## 2008 TERC Math vs. 2008 National Math Panel Recommendations

Bill Quirk is a graduate of Dartmouth College and holds a Ph.D. in Mathematics from the New Mexico State University. He co-authored The State of State Math Standards 2005, a report published by the Fordham Foundation.

### by Bill Quirk ( wgquirk@wgquirk.com)

A major objective of elementary math education is to provide the foundations for algebra, the gateway to higher math education. Although we call it “elementary math,” K-5 math content is quite sophisticated and not easy to master. But constructivist math educators believe that concrete methods, pictorial methods, and learning by playing games are the keys to a stress-free approach. This is the approach found in the second edition of TERC’sInvestigations in Number, Data, and Space (TERC 2008). Unfortunately, as we will explain below, TERC has achieved their “easy to learn” objective by eliminating the content that’s necessary for later success in algebra. What is this necessary content? That question has been at the heart of the “math wars” debate. For many years opposing sides have failed to communicate. But a 4-year search for common ground has now reached consensus. The March 2008 Final Report of the National Mathematics Advisory Panel (NMP 2008) clearly identifies the “Critical Foundations for Algebra.” The primary purpose of this paper is to show how TERC 2008 misdirects students and fails to provide the “foundations of algebra” K-5 math content identified in NMP 2008.

The complete set of TERC 2008 5th grade materials, provided by NYCHOLD, served as the primary source for this paper. The reader will find a much more limited view by clicking on TERC 2008 Curriculum by Content and following links to PDF documents. Some of these links will be given in context below.

The following clickable links also serve as an outline for this paper:

- The NMP 2008 View of the Foundations of Algebra
- The TERC 2008 View of the Foundations of Algebra
- How TERC Misdirects Children Away from the Foundations of Algebra
- Faced with TERC 2008? A Compact Guide for Parents

### The NMP 2008 View of the Foundations of Algebra

The March 2008 Final Report of the National Mathematics Advisory Panel emphasizes the critical importance of preparing students for traditional algebra. The report first defines “school algebra” as the “term chosen to encompass the full body of algebraic material that the Panel expects to be covered through high school, regardless of its organization into courses and levels.” Then, on NMP PDF page 44, the report lists The Major Topics of School Algebra. There are 27 major topics, organized into the following six categories:

- Symbols and Expressions
- Linear Equations
- Quadratic Equations
- Functions
- Algebra of Polynomials
- Combinatorics and Finite Probability

NMP 2008 makes several observations and/or recommendations about how students need to be prepared for school algebra. Here are key “foundations of algebra” quotes from this report. [Bold and emphasis added].

- Although our students encounter difficulties with many aspects of mathematics, many observers of educational policy see Algebra as a central concern. The sharp falloff in mathematics achievement in the U.S. begins as students reach late middle school, where, for more and more students, algebra course work begins. Questions naturally arise about how students can be best prepared for entry into Algebra. These are questions with consequences, for Algebra is a demonstrable gateway to later achievement. Students need it for any form of higher mathematics later in high school; moreover, research shows that completion of Algebra II correlates significantly with success in college and earnings from employment. In fact, students who complete Algebra II are more than twice as likely to graduate from college compared to students with less mathematical preparation. [NMP PDF page 13]
- Proficiency with whole numbers, fractions, and particular aspects of geometry and measurement should be understood as the Critical Foundations of Algebra. Emphasis on these essential concepts and skills must be provided at the elementary and middle grade levels. [NMP PDF page 46]
- The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected. [NMP PDF page 46]
- Proficiency with whole numbers is a necessary precursor for the study of fractions. [NMP PDF page 17]
- Computational proficiency with whole number operations is dependent on sufficient and appropriate practice to develop automatic recall of addition and related subtraction facts, and of multiplication and related division facts. It also requires fluency with the standard algorithms for addition, subtraction, multiplication, and division. Additionally it requires a solid understanding of core concepts, such as the commutative, distributive, and associative properties. [NMP PDF page 19]
- By the term proficiency, the Panel means that students should understand key concepts, achieve automaticity as appropriate (e.g., with addition and related subtraction facts), develop flexible, accurate, and automatic execution of the standard algorithms, and use these competencies to solve problems. [NMP PDF pages 17 and 50]
- The Panel cautions that to the degree that calculators impede the development of automaticity, fluency in computation will be adversely affected. [NMP PDF pages 24 and 78]

- Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra. A nationally representative sample of teachers of Algebra I who were surveyed for the Panel rated students as having very poor preparation in “rational numbers and operations involving fractions and decimals.” [PDF page 19]
- Before they begin algebra course work, middle school students should have a thorough understanding of positive as well as negative fractions. They should be able to locate positive and negative fractions on a number line; represent and compare fractions, decimals, and related percents; and estimate their size. They need to know that sums, differences, products, and quotients (with nonzero denominators) of fractions are fractions, and they need to be able to carry out these operations confidently and efficiently. They should understand why and how (finite) decimal numbers are fractions and know the meaning of percentages. They should encounter fractions in problems in the many contexts in which they arise naturally, for example, to describe rates, proportionality, and probability. Beyond computational facility with specific numbers, the subject of fractions, when properly taught, introduces students to the use of symbolic notation and the concept of generality, both being integral parts of algebra. [NMP PDF page 46]
- Furthermore, students should be able to analyze the properties of two- and three-dimensional shapes using formulas to determine perimeter, area, volume, and surface area. [NMP PDF page 46]

Comments:

- Notice the emphasis on automaticity. The power of our brains to carry out effective conscious thought is significantly enhanced by what our brains can do automatically in the background, without conscious thought.
- Practice allows students to achieve automaticity of basic skills—the fast, accurate, and effortless processing of content information— which frees up working memory for more complex aspects of problem solving.[NMP PDF page 58]

- NMP 2008 in a nutshell: Success in school algebra depends primarily on prior mastery of standard arithmetic.
- The NMP recommendations echo recommendations made by the NCTM. See their 5th Grade Focal Points. See also David Klein’s discussion on PDF pages 16-18 of our 2005 Fordham Report. Finally see What Should Children Learn in Elementary Math, published at this website in May, 2007.

### The TERC 2008 View of the Foundations of Algebra

Here is the introduction to a section found in Implementing Investigations in Grade 5 [the summary level TERC 2008 5th grade teacher’s guide].

#### Foundations for Algebra in the Elementary Grades

In the Investigations curriculum, these areas of early algebra are addressed in two major ways: (1) work within the counting, number, and operations units focusing on generalizations that arise in the course of students’ study of numbers operations and (2) a coherent strand, consisting of one unit in each grade, K-5, that focuses on patterns, functions, and change.

For “major way (1)” the TERC 2008 PDF document Early Algebra: Numbers and Operations reveals the following learning expectations for “generalizations that arise in the course of the students’ study of numbers and operations:”

- How whole numbers can be composed and decomposed.
- The commutative, associative, and distributive properties.
- Properties of equalities and inequalities.
- Addition and subtraction are inverse operations.
- Multiplication and division are inverse operations.

For “major way (2)” the TERC 2008 PDF document Patterns, Functions, and Change reveals the following learning expectations for “patterns, functions, and change:”

- Make predictions about “repeating patterns,” where the pattern may involve numbers, letters, pictures, or physical objects.
- Develop 2-column tables and corresponding line graphs.
- Write equations involving one variable.

Comments:

- NMP 2008 carefully defined “school algebra.” TERC counters with “algebra is a multifaceted area of mathematics content.”
- As stated by David Kline on PDF page 19 of our 2005 Fordham Report, “the attention given to patterns is excessive, sometimes destructive, and far out of balance with the actual importance of patterns in K-12 mathematics.”
- Other than too much focus on patterns, the TERC 2008 topics listed above are appropriate for an elementary program and they are all “foundational to the study of algebra.” This is not to suggest that these topics are nicely covered in TERC 2008. Only that it’s not a distraction to include them. This point needs to be made because much of TERC 2008 content is a time-consuming distraction.
- As we will show below, the key NMP 2008 foundations of algebra topics are only given token recognition in TERC 2008.

