TERC Hands-On Math: The Truth is in
An Analysis of The First Edition of
These Clickable Links Serve as an Outline of This Paper:
Program: A Summary View
Developed by TERC,
with funding from the National Science Foundation (NSF), Investigations
in Number, Data, and Space purports to be "a complete K-5
curriculum that supports all students as they learn to think
Here, using extensive quotes from TERC's book for teachers, Beyond
Arithmetic, and TERC's Fifth Grade Teaching
we'll reveal how TERC has redefined the meaning of "think
The NSF is now spending millions to
implementation of the TERC program. School Boards find it
to say no. They rationalize: "it's just a different way to teach
math, and the NSF backs it, so how bad can it be?" This
is very bad because it omits standard computational methods, standard
and standard terminology. TERC says most of this is now obsolete,
due to the power of $5 calculators. They claim their program
"beyond arithmetic" to offer "significant math," including important
from probability, statistics, 3-D geometry, and number theory.
But math is a vertically-structured
domain. Learning more advanced math isn't possible without first
mastering traditional pencil-and-paper arithmetic. This truth is
demonstrated by the shallow details of the TERC fifth grade
Their most advanced "Investigations" offer probability without
fractions, statistics without the arithmetic mean, 3-D geometry without
formulas for volume, and number theory without prime numbers.
TERC Rejects Standard Knowledge and
Major Characteristics of TERC's
Investigations in Number, Data, and Space
- TERC regularly suggest that
better than "standard." This may be true for arts and crafts, but
standards and conventions are essential for mutual understanding and
communication in business, science, and professional life.
- TERC says that each student must
collaboratively" with other students in a small group. Progressive
believe this helps to prepare kids for the way teams function in modern
business. But teams in business don't watch Jamal do it. Business
team members typically work alone, functioning as specialists, as they
carry out their assigned portion of the general task. They may
entirely in writing and never physically meet other members of a
team. They may participate in multiple teams at the same time.
- TERC insists on the ongoing use of
tools (manipulatives, "models," and calculators).
TERC rejects the need for
- They say concrete tools must
and regularly used.
- TERC strongly rejects the idea
must eventually migrate from hands-on to abstract thinking.
TERC fails to clearly define terms.
- They say that
familiarity with single-digit number facts must "grow out of
lots of experience with constructing these facts on their own."
BA, Page 72 (emphasis added)
on References for the meaning of the
- There's no additional gain in
associated with the task of trying to "construct" one more basic number
- TERC doesn't think it's
memorized information. But knowledge must first be loaded into the
before it can connected to other knowledge and "understood."
memorization is sometimes the most efficient way to get it there.
- TERC fails to understand that
to move to automatic use of knowledge. The mind must be free to think
higher levels of complexity, without consciously revisiting underlying
details. For example, the key idea of the standard algorithms is that
calculations are reduced to multiple single-digit calculations.
children don't have instant recall of the single-digit number facts,
aren't equipped with the essential pre-knowledge for easily carrying
TERC emphasizes "familiar numbers."
- They regularly state: "We don't
to learn definitions of new terms."
- They offer some "definitions,"
multiple undefined terms to "define" a new term.
- They favor "natural language"
TERC omits standard formulas.
TERC emphasizes estimation and
- The "landmark
numbers" are 5, multiples of 10, and multiples of 25.
- Landmark number are also known
- The "familiar
are limited to proper fractions with denominator equal to 2, 3, 4, 5,
8, 10 or 12. Thus 7 and 11 are not familiar denominators.
TERC is opposed to gambling.
- Note: 12 is included because
TERC's clock face method for adding fractions.
- TERC doesn't believe in defining
you won't find the preceding definitions in TERC materials. This
is what they appear to mean by these phrases. We welcome their
- Although TERC rejects explicit
of basic single-digit number facts, they expect students to remember
non-basic facts about landmark numbers and familiar fractions.
TERC proudly rejects standard
- They suppress the concepts of
TERC attempts to directly teach
and misleading content.
- No standard algorithms for
- No standard methods for
- No general methods for
- TERC emphasizes special case
landmark numbers and familiar fractions.
- They claim to offer a
where students discover math as they play games and carry out
But they provide thousands of pages of teaching instructions and
scripts that identify the content they expect kids to "discover."
- Thousands of pages for the
teacher, but no
text for the student.
- What's the TERC content?
this paper provides the answer.
Philosophy of Elementary Math Education
their text, Beyond
Arithmetic (BA), for teachers who want to learn about InvestigaItions
in Number, Data, and Space. TERC uses the term "traditional"
to characterize what they oppose. They say their program offers a
Math Must Be Discarded Because It:
- Was "developed to meet the
the 19th century." BA, Page 2
- Requires that students "memorize
facts, procedures, definitions, and formulas."
BA, Page 2
- "Focuses on learning a
of procedures for addition, subtraction, multiplication, and division
whole numbers, fractions, and decimals."
BA, Page 2
- Results in "overpracticed
BA, Page 3
- Ignores the fact that "today's
have an important tool available to them: the calculator."
BA, Page 77
Manipulatives and Calculators Are Essential as Students Work
- In the "constructivist
mathematics class today", students "work together, consider their own
and the reasoning of others, and communicate about mathematics in
and by using pictures, diagrams, and models. They carry out a
number of problems thoughtfully during a class session or perhaps work
on a single problem for one or several sessions. They use more
one strategy to double check, and they use blocks, cubes, measuring
calculators and a variety of other materials to help them solve
BA, Page 42
- "Students need to do
By this, we do not mean that they are assigned specified roles, as they
are in many cooperative learning approaches. Rather, all students
in the group participate in joint problem solving."
BA, Page 54
- "Constructivist curricula
that tools and manipulatives be available to students at all
Students at all levels need to be able to use whatever tools they
to solve problems, and should make their own decisions about which
might best help them solve a problem."
BA, Page 62
- " 'Having materials readily
should convey to students the expectation that they will use these
to solve problems. Traditionally, concrete objects like blocks and
and even fingers were considered babyish; although sometimes used
to introduce a new concept, they were quickly dismissed in favor of
and paper. Even worse, those students who needed extra help were
often stigmatized by 'having to use materials' because they 'didn't get
it' as quickly as others. In the traditional mathematics
even the youngest student quickly gets the message that the goal of
math is to use symbols and do it on paper. In a constructivist
all students - as well as the teacher - use tools, drawings, and
to solve problems. Manipulatives offer more ways for a range of
to enter and persist at difficult problems, a way of keeping track of
and a way of representing solutions to problems."
BA, Page 63
- Note: The
track" occurs regularly in TERC teaching materials. Because TERC has
traditional computational methods, they have lost the "keeping track"
aspects of pencil-and-paper methods. If the student is taught to
add 56 + 43 by "counting on" from 56, the student needs to keep
of (56 + N) and (43 - N), where N runs from 1 to 43 . Manipulatives
the day. Put down 43 plastic discs and count 57 for the fist
58 for the second, until you reach the last disc.
TERC Says Their
- Creates an environment where "students
are doing, thinking, and talking about significant mathematics."
