TERC Hands-On Math: The Truth is in
the Details
An Analysis of The First Edition of
TERC's Investigations
These Clickable Links Serve as an Outline of This Paper:
The
TERC
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
mathematics
curriculum that supports all students as they learn to think
mathematically."
Here, using extensive quotes from TERC's book for teachers, Beyond
Arithmetic, and TERC's Fifth Grade Teaching
Materials,
we'll reveal how TERC has redefined the meaning of "think
mathematically."
The NSF is now spending millions to
promote
implementation of the TERC program. School Boards find it
difficult
to say no. They rationalize: "it's just a different way to teach
elementary
math, and the NSF backs it, so how bad can it be?" This
program
is very bad because it omits standard computational methods, standard
formulas,
and standard terminology. TERC says most of this is now obsolete,
due to the power of $5 calculators. They claim their program
moves
"beyond arithmetic" to offer "significant math," including important
ideas
from probability, statistics, 3-D geometry, and number theory.
But math is a vertically-structured
knowledge
domain. Learning more advanced math isn't possible without first
mastering traditional pencil-and-paper arithmetic. This truth is
clearly
demonstrated by the shallow details of the TERC fifth grade
program.
Their most advanced "Investigations" offer probability without
multiplying
fractions, statistics without the arithmetic mean, 3-D geometry without
formulas for volume, and number theory without prime numbers.
TERC Rejects Standard Knowledge and
Working Alone
- TERC regularly suggest that
"personal"
is
better than "standard." This may be true for arts and crafts, but
standards and conventions are essential for mutual understanding and
effective
communication in business, science, and professional life.
- TERC says that each student must
always "work
collaboratively" with other students in a small group. Progressive
educators
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
communicate
entirely in writing and never physically meet other members of a
project
team. They may participate in multiple teams at the same time.
Major Characteristics of TERC's
Program:
Investigations in Number, Data, and Space
- TERC insists on the ongoing use of
hands-on
tools (manipulatives, "models," and calculators).
- They say concrete tools must
always
be available
and regularly used.
- TERC strongly rejects the idea
that
children
must eventually migrate from hands-on to abstract thinking.
- TERC rejects the need for
memorization
and
practice.
- They say that
student's
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)
[Please
click
on References for the meaning of the
BA
code.]
- There's no additional gain in
conceptual understanding
associated with the task of trying to "construct" one more basic number
fact.
- TERC doesn't think it's
possible
to understand
memorized information. But knowledge must first be loaded into the
brain
before it can connected to other knowledge and "understood."
Explicit
memorization is sometimes the most efficient way to get it there.
- TERC fails to understand that
it's
often desirable
to move to automatic use of knowledge. The mind must be free to think
at
higher levels of complexity, without consciously revisiting underlying
details. For example, the key idea of the standard algorithms is that
multi-digit
calculations are reduced to multiple single-digit calculations.
If
children don't have instant recall of the single-digit number facts,
they
aren't equipped with the essential pre-knowledge for easily carrying
out
multi-digit computations.
- TERC fails to clearly define terms.
- They regularly state: "We don't
ask
students
to learn definitions of new terms."
- They offer some "definitions,"
typically using
multiple undefined terms to "define" a new term.
- They favor "natural language"
and
"personal
language."
- TERC emphasizes "familiar numbers."
- The "landmark
numbers" are 5, multiples of 10, and multiples of 25.
- Landmark number are also known
as anchors.
- The "familiar
fractions"
are limited to proper fractions with denominator equal to 2, 3, 4, 5,
6,
8, 10 or 12. Thus 7 and 11 are not familiar denominators.
Perhaps
TERC is opposed to gambling.
- Note: 12 is included because
it's
needed for
TERC's clock face method for adding fractions.
- TERC doesn't believe in defining
terms, so
you won't find the preceding definitions in TERC materials. This
is what they appear to mean by these phrases. We welcome their
clarification.
- Although TERC rejects explicit
memorization
of basic single-digit number facts, they expect students to remember
many
non-basic facts about landmark numbers and familiar fractions.
- TERC omits standard formulas.
- TERC emphasizes estimation and
many
right
answers.
- They suppress the concepts of
precision and
accuracy.
- TERC proudly rejects standard
computational
methods.
- No standard algorithms for
multi-digit computation.
- No standard methods for
operations
with fractions
and decimals.
- No general methods for
calculating
with numbers.
- TERC emphasizes special case
methods involving
landmark numbers and familiar fractions.
- TERC attempts to directly teach
their
shallow
and misleading content.
- They claim to offer a
"constructivist" approach
where students discover math as they play games and carry out
investigations.
But they provide thousands of pages of teaching instructions and
recommended
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?
The
bulk of
this paper provides the answer.
The
TERC
Philosophy of Elementary Math Education
TERC
recommends
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
"constructivist" approach.
TERC Says
Traditional Elementary
Math Must Be Discarded Because It:
- Was "developed to meet the
needs of
the 19th century." BA, Page 2
- Requires that students "memorize
many
facts, procedures, definitions, and formulas."