### How TERC Misdirects Children Away From the Foundations of Algebra

In our review of the first edition of TERC’s Investigations, we didn’t find the “foundations of algebra” K-5 math content identified in NMP 2008. We found open hostility to standard language, standard formulas, and standard arithmetic. The hostility is now missing, and the standard algorithms for addition and subtraction are now mentioned. But this is minimal treatment, designed to quiet critics. Nonstandard language still dominates, standard formulas are still missing, and it’s clear that TERC still has no interest in giving classroom time to standard arithmetic. They continue to promote nonstandard computational methods. This is a major misdirection for elementary school children because:

- TERC’s nonstandard computational methods are substituted for the standard methods that children need to master to prepare for algebra.
- TERC confuses children by claiming to offer several “strategies” for each operation. Why so many? Constructivists place a very high value on personal choice. Ideally, each child chooses their own personal way to do math and communicate math ideas.
- Standards and conventions are essential for effective communication in math and science. TERC should be concerned about how their “graduates” will effectively communicate math ideas with others after the K-5 years.

- TERC no longer claims that students “invent” these methods. Now they are “constructed” and “named” with the assistance of the teacher. Apparently every class chooses the same names. They’re really “standard” TERC 2008 names, but not standard elsewhere.
- Generally speaking, the TERC computational methods are cumbersome, inefficient, and only work for carefully selected simple problems. They seriously mislead children because they attempt to avoid the concepts of carrying, borrowing, and common denominators.
- Traditionally, children first experience the power of automaticity as they migrate up the elementary math learning curve. But there is no possibility for automaticity with the TERC 2008 methods. By attempting to suppress carrying, borrowing, and common denominators, TERC eliminates the keys to automaticity for basic arithmetic. Conscious thought is regularly required for both TERC method selection and TERC method execution. TERC’s authors are openly proud of this. They believe that maximum conscious thought indicates maximum conceptual understanding. They fail to recognize that automaticity at lower learning levels helps to maximize the effectiveness for conscious thought at higher learning levels.

### TERC 2008 Whole Number Arithmetic

In this section you will find a compact summary of the whole number computational methods found in TERC 2008. These methods are always presented as samples of student’s work. TERC 2008 teacher guides don’t explain how students are to learn these methods. Although TERC offers several methods for each operation, most of these fall into the category of “mental math.” TERC only offer one “general” whole number computational strategy for addition, subtraction, and multiplication. This is always TERC’s “transparent” version of the standard algorithm. As we will see below, TERC sacrificed “generality” when then emphasized “transparency.”

Two TERC 2008 internet documents, Addition and Subtraction PDF and Multiplication and Division PDF, will help orient the reader to TERC 2008 nonstandard approach to whole number arithmetic. Sources for TERC examples found below are given in context. TERC 2008 Addition Strategies for 6,831 + 1,897 [Source: TERC 2008 Grade 5 Student Handbook, pages 8 – 9.] Adding in Parts Change the Numbers [2 Examples]

Comments to the Strategies used in the Grade 5 Student handbook, pages 8-9:

- “Adding in Parts” and “Change the Numbers” demonstrate mental math methods, not general addition strategies.
- The third “Adding by Place” example given above is simply identified as Janet’s solution. There’s no mention that this is the standard algorithm for multidigit addition. TERC presents it as just another “Adding by Place” method.
- The second “Adding by Place” example demonstrates TERC’s most general strategy for addition. TERC promotes this as more “transparent” than the standard algorithm, because the partial sums [7,000, 1,600, 120, and 8] more openly reveal the underlying place value details. Also, for the case of adding just 2 whole numbers, carrying isn’t necessary. The student can add one column at a time, moving left to right, and never find a case where the column sum is greater than 9. This works smoothly for the case of adding just 2 numbers, but not necessarily when more than 2 numbers are involved. [See examples below for cases requiring carrying.]
- The first “Adding by Place” method is similar to the second method, but more “transparent.” That is, more inefficient. Notice the inappropriate use of the equal sign following 1,897 in the first line. This is presented as a model in the student handbook!
- Notice that subtraction is used in the “Change the Numbers” examples. These may be good illustrations of mental math, but both examples are presented as “addition strategies” in the TERC 2008 Grade 5 Student Handbook. Elementary math students shouldn’t get the idea that subtraction is needed for addition.
- TERC would have us believe that their methods avoid carrying and borrowing. But how did the student carry out the details in the “Adding in Parts” and “Change the Numbers” examples? TERC never explains such details.