BA, Page vii
- Carries out NCTM recommendations "for
a shift from teaching students procedures to teaching them to think and
reason mathematically. This shift is required by the more complex
demands of today's society. Employers no longer look for
who can apply memorized procedures to do rote calculations - everyone's
pocket calculator takes care of these quite efficiently."
BA, Page 4
- "Provides a coherent set of
that allow all students at each grade level to explore important
mathematical ideas in the use of number, data, geometry, and the
of change." BA, Page 18
- Recognizes that it isn't "possible
to present all the worthwhile mathematics that might be included at a
grade level," and is is better to "serve students by helping them to
some topics in depth." BA, Page 32
- Allows students to "revisit
ideas of the elementary curriculum year after year, from different
BA, Page 32
- Supports the fact that
developers "do not expect students to be able to to do difficult
computation with fractions and decimals, multidigit multiplication, or
long division." BA, Page 33
- Recognizes that "whole
is an area in which the elementary mathematics curriculum needs to be
down and deepened Rather than being hurried into complex
students need time to develop strategies based on numerical reasoning."
BA, Page 34
Searching For "More
TERC points to Steven Leinwand's 1994
Time to Abandon Computational Algorithms. They say he
"eloquently describes the changes that calculators should be bringing
our mathematics classrooms" when he wrote: A few short years
we had few or no alternatives to pencil-and-paper computation. A
few short years ago we could even justify the pain and frustration we
in our classes as necessary parts of learning what were then important
skills. Today there are alternatives and there is no honest way
justify the psychic toll it takes. We need to admit that drill
practice of computational algorithms devour an incredibly large
instructional time, precluding any real chance for actually applying
and developing the conceptual understanding that underlies mathematical
literacy. BA, Page 78
TERC says "students must learn a
great deal more mathematics than what we considered sufficient in the
and that we must make room for more and deeper mathematics in the
In the past, the elementary curriculum focused almost entirely on
work with arithmetic, while the study of geometry, data, number theory,
and other important aspects of mathematics were relegated to a few
exercises or chapters that were never reached."
BA, Page 4
There you have it. Now that
drudgery is out, there's time to do the interesting stuff. TERC
a K-5 math learning environment where students are doing "significant
dealing with "big ideas," and "learning a great deal more mathematics
what we considered sufficient in the past." TERC offers "more and
deeper mathematics." Their students "explore important
ideas in the use of number, data, geometry, and the mathematics of
But significant math isn't possible
the facts and skills that TERC has excluded. As evidence
the TERC Fifth Grade Teaching Materials shows
TERC's "everyday math" emphasis has led them to offer:
claims that their fifth grade materials are "Also appropriate for
- One-half as a big idea, while not
the major ideas of place value and the distributive law.
- Statistics without the concept of
- Probability without operations
- 3-D geometry without formulas for
- The "mathematics of change," where
they're helping to prepare kids for calculus by teaching them their
for describing change.
is a Big Idea for TERC
"Students should revisit the big ideas of the elementary curriculum
year after year, from different perspectives. Consider the idea
fractions are equal parts of a whole. What is one-half?
It is a fraction of a pizza, or of the distance from my house to yours,
or of a group of people. Half is the result of dividing
cookie among two people, or it is one out of every two cookies, or it
the number of cookies in one of two equal piles. Half the money
in my pocket is a different amount than half the money in your
BA, Page 32
Isn't a Big Idea for TERC
"This difficult idea needs to be addressed over many years as
thinking about one-half becomes deeper and more complex ... in the
grade, students might use fractions to describe data they've
14 out of 26 students were born in this state, and that's a little more
than half of our class. Through experiences like these, students
a model of 1/2, and its relationship to other fractions and to whole
BA, Pages 32-33
TERC and the NCTM rave on about "the
and beauty" of mathematics, but they both fail to see it in the concept
of place value. The topic isn't discussed in TERC classrooms, but
mentioned in some teaching units and on the web in the essay CESAME:
Place Value. The essay author illustrates TERC's
way to discuss place value: "The '2' in 24 represents 20, the '2' in
represents 200." (PV, Page 1) True, but this isn't the
that gets students to appreciate "the power and beauty" of our decimal
number system and the standard algorithms for multi-digit computation.
Students must eventually come to
that every real number has a unique, beautifully compact representation
in terms of powers of 10. First we want students to think: 247 =
2 x 100 + 4 x 10 + 7. Next we want them to think: 247 = 2 x
102 + 4 x 10 + 7. Eventually we want them to think:
= 8 x 103 + 2 x 102 + 4 x 10 + 7 + 9 x 10-1
4 x 10-2 + 2 x 10-3. Then, students
ready to appreciate the powerful fact that any complex
can be reduced to multiple single digit calculations, where the single
digits are the coefficients of the powers of 10.
For example, 54 x 247 can be reduced to
six single digit products (4 x 7, 4 x 4, 4 x 2, 5 x 7, 5 x 4, and 5 x
To fully appreciate this simplification and what carrying is all about
, students need to first work with numbers in expanded form
54 = 5 x 10 + 4 and 247 = 2 x 102 + 4 x 10 + 7). They
need to be comfortable, using repeated applications of the distributive
law, to multiply out "the long way," with a chain of
equalities and regular rearrangements of component expressions using
commutative and associative laws. Eventually they will come to
the power and generality of "carrying" and respect the compact and
standard algorithm for multiplication.
Once students move
to fully automatic use of a standard algorithms, they may forget the
rational. This is perfectly natural. Our brains regularly push
into the background in order to free the mind to focus on the next
of knowledge. A good teacher recognizes when it's appropriate to pull
knowledge back into the foreground. The knowledge can then be
and connected to knowledge at the next level of
For example, the student might be reminded of the underlying rational
multiplying 54 times 247, when learning how to multiply (5x
+ 4) times (2x2 + 4x + 7).
The Distributive Law
a Big Idea for TERC
states: A x (B +C) = (A x B) + (A x C) [For any three real
A, B, and C.] This major idea isn't covered by TERC, but it
explains many of the simple case computation methods they recommend.
distributive law is of fundamental importance in arithmetic and algebra
because it's the property that "connects" multiplication and
TERC is excited about "connections," but fails to see any in the
Calls It Data: Statistics Without The Arithmetic Mean
"You are likely to have students who suggest the arithmetic
mean, or as they may call it, the average. They may know
to find it with the "add-'em-all-up-and-divide-by-the-number"
Although this algorithm is often taught in elementary school, research
has shown that it is often not understood, even by older students and
At this point, it is better to stay away from the mean and the
it may introduce." U9, Page 6
TERC Recommends Natural Language and
Words, Not Standard Terms.
[Please click on References for the
of the U9 code.]
A Major Statistics Investigation - At
End of the Fifth Grade
- Each TERC curriculum unit book
teachers: "In the Investigations curriculum,
vocabulary is introduced naturally during the activities. We
ask students to learn definitions of new terms."
U9, Page I-22 (as one reference)
language for statistics: "To help students pay
to the shape of the data - the patterns and special features - we have
found useful such words as clumps, clusters, bumps, gaps, holes, spread
out, and bunched together."
The following example of TERC
math" is a compact version of Investigation 4: A Sample of Ads
in TERC's "Data: Kids, Cats, and Ads, the 159 page "Statistics"
unit, the ninth and final unit in the TERC Grade 5 program.