BA, Page 2
- "Focuses on learning a
particular set
of procedures for addition, subtraction, multiplication, and division
of
whole numbers, fractions, and decimals."
BA, Page 2
- Results in "overpracticed
students."
BA, Page 3
- Ignores the fact that "today's
students
have an important tool available to them: the calculator."
BA, Page 77
TERC
Says
Manipulatives and Calculators Are Essential as Students Work
Collaboratively
- In the "constructivist
mathematics class today", students "work together, consider their own
reasoning
and the reasoning of others, and communicate about mathematics in
writing
and by using pictures, diagrams, and models. They carry out a
small
number of problems thoughtfully during a class session or perhaps work
on a single problem for one or several sessions. They use more
than
one strategy to double check, and they use blocks, cubes, measuring
tools,
calculators and a variety of other materials to help them solve
problems."
BA, Page 42
- "Students need to do
mathematics collaboratively.
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
requires
that tools and manipulatives be available to students at all
times.
Students at all levels need to be able to use whatever tools they
choose
to solve problems, and should make their own decisions about which
materials
might best help them solve a problem."
BA, Page 62
- " 'Having materials readily
available'
should convey to students the expectation that they will use these
materials
to solve problems. Traditionally, concrete objects like blocks and
counters
and even fingers were considered babyish; although sometimes used
to introduce a new concept, they were quickly dismissed in favor of
pencil
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
classroom,
even the youngest student quickly gets the message that the goal of
doing
math is to use symbols and do it on paper. In a constructivist
classroom,
all students - as well as the teacher - use tools, drawings, and
materials
to solve problems. Manipulatives offer more ways for a range of
students
to enter and persist at difficult problems, a way of keeping track of
work,
and a way of representing solutions to problems."
BA, Page 63
- Note: The
theme
of "keeping
track" occurs regularly in TERC teaching materials. Because TERC has
discarded
traditional computational methods, they have lost the "keeping track"
recording
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
track
of (56 + N) and (43 - N), where N runs from 1 to 43 . Manipulatives
save
the day. Put down 43 plastic discs and count 57 for the fist
disc,
58 for the second, until you reach the last disc.
TERC Says Their
"Constructivist
Math" Approach:
- 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
employees
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
investigations
that allow all students at each grade level to explore important
mathematical ideas in the use of number, data, geometry, and the
mathematics
of change." BA, Page 18
- Recognizes that it isn't "possible
to present all the worthwhile mathematics that might be included at a
particular
grade level," and is is better to "serve students by helping them to
study
some topics in depth." BA, Page 32
- Allows students to "revisit
the
big
ideas of the elementary curriculum year after year, from different
perspectives."
BA, Page 32
- Supports the fact that
middle-school
curriculum
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
number
computation
is an area in which the elementary mathematics curriculum needs to be
slowed
down and deepened Rather than being hurried into complex
computation,
students need time to develop strategies based on numerical reasoning."
BA, Page 34
Searching For "More
and
Deeper Mathematics"
TERC points to Steven Leinwand's 1994
article,
It's
Time to Abandon Computational Algorithms. They say he
"eloquently describes the changes that calculators should be bringing
to
our mathematics classrooms" when he wrote: A few short years
ago
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
witnessed
in our classes as necessary parts of learning what were then important
skills. Today there are alternatives and there is no honest way
to
justify the psychic toll it takes. We need to admit that drill
and
practice of computational algorithms devour an incredibly large
proportion
of
instructional time, precluding any real chance for actually applying
mathematics
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
past,
and that we must make room for more and deeper mathematics in the
curriculum.
In the past, the elementary curriculum focused almost entirely on
paper-and-pencil
work with arithmetic, while the study of geometry, data, number theory,
and other important aspects of mathematics were relegated to a few
special
exercises or chapters that were never reached."
BA, Page 4
There you have it. Now that
computational
drudgery is out, there's time to do the interesting stuff. TERC
promises
a K-5 math learning environment where students are doing "significant
math,"
dealing with "big ideas," and "learning a great deal more mathematics
than
what we considered sufficient in the past." TERC offers "more and
deeper mathematics." Their students "explore important
mathematical
ideas in the use of number, data, geometry, and the mathematics of
change."
But significant math isn't possible
without
the facts and skills that TERC has excluded. As evidence
from
the TERC Fifth Grade Teaching Materials shows
below,
TERC's "everyday math" emphasis has led them to offer:
- One-half as a big idea, while not
mentioning
the major ideas of place value and the distributive law.
- Statistics without the concept of
the
mean.
- Probability without operations
with
fractions.
- 3-D geometry without formulas for
volume.
- The "mathematics of change," where
TERC believes
they're helping to prepare kids for calculus by teaching them their
language
for describing change.
Note: TERC
claims that their fifth grade materials are "Also appropriate for
grade
6."
One-Half
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
that
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
one
cookie among two people, or it is one out of every two cookies, or it
is
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
pocket."
BA, Page 32
"This difficult idea needs to be addressed over many years as
students'
thinking about one-half becomes deeper and more complex ... in the
fourth
grade, students might use fractions to describe data they've
collected:
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
develop
a model of 1/2, and its relationship to other fractions and to whole
numbers."
