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### Washington’s Math Standards

### Mathematicians on Textbooks

### NCTM Reply on Textbooks

David Klein

Professor of Mathematics

California State University, Northridge

Why do standards matter?

- goal posts for teaching and learning
- coherence across grade levels
- determine the content and emphasis of tests
- influence the selection of textbooks
- form the core of teacher education programs

The State of State Math Standards 2005

Fordham Foundation

Co-authors of the Fordham Foundation Report:

- Bastiaan Braams, Emory University
- Thomas Parker, Michigan State University
- William Quirk, Ph.D. in Mathematics
- Wilfried Schmid, Harvard University
- W. Stephen Wilson, Johns Hopkins University

What’s Wrong with Washington's Standards?

Fordham Foundation grade: F

Excessive use of calculators, standard algorithms missing, poor development of fractions and decimals, weak algebra standards (little more than linear equations), very little geometric reasoning and proofs, weak problem solving standards, too many standards unrelated to math

Standards with little relationship to math:

- Determine the target heart zone for participation in aerobic activities.
- Determine adjustments needed to achieve a healthy level of fitness.
- Explain or show how height and weight are different.
- Explain or show how clocks measure the passage of time.
- Explain how money is used to describe the value of purchased items.
- Explain why formulas are used to find area and/or perimeter.
- Explain a series of transformations in art, architecture, or nature.
- Recognize the contributions of a variety of people to the development of mathematics (e.g. research the concept of the golden ratio).

Focus:talking about solving problems, rather than actually solving problems.Long lists of vague, generic tasks: "Gather and organize the necessary information or data from the problem," "Use strategies to solve problems," "Describe and compare strategies and tools used," "Generate questions that could be answered using informational text". Misleading: one does not learn how to solve problems by following these outlines. Useless: little indication of which types of problems students are expected to solve.

First Grade Sample Problem:A classroom is presenting a play and everyone has invited two guests. Enough chairs are needed to seat all the guests. There are some chairs in the classroom.Grade 9/10 sample problem asks if it is “reasonable to believe that the women will run as fast as the men” in the Olympics. Given: a list of running times of men and women, for an unspecified distance for several years of Olympic games. No further information.

Calculators“Technology should be available and used throughout the K–12mathematics curriculum. In the early years, students can usebasic calculators to examine and create patterns of numbers.” Calculators introduced in 1st grade2nd grade standard: Solve problems involving addition and subtraction with two or three digit numbers using a calculator and explaining procedures used.

FractionsIntroduced for the first time in 4th grade:Explain how fractions (denominators of 2, 3, 4, 6, and 8) represent information across the curriculum (e.g., interpreting circle graphs, fraction of states that border an ocean).Fifth graders use calculators to multiply decimal numbers before they learn meaning of fraction multiplication. What does it mean to multiply fractions, in particular, decimals? The answer comes a year later. This is rote use of technology without mathematical reasoning.

Fractions Grade 6:Explain the meaning of multiplying and dividing non-negative fractions and decimals using words or visual or physical models (e.g., sharing a restaurant bill, cutting a board into equal-sized pieces, drawing a picture of an equation or situation).Division of fractions is often incorrectly defined as repeated subtraction. E.g. “cutting a board into equal sized pieces.” Widely used CMP 6th grade textbook treats fraction multiplication and division poorly, but is considered to be aligned to Washington's standards

Patterns“What is Mathematics? - Mathematics is a language and science of patterns.”“As a language of patterns, mathematics is a means for describing the world in which we live. In its symbols and vocabulary, the language of mathematics is a universal means of communication about relationships and patterns.”“As a science of patterns, mathematics is a mode of inquiry that reveals fundamental understandings about order in our world. This mode of inquiry relies on logic and employs observation, simulation, and experimentation as means of challenging and extending our current understanding.”-- Office of the Superintendent of Public Instructionwww.k12.wa.us/curriculumInstruct/mathematics/default.aspx

Patterns: 6th Grade

- Recognize or extend patterns and sequences using operations that alternate between terms.
- Create, explain, or extend number patterns involving two related sets of numbers and two operations including addition, subtraction, multiplication, or division.
- Use rules for generating number patterns (e.g., Fibonacci sequence, bouncing ball) to model real-life situations.
- Use technology to generate patterns based on two arithmetic operations. Supply missing elements in a pattern based on two operations.

More Patterns, 6th Grade

- Select or create a pattern that is equivalent to a given pattern.
- Describe the rule for a pattern with combinations of two arithmetic operations in the rule.
- Represent a situation with a rule involving a single operation (e.g., presidential elections occur every four years; when will the next three elections occur after a given year).
- Create a pattern involving two operations using a given rule.
- Identify patterns involving combinations of operations in the rule, including exponents (e.g., 2, 5, 11, 23).*

*Note: 3 x 2n– 1 and 1/2 (4 + 5n + n3) both give these values starting with n = 0

6th Grade WASL

Karen made a triangle out of number tiles. She used a rule to create the pattern in the number tiles.