TERC 2008 Subtraction Strategies for 3,451 – 1,287 [Source: TERC 2008 5th grade Teacher’s Guide for Unit 3, Pages 119-123]

Subtracting in Parts Adding Up Subtracting Back 3,451 – 1000 = 2,451

1,287 + 13 = 1,300

3,451 – 51 = 3,400

2,451 – 100 = 2,351

1,300 + 2,151 = 3,451

3,400 – 2,100 = 1,300

2,351 – 100 = 2,251

13 + 2,151 = 2,146

1,300 – 13 = 1,287

2,251 – 50 = 2,201

2,201 – 30 = 2,171

51 + 2,100 + 13 = 2,164

2,171 – 7 = 2,164

:

- These 5 methods are demonstrated [for 3,726 – 1,584] on pages 10-13 in the Grade 5 Student Handbook. See also pages 15-16 in the Addition and Subtraction PDF.
- Except for “Subtract by Place,” these are mental math methods, not general subtraction strategies. TERC’s “Subtract by Place” method is their attempt at a more “transparent” version of the standard algorithm for subtracting whole numbers.
- As indicated by the borrowing note above, the standard algorithm for subtraction is demonstrated [on page 122 of Unit 3] as an example of a student’s work. It’s also demonstrated on page 13 in the TERC 2008 Grade 5 Student Handbook. In both cases it’s positioned as an alternative way to “Subtract by Place.” It’s not identified as the standard algorithm for subtraction.
- Notice the – 30 and – 6 in TERC’s “Subtract by Place” example. The 5th grade student somehow knew that 50 – 80 = – 30 and 1 – 7 = – 6, and the student then knew how to add 2000 + 200 + (-30) + (-6). These same skills are required for the 3,726 – 1,584 subtraction problem on page 13 in the Grade 5 Student Handbook. But the topic of negative numbers isn’t mentioned elsewhere in TERC 2008, and that’s entirely appropriate. Negative numbers and the extension of whole number arithmetic to integer arithmetic should not be introduced until middle school. The inappropriate use of integer arithmetic here shows how far TERC will go to suppress the concept of borrowing. Will a 5th grader somehow find the advanced topics easier than the concept of borrowing?
- Notice that for the “Subtracting in Parts” 5th line [2,201 – 30], borrowing is necessary. It’s also necessary for the 2,226 – 80 step in the Subtracting in Parts example (3,726 – 1,584) found on page 10 of the Grade 5 Student Handbook.

TERC 2008 Multiplication Strategies for 48 x 42 [Source: TERC 2008 5th grade Teacher’s Guide for Unit 1, Pages 161-162]

Breaking Numbers Apart Change one Number

Create an Equivalent Problem

Comments:

- “Changing One Number” and “Create an Equivalent Problem” demonstrate mental math, not general strategies for multiplication. “Breaking Numbers Apart” is TERC’s attempt for a “transparent” version of the standard algorithm for whole number multiplication.
- Why is the 96 x 21 “equivalent problem” easier than 48 x 42? TERC never explains, but TERC 5th grade students construct “multiple towers” for many 2-digit numbers. The first 49 multiples of 21 are listed on page 124 of the TERC 2008 Teacher’s Guide for Unit 1.
- TERC” doesn’t explain how 1,600, 80, 320, and 16 are actually added. They just show the second column as it appears above. If TERC used their only general addition strategy, “Adding by Place,” the problem is converted to adding 1000, 900, 110, and 6. If positioned conventionally as in the third column above, notice that the student can’t smoothly add one column at a time, moving left to right, because 9 + 1 is not less 10. This explains why TERC omits the details. “Adding by Place” often doesn’t work when there are more than 2 numbers to be added.
- The standard algorithm for multiplication is not found in the Grade 5 Student Handbook or in the Multiplication and Division PDF, but it is mentioned in a Dialogue Box on pages 146-147 of the TERC 2008 5th grade Teacher’s Guide for Unit 7. Here it described by the teacher, and not presented as a sample of a student’s work. This is the first suggestion that it is not being recommended for student use. The teacher asks for student comments. Felix said “I sort of like it,” Georgia then hotly responded with “I disagree! I think it’s horrible. Carrying the numbers is so hard to do on paper or in your head.” TERC 2008 authors then observe “for some students the shortcut notation of the U.S. Conventional Algorithm may pull them away from making sense of the problem and keeping track of all parts of the problem.”
- This is the TERC justification for attempting to suppress the powerful concept of carrying. It’s powerful on two levels for multiplication. For example consider computing 48 x 42 using the standard algorithm. Carrying is first used to efficiently create the two partial products, 336 and 1,680, and then carrying is used to efficiently add these two partial products.