BA, Pages 32-33
Place
Value
Isn't a Big Idea for TERC
TERC and the NCTM rave on about "the
power
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
it's
mentioned in some teaching units and on the web in the essay CESAME:
Place Value. The essay author illustrates TERC's
conventional
way to discuss place value: "The '2' in 24 represents 20, the '2' in
247
represents 200." (PV, Page 1) True, but this isn't the
conceptualization
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
understand
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
10^{2 }+ 4 x 10 + 7. Eventually we want them to think:
8,247.942
= 8 x 10^{3} + 2 x 10^{2 }+ 4 x 10 + 7 + 9 x 10^{-1
}+
4 x 10^{-2} + 2 x 10^{-3}. Then, students
are
ready to appreciate the powerful fact that any complex
calculation
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
2).
To fully appreciate this simplification and what carrying is all about
, students need to first work with numbers in expanded form
(Example:
54 = 5 x 10 + 4 and 247 = 2 x 10^{2 }+ 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
extended
equalities and regular rearrangements of component expressions using
the
commutative and associative laws. Eventually they will come to
understand
the power and generality of "carrying" and respect the compact and
efficient
standard algorithm for multiplication.
Once students move
to fully automatic use of a standard algorithms, they may forget the
underlying
rational. This is perfectly natural. Our brains regularly push
knowledge
into the background in order to free the mind to focus on the next
level
of knowledge. A good teacher recognizes when it's appropriate to pull
background
knowledge back into the foreground. The knowledge can then be
reinforced
and connected to knowledge at the next level of
abstraction.
For example, the student might be reminded of the underlying rational
behind
multiplying 54 times 247, when learning how to multiply (5x
+ 4) times (2x^{2 } + 4x + 7).
The Distributive Law
Isn't
a Big Idea for TERC
The distributive
law
states: A x (B +C) = (A x B) + (A x C) [For any three real
numbers:
A, B, and C.] This major idea isn't covered by TERC, but it
frequently
explains many of the simple case computation methods they recommend.
The
distributive law is of fundamental importance in arithmetic and algebra
because it's the property that "connects" multiplication and
addition.
TERC is excited about "connections," but fails to see any in the
distributive
law.
TERC
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
how
to find it with the "add-'em-all-up-and-divide-by-the-number"
technique.
Although this algorithm is often taught in elementary school, research
has shown that it is often not understood, even by older students and
adults.
At this point, it is better to stay away from the mean and the
confusion
it may introduce." U9, Page 6
[Please click on References for the
meaning
of the U9 code.]
TERC Recommends Natural Language and
Personal
Words, Not Standard Terms.
- Each TERC curriculum unit book
reminds
teachers: "In the Investigations curriculum,
mathematical
vocabulary is introduced naturally during the activities. We
don't
ask students to learn definitions of new terms."
U9, Page I-22 (as one reference)
- TERC's
recommended natural
language for statistics: "To help students pay
attention
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."
U9,
Page 10
A Major Statistics Investigation - At
the
End of the Fifth Grade
The following example of TERC
"significant
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
going
to look at sample issues of USA Today to find out whether they
achieve
their goal of two-fifths ads." U9, Page 70
- The teacher divides the class into
"groups
of two or three students" and says: "First your group should choose a
page
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
least
a couple of ways. Work with familiar
fractions
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
- TERC advises the teacher to "expect
to see a variety of strategies for figuring out fractions. Some
students
may cut up the page and rearrange it like puzzle parts to figure out
what
fraction is ads. Others may use markers to color the ads and
visualize
the fraction of area they take up."
U9, Page 72
- The teacher begins the second
session
by giving
each group several paper sheets, each containing six Recording
Strips.
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
these
sheets
gives you six Recording Strips. For each page of the newspaper you
examine,
you'll need to record what fraction of that pages is ads. Use a
separate
Recording Strip for each page. On the strip, color the fraction
of
the page that is ads. If a page has no ads, you still need a
strip
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
page
has one-eighth ads?" U9, Page 74
- The teacher gives each group a
complete issue
of USA Today and says: "Each group will get a whole
newspaper,
with all the sections, and your job is to figure out what fraction of
the
paper is ads. You won't have enough time to do this for every
page,
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
- A Dialogue
Box
(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
pages,
we'd have a sample of... Does someone have a calculator?" U9,
Page 76
- "Tai: [Entering 48 ÷ 3 on a calculator]
It would be
16. That's pretty close to 15. It wouldn't matter so much
if
we didn't do the last page." U9,
Page
76
- The teacher begins the third
session
by saying:
"Today, each group will put together their Recording Strips to
figure
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
were
grouped together, how many pages would they fill up? What
fraction
of the pages in your sample would that be?"
U9, Page 77
- TERC advises the teacher: "Explain
that while there are many different ways of finding the combined
fraction,
everyone in the class will use the same method so that it will be
easier
to compare the findings. Demonstrate this method, by taping your
30-inch piece of adding machine tape to the board. Then fasten
your
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 instructs the teacher to
ask: "What
does this whole paper tape now represent (all the sampled pages added
together)
How might we use these Recording Strips to figure out what fraction of
these pages is covered with ads?"