- Extend the pattern to complete the next row of the triangle.
- Describe the rule you used to extend the pattern.

Why are most state standards, including Washington's, so bad?

The National Council of Teachers of Mathematics (NCTM) has immense influence on state education departments and K-12 mathematics education in general.

Most state standards adhere closely to guidelines published by the NCTM:

- AnAgenda for Action (1980),
- Curriculum and Evaluation Standards for School Mathematics (1989)
- Principles and Standards for School Mathematics (2000).

An Agenda for Action

- problem solving should be the focus of school mathematics.
- “difficulty with paper-and-pencil computation should not interfere with the learning of problem-solving strategies.”
- “All students should have access to calculators... throughout their school mathematics program”
- “decreased emphasis on...performing paper and pencil calculations with numbers of more than two digits.”
- de-emphasis of calculus

1989 NCTM Standards

- “The new technology not only has made calculations and graphing easier, it has changed the very nature of mathematics . . .”
- “appropriate calculators should be available toall students at all times”
- More emphasis: “collection and organization of data,”“pattern recognition and description,” and “use of manipulative materials”
- Less emphasis (K-4):“long division,”“paper and pencil fraction computation,”“rote practice,”“rote memorization of rules,” and “teaching by telling”
- Less emphasis (5-8):“manipulating symbols,”“memorizing rules and algorithms,”“practicing tedious paper-and-pencil computations,”“finding exact forms of answers”

2000 NCTM Standards decreased the extreme rhetoric but continued to promote the same themes: calculators, patterns, manipulatives, estimation over exact calculation and standard algorithms and coherent development of math.But,2006 NCTM Focal Points are a step in the right direction.

November 1999: more than 200 university mathematicians added their names to an open letter to the U.S. Education Secretary calling upon him to withdraw recommendations for NCTM aligned textbooks, including Connected Math, Core-Plus, and IMP.

The list of signatories included seven Nobel laureates and winners of the Fields Medal, as well as math department chairs of many of the top universities in the U.S., and several state and national education leaders. Seven of the signers of this letter now serve on the National Mathematics Panel.

NCTM President Johnny Lott in 2004 posted a denunciation of the open letter on the NCTM website, under the title, “Calling Out” the Stalkers of Mathematics Education:

Consider people who use half-truths, fear, and innuendo to control public opinion about mathematics education. As an example, look at Web sites that continue to use a public letter written in 1999 to then Secretary of Education Richard Riley by a group of mathematicians and scientists defaming reform mathematics curricula developed with National Science Foundation grants. . . A small group continues to use the letter in an attempt to thwart changes to mathematics curricula.

TERC 5th grade 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/subtraction problems for fractions I could find in the TERC 5th grade curriculum (which is described as ‘also suitable for 6th grade’)”--Wilfried Schmid, Dept. of Mathematics, Harvard University

Mathematical Content

Skills-based

cohesion, clarity

proof

Classical tradition

whole class instruction

teacher centered

model: university

parents, mathematicians, academics

Pedagogy

learn skills as needed for “real world” problems

discovery learning

learning styles

Romantic tradition

small group learning

child centered

model: kindergarten

colleges of education, school administrators, corporate leaders

Math Wars: Points of EmphasisMathematical CompetenceMathematical ignorance of district and state K-12 math leadership is the single greatest barrier to effective education.Most math curriculum experts are mathematically weak. They make bad decisions, not only because they blindly follow the NCTM standards, but also because they don't understand mathematics very well, generally at a level far below that of classroom teachers in the schools they serve.

The mechanism leading to this paradoxical state of affairs is that the weakest math teachers are usually the first to embrace the latest education fads, and are consequently rewarded by principals and other administrators for their willingness to be innovative.This kind of “innovation” has a higher priority than proven effectiveness. The weakest teachers rise through the administrative ranks in this way. The least competent teachers end up advising senior administrators and gain authority over mathematics programs at all levels.

The result is bad standards, bad books, bad tests, and bad teacher training. Would anyone want to leave in charge of writing new state standards the same people who wrote Washington's current standards? Does anyone want to leave in charge of textbook selection the same people who chose TERC, CMP, IMP, and Core-Plus? By way of analogy, should the surgeon who consistently amputates the wrong leg be put in charge of the hospital? Mathematically competent people must be given actual decision making power within state and district bureaucracies.

Recommendations

- Completely rewrite state standards or adopt already existing high quality standards: California, Indiana, Massachusetts
- Appoint university mathematicians (not math education professors) and experienced classroom teachers (not math administrators) to high level positions with decision making power over standards, textbooks, pacing plans, state assessments, inservices
- Shift control of college teacher education courses away from colleges of education into subject matter departments

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