded, "Yes." I reminded the students that we were trying to make as many groups of one-half as

possible, and we had to compare what was left to the divisor of 1/2.1 then asked, "Is the remaining quar ter enough to make another unit of size one-half?" One student answered, "The quarter is only half of what we need." I then summarized, "So we have 7 full half-pound containers of sugar and one half

pound container that is only half full. Therefore,

3 1 1 3? divided by

? equals 7 plus one-half, or 7?." 4 2 2

At this juncture, students were able to look at division of fractions as more than simply an algo rithm to be followed. It was still important, how

ever, for them to make sense of the procedure, so we continued the discussion, focusing on justifying the steps of the accepted algorithm by following them through with the same example.

We wrote

3 1 3? divided by

? 4 J 2

as

3* 4 1

'

2

In order to simplify this, we knew that we could

multiply the entire fraction by 1, since 1 is the iden

tity for multiplication. The question was then to determine the best form of 1 to use. Since we wanted to replace the 1/2 in the denominator with 1, we

decided to use 2/2. Multiplying the entire fraction

3? was multiplied by 2

and ^

was multiplied by 2.

Once we performed this operation,

we saw that the results could be simplified to

It was then clear that this was the same as multiply ing 3 3/4 by the reciprocal of 1/2.

We also recalled that division and multiplication are inverse operations, so if we knew that 6/2 = 3, we would then know that 2*3 = 6. We applied the same transformation to the statement

which resulted in the statement

3 1 3- = -x. 4 2

In order to solve for x, we multiplied both sides of the equation by 2. This yielded

3 1 3-.2 =

-x-2, 4 2

which simplified to

3--2 = jc. 4

Once again, it was apparent that multiplying 3 3/4

by the reciprocal of 1/2 resulted in the solution to the division example.

Working the problem in this way, we can see how mathematics educators want the understanding of frac tions to evolve. NCTM's Principles and Standards states that in grades 3-5, students "develop understanding of fractions as parts of unit wholes, as parts of a col

lection, as locations on number lines, and as divisions of whole numbers" (p. 148). In grades 6-8, students "work flexibly with fractions, decimals, and percents to solve problems" (p. 214). Having taught at the middle school level, I know that many students in grades 6-8 have not yet met the grades 3-5 standard. How will middle school students work flexibly with fractions if

they have never developed an understanding of frac tions? How will secondary mathematics teachers be able to help students who do not understand and can not remember the algorithm for division of fractions if

they have no further understanding themselves? It is the obligation of preservice programs for sec

ondary mathematics teachers to fill in the conceptual deficiencies that exist in all topics, not simply division of fractions, and to expose preservice teachers to the

underlying concepts of the elementary school mathe matics they may have forgotten or never learned. Sec

ondary preservice teachers should have exposure to the manipulative materials that their future students

may have seen and used in their elementary school

experiences. Our middle school and high school teachers should be using conceptual phrases such as

"regrouping" or "trading" instead of borrowing," and "simplifying" instead of "reducing."

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It was the recognition of this gap that led to the

development of "Number Systems from an Advanced

Viewpoint." Many of the numerous excellent text

books designed for elementary mathematics methods courses could be used to support any course like this. I have used A Problem Solving Approach to Mathematics

for Elementary School Teachersby Billstein, Libeskind, Lott, which is a textbook, and Mathematics Activities

for Elementary School Teachers by Dolan, William

son, Muri, which is a workbook. Topics included in this course are sets, functions, numeration systems,

integers, number theory, fractions, decimals, per cents, ratio and proportion, exponents, introductory

geometry, measurement, and motion geometry. I am

able to include so many topics because our secondary preservice teachers already know the procedures, and

many of the concepts, for these topics. This course concentrates on teaching the preservice teachers how

to explain these concepts to their future secondary students. I use many of the activities in the associ ated workbook to do this, since they support the ideas from the textbook extremely well.

This course also requires each preservice teacher to visit an elementary mathematics classroom and to give a presentation to our class that describes

what was observed. When asked to reflect on the

visit, one of the preservice teachers wrote:

I enjoyed visiting an elementary classroom. It helped me see how it all begins. It was very beneficial because I could see what kind of level of thinking children are at in that grade. It was surprising to see

how much children really know and understand.

Another of the reflections included this comment:

Math education is cumulative. Students don't

just appear one day in Algebra I. They bring years of math instruction and experiences with them. It was great to see some of the foundation

being laid and [it] gave me a better understand

ing of what students bring to the table.

When asked if listening to the presentations of their classmates was a beneficial experience for

them, one preservice teacher responded:

I never knew there were so many ways to go about lessons. Hearing other presentations was

very interesting. Even though I will be teaching high school, some of these ideas could work there.

Another preservice teacher said,

I went to one fifth-grade class for one hour, but

today I was able to experience kindergarten, first, second, and third grade classrooms. My fellow

students' insights and analyses helped me think about math education in ways I hadn't previously.

In the five years that "Number Systems from an Advanced Viewpoint" has been offered, a num ber of preservice teachers who have taken it have

graduated and are teaching mathematics in local

secondary schools. Whenever I am in those schools, I visit my former students and make a point to look at the collection of textbooks on their desks. Of all the textbooks they purchased as an undergraduate, the ones they still use most often are those from this course. When asked why they have these books

easily accessible, my former students give me the same answer: The difficulties their middle school or high school students have in understanding con

cepts often stem from not having mastered elemen

tary topics. My former students find that they are

able to use concepts discussed in "Number Systems from an Advanced Viewpoint" to help their own

students gain the information they need to con

tinue successfully in mathematics.

REFERENCES Billstein, R., et al. A Problem Solving Approach to

Mathematics for Elementary School Teachers. Bos

ton: Pearson/Addison Wesley, 2004.

Conference Board of the Mathematical Sciences (CBMS). The Mathematical Education of Teachers. Washington,

DC: American Mathematical Society in cooperation

with Mathematical Association of America, 2001.

Dolan, D., et al. Mathematics Activities for Elementary

School Teachers. Boston: Addison Wesley Longman,

2001.

Donoghue, E. F. "Mathematics Education in the United

States: David Eugene Smith, Early Twentieth Cen

tury Pioneer." Paedagogica Historica 42 (August 2006): 4-5.

National Council of Teachers of Mathematics

(NCTM). Principles and Standards for School Math ematics. Reston, VA: NCTM, 2000.

Ma, Liping. Knowing and Teaching Elementary Math ematics: Teachers' Understanding of Fundamental

Mathematics in China and the United States. Mah

wah, NJ: Lawrence Erlbaum Associates, 1999. ?>

For more on the "flip," see Karen Stohlman, "Why Do We Flip and Multiply?" in "Reader Reflections, November 2006, p. 237.?Ed.

ROBIN S. KALDER, kalderr@ccsu.edu, is an associate

y~ professor in the Mathematical Sciences Department at " Central Connecticut State University, in New Britain, CT 06050. Having taught previously at the middle school and

high school levels, she is mostly involved in teacher preparation class

es, with her present research focus on reading in mathematics.

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