U9,
Page 78
- 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.
Take
the Recording Strips down and cut off the colored-in part of each
one.
Save the blank pieces to show later how they fill in the rest of the
strip.
Tape the colored-in pieces onto the paper tape, starting from one end
of
the tape and putting the pieces right next to each other."
U9, Page 78
- TERC instructs the
teacher: "Give
each group a piece of adding machine tape that is 3 inches times the
number
of pages they have sampled. For example, if they have sampled 12
pages, give them a 36-inch length of adding machine tape. The
group
cuts each of their Recording Strips into two pieces and fastens down
the
colored-in parts, starting at one end of the tape."
U9, Page 78
- Note: Recording Strips are 3
inches long.
- TERC instructs the
teacher: "As
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
strip.
They can fold their strips to figure out this fraction, or use any
other
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
Fifth
Grade?
- Eyeball estimates to associate a
"familiar"
fraction with the (combined area) of the ads on a page.
- The central role of manipulatives
(Recording
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
coloring,
cutting,
taping, and other kindergarten activities.
Probability
Without Fractions: It's Guess and Check
Elementary probability is a math topic
that's "beyond arithmetic," but quite accessible to fifth grade
students.
This looks like a natural! TERC emphasizes counting, and advanced
counting techniques are naturally connected to elementary
probability.
First the fundamental
counting principle, then permutations and combinations, all with
the
excitement of factorials. This only requires prior mastery
of
fractions. Ooops! There's the problem. TERC
doesn't
teach kids how to compute with fractions. No problem! TERC
discards the traditional content of elementary probability and
substitutes
the "language of probability" and "predict and check." Here are
some
illustrations from Between Never and Always, the
"probability"
unit in TERC's fifth grade program.
- The teacher speaks to the
class:
"Over
the next couple of weeks we're going to be studying probability.
Learning about probability helps you figure out how likely it is that
an
event will happen. We'll be particularly interested in studying
events
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
classifying
events into the five categories.
- TERC provides two "definitions"
for
the teacher:
"theoretical probability ...describes what would happen
'in
theory' ... .experimental probability ... describes what happens
when we do an experiment." U4, Page
14 Simple as that!
- Using a variety devices, including
"spinners"
and bottle caps, students are repeatedly asked to first predict and
then
test their predictions. U4, Pages 15-26
- Example: Spin the "one-fourth
green
spinner"
50 times.
- TERC
instructs the teacher:
"ask students to jot their predictions on scrap paper.
Compile
their predictions on the board or overhead. Since 1/4 of 50 is 12^{1}/_{2
},
both 12 and 13 are likely predictions, but be prepared for a variety of
responses." U4, Page 19
- TERC introduces their concept of
Expected
Number.
- "If we flip the coin 10
times, we have
an expectation that 1 out of 2 flips will be heads, so that we expect
to
get 5 heads out of 10. We call 5 heads in this case the expected
number." U4, Page 24
- "Since the probability of
correctly
guessing a spin is ^{1}/_{4 }, the expected number of
correct
guesses in 20 spins is ^{1}/_{4 }of 20, or 5."
U4, Page 30
- Then more experiments to test
"expected" versus
"experimental."
- Now, armed with this new
theoretical
tool,
students predict with more confidence as they experience sessions in
"Testing
Guessing Skills" and "Guessing Skills Distributions." But they
end
up losing faith in the power of theory when they continually discover
"expected
number" predictions that differ greatly from experimental results. U4,
Pages 27-40
- Investigation 2, Fair and
Unfair
Games,
concludes unit 4. Here kids learn that "not fair" means that "players
have
different chances of winning." U4, Page 46
Too Far "Beyond Arithmetic" for TERC:
If we flip the coin two times, what 's the probability that both flips
will be heads?
TERC
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
program.
Co-author Michael T. Battista has distinguished himself with his
essay The
Mathematical
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,
inventing
strategies for finding the volume of small paper boxes and larger
spaces
such as their classroom. They investigate volume relationships
between
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
layers:
'A layer contains 3 x 4 or 12 cubes, and there are 3 layers so there
are
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
calculations
without thinking about why or how the formula works. For
meaningful
use of the formula, students need to first understand the structure of
3-D arrays of cubes. We strongly discourage teaching this
formula
to students; the layering strategies that they invent will
be
more powerful." U8, Page
I-18
(bold and underline emphasis added)
What Happens in the Classroom: A Month
of Geometry
- Students don't "invent." As
the
month
begins they are taught to fill a box with layers of plastic cubes and
then
count the cubes. Kids make the boxes, using graphing paper,
scissors, and tape. The term volume is not yet mentioned. After
"exploring"
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
rows
in a layer, so the number of cubes in a layer is the product of the
length
and width. Because the height gives the number layers, they
multiply
the number of cubes in a layer by the height to find the total number
of
cubes in the array." U8, Page 14
- Good idea! But more TERC
teaching, not
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
restricted
to the very special case of the supplied plastic cubes exactly filling
a box.