- The teacher tells the
students: "at USA
Today, the goal is to sell ads to fill two-fifths of the paper."
U9, Page 70
- The teacher says: "we're
to look at sample issues of USA Today to find out whether they
their goal of two-fifths ads." U9, Page 70
- The teacher divides the class into
of two or three students" and says: "First your group should choose a
that has some ads, but that isn't all ads. Then use any strategy
you want to figure out the fraction of ads on the page. Try at
a couple of ways. Work with familiar
like tenths, sixths, eighths, fourths, thirds, and other fractions you
can easily figure out. Your job is to get a good estimate, not an
exact number." U9, Page 71
The teacher begins the second
each group several paper sheets, each containing six Recording
These paper strips are 3 inches long and pre-marked with familiar
fractions, such as 1/4, 2/5, and 7/8.
The teacher says "each of
gives you six Recording Strips. For each page of the newspaper you
you'll need to record what fraction of that pages is ads. Use a
Recording Strip for each page. On the strip, color the fraction
the page that is ads. If a page has no ads, you still need a
for it; just leave the whole strip uncolored. If a page is all
ads, you would color the whole strip. What would you color if a
has one-eighth ads?" U9, Page 74
The teacher gives each group a
of USA Today and says: "Each group will get a whole
with all the sections, and your job is to figure out what fraction of
paper is ads. You won't have enough time to do this for every
so you need to choose a sample of pages from the paper. You will
have time to figure out the fraction of ads for 10 to 15 pages of your
paper. What are some reasonable ways of sampling the paper?
How will you get a representative sample?"
U9, Page 75
- TERC advises the teacher to "expect
to see a variety of strategies for figuring out fractions. Some
may cut up the page and rearrange it like puzzle parts to figure out
fraction is ads. Others may use markers to color the ads and
the fraction of area they take up."
U9, Page 72
The teacher begins the third
"Today, each group will put together their Recording Strips to
out what fraction of their sample of the newspaper is ads. Here's
the main question we're asking: If all the ads in your sample
grouped together, how many pages would they fill up? What
of the pages in your sample would that be?"
U9, Page 77
- A Dialogue
(U9, Page 76) illustrates the sample selection analysis:
- "Shakita: How about if we took every third page? Then
if it were
36 pages long, we'd have a sample of 12. If it's 48
we'd have a sample of... Does someone have a calculator?" U9,
- "Tai: [Entering 48 ÷ 3 on a calculator]
It would be
16. That's pretty close to 15. It wouldn't matter so much
we didn't do the last page." U9,
TERC instructs the teacher to
does this whole paper tape now represent (all the sampled pages added
How might we use these Recording Strips to figure out what fraction of
these pages is covered with ads?"
- TERC advises the teacher: "Explain
that while there are many different ways of finding the combined
everyone in the class will use the same method so that it will be
to compare the findings. Demonstrate this method, by taping your
30-inch piece of adding machine tape to the board. Then fasten
ten colored-in Recording strips so that they cover it. Glue stick
with a removable adhesive or removable tape is best, since you will be
removing the Recording Strips, cutting them, and retaping them."
U9, Page 77
- TERC informs the teacher: "The
idea is that the colored-in part and the blank part of each strip need
to be cut apart so the colored parts can be grouped together.
the Recording Strips down and cut off the colored-in part of each
Save the blank pieces to show later how they fill in the rest of the
Tape the colored-in pieces onto the paper tape, starting from one end
the tape and putting the pieces right next to each other."
U9, Page 78
- TERC instructs the
each group a piece of adding machine tape that is 3 inches times the
of pages they have sampled. For example, if they have sampled 12
pages, give them a 36-inch length of adding machine tape. The
cuts each of their Recording Strips into two pieces and fastens down
colored-in parts, starting at one end of the tape."
U9, Page 78
- Note: Recording Strips are 3
- TERC instructs the
the groups finish, ask them to write the day of the week, the date, and
the fraction of ads in their sample on the blank part of their
They can fold their strips to figure out this fraction, or use any
strategy that makes sense to them. To use the same technique they
used with Data Strips, students may make another strip to fold and use
as a fraction strip to compare with their combined Recording Strip."
U9, Page 78
Significant Statistics At The End of the
Without Fractions: It's Guess and Check
- Eyeball estimates to associate a
fraction with the (combined area) of the ads on a page.
- The central role of manipulatives
Strips and Data Strips)
- Fifth graders who can't add familiar
fractions and need a calculator to divide 48 by 3.
- Fifth graders engaging in
taping, and other kindergarten activities.
Elementary probability is a math topic
that's "beyond arithmetic," but quite accessible to fifth grade
This looks like a natural! TERC emphasizes counting, and advanced
counting techniques are naturally connected to elementary
First the fundamental
counting principle, then permutations and combinations, all with
excitement of factorials. This only requires prior mastery
fractions. Ooops! There's the problem. TERC
teach kids how to compute with fractions. No problem! TERC
discards the traditional content of elementary probability and
the "language of probability" and "predict and check." Here are
illustrations from Between Never and Always, the
unit in TERC's fifth grade program.
Too Far "Beyond Arithmetic" for TERC:
If we flip the coin two times, what 's the probability that both flips
will be heads?
- The teacher speaks to the
the next couple of weeks we're going to be studying probability.
Learning about probability helps you figure out how likely it is that
event will happen. We'll be particularly interested in studying
that fall somewhere between the points marked impossible and certain,
so we'll need some words to describe the middle ground."
U4, Page 5
- After much discussion, the word unlikely
is associated with the probability of 1/4, maybe with 1/2, and
likely with 3/4. Then students spend considerable time
events into the five categories.
- TERC provides two "definitions"
"theoretical probability ...describes what would happen
theory' ... .experimental probability ... describes what happens
when we do an experiment." U4, Page
14 Simple as that!
- Using a variety devices, including
and bottle caps, students are repeatedly asked to first predict and
test their predictions. U4, Pages 15-26
- Example: Spin the "one-fourth
instructs the teacher:
"ask students to jot their predictions on scrap paper.
their predictions on the board or overhead. Since 1/4 of 50 is 121/2
both 12 and 13 are likely predictions, but be prepared for a variety of
responses." U4, Page 19
- TERC introduces their concept of
- "If we flip the coin 10
times, we have
an expectation that 1 out of 2 flips will be heads, so that we expect
get 5 heads out of 10. We call 5 heads in this case the expected
number." U4, Page 24
- "Since the probability of
guessing a spin is 1/4 , the expected number of
guesses in 20 spins is 1/4 of 20, or 5."
U4, Page 30
- Then more experiments to test
- Now, armed with this new
students predict with more confidence as they experience sessions in
Guessing Skills" and "Guessing Skills Distributions." But they
up losing faith in the power of theory when they continually discover
number" predictions that differ greatly from experimental results. U4,
- Investigation 2, Fair and
concludes unit 4. Here kids learn that "not fair" means that "players
different chances of winning." U4, Page 46
Calls it Space: Volume Without Formulas
Containers and Cubes: 3-D Geometry:
Volume is Unit 8, the next to last unit in TERC's fifth grade
Co-author Michael T. Battista has distinguished himself with his
Miseducation of America's Youth Here you will lean how
he thinks fifth (or sixth) graders should be educated in 3-D geometry.