- “Keeping track” is actually handled nicely by the standard algorithms. Once mastered, the steps can be carried out automatically, and most of us eventually feel no need to write down carrying and borrowing notations. But the same can’t be said for TERC’s methods. TERC regularly warns teachers that students often lose track of what they are doing. This is somewhat hidden by a simple problem, such as 48 x 42. But consider all the steps required for 485 x 425. The standard method calls for computing and adding 3 partial products. The TERC method requires 9 partial products, first to be (consciously) identified and then computed. These 9 must then be added using TERC’s “Adding by Place” method. Notice the further complication: “Adding by Place” doesn’t work. Carrying is needed to get the leading 20 in 206,125.

400 x 400 = 160,000 160,000 100,000

400 x 20 = 8,000 8,000 90,000

400 x 5 = 2,000 2,000 15,000

80 x 400 = 3,200 32,000 1,100

80 x 20 = 1,600 1,600 20

80 x 5 = 400 400 5

5 x 400 = 2,000 2,000 1?6,125

5 x 20 = 100 100 ^

5 x 5 = 25 25 ^

An example like this is not found anywhere in the TERC 2008 materials. TERC recommends a calculator for such computations. TERC 2008 Division Strategies for 1,780 ÷ 32 [Source: TERC 2008 Grade 5 Student Handbook, pages 38 – 39]

Multiplying Groups of 32 Subtracting Groups of 32

Comments:

- TERC 5th grade students construct “multiple towers, and use “multiplication clusters” to help them solve problems in multiplication and division. It’s of particular interest here to know that TERC students construct “multiple towers” for 32 and 21. We first reported these practices in our review of TERC’s first edition [click here and see list items 2 and 10]. Note the use of 21 and 32 back then.
- The “multiple tower for 21” explains why TERC wanted to convert 48 x 42 to 96 x 21 (just above) and it explains why TERC chose 1,275 ÷ 21 to demonstrate their division strategies in the Multiplication and Division PDF. But TERC authors don’t mention this.
- The “multiple tower for 32” and “multiplication cluster for 32” explains why TERC chose 1,780 ÷ 32 to demonstrate their division strategies on pages 38-39 of the TERC 2008 Grade 5 Student Handbook. But TERC authors don’t mention this.
- Notice that TERC appears to be using the standard algorithms for addition and subtraction. If these student handbook division examples were consistent with handbook addition and subtraction strategies, students should see “Add by Place” and “Subtract by Place” methods used at this later point in the handbook. For “Add by Place,” the 960 + 640 + 160 addition should be shown converted to 1,600 + 160, For “Subtract by Place,” the 1,140 – 640 subtraction should be converted to 1,000 + (- 500), and the 500 – 320 subtraction should be converted to 200 + (-20). But this is all hidden. Guess they didn’t want it to be too messy.
- Long division, the standard algorithm for division, is not found in the Grade 5 Student Handbook or in the Multiplication and Division PDF. But it is mentioned in the TERC 2008 materials. More about that in the last section below.

### TERC 2008 Rational Number Arithmetic

Rational number arithmetic includes fractions, decimals, and percents. The methods for TERC fraction arithmetic are covered in the TERC 2008 5th grade Teacher’s Guide for Unit 4. In a page 100 “Math Note” TERC’s authors inform us that “one useful strategy students encounter in later grades for adding and subtracting fractions is finding equivalent fractions with a common denominator.” This suggests that the concept of common denominator is too difficult for elementary math. It also suggests that TERC’s methods don’t use common denominators. Now TERC may be able to avoid carrying and borrowing by limiting to carefully chosen problems, but fractions can’t be added or subtracted without the concept of common denominator.

TERC offers pictorial aids for adding and subtracting fractions. They emphasize rectangular area models and the “clock face” model. Beginning on page 160 of Unit 4 we have a 2 page “Teacher Note” titled “Adding and Subtracting Fractions.” The TERC authors say “there are two basic strategies students are developing during this unit.” They then demonstrate these two strategies by presenting two examples of student’s work for the problem 1/4 + 2/3.

- Rectangular Representation Method: TERC emphasizes 4 x 6 and 5 x 12 rectangular grid area models. In this case the student chose to use the 5 x 12 rectangle. The student says “I know the whole rectangle is 60 square units. 1/2 of that is 30, so 1/4 is 15. If we split the 60 into 3, that’s 20, so 2/3 is 40. When I add them up, that’s 55 square units altogether. So then I said the answer is 55/60.” The teacher said to “think about 5s.” The student said “I finally realized that every 5 is 1/12, and so it’s 11/12.” Nothing about reducing 55/60 to lowest terms by dividing both the numerator and denominator by the common factor, 5.
- Clock Face Method: The student says “One fourth of the way around is 3 hours. One third around is four hours, because it’s 1/3 of 12, that’s 4. So 2/3 is eight hours. That’s three plus eight is 11 hours. The answer is 11/12.

Comments:

- There are no accompanying pictures for the 5 x 12 rectangle or the clock face, because these are not really pictorial arguments. There’s no shading areas of the rectangle, and there’s no tracing movement around the clock face [12 to 3 and then 3 to 11]. Both students are actually using common denominators. The first student is using 60 as a common denominator, and the second student is using 12 as a common denominator.
- TERC suggests that they are offering pictorial methods for adding fractions, but they never offer a true pictorial example. To add fractions using one of their rectangular area models may look easy, looking back from the finished product, but carrying out a pictorial argument requires several (conscious) choices. The student must first choose the right rectangular model, then choose how to best shade two areas to represent each of the fractions being added, and then use the visual sum of the two areas to somehow see the fraction that the sum of the two areas represents.
- Each TERC model has limited use as an aid for adding fractions. The fractions to be added need a common denominator of 24 for the 4 x 6 rectangle, they need a common denominator of 60 for the 5 x 12 rectangle, and a common denominator of 60 (or possibly just 12) is necessary for the clock face. TERC avoids examples where either of the fractions to be added has a denominator that contains a factor of 7, 9, or 11.
- TERC offers a “shaded strip” linear model as an aid for adding fractions. The only example is found on page 53 of the Grade 5 Student Handbook. The handbook example uses two strips, each divided into 8 segments. If you look closely, the student adds 3/4 + 5/8 + 1/2 by converting to 6/8 + 5/8 + 4/8. There’s no mention of the fact that 8 is used as a common denominator.
- TERC devotes little space to the subtraction of fractions. On page 161 of Unit 4 they say “students approach subtraction of fractions in same way they solve addition problems, through using fraction equivalencies and representations.” By “fraction equivalencies,” they appear to be quietly recognizing the need to convert to a common denominator.
- TERC does not cover multiplying two fractions, but this should be a 5th grade topic.
- TERC does not cover dividing two fractions. It is acceptable that this be delayed until the 6th grade. [Note: TERC promoted the first edition as a K-6 math program, but it’s just K-5 for TERC 2008.]
- TERC says that they do not cover multiplying a fraction and a whole number, but they regularly use phrases such as “1/3 of 12.” Now the elementary math interpretation of “1/3 x12” is “1/3 of 12.” So TERC is further along than they think.
- TERC immediately moves to fractions as division, with the division carried out by calculator. A lengthy section, titled “Fractions to Decimals on the Calculator” begins on page 58 of Unit 6. The teacher is instructed: “all students need access to a calculator during this discussion. Write these fractions on the board: 1/2 1/4” The teacher is told to ask the students “how I could use a calculator” to “find the equivalent decimals for these fractions.” Nothing about 1/2 = 5/10 = .5 or 1/4 = 25/100 =.25.

TERC does offer methods for comparing fractions without converting to common denominators. Here are two examples of student’s work from pages 152-153 of Unit 4.

- Comparing Fractions to 1: Samantha compares 4/5 to 7/8 by arguing “7/8 is only 1/8 away from 1. But 4/5 is one fifth away from one. An eighth is smaller than a fifth, so 7/8 is just a little smidge away from one. 7/8 is closer to one, so it’s bigger.”
- Comparing Fractions to 1/2: Shandra compares 2/5 to 3/8 by arguing “For 3/8, you need another 1/8 to make a half. For 2/5, you need half of a fifth to make a half. That’s the same as 1/10, so 1/10 is smaller than 1/8, so 2/5 is closer to 1/2. This means that 2/5 is more.”

Comments:

- If Samantha converted 4/5 to 32/40 and 7/8 to 35/40, she would easily see that 7/8 is exactly 3/40 more than 4/5. Similarly, if Shandra converted 2/5 to 16/40 and 3/8 to 15/40, she would easily see that 2/5 is exactly 1/40 more than 3/8.
- TERC’s methods require considerable time and considerable conscious thought. Comparing two fractions by converting both to a common denominator is quick and becomes an automatic skill. This method also yields the exact difference between the two fractions.

Although they don’t shout it from the rooftops, TERC 2008 clearly encourages the use of class cheat sheets. Here are some examples:

- The “Class Equivalents Chart” includes a list of all fraction equivalents for fractions between 0 and 1 with denominators equal to 2, 3, 4, 5, 6, 8, and 10. The denominators 12, 24, and 60 are also included for convenience when working with the 4 x 6 rectangle, 5 x 7 rectangle, and clock face model.
- In Unit 4 students create a table fraction-percent equivalents, and in Unit 6 students create a table of fraction-decimal equivalents.
- Here’s one use of the fraction-percent equivalent chart. TERC students add fractions by converting to percents, adding the percents, and then converting the percent sum to a fraction.

- The “Fraction Track” is a “linear model” consisting of “a set of seven parallel number lines that show the relationship between fractions with the denominators 2, 3, 4, 5, 6, 8, and 10.” This model gives a visual presentation for all fractions with one of these 7 denominators [and also between 0 and 1]. Thus, for example, it is possible to visually compare 4/5 to 7/8 and 2/5 to 3/8. [Recall these two example from the comparing fractions examples given just above.]
- On page 105 of Unit 4, teachers are told to add the following equations to the “Faction Addition Equations” class chart.
- 1/3 + 1/6 = 1/2 1/4 + 2/4 = 3/4 1/4 + 1/3 = 7/12 1/4 + 2/3 = 11/12
- 5/6 + 1/4 = 1 1/12 1/2 + 5/12 = 11/12 1/2 + 1/3 = 5/6 3/4 + 1/6 = 11/12

- On page 130 of Unit 4, teachers are to add the following equations to the “Fraction Subtraction Equations” class chart.
- 8/10 – 3/10 = 4/8 8/10 – 4/8 = 3/10 8/10 – 1/2 = 3/10
- 8/10 – 3/10 = 1/2 8/10 – 3/5 = 2/10 8/10 – 2/10 = 3/5
- These are all for 8/10. How long is this chart?

TERC 5th grade students spend considerable time working with fraction, decimal, and percent equivalents. But there are some fundamental problems about what students come to understand about the meaning of “equivalence.”

- On page 47 of Unit 4, the teacher asks students to explain how they would determine that a fraction is equal to 1/2. One student says ” If you divide the numerator by the denominator on the calculator, like if you had 8/16, if you divide 8 by 16 you get 0.5 which is the same as 1/2.” There’s no further comment, so this explanation is satisfactory for TERC. But the TERC student shouldn’t need to convert to a decimal. By the 4th grade, the student should know that two fractions are equivalent if the numerator and denominator of one fraction can both be computed by multiplying (or dividing) both the numerator and denominator of the other fraction by the same non-zero whole number. Thus 1/2 is obtained from 8/16 by dividing both 8 and 16 by 8. Alternatively, 8/16 is obtained from 1/2 by multiplying both 1 and 2 by 8.
- It’s not clear that TERC students have a conceptual understanding of the basic relationship between fractions and decimals and fractions and percents. Fifth graders should understood 0.1 as another way to write 1/10, and they should understand 1% as another way to write 1/100.

TERC offers 10x 10 rectangular grids as a pictorial aid for adding decimals. They also promote using their” Adding by Place” method for whole number addition. Here are some examples from pages 63-65 in the TERC 2008 Grade 5 Student Handbook. The problem is 0.4 +.25.

- Deon said “I used different colors to shade the decimals on a 10 x 10 square. The total is 6 tenths and 5 hundreds, or 0.65.
- Alicia said “0.4 is close to 1/2 and 0.25 is the same as 1/4, so I knew the answer should be close to 3/4 or 0.75.”
- Zachary said “Since 25 + 4 = 29, at first I thought the answer would be 0.29, but I could tell from Deon’s picture that 0.29 didn’t make sense. So, I added by place. I added the tenths, and then the hundredths. 0.4 is 4 tenths and 0 hundredths. 0.25 is 2 tenths and 5 hundredths. 0.4 + 0.2 = 0.6. 6 tenths and 5 hundredths is 0.65.”

### Faced with TERC 2008? A Compact Guide for Parents

The major objective of this paper has been to arm parents with the specific information needed to effectively oppose the implementation of TERC 2008. Short of that, this information should help parents achieve maximal class time for the mastery of standard arithmetic. Remember the following points when you communicate with those who control your child’s math education.

- NMP 2008 is very clear about the foundations of algebra. Mastery of standard arithmetic is essential.
- Mastery isn’t possible without practice. TERC provides no practice for standard arithmetic. Suggest that Singapore math materials be used as a supplement. Singapore textbooks and workbooks are great for practice and the textbooks offer a clear, child-friendly development of elementary math ideas. And they’re reasonably priced!
- The National Council of Teachers of Mathematics (NCTM) is now clear about the fundamental importance of standard arithmetic. In their 5th Grade Focal Points, they say that 5th grade students should develop fluency with “the standard algorithm for dividing whole numbers” (long division). Also, they should learn to “represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators” (convert to a common denominator). And they “should develop proficiency with standard procedures for adding and subtracting fractions and decimals.”
- TERC knows this! The 5th grade NCTM Focal Points are found on page 109 of the TERC 2008 5th Grade Implementation Guide for Teachers. TERC offers a table that indicates where these topics are (supposedly) covered in TERC 2008.

- TERC promotes the weaker “fluency” for single digit number facts. You want your child to know these facts automatically, without conscious thought. You want instantly known, not quickly figuring it out.
- TERC promotes “mental math” methods and the use of calculators. They offer one “general” whole number computational strategy for addition, subtraction, and multiplication. Relative to the standard algorithms, TERC’s alternative methods more openly reveal underlying place value details. The price for this “transparency” benefit is significantly reduced computational efficiency. The TERC alternatives also reduce the need for carrying and borrowing. But TERC is not satisfied with reducing the need, they want elimination of the need. The price for this “total avoidance of carrying and borrowing” benefit is significantly reduced generality. TERC must carefully limit to special case problems to achieve this objective. As we’ve noted above, TERC doesn’t want to admit the special case limits, so we get the suppression of the embarrassing carrying and borrowing details. We also see borrowing avoided by the premature introduction of negative numbers and integer arithmetic. Somehow these advanced concepts are easier than the important concepts of carrying and borrowing.
- On page 74 of the TERC 2008 5th grade Teacher’s Guide for Unit 7, a “Math Note” includes this quote: “just as students may use standard algorithms for other operations, they may use long division to solve division problems.” Key point: TERC says that students may choose to use the standard algorithms for all operations. TERC also says that each student should choose their own computational methods. This primacy of personal choice philosophy was also promoted in TERC’s first edition, but teachers were then advised how to discourage choosing a standard algorithm. With TERC 2008, the attempt to discourage may still be present, so parents should make sure that their child can explain carrying and borrowing. Have them ready to explain for simple problems, such as 28 + 14 and 54 – 27.
- TERC misleads students when they they attempt to suppress the concept of common denominator and when they suggest the use of a calculator to convert 1/4 to 0.25.
- TERC dwells on concrete and pictorial methods. At most, such methods may be helpful to demonstrate an idea, but they have no long term value. Children need to move to abstract level, sooner rather than later.
- TERC emphasizes the use of calculators. But easily acquired calculator skills will not help when it comes to learning algebra.

Bill Quirk is a graduate of Dartmouth College and holds a Ph.D. in Mathematics from the New Mexico State University. He co-authored The State of State Math Standards 2005, a report published by the Fordham Foundation.

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