- Generalize later
perhaps?
They never
get to dimensions that aren't an exact multiple of the length of the
edge
of one of their plastic cubes. Forget about dimensions that
aren't
expressed in whole numbers.
- TERC then changes the focus to box
relationships
and links to the big idea of one-half.
U8, Pages 16-23
- "Student pairs determine
the
dimensions
of boxes that will hold twice as many cubes as a box that is 2 by 3 by
5."
- Shakita's group "found a
concrete way
to generate several spatially meaningful solutions" to the doubling
problem.
She explained "we made two packages of cubes that were 2 by 3 by
5.
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
got
a 2 by 6 by 5, then we got a 2 by 3 by 10."
U8, Page 23
- TERC likes this
strategy.
It's beautifully
hands-on and it demonstrates multiple correct answers.
- TERC fails to point out that
there
are many
other solutions that can't be represented this way.
- Next it's filling boxes with
identical
packages
of 2 or more cubes. U8, Pages 24-37
- The cubes are
"interlocking."
This is
high tech stuff!
- Kids learn that multiple copies
of a
given
package may not always exactly fill a particular box.
- Solution? Design the box
using
the generalization
of the method of Shakita's group.
- Enter plastic "centimeter cubes."
- TERC instructs the teacher to "show
a centimeter cube as you introduce the first activity."
U8, Page 40
- Then the term "volume" is
defined: "We
call the amount of space inside this box its volume. The
volume
of a three-dimensional object is the amount of space enclosed by its
outer
boundary." U8, Page 42
(emphasis in the original)
- Now we know why it's
Investigations
in Number,
Data, and Space.
- Using multiple undefined terms,
TERC's definition
is general, not limited to boxes, and avoids formulas.
- Students build "models of
volume units."
They "build a cubic meter using 12 meter sticks, joined at the
corners
with masking tape." U8, Page 44
- Students measure "the space
in
the
classroom." U8, Pages 38-60
- The teacher says: "You
can
use meter
sticks, string, calculators, and any other tools we have."
U8, Page 46
- Team answers ranged from 220 to
290
cubic
meters. U8, Page 52
- Students start "Comparing
Volumes."
"After measuring how much rice or sand small household
containers
will hold, students order them from least to greatest volume."
U8, Page 62
- Students are told to "start
by predicting
the order, then measure to check."
U8,
Page 63
- "To test their
predictions,
most students
directly compare two containers by pouring rice from one container into
another. They might pour from a smaller container into a larger
and
see that the larger is not filled. Or, they might pour from a
larger
container into a smaller and see an overflow."
U8, Page 65
- Next it's "Comparing Volumes of
Related Shapes."
U8, Pages 66-74
- Students are given patterns for
"11
solids."
They "use scissors and tape to create the prisms, pyramids,
cylinders,
and cones from the patterns." U8,
Page
64
- "Each solid is paired
with
another
that has equal base and/or height measurements - pyramids are paired
with
related prisms, and cones with related cylinders."
U8, Page 66
- TERC advises: "If the
cutting
and taping
is fairly accurate, students should discover that the volume of each
larger,
flat-topped solid is about 3 times the volume of the smaller, pointed
solid,
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
pouring rice,
of course.
- Enter "a special measuring tool -
a
see-through
graduated prism." U8, Pages 74-78
- TERC graduates don't need
formulas
for volume.
Just plenty of rice and this plastic tool.
- Students use it to determine "the
volume,
in cubic centimeters, of each of their 11 solids."
U8, Page 74
- Rice is poured, from each of
the
"11 solids,"
into the plastic measuring tool.
- TERC advises: "Loosely
taped
or bulging
sides can cause surprisingly large discrepancies. Thus, it is
important
to evaluate students' methods, rather than their answers."
U8, Page 78
- "As a final project,
students
design
and create a model made from geometric solids. The model must
include
prisms, pyramids, cylinders, and cones, all of which students can
create
from patterns." U8, Page 81
- "When we had to find the
volume of
the robot we started with the body because it was a rectangular prism
on
centimeter paper and we could count the dimensions. We multiplied
the dimensions and found the volume. With the other shapes we
couldn't
count the dimensions , so we filled the shapes and then poured that
into
the see-through prism with the centimeters measured on it. We
counted
those dimensions and multiplied them together the same way we did for
the
rectangular prism." U8, Page 86
- Student thinking has moved to
a
higher level
of abstraction. They now count using centimeter paper. They
are free of the centimeter cube crutch.
Significant Geometry at the End of the
Fifth Grade?
- Discovering exact relationships by
pouring
sand. That is significant! Let's call the New York Times.
- More kindergarten activities:
cutting,
folding,
taping, pouring, and building a paper robot.
- Relative, not absolute
information:
The volume of this container is less than the volume of that container,
but no way to compute the absolute volume of either container.
- Wait, there is a way!
"See-through graduated
prisms" are always conveniently available these days.
- No mention of formulas for the
volume
of the
"11 solids."
