A CRITICAL LOOK AT MATHEMATICS EDUCATION

IN THE UNITED STATES

Wilfried Schmid

Department of Mathematics

Harvard University

Public schools (“K-12”) in the US

• Administered and financed (mostly) by the communities, therefore highly non-uniform

• Communities levy taxes on real estate (i.e., houses, commercial buildings, land)

• Per-student-spending depends on the wealth of the community

• The quality of the schools affects the value of real estate

• Details vary by state

Role of the states

• Set standards for teacher training and “certify” teachers

• Set broad standards for the schools and for textbooks

• Provide some financing

• Most states now conduct “assessment tests”

• The assessment tests require “curriculum frameworks” (documents defining what the schoolchildren are supposed to know)

Role of the federal government

• Provides additional funding for financially disadvantaged school districts

• Pays for educational research and experimental curricula

• Funds special projects, e.g., “in service teacher development” and internet access

• Runs American schools abroad, for children of diplomats and US military personnel

• Recently: publishes non-binding suggestions on curriculum content and textbooks

How is mathematics taught?

• Typically, no “streaming” before 8th grade

• Beginning in 8th grade, two parallel tracks: single-discipline courses (Algebra I, Geometry, Algebra II, Pre-calculus, Calculus), and an “integrated” track, usually taught at a lower level

• Typically, mathematics is taught by specialized mathematics teachers in 5th grade or later

• Textbooks are loaned to students

Teacher training

• Teachers (and “teachers of teachers”) are trained in “Schools of Education”

• Extreme emphasis on pedagogy over subject knowledge

• Future teachers learn mathematics mostly from mathematics educators; even future high school mathematics teachers are often taught mathematics by mathematics educators, rather than by mathematicians.

Third International Mathematics and Science Study (TIMSS, 1995)

• Elaborate international comparison of mathematics and science education

• Large amount of data, unusually careful methodology

• Comparison of student performance, teacher preparation, textbooks, teaching styles

300

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4th grade8th grade12th grade

TIMSS Mathematics Scores

TIMSS sample questions, 8th grade

Solve linear equation for x: if 3(x+5) = 30, then

a) x = 2 b) x = 5 c) x = 10 d) x = 95

Correct responses: Singapore 96%; Japan 90%; France 82%; Russia 88%; Germany 79%; England 61%; USA 73%

If m represents a positive number, which of these is equivalent to m + m + m + m ?

a) m + 4 b) 4m c) m4 d) 4(m+1)

Correct responses: Singapore 82%; Japan 75%; France 65%; Russia 75%; Germany 57%; England 42%; USA 46%

TIMSS sample questions, 12th gradeIf there are 300 calories in 100 grams of a certain food, how many calories are there in a 30 gram portion of that food?

a) 90 b) 100 c) 900 d) 1000 e) 9000

Correct responses: Netherlands 84%; France 80%; Russia 71%; Germany 74%; USA 68%

Brighto soap powder is packed in cube-shaped cartons. A carton measures 10 cm on each side. The company decides to increase the length of each side by 10%. How much does the volume increase?

a) 10 cm3 b) 21 cm3 c) 100 cm3 d) 331 cm3

Correct responses: Netherlands 50%; France 31%; Russia 30%; Germany 25%; USA 17%

Conclusions drawn from TIMSS: US student performance

• Relative performance declines drastically in later grades

• Students do relatively well on one-step problems, but not well on multi-step problems

• Students do relatively well on “data analysis” problems

• Students do badly on problems requiring conceptual thinking

Conclusions drawn from TIMSS: US textbooks

• Number of pages is much larger than in other countries

• Number of topics covered in any one year is greater

• Topics remain in the curriculum (from year to year) much longer than in other countries

• Less cohesion: textbooks have more frequent breaks between sub-topics

Problems documented by TIMSS were at least partially understood already in 1980’s

1983 report by US government commission: “A Nation at Risk”

1989 “Everybody Counts”, recommendations by the National Research Council (NRC)

1980 - 85, a new player emerges: National Council of Teachers of Mathematics (NCTM)

National Council of Teachers of Mathematics (NCTM)

• Professional organization of mathematics teachers

• Many teachers are required to become members and to pay dues

• Relatively inactive until the eighties, now very active

• In recent years, most leaders of the organization have been mathematics educators, not teachers

NCTM 1989 Curriculum Guidelines

• Elaborate document, written by a large committee of mathematics educators and teachers

• Promoted by supporters as de-facto national mathematics curriculum guidelines

• Includes social agenda: make mathematics likable and approachable, involve boys and girls equally, address needs of disadvantaged students

NCTM 1989 Curriculum Guidelines

• Emphasis on “group learning” and “discovery learning”

• Problem solving as key to mathematical learning• Method of solution is more important than the

answer• Heavy emphasis on the use of calculators• Computational skills are downgraded• Proofs are (almost completely) eliminated• “Data analysis” and “statistics” become important

topics

After NCTM 1989 guidelines, a debateReformers demand:

• develop students’ “mathematical thinking”

• less emphasis on paper-and-pencil computations

• use calculators at all times• much less memorization• reduce or eliminate direct

instruction• emphasize “group learning”

and “discovery learning”

Skeptics of reform:• computational practice and

memorization are still very important

• use calculators sparingly, after computational competence is attained

• use mix of instructional techniques, including direct instruction

• don’t neglect mathematical content in race to reform

Example: TERC’s “Investigations in Number, Data and Space”, K-5

• Developed by “TERC” (non-profit educational think tank in Cambridge, MA)

• Supported by $7,000,000 grant by the National Science Foundation (NSF)

• Several school districts have multi-year NSF grants to implement this program

• Used by roughly 3% of US elementary schools

• Typically used in liberal, affluent school districts

Example: TERC• Program consists of teacher’s manuals -- roughly 10 per

year, of up to 120 pages each, plus training sessions for teachers, conducted by TERC

• No textbooks for students; students receive copies of work sheets provided by TERC

• Program is highly scripted• Authors have published a volume of educational theory

(“Beyond Arithmetic: Changing Mathematics in the Elementary classroom”) as intellectual underpinning

• Authors claim adherence to the 1989 NCTM guidelines• Stated aims include “teacher development”• Authors oppose the teaching of standard algorithms

Quotes from TERC manuals

• worked alone

• focused on getting the right answer

• recorded by only writing down numbers

• used a single prescribed procedure for each type of problem

• used only pencil and paper, chalk and chalkboards as tools

• work in a variety of groupings

• consider their own reasoning and the reasoning of other students

• communicate about mathematics orally, in writing, and by using pictures, diagrams and models

• use more than one strategy to double-check

• use cubes, blocks, measuring tools, calculators, and a large variety of other materials

In old-style class, students: In new-style class, students:

Quotes from TERC manuals

• to observe and listen carefully to students• to try to understand how students are thinking• to help students articulate their thinking, both orally

and in writing• to establish a classroom atmosphere in which high

value is placed on thinking hard about a problem• to ask questions that push students’ mathematical

thinking further• to facilitate class discussion about important

mathematical ideas

The teacher’s role is:

Quote from TERC teachers’ manual

If you have students who have already memorized the traditional right-to-left algorithm (of addition) and believe that this is how they are “supposed” to do addition, you will have to work hard to instill some new values -- that estimating the result is critical, that having more than one strategy is a necessary part of doing computation, and that using what you know about the numbers to simplify the problem leads to procedures that make more sense, and are therefore used more accurately

TERC work sheet, 3rd grade

Solve this problem in two different ways, and write about how you solved it: 234 + 123 =

Here is the first way I solved it:

Here is the second way I solved it:

TERC work sheet, 3rd gradeSolve this problem three different ways. Using a calculator can be one way. Make notes about how you solved the problem. Be sure that others can understand what you did: 42 + 36 + 18 =

First way:

Second way:

Third way:

TERC work sheet, 3rd grade

Suppose you can hold 150 beans in your right hand and 217 beans in your left hand.

How many more beans are in your right hand?

Write down how you figured this out.

From 3rd grade manual

Birthday: Pantomime holding a newborn baby in your arms. Tell students that the baby was just born, and write today’s date on the board. Explain that this is the baby’s birthday. Sing “Happy Birthday”, and encourage students to sing with you. Ask for volunteers to sing the song in their native languages. Students might also make a poster with the words “Happy Birthday” in all the languages that are spoken in the class. Have each student point to his or her birthday on a calendar. This is also a good opportunity to make a graph of the months of students’ birthdays.

From 5th grade manual (fractions)

Teacher: Now let’s use the clock face to add fractions. Say the hand moved one third of the way around the clock and then it moved one sixth more. Where will it end up?

Write the problem on the board: 1/3 + 1/6 =

Encourage students to talk together and find more than one way to think about the problem. Some might find it helpful to look at the clock faces on their (student work sheet).

From 5th grade manual (fractions)

Suggested problems for the students:

1/5 + 1/4 = 3/8 + 3/4 =

5/6 - 1/3 = 3 - 11/4 =

These are the most difficult addition problems for fractions I could find in the TERC 5th grade curriculum (which is described as “also suitable for 6th grade”)

Problems with TERC• Too much play, too little substance

• Memorization is discouraged

• Students are kept dependent on mental crutches (fingers, blocks, clock faces, etc.)

• Practice problems are far too easy, far too few in number

• Students with poor verbal skills are greatly disadvantaged

• Intellectual level is demeaning to bright students

Singapore mathematics textbooks in the US

• Are imported by a family enterprise, can be ordered easily from www.singaporemath.com

• Widely used by home schoolers

• Used on an experimental basis in a significant number of public and private schools

• Experiments so far have been generally encouraging

• Useful as point of reference in the debate about mathematics education

Obstacles to widespread adoption of Singapore textbooks in the US

• Requires extensive training of teachers

• To work really well, needs to be started in first grade and requires a long-term commitment

• Not enough coverage of certain topics, e.g., “data analysis”, which are emphasized in the US

• Incentives in the mathematics education community and textbook publishing industry are slanted towards domestic curricula

Ingredients of a good mathematics education

• Well-trained teachers

• Balance between computational practice, problem solving, and conceptual understanding

• Sensible balance between direct instruction and “discovery learning”

• Good textbooks

• Addressing the needs of students with various degrees of mathematical talent

• Some pressure on students, but not too much!