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).

Calculators

“Technology should be available and used throughout the K–12 mathematics curriculum. In the early years, students can use basic calculators to examine and create patterns of numbers.”

1st grade

:

Use strategies and appropriate tools from among mental math, paper and pencil, manipulatives, or calculator to compute in a problem situation.

2nd grade

:

Solve problems involving addition and subtraction with two or three digit numbers using a calculator and explaining procedures used.

4th Grade:

Use calculators to compute with large numbers (e.g., multiplying two digits times three digits; dividing three or four digits by two digits without remainders).

No requirement to learn the standard algorithms of arithmetic.

Fractions

Introduced 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

Definition “

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 Instruction www.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 2 n – 1 and 1/2 (4 + 5n + n 3 ) 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.

NSF-Funded Connected Math (CMP)

Question from a unit quiz for grade 7: Find the slope and the y-intercept for the equation 10 = x − 2.5.

Middle School Mathematics Comparisons for Singapore Mathematics, Connected Mathematics Program, and Mathematics in Context: A Summary (Including Comparisons with the NCTM Principles and Standards 2000)

by Loyce Adams, K. K. Tung, Virginia Warfield and others (2001) “ We also expect that in the next edition [of CMP] some of the typographical errors, on which much has been written and which are usually interpreted by critics of these curricula as mathematically incorrect reasoning, will also be corrected (e.g. Complaints have been raised about ‘ Find the slope and y-intercept for the equation

10 = x – 2.5

’ , CMP 7th grade

Moving Straight Ahead

,

where y was mistakenly printed as 10

.) ” – page 49

Connected Math (CMP)

The answer given in the CMP Teacher ’ s Guide: “

The equation 10 = x − 2.5 is a specific case of the equation y = x − 2.5, which has a slope of 1 and a y-intercept of −2.5.

”

In addition to the answer, the Teacher ’ s Guide contains two student papers and teacher ’ s comments on them. First, the work of two pairs of students.

Next,in the Teachers Guide, what a teacher wrote about this student work: “ Beth and Kim ’ s work for question 12 makes it clear how they found the slope for the given equation. Their work even suggests that they may have learned something from doing this problem. By constructing and finding a couple of values for a table related to the equation, they found the rise and run between two points and thus the slope. It appears that they could not just use the equation to give slope. The question I have as a teacher is, after finding slope as they did, do the students now see how they could have found the slope for the given equation?

” “ Susy and Jeff received 1 point for the correct y-intercept.

”

From

Good intentions are not enough

(2001) by Richard Askey, Dept. of Mathematics, University of Wisconsin at Madison Regarding this example, Askey writes: “ If the students have learned anything they have learned that pattern matching of a simple type will give you a good grade in a math quiz. They will have also have learned some incorrect mathematics. The first pair of students did a correct calculation for a different problem.

” “

I was told about this problem by a parent whose child took this quiz. The marking was exactly as in the text

.

This is far from the only error in these books.

”

Quote from TERC manuals

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

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 - 1 1 / 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