TERC's
"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
until
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
"Figuring out how something grows or declines is essential, not
just
in higher mathematics, but in the sciences and social sciences as well."
U7, Page I-17
TERC's Recommended Natural Language For
Describing Change: Grow, shrink, faster, slower, steep,
flat,
slow, steady, speed up, slow down, grows steadily, grows faster and
faster,
grows slower and slower, shrinks steadily, shrinks slower and slower,
shrinks
faster and faster, grows and then shrinks, oscillates between growing
and
shrinking. U7, Pages 21, 54, and 98
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
Found
in TERC's Book For Teachers
Here we present all the math
ideas
found (in page sequence) in Beyond
Arithmetic. Why all? It's another way to
appreciate
how much is missing and the total lack of big ideas and significant
math.
- "I have ^{1}/_{2
}cup
of flour and need 1^{1}/_{4 }cups of flour; how much
more
flour do I need? If I have good number sense of these
familiar fractions, their magnitudes, and their relationships to
each
other and to 1, I would be unlikely to use the traditional subtraction
algorithm (1^{1}/_{4} - ^{1}/_{2
}),
which requires that I find common denominators, transform the mixed
number
into an improper fraction, then subtract. Rather, I immediately
"see"
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
fact
I need ^{1}/_{2 }cup and ^{1}/_{4 }cup,
or ^{3}/_{4 }cups."
BA, Page 6
- Agreed, but now I have ^{2}/_{5
}of
a cup and want 1^{1}/_{4 }cups.
- With this simple change, we have
a
problem
that can't be solved by TERC students. They don't know about
common
denominators, and their knowledge of calculating with fractions is
limited
to a small subset of the familiar fractions.
Even if the student recognizes the need to calculate ^{3}/_{5
}+ ^{1}/_{4}
, the sum, ^{17}/_{20 }, is an unfamiliar
fraction
fact, not likely to be remembered by the TERC student. What
next? TERC offers their "fraction strip" addition method for
halves
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
three-fifths
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
that acceptable.
- Note: Given a TERC
illustration, always
think "what if we change one detail?"
- Add 58 + 57 by first adding 60 +
60,
and then subtracting 5 from 120. BA, Page 6
- What's the point? TERC has
converted
the problem to an equivalent problem involving the landmark
numbers, 5 and 60. They expect students to remember
sums
and differences for landmark numbers.
- Note that TERC has a genuine
challenge.
It isn't easy to fully convert to landmark numbers.
- Why not first add 50 + 50?
That's very
different. The student would then be faced with 8 +
7.
This single digit addition fact is not a familiar sum for many TERC
students.
They haven't yet constructed this fact.
- Fourth graders using TERC's
recommended language
for describing mathematical change: "It started out fast
then
it slowed down but now its (sic) growing faster again."
BA, Page 12
- Challenge in a second grade
classroom:
"Can
you land exactly on 100 if you count by fives?"
BA, Page 14
- Hanna explained: "Well I
first went
5, 10, 15, 20, 25 like we learned, and I did land on 100.
So
it works. But Jamie [her partner] didn't get it because maybe I
was
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
100.
But then, you know, Jamie said 'How much do you think we have?' At
first
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
skip
counting
by fives is the same process as repeated addition. She realizes
that
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,
Page
14
- Computing 42 x 37 in a fifth grade
classroom:
"It's the same as multiplying 42 by 40, then subtracting three
of
them. Ten 42's is 420. Double that to get 840.
Then you double that, so it's 1680. Then you have to subtract
three
42's. Two 40's down from 1680 is 1600, then another 40 off is
1560,
then subtract 6 more. So it's 1554."
BA, Page 20
- Fifth graders synthesizing data
and
making
generalizations: After "a survey on what occupations
interested
their classmates" they "began to
get
different views of their data" Two
examples
are given: BA, Page 23
- "About a quarter of our
class
wants
to help sick people or animals."
- "Most of us want to be in
a
profession
where we could be a star."
- Challenge in kindergarten: "How
many
eyes are there in our classroom?"
BA,
Page 28
- "We knew 26 to start,
then we
counted
27, 28, 29, 30. And kept on going. I counted and Abdulah
tried
to keep track. But it was too hard, so we got out some cubes and made
piles
of 26. Then we counted them all up."
BA, Page 28 [Recall keep track
comment
above]
- Math time in a third grade
classroom:
Using "any height that you think is reasonable for a third
grader
. . . figure out how the heights of six third graders can add up to 318
inches." BA, Pages 44-58
- "If we have two kids at
50
inches and
two kids at 53 inches, that's 206 inches."
One team member used a calculator to subtract 206 from 318, but Jamal
"counted
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,
Page
44
- 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
three
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
150
+ 18 = 168 168 + 1 = 169."
The team decided to use 3 at 55 inches and change the 49 to 53 to get
the
169 total. BA, Page 45 [So much for using "kids from our
class."]
- The teacher was impressed with
"Jamal's
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
demonstrating
how it's often easier to break numbers apart into
more familiar components and then add from left to right."
BA, Page 54
- A first grader indicated "how
she used
groups of 5 to count her set of objects."