"Content of This Unit: By packing rectangular boxes
with cubes, students develop strategies to determine how many cubes or
packages fit inside. They explore the concept of volume,
strategies for finding the volume of small paper boxes and larger
such as their classroom. They investigate volume relationships
cylinders and cones and between pyramids and prisms with the same base
and height. They also learn about the structure of geometric solids and
improve their visualization skills."
U8, Page I-12
"As they work through the unit, most students will come to determine
the number of cubes in rectangular boxes by thinking in terms of
'A layer contains 3 x 4 or 12 cubes, and there are 3 layers so there
36 cubes altogether.' Traditionally, students have been taught to
solve such problems with a formula learned by rote: Volume = length
x width x height. They plug in the numbers and perform the
without thinking about why or how the formula works. For
use of the formula, students need to first understand the structure of
3-D arrays of cubes. We strongly discourage teaching this
to students; the layering strategies that they invent will
more powerful." U8, Page
(bold and underline emphasis added)
What Happens in the Classroom: A Month
- Students don't "invent." As
begins they are taught to fill a box with layers of plastic cubes and
count the cubes. Kids make the boxes, using graphing paper,
scissors, and tape. The term volume is not yet mentioned. After
cube counting strategies, TERC expects kids to "reason that the length
gives the number of cubes in a row and the width gives the number of
in a layer, so the number of cubes in a layer is the product of the
and width. Because the height gives the number layers, they
the number of cubes in a layer by the height to find the total number
cubes in the array." U8, Page 14
TERC then changes the focus to box
and links to the big idea of one-half.
U8, Pages 16-23
- Good idea! But more TERC
inventing. Note that they've introduced the idea of the volume formula,
but presented it as a way to efficiently count plastic cubes, with the
subtle detail that the use of this "cube counting" formula is
to the very special case of the supplied plastic cubes exactly filling
- Generalize later
get to dimensions that aren't an exact multiple of the length of the
of one of their plastic cubes. Forget about dimensions that
expressed in whole numbers.
Next it's filling boxes with
of 2 or more cubes. U8, Pages 24-37
- "Student pairs determine
of boxes that will hold twice as many cubes as a box that is 2 by 3 by
- Shakita's group "found a
to generate several spatially meaningful solutions" to the doubling
She explained "we made two packages of cubes that were 2 by 3 by
When we put them next to each other one way, we got a package that's 4
by 3 by 5. When we put them next to each other another way, we
a 2 by 6 by 5, then we got a 2 by 3 by 10."
U8, Page 23
- TERC likes this
hands-on and it demonstrates multiple correct answers.
- TERC fails to point out that
other solutions that can't be represented this way.
Enter plastic "centimeter cubes."
- The cubes are
high tech stuff!
- Kids learn that multiple copies
package may not always exactly fill a particular box.
- Solution? Design the box
of the method of Shakita's group.
Then the term "volume" is
call the amount of space inside this box its volume. The
of a three-dimensional object is the amount of space enclosed by its
boundary." U8, Page 42
(emphasis in the original)
- TERC instructs the teacher to "show
a centimeter cube as you introduce the first activity."
U8, Page 40
Students build "models of
They "build a cubic meter using 12 meter sticks, joined at the
with masking tape." U8, Page 44
Students measure "the space
classroom." U8, Pages 38-60
- Now we know why it's
Data, and Space.
- Using multiple undefined terms,
is general, not limited to boxes, and avoids formulas.
Students start "Comparing
"After measuring how much rice or sand small household
will hold, students order them from least to greatest volume."
U8, Page 62
- The teacher says: "You
sticks, string, calculators, and any other tools we have."
U8, Page 46
- Team answers ranged from 220 to
meters. U8, Page 52
Next it's "Comparing Volumes of
U8, Pages 66-74
- Students are told to "start
the order, then measure to check."
- "To test their
directly compare two containers by pouring rice from one container into
another. They might pour from a smaller container into a larger
see that the larger is not filled. Or, they might pour from a
container into a smaller and see an overflow."
U8, Page 65
Enter "a special measuring tool -
graduated prism." U8, Pages 74-78
- Students are given patterns for
They "use scissors and tape to create the prisms, pyramids,
and cones from the patterns." U8,
- "Each solid is paired
that has equal base and/or height measurements - pyramids are paired
related prisms, and cones with related cylinders."
U8, Page 66
- TERC advises: "If the
is fairly accurate, students should discover that the volume of each
flat-topped solid is about 3 times the volume of the smaller, pointed
if it has the same base and height. (Because of inevitable measurement
and construction errors, the 3-to-1 relationships won't be exact.)"
U8, Page 67
- How do they discover? By
"As a final project,
and create a model made from geometric solids. The model must
prisms, pyramids, cylinders, and cones, all of which students can
from patterns." U8, Page 81
- TERC graduates don't need
Just plenty of rice and this plastic tool.
- Students use it to determine "the
in cubic centimeters, of each of their 11 solids."
U8, Page 74
- Rice is poured, from each of
into the plastic measuring tool.
- TERC advises: "Loosely
sides can cause surprisingly large discrepancies. Thus, it is
to evaluate students' methods, rather than their answers."
U8, Page 78
- "When we had to find the
the robot we started with the body because it was a rectangular prism
centimeter paper and we could count the dimensions. We multiplied
the dimensions and found the volume. With the other shapes we
count the dimensions , so we filled the shapes and then poured that
the see-through prism with the centimeters measured on it. We
those dimensions and multiplied them together the same way we did for
rectangular prism." U8, Page 86
- Student thinking has moved to
of abstraction. They now count using centimeter paper. They
are free of the centimeter cube crutch.
Significant Geometry at the End of the
- Discovering exact relationships by
sand. That is significant! Let's call the New York Times.
- More kindergarten activities:
taping, pouring, and building a paper robot.
- Relative, not absolute
The volume of this container is less than the volume of that container,
but no way to compute the absolute volume of either container.
No mention of formulas for the
- Wait, there is a way!
prisms" are always conveniently available these days.
"Mathematics of Change" >>> Moving Towards Calculus
"In calculus, students
learn that ... rates of change may not be constant. For example,
a baby grows fastest right after it is born, then growth slows down
adolescence, at which point it speeds up. Describing the rate at
which something is speeding up or slowing down is an important part of
calculus." BA, Pages 10-11
TERC's Recommended Natural Language For
Describing Change: Grow, shrink, faster, slower, steep,
slow, steady, speed up, slow down, grows steadily, grows faster and
grows slower and slower, shrinks steadily, shrinks slower and slower,
faster and faster, grows and then shrinks, oscillates between growing
shrinking. U7, Pages 21, 54, and 98
"Figuring out how something grows or declines is essential, not
in higher mathematics, but in the sciences and social sciences as well."
U7, Page I-17
There you have the essence of Unit 7, Patterns
of Change in TERC's fifth grade program. It's all about using
TERC's language to describe growth and movement. Enough said.
The Math Ideas
in TERC's Book For Teachers
Here we present all the math
found (in page sequence) in Beyond
Arithmetic. Why all? It's another way to
how much is missing and the total lack of big ideas and significant
- "I have 1/2
of flour and need 11/4 cups of flour; how much
flour do I need? If I have good number sense of these
familiar fractions, their magnitudes, and their relationships to
other and to 1, I would be unlikely to use the traditional subtraction
algorithm (11/4 - 1/2
which requires that I find common denominators, transform the mixed
into an improper fraction, then subtract. Rather, I immediately
that if I need 1 cup of flour, I would need 1/2 cup
more, but I need 1/4 cup more than 1, so in
I need 1/2 cup and 1/4 cup,
or 3/4 cups."
BA, Page 6
Add 58 + 57 by first adding 60 +
and then subtracting 5 from 120. BA, Page 6
- Agreed, but now I have 2/5
a cup and want 11/4 cups.
- With this simple change, we have
that can't be solved by TERC students. They don't know about
denominators, and their knowledge of calculating with fractions is
to a small subset of the familiar fractions.
Even if the student recognizes the need to calculate 3/5
, the sum, 17/20 , is an unfamiliar
fact, not likely to be remembered by the TERC student. What
next? TERC offers their "fraction strip" addition method for
through sixths. Thus, since there are fraction strips for fourths
and fifths, this may look promising. But the sum 17/20
isn't found on another fractions strip, so the student can't fold
of the fifths strip, one-fourth of the fourths strip, put them together
and find a matching point on another strip. The best the student
can do is estimate the answer, probably as 4/5
. Of course, the student might say that the answer is 4/5,
without suggesting that this is an estimate. TERC might well find
- Note: Given a TERC
think "what if we change one detail?"
Fourth graders using TERC's
for describing mathematical change: "It started out fast
it slowed down but now its (sic) growing faster again."
BA, Page 12
Challenge in a second grade
you land exactly on 100 if you count by fives?"
BA, Page 14
- What's the point? TERC has
the problem to an equivalent problem involving the landmark
numbers, 5 and 60. They expect students to remember
and differences for landmark numbers.
- Note that TERC has a genuine
It isn't easy to fully convert to landmark numbers.
- Why not first add 50 + 50?
different. The student would then be faced with 8 +
This single digit addition fact is not a familiar sum for many TERC
They haven't yet constructed this fact.
Computing 42 x 37 in a fifth grade
"It's the same as multiplying 42 by 40, then subtracting three
them. Ten 42's is 420. Double that to get 840.
Then you double that, so it's 1680. Then you have to subtract
42's. Two 40's down from 1680 is 1600, then another 40 off is
then subtract 6 more. So it's 1554."
BA, Page 20
Fifth graders synthesizing data
generalizations: After "a survey on what occupations
their classmates" they "began to
different views of their data" Two
are given: BA, Page 23
- Hanna explained: "Well I
5, 10, 15, 20, 25 like we learned, and I did land on 100.
it works. But Jamie [her partner] didn't get it because maybe I
going too fast. So I got out the play nickels and I put them down
and I said SLOWLY 5, 10, 15, 20 each time. And I got to
But then, you know, Jamie said 'How much do you think we have?' At
I didn't know. So we did 5, 10, 15 again. But in the middle
of it, I just knew it was going to be $1.00. Because counting by
fives is like adding it all up. You land on the place that is the
same as how much you have altogether."
BA, Page 14
- "Hanna has proved that
by fives is the same process as repeated addition. She realizes
each time she counts, she is actually accumulating another five.
Her proof is based on a combination of logic, work with manipulatives,
and use of a number pattern." BA,
Challenge in kindergarten: "How
eyes are there in our classroom?"
- "About a quarter of our
to help sick people or animals."
- "Most of us want to be in
where we could be a star."
Math time in a third grade
Using "any height that you think is reasonable for a third
. . . figure out how the heights of six third graders can add up to 318
inches." BA, Pages 44-58
- "We knew 26 to start,
27, 28, 29, 30. And kept on going. I counted and Abdulah
to keep track. But it was too hard, so we got out some cubes and made
of 26. Then we counted them all up."
BA, Page 28 [Recall keep track
A first grader indicated "how
groups of 5 to count her set of objects."
BA, Page 50
Martin Luther King was born in
old would he have been be in 1996? BA, Pages 68-71
- "If we have two kids at
two kids at 53 inches, that's 206 inches."
One team member used a calculator to subtract 206 from 318, but Jamal
up." ("See if you add 100, that's 306. Then 10 more
is 316, plus 2 - that's 112 in all.")
Then Pete used a calculator to divide 112 by 2. But Jamal divided
112 by 2 by recognizing that "half of 100 is 50, then half of 12
is six," and then adding 50 + 6. The team
decided to use two at 56 inches to add to 206 to get 318. BA,
- Jess, a member of another team "wanted
to use kids from our class, so we started out trying to use 50 inches,
because 50's are easy to add, and I'm 50 inches tall. So we made
two people be 50 inches - that's 100. Then we used Kaiya's height
because that's 49 inches and that's close to 50. So with these
people we had 150 - no, 149 inches, that's less than half of 318."
To determine "how much less than 318 is 149 inches" ...Elena
writes: 150 + 150 = 300 300 + 18 = 318
+ 18 = 168 168 + 1 = 169."
The team decided to use 3 at 55 inches and change the 49 to 53 to get
169 total. BA, Page 45 [So much for using "kids from our
- The teacher was impressed with
knowledge of landmark numbers like 100"
and noted that "when Jess totaled the six heights by first
calculating the 50's and then adding what was left, she was
how it's often easier to break numbers apart into
more familiar components and then add from left to right."
BA, Page 54
- Example A: "1929
- 0 years old 1930 - 1 year old
- 2 years old 1932 - 3 years old (and so
forth)" BA, Page 69
- Example B:
1930 - 1 year old 1940 - 11 (1 year + 10 years) 1950
- 21 (and so forth)"
- Teacher notes that this
good use of decades or 'landmarks' in the number
- Example C: "Some
represent this data on a horizontal line, and show jumps from year to
or from decade to decade." BA, Page
- "Some students develop
strategies that solve the problem with very few intermediate
Here's a strategy from a student who began with the standard vertical
"Faced with a problem like
25, we are likely to write out the long division procedure - figure out
how many times 25 will fit into 37, do the multiplication, then the
(37 - 25 = 12), then bring down the final 5. But does this
make sense in this context? It is neither the most elegant, the
nor the most efficient way of doing the problem. In the time it took
to write the problem down, you could easily have said, "I know there
four 25's in 100, so there are sixteen 25's in 400. There's one less in
375, so it's 15." Furthermore, the algorithm for long division is one
very few adults can explain (Simon, 1993). How then can elementary
hope to explain how the long division algorithm works"?
BA, Pages 73-74
This child never learned to borrow. Her approach was to subtract
9 from 6 and correctly arrive at -3. Then she subtracted 20 from
90 and arrived at 70. She combined these two figures, getting the
correct result . . . She used what she knew about 'counting below zero'
to do the subtraction. In order to check her solution, she
on in decades from 1929, saying quietly to herself '1939 is 10,
is 20 . . .1989 is 60, and from 1989 to 1996 is 7 more, so 67."
BA, Page 71
70 - 3 = 67
- Note: TERC believes it's good
never learned to borrow."
- Note: Negative numbers and
zero" don't appear in the actual TERC K-5 program.
"It's easy to think of many
in which the standard American algorithm is an inefficient way of doing
an operation. Adding 1987 + 1013 is another good example. Here's the
- Agreed. Most of us would
15, without using long division. [Recall related discussion
The algorithm involves three rounds of carrying - an inefficient
under the circumstances. We should know enough about number relations
look at the 13 + 87 and immediately conclude that it's 100, or to look
at the 987 + 13 and conclude that it's 1000. The standard approach to
this problem is cumbersome; it breaks the problem into little pieces,
the most efficient way of solving it is to work with the big picture."
BA, Page 74
Students are expected to learn
strategies are efficient in different situations. Multiplying by 9 may
well involve a different strategy than multiplying by 4."
BA, Page 76
- Agreed. It's immediately
the answer is 3000. But what about 867 + 1889? Most
us would still do this mentally (2700 + 67 - 11), perhaps jotting down
a detail, but we're beginning to move towards problems that aren't so
TERC carefully picks the problems. Just change a number.
"Multiplying 1346 x 231 is
that is best solved with a calculator."
BA, Page 76
"You would probably not use
to figure out 35 x 11, 21 - 19, or even $10.94 + $1.07. You
use mental arithmetic to come up with quick and accurate answers.
We want students to be able to do this to, too. But calculators
be available nearly all the time, so that students can do more
calculations or check the answers they arrived at using their own
BA, Page 78
- Exactly what we don't
standard methods, which can be used automatically, so we can move
on to the next level of complexity.
a mathematical quest to find all possible ways to subtract one number
another to make 24. He chose this problem on his own. He started with
- 1, 26 - 2, 27 - 3, and continued with this pattern for some time. The
fact that he would still get 24, even as the numbers got bigger and
fascinated him. After half an hour, he had three pages of systematic
to proudly show to his parents and, later, to his class. He continued
this problem for several more days, at which point he confidently
to his mother, 'I think there's a thousand ways to make 24, and I'm
to find them all.' " BA, Page 82
- TERC emphasizes the importance
a problem in multiple ways. Can't they find a second method to
answers, without using a calculator?
- As for "more difficult
don't appear in the TERC program.
"Suppose the problem
two sets of objects---one numbering 46, the other 64---to see which set
has more. A student who adds up all the objects in both sets is, quite
simply, employing a strategy that doesn't work. This child either
understood the problem or hasn't seen that comparison involves a
other than addition. A second student, who compares by "counting on"
46 to 64 and keeping track of the number of numbers along the way, has
a more effective strategy, one that shows understanding of the problem.
A third student, who says "46, 56, 66, that's 20, take away 2 to get
to 64, that's 18," is employing an even more elegant strategy
figuring out the difference." BA,
97 (emphasis added) [Found in Chapter 5, A New Kind of
- What was the mother thinking?
Did You Notice Any Elegant Strategies
or Significant Math?
Most of these techniques are familiar
anyone who has mastered traditional arithmetic. Transforming a problem
into one that can be solved using "familiar" facts is a good
strategy. The error here is the limited applicability of
techniques presented, the misrepresentation of these techniques
powerful strategies, and the complete omission of general computational
methods. But TERC doesn't just omit standard computational
they do it proudly, with a clear hostility. Please read on to
more about that.
Calls it Number: Arithmetic Without Standard Methods
Trouble in TERC City: Kim is Using a Standard Algorithm!
TERC and the Standard Algorithms: As
in Their Third Grade Teaching Materials
"If you have students who have already memorized the
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
new values -- that estimating the result is critical, that having more
than one strategy is a necessary part of doing computation, and that
what you know about the numbers to simplify the problem leads to
that make more sense, and are therefore used more accurately." From Combining
and Comparing (Addition and Subtraction), Page 38
TERC and the Standard
As Stated in Beyond Arithmetic
TERC and The Standard Algorithms: As
Algorithm Issue (Essay at the CESAME Website)
- "Everyone needs to know a
good ways to add, subtract, multiply, and divide. However, the
we were taught to do these operations aren't necessarily the best or
efficient ways. Believe it or not, these traditional algorithms
largely the result of historical accident. Borrowing, carrying,
the procedure for long division - they're not universal. In other
countries and at other times in our own, students have been taught
and equally effective algorithms for the basic operations."
BA, Page 73
- "In the Investigations
standard algorithms are not taught because they interfere with a
growing number sense and fluency with the number system."
BA, Page 74
- "There is no longer a
for asking students to do pages of calculations - especially more
calculations like adding series of numbers or multiplying three-digit
by other three-digit numbers. We do not want to waste students'
on frustrating tasks that involve the rote application of memorized
Before asking your students to do problems without a
ask yourself how you'd do the problem yourself. Would you grab a
calculator if it was available? If so, let your students do the
BA, Page 78
- Note: This is a "whole
of argument that fails to differentiate between the learning needs of
novice and the skills of the expert.
- "There is growing evidence
unteach place value and prevent children from developing number sense'
(Kamii, Lewis, and Livingston, 1993, P. 202)."
BA, Page 79
- TERC doesn't need to worry about
place value. They don't teach it.
the conventional American algorithm for each operation as just one more
way to perform the operation." AI,
- "There are other efficient,
algorithms that students understand better."
AI, Page 1
- "While the developers of Investigations
recommending forcing students to use the historically taught American
the don't recommend hiding these algorithms either ... the criteria for
using them . the student must be able to explain it and must have more
than one approach available." AI,
The Truth About the
- TERC doesn't teach the standard
but they can't hide them because parents and tutors teach them.
The standard algorithms are true
"algorithms." That is, they are (potentially programmable) general
for carrying out a (potentially complex) calculation by repeating
a sequence of simple steps.
The "American algorithms" are the
algorithms used in Europe, Asia, Africa, and the rest of the
Despite claims to the contrary, there are no alternative algorithms
can match the efficiency, accuracy, and generality of the standard
TERC offers special case methods,
Their "strategies" require conscious student observations that differ
problem to problem. Such observations are not guaranteed to
and they don't involve repeating a sequence of steps.
- TERC regularly instructs
that difficult day - Kim is using a standard
What to do? The TERC solution? Ask Kim to explain her
Regardless of what she says, tell her to try another method. TERC
has never met a child who can adequately explain a standard algorithm.
only work for a very small subset of whole number problems. The
algorithms work for arbitrary real numbers. No problem with
numbers or digits to the right of the decimal point.
TERC is opposed to explicit
the basic (single-digit) number facts, but TERC students must remember
many non-basic number facts as a necessary condition for successfully
out the TERC alternative methods.
- The student is asked to multiply
99 and notices that this can be accomplished by multiplying 5 x
and then subtracting 5. Good TERC think, nicely transformed to an
equivalent problem involving only landmark numbers. But not a
method. Certainly not an algorithm.
TERC and the NCTM want kids to
"the power and beauty" of mathematics, but they're blind to the power
beauty of the standard algorithms. With a very small amount of
(remembering the single-digit number facts and knowing how to carry and
borrow relative to the ingenious design of our decimal system), the
can carry out any calculation involving the four basic operations,
cases with digits to the right of the decimal point. With the
of long division, the calculation can be carried out automatically,
of the complexity of the calculation. Long division also requires
skills. Although TERC emphasizes estimation, they omit long
and thereby miss the perfect opportunity to demonstrate estimation as a
- Remembered, but never
wants that to be clear.
Algorithms in The New York Times: An
Introduction to the New Math ( Link to NYT Article)
- More generally, mathematical
involve leveraging simple case facts to solve complex problems.
example, in differential calculus we use properties of tangent
to study the local behavior of arbitrary continuous curves, and in
calculus we determine the area of complex regions by using a limiting
that involves summing the area of rectangles.
- TERC sometimes recognizes the
the complex to the simple, but they seriously miss the boat when they
the standard algorithms.
Lucy West, Director
of Mathematics for Community School District 2 in New York City,
is identified as the source of An
Introduction to the New Math in the New York Times. Using a
by side comparison, Ms. West compares "constructivist new math" to the
"traditional method." The casual reader may think: Is that all
is about? TERC will be pleased. They want readers to go
thinking that the "math wars" are caused by purists quibbling about
They want you to be impressed that they nicely avoided "carrying." They
hope you won't know or notice that this NYT illustration is
Number Computation in the TERC Fifth (or Sixth) Grade
- Deception 1: What's the
constructivist methods? It appears that they have eliminated
But they've only suppressed it and hidden the power of place
value. How did they compute "18 plus 80 plus 630 plus 2,800"?
- Deception 2: The
look attractive with the simple cases presented, but they quickly
labor intensive and unwieldy when you use them to to multiply two
numbers, multiply two 4-digit numbers, or add a column of five 5-digit
numbers. For the final defeat, try a case with digits to the
of the decimal point.
- Deception 3: These
constructivist methods don't
appear in the TERC curriculum! If they did, the problem would
be much less severe, since students would eventually recognize their
special-case limitations and thus be ready to appreciate the
accuracy, and generality of the standard algorithms. They would
look back at the "constructivist" methods as inefficient, preliminary
of the standard algorithms.
[TERC instructs the teacher] Write
the following two problems on the board: 253 x 46 701
"As you've been playing the Estimation Game, you've had
to work with problems that are sometimes very difficult to multiply or
divide without a calculator. These problems are like that.
In the game, you got to use a calculator to find the exact
Now we're going to try finding the answers to these harder problems
U5, Page 128 (Bold in the original indicates TERC's script for
The teacher goes on to tell the students
they are to solve these problems, but not with standard algorithms.
TERC must state this, since they don't teach these algorithms.
some kids have learned them from parents and/or tutors, and all TERC
are trivial for kids who know the standard algorithms and are allowed
We next list the whole number
methods that TERC teaches to fifth graders. You won't find
this list anywhere in the TERC materials. It offers a
of the ideas found in those materials, but in a much more compact form.
- Remember facts about
numbers (anchor numbers)
Remember or reference (non-basic)
facts that have been recorded in "Multiple Towers."
- Example: The student should
10 x 25 = 250 and that 20 x 25 = 500
- Note that TERC de-emphasizes
of basic number facts, but they expect kids to remember many other
Skip-counting, by 5, a
a large one digit number, or a small two-digit number.
Addition by "counting on"
first number, while simultaneously subtracting 1 from the second number.
- "The class builds a
on a long strip of adding machine tape, listing multiples of 21 in
and looking for patterns in the sequence. They use the patterns
find to solve multiplication and division problems involving multiples
of 21." U5, Page 2
- The class builds other Multiple
they are expected to use information recorded in these towers to help
solve problems. An example involving 32 is given below in this
Subtraction by "adding on."
- 24 + 17 is computed as 24 + 1,
+ 1 ....... 40 +1, with 17 reduced to 16, then 15, then 14, etc..
- Note that students need a method
to place value. (TERC says break , not decompose)
Decompose numbers relative to
(TERC says "break," not decompose)
- 212 - 98 is calculated by adding
Use the distributive law.
mentions the "distributive law," but they can't avoid using it.)
- 27 = 25 + 2
- 355 = 350 + 5
Multiplication as repeated
with the possible efficiency of adding multiples of 10 or 100.
- 25 x 21 = 25 x (20 + 1) = (25 x
x 1) = 500 + 25 = 525
- Note also the conversion to
Multiplication (or division) by
facts in a "multiplication cluster."
- 27 x 34 is calculated as (10 x
34) + (7 x 34)
- Note the application of the
and the remaining difficulty with 7 x 34.
Division as repeated subtraction,
the possible efficiency of subtracting multiples of 10 or 100.
- Example of a multiplication
32, 20 x 32 , 30 x 32, 5 x 32, 35 x 32] U5, Page 77
- "How can 10 x 32
you find 5
x 32? How can 10 x 32 help you to solve 20 x 32? Which of
problems in this cluster helped you to figure out the answer to 35 x 32?"
U5, Page 77 (Bold in the original indicates TERC's script for the
- Students are allowed to use
Tower" for 32 U5, Page 77
Division (or multiplication) by
facts in a "division cluster
- 510 ÷ 24 is calculated by
240 and then subtracting 24 from 30 to get 21 R6
TERC says these are examples of
work at the end of the TERC K-5 program.
- "The teacher presents
[12 ÷ 12 120 ÷ 12 132 ÷ 12
133 ÷ 12]." U5,
- The student is to solve 133
÷ 12 by breaking
the problem down in order to use the other facts in the cluster.
- "Well, 120 ÷ 12
10, and 12
÷ 12 is 1. Put them together and get 132 ÷ 12 is
Between 132 and 133 there's only 1 difference. We're looking for
12 difference, so it's 11 R 1." U5,
- Note the hidden use of the
- 2015 - 598
6029 - 4873
- "I found the answer
1000 + 100 + 100 +100 +10 +7. Start with 598. Add 1000 to
1598. Add 100 to get to 1698. Keep adding 100's to get to
Then add 10 to get to 2008. Then add 7 more."
U5, Page 10
- We would mentally add 2 + 1400
26 x 31
- The student wrote 1000 + 100 +
25 + 2.
Then wrote 1000 + 100 + 25 + 4 + 25 + 2. This was then rewritten
as 1000 + 100 + 50 + 6. Nothing more. U5, Page
- Note the conversion to
- We would mentally add 27 +
767 ÷ 36
- The student wrote 10 x 31
20 x 31 = 620 25 x 31 = (620 + 155) =
The student placed an arrow, pointing to 155. At the other
end of the arrow the student wrote 5 x 31 = 155 and 1 x 31 =
Finally, below these calculations, the student wrote 26 x 31 = (775 +
- The details are not given for
31 = 155
and 775 + 31 = 806. U5, Page 80
- The student used a
for 31 and perhaps a Multiple Tower for 31.
- TERC instructs the teacher how
students toward different strategies."
U5, Page 82
- "There are ten 36's in
are twenty in 720. In 767, there are twenty-one 36's and a
of 11 (767 - 756 = 11), so one way of expressing the solution to 767
36 is 21 R 11." U5, Page 82
- "I know 36 x 2 is 72,
x 20 is
720 (or 720 ÷ 20 = 36). Then 767 - 720 is 47. Take
1 more 36 (or twenty-one 36's in all) and you're left with 11."
U5, Page 82
- "The problem 767
36 is the
same as twice 360 ÷ 36 plus 47 ÷ 36. That's 10 + 10
+ 1, with 11 out of a group of 36 left over."
U5, Page 82
What's Wrong With TERC's Methods
for Whole Number Calculations?
- They're limited to simple problems
small whole numbers.
They work best with problems that
up" to make them look good.
They're slow and require conscious
to classify the problem relative to one or more of the relevant
They use non-standard
landmark numbers and familiar fractions.]
They use non-standard
multiplication cluster and division cluster.]
They use tools that are only
TERC classrooms. [Examples: multiple towers and manipulatives.]
They can be difficult to master
has been denied the necessary orienting framework (the complete facts
skills of traditional arithmetic). The TERC techniques are not
for us who have mastered traditional arithmetic. We use some of them
They're easy for us because they actually form a small subset of the
knowledge we have stored in our brains.
- Kids need to learn about
They need to learn about efficient, accurate, and general methods for
computation. They need methods that work for arbitrary real numbers,
just small whole numbers. They can forgot about algebra if they haven't
mastered the standard algorithms. Why? See links.
Hands-On Methods For Fractions, Decimals, and Percents
"The proper study
fractions provides a ramp that leads students gently from arithmetic to
algebra. But when the approach to fractions is defective, that ramp
and students are required to scale the wall of algebra not as a gentle
slope but at a ninety degree angle. Not surprisingly, many can't."
-WU, Page 11
TERC doesn't even attempt to discuss
with decimals. Their special case whole number strategies don't look so
attractive to the right of the decimal point. TERC kids never get
to appreciate the (truly) elegant fact that carrying and borrowing work
identically for every column, regardless of the column's location, left
or right of the decimal point.
"This unit does not concentrate on procedures for either
or fraction computation. Students solve computation problems
good number sense, based on their understanding of the quantities and
relationships. They carry out addition and subtraction of
amounts in their own ways and in more than one way, using fractions,
or percents, and using any models that make sense to them."
U3, Page I-18
As for fractions, TERC students learn
nothing about multiplying and dividing fractions. They learn how
to add and subtract a small subset of the "familiar
fractions", using fraction strip and clock face
for fractions," not common denominators.
Tai don't know that 3/48 = 1/16, but they should remember
4/8 = 3/6 = 2/4 = 1/2 because these "familiar" fraction facts have been
recorded on their Fraction Equivalent Chart. Shakita
know about common denominators. To add 1/4 + 1/5, she
her blue (fourths) and yellow (fifths) fraction strips, put them
together, and compared to the the pink (halves) fraction strip.
discovered that 1/4 + 1/5 = 1/2. Tai tried the Large Clock
Face for that one, but Shakita knew that wasn't a good choice of model
for fifths. Then Tai borrowed Shakita's folded blue strip, put it
together with his folded blue strip, and compared to the pink strip. He
discovered that 1/4 + 1/4 = 1/2. He told Shakita. She was
but convinced after checking Tai's work. She then suggested that both
should be recorded on the Fraction Equivalent Chart. Tai wasn't
The Fraction Equivalent Chart didn't currently show fraction sums, just
equivalent fractions. But surely these facts were acceptable.
all, they were about the big idea of one-half!
OK, this story isn't found in the TERC
materials, but parents in New York City aren't laughing. It's too
close to the truth of their "Saturday morning live" experience ("mom,
not the way I'm supposed to do it"). The story contains no
of fact and is quite plausible, especially when you consider:
Later that day, Shakita shared their
with Sarah, her home-schooled friend. Sarah said, if that's true,
1/4 = 1/5. Now Shakita was excited. This new equivalent
fact definitely belonged on the Fraction Equivalent Chart. But
needed to be sure, so she asked Sarah to explain. "It's
if 1/4 + 1/5 = 1/2 and 1/4 + 1/4 = 1/2, then 1/4 + 1/5 = 1/4 +
by substitution. Then cancel 1/4 from both sides of the equation,
and viola!" She was smiling when she added "I'm sure you can
the obvious corollary." Shakita really didn't know what Sarah meant by
substitution, cancel, equation, viola, prove, and corollary, but Sarah
appeared confident. Now she was laughing.
- TERC insists that children use
strategies (only) to make their colored fraction strips.
"Fraction on Clocks" and
are the only "Models for Fractions" available for adding and
- TERC instructs the
the five colors, each paired with a fraction, on the board or chart
Include a model showing students how to write the corresponding
name at the bottom of each sheet For example: pink -
green - thirds blue - fourths yellow -
white - sixths." U3, Page 42
- "The students' challenge
the strips accurately to show different fractions."
U3, Page 42
- "When marking their
write only in pencil so they can erase and move the marks if they need
to. They check all five strips with their neighbors and adjust
marks until they think they are quite accurate."
U3, Page 42
The problem 1/4 + 1/5 actually
"more difficult problem" in the TERC materials (U3, Page 99).
- TERC teaches kids to solve
the available models. It's exclusively "hands on."
TERC heavily promotes estimation,
than accurate answers. 9/20 is about 1/2 in TERC-think.
TERC does want children to learn how to
fractions and understand fraction "relationships." But once again they
want to avoid the complexity of "common denominators."
Students work with models and play games until they remember (some of
the relationships. [4/5 is less than 7/8. Isn't
Let's check the chart.]
- Children would naturally expect
that they have been taught are adequate for solving the problems given
to them, unless they've been told otherwise. But TERC never instructs
to tell kids that "strip addition" doesn't "work" for many pairs of
fractions. The familiar fractions aren't closed under
That is, the sum of two familiar fractions may not be familiar.
9/20 (1/4 + 1/5) is very unfamiliar to TERC fifth graders.
Finally, TERC wants kids to remember
fact equivalents." For familiar
only, students complete strips, charts and tables for "Fraction
Percent Equivalents," "Fraction to Decimal Division," and
Decimal, Percent Equivalents." Students are eventually given
strips that show the relationship between familiar fractions and whole
number percents between 0 and 100. For example, the mark for 1/8
th is shown between 12 percent and 13 percent.
TERC Unit 3 contains 193 pages for
decimals, and percents. There are many more "models," but nothing
else that could be called content. There you have it: TERC
fractions, decimals, and percents in a nutshell,. Still room for
the nut. References
Changing Mathematics in the Elementary Classroom
IM: Implementing the
Investigations in Number, Data, and Space
Investigations Grade 5 Teacher Books (Units 1 - 9)
at Grade 5 (Introduction and Landmarks in the Number System)
U3: Name That
(Fractions, Percents, and Decimals)
U4: Between Never
and Always (Probability)
U5: Building On
You Know (Computation and Estimation Strategies)
(Estimating and Measuring)
U7: Patterns of
(Tables and Graphs)
U8: Containers and
Cubes (3-D Geometry: Volume)
U9: Data: Kids,
and Ads (Statistics)
Click for link to
The Algorithm Issue
to Prepare Students For Algebra by H. Wu, (American Educator,
2002-2011 William G. Quirk, Ph.D.