For a brief analysis of TERC 2008 math, click
on TERC 2008 Math vs.
NMP 2008 Math: A Snapshot View.

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's Investigations 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

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

- 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]

- 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.

Algebra is a
multifaceted area of
mathematics content that has been described and classified in different
ways. Across many of the classification schemes, four areas
foundational to the study of algebra stand out: (1)
generalizing and formalizing patterns; (2) representing and analyzing
the structure of numbers and operations; (3) using symbolic notation to
express functions and relations; and (4) representing and analyzing
change.

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.

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.

- 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.

- 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.

- 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.

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]

6,831 6,831 6,828 (-3)

+ 1,000 + 2,000 + 1,900 (+3)

7,831 8,831 8,728

+ 800 - 103

8,631 8,728

+ 90

8,721

+ 7

8,728

Adding by Place >>>>

1 1 << Carrying

6,831 + 1,897 = 6,831 6,831

6,000 + 1,000 = 7,000 + 1,897 + 1,897 <<< Standard

800 + 800 = 1,600 7,000 8,728 Algorithm

30 + 90 = 120 1,600

1 + 7 = 8 120

8,728 8

8,728

Comments:

- "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.

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

Change the Numbers Subtract by Place

34 <<Borrowing

3,451 - 1,300 = 2,151 3,451 3,451

2,151 + 13 = 2,164 - 1,287 - 1,287

2,000 2,164

200

- 30

- 6

2,164

Comments:

- 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.

Breaking Numbers Apart Change one Number Create an

Equivalent Problem

40 x 40 = 1,600 1,600 1,000 48 x 42 = 96 x 21

40 x 2 = 80 80 900 50 x 42 = 2,100

8 x 40 = 320 320 110

8 x 2 = 16 16 6 2,100 - 84 = 2,016

2,016 1,?16

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. Carrying is often needed in such cases.

- 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 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

30 x 32 = 960 << 30 times 32 1,780

20 x 32 = 640 << 20 times 32 - 640 << 20 times 32

5 x 32 = 160 << 5 times 32 1,140

1,760 - 640 << 20 times 32

500

1,780 ÷ 32 = 55 R20 - 320 << 10 times 32

180

- 160 << 5 times 32

20

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 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.

- 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.

- 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."

- 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.

- 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?

- 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.

- 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."

- 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.