BA, Page 50
- Martin Luther King was born in
1929.
Hold
old would he have been be in 1996? BA, Pages 68-71
- Example A: "1929
(just
born)
- 0 years old 1930 - 1 year old
1931
- 2 years old 1932 - 3 years old (and so
forth)" BA, Page 69
- Example B:
"1929
- 0
1930 - 1 year old 1940 - 11 (1 year + 10 years) 1950
- 21 (and so forth)"
- Teacher notes that this
student is
"making
good use of decades or 'landmarks' in the number
system."
- Example C: "Some
children might
represent this data on a horizontal line, and show jumps from year to
year
or from decade to decade." BA, Page
69
- "Some students develop
more
complex
strategies that solve the problem with very few intermediate
steps.
Here's a strategy from a student who began with the standard vertical
format:
1996
-1929
-3
70
70 - 3 = 67
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
counted
on in decades from 1929, saying quietly to herself '1939 is 10,
1949
is 20 . . .1989 is 60, and from 1989 to 1996 is 7 more, so 67."
BA, Page 71
- Note: TERC believes it's good
that
"this child
never learned to borrow."
- Note: Negative numbers and
"counting below
zero" don't appear in the actual TERC K-5 program.
- "Faced with a problem like
375
÷
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
subtraction
(37 - 25 = 12), then bring down the final 5. But does this
procedure
make sense in this context? It is neither the most elegant, the
easiest,
nor the most efficient way of doing the problem. In the time it took
you
to write the problem down, you could easily have said, "I know there
are
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
that
very few adults can explain (Simon, 1993). How then can elementary
students
hope to explain how the long division algorithm works"?
BA, Pages 73-74
- Agreed. Most of us would
quickly figure
15, without using long division. [Recall related discussion
above.]
- "It's easy to think of many
situations
in which the standard American algorithm is an inefficient way of doing
an operation. Adding 1987 + 1013 is another good example. Here's the
standard
algorithm:
1 1
1987
+1013
3000
The algorithm involves three rounds of carrying - an inefficient
procedure
under the circumstances. We should know enough about number relations
to
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
doing
this problem is cumbersome; it breaks the problem into little pieces,
when
the most efficient way of solving it is to work with the big picture."
BA, Page 74
- Agreed. It's immediately
obvious that
the answer is 3000. But what about 867 + 1889? Most
of
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
trivial.
TERC carefully picks the problems. Just change a number.
- Students are expected to learn
that "different
strategies are efficient in different situations. Multiplying by 9 may
well involve a different strategy than multiplying by 4."
BA, Page 76
- Exactly what we don't
want. We
want
standard methods, which can be used automatically, so we can move
on to the next level of complexity.
- "Multiplying 1346 x 231 is
problem
that is best solved with a calculator."
BA, Page 76
- "You would probably not use
a
calculator
to figure out 35 x 11, 21 - 19, or even $10.94 + $1.07. You
easily
use mental arithmetic to come up with quick and accurate answers.
We want students to be able to do this to, too. But calculators
should
be available nearly all the time, so that students can do more
difficult
calculations or check the answers they arrived at using their own
strategies."
BA, Page 78
- TERC emphasizes the importance
of
solving
a problem in multiple ways. Can't they find a second method to
check
answers, without using a calculator?
- As for "more difficult
calculations," they
don't appear in the TERC program.
- "Seven-year-old Jacob
recently
began
a mathematical quest to find all possible ways to subtract one number
from
another to make 24. He chose this problem on his own. He started with
25
- 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
bigger,
fascinated him. After half an hour, he had three pages of systematic
calculations
to proudly show to his parents and, later, to his class. He continued
with
this problem for several more days, at which point he confidently
announced
to his mother, 'I think there's a thousand ways to make 24, and I'm
going
to find them all.' " BA, Page 82
- What was the mother thinking?
- "Suppose the problem
involves
comparing
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
hasn't
understood the problem or hasn't seen that comparison involves a
process
other than addition. A second student, who compares by "counting on"
from
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
down
to 64, that's 18," is employing an even more elegant strategy
for
figuring out the difference." BA,
Page
97 (emphasis added) [Found in Chapter 5, A New Kind of
Assessment].
Did You Notice Any Elegant Strategies
or Significant Math?
Most of these techniques are familiar
to
anyone who has mastered traditional arithmetic. Transforming a problem
into one that can be solved using "familiar" facts is a good
problem-solving
strategy. The error here is the limited applicability of
the
techniques presented, the misrepresentation of these techniques
as
powerful strategies, and the complete omission of general computational
methods. But TERC doesn't just omit standard computational
methods,
they do it proudly, with a clear hostility. Please read on to
learn
more about that.
TERC
Calls it Number: Arithmetic Without Standard Methods
There's
Trouble in TERC City: Kim is Using a Standard Algorithm!
TERC and the Standard Algorithms: As
Stated
in Their Third Grade Teaching Materials
"If you have students who have already memorized the
traditional
right-to-left algorithm (of addition) and believe that this is how they
are "supposed" to do addition, you will have to work hard to instill
some
new values -- that estimating the result is critical, that having more
than one strategy is a necessary part of doing computation, and that
using
what you know about the numbers to simplify the problem leads to
procedures
that make more sense, and are therefore used more accurately." From Combining
and Comparing (Addition and Subtraction), Page 38
TERC and the Standard
Algorithms:
As Stated in Beyond Arithmetic
- "Everyone needs to know a
couple of
good ways to add, subtract, multiply, and divide. However, the
ways
we were taught to do these operations aren't necessarily the best or
most
efficient ways. Believe it or not, these traditional algorithms
are
largely the result of historical accident. Borrowing, carrying,
and
the procedure for long division - they're not universal. In other
countries and at other times in our own, students have been taught
different
and equally effective algorithms for the basic operations."
BA, Page 73
- "In the Investigations
curriculum,
standard algorithms are not taught because they interfere with a
child's
growing number sense and fluency with the number system."
BA, Page 74
- "There is no longer a
justifiable reason
for asking students to do pages of calculations - especially more
time-consuming
calculations like adding series of numbers or multiplying three-digit
numbers
by other three-digit numbers. We do not want to waste students'
time
on frustrating tasks that involve the rote application of memorized
algorithms.
Before asking your students to do problems without a
calculator,
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
same!"
BA, Page 78
- Note: This is a "whole
language" type
of argument that fails to differentiate between the learning needs of
the
novice and the skills of the expert.
- "There is growing evidence
that
'Algorithms
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
"unteaching"
place value. They don't teach it.
TERC and The Standard Algorithms: As
Stated
in The
Algorithm Issue (Essay at the CESAME Website)
- "Investigations
advocates treating
the conventional American algorithm for each operation as just one more
way to perform the operation." AI,
Page
1
- "There are other efficient,
accurate
algorithms that students understand better."
AI, Page 1
- "While the developers of Investigations
don't
recommending forcing students to use the historically taught American
algorithms,
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,
Page
2
The Truth About the
Standard Algorithms
- TERC doesn't teach the standard
algorithms,
but they can't hide them because parents and tutors teach them.
- TERC regularly instructs
teachers
regarding
that difficult day - Kim is using a standard
algorithm!
What to do? The TERC solution? Ask Kim to explain her
method.
Regardless of what she says, tell her to try another method. TERC
has never met a child who can adequately explain a standard algorithm.
- The standard algorithms are true
computational
"algorithms." That is, they are (potentially programmable) general
methods
for carrying out a (potentially complex) calculation by repeating
a sequence of simple steps.
- The "American algorithms" are the
standard
algorithms used in Europe, Asia, Africa, and the rest of the
world.
Despite claims to the contrary, there are no alternative algorithms
that
can match the efficiency, accuracy, and generality of the standard
algorithms.
- TERC offers special case methods,
not
algorithms.
Their "strategies" require conscious student observations that differ
from
problem to problem. Such observations are not guaranteed to
occur,
and they don't involve repeating a sequence of steps.
- The student is asked to multiply
5 x
99 and notices that this can be accomplished by multiplying 5 x
100
and then subtracting 5. Good TERC think, nicely transformed to an
equivalent problem involving only landmark numbers. But not a
general
method. Certainly not an algorithm.
- The TERC
alternative
methods
only work for a very small subset of whole number problems. The
standard
algorithms work for arbitrary real numbers. No problem with
non-landmark
numbers or digits to the right of the decimal point.
- TERC is opposed to explicit
memorization of
the basic (single-digit) number facts, but TERC students must remember
many non-basic number facts as a necessary condition for successfully
carrying
out the TERC alternative methods.
- Remembered, but never
memorized. TERC
wants that to be clear.
- TERC and the NCTM want kids to
appreciate
"the power and beauty" of mathematics, but they're blind to the power
and
beauty of the standard algorithms. With a very small amount of
knowledge
(remembering the single-digit number facts and knowing how to carry and
borrow relative to the ingenious design of our decimal system), the
student
can carry out any calculation involving the four basic operations,
including
cases with digits to the right of the decimal point. With the
exception
of long division, the calculation can be carried out automatically,
regardless
of the complexity of the calculation. Long division also requires
estimation
skills. Although TERC emphasizes estimation, they omit long
division
and thereby miss the perfect opportunity to demonstrate estimation as a
necessary skill.
- More generally, mathematical
techniques frequently
involve leveraging simple case facts to solve complex problems.
For
example, in differential calculus we use properties of tangent
lines
to study the local behavior of arbitrary continuous curves, and in
integral
calculus we determine the area of complex regions by using a limiting
process
that involves summing the area of rectangles.
- TERC sometimes recognizes the
power
of reducing
the complex to the simple, but they seriously miss the boat when they
reject
the standard algorithms.
Algorithms in The New York Times: An
Introduction to the New Math ( Link to NYT Article)
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
side
by side comparison, Ms. West compares "constructivist new math" to the
"traditional method." The casual reader may think: Is that all
this
is about? TERC will be pleased. They want readers to go
away
thinking that the "math wars" are caused by purists quibbling about
details.
They want you to be impressed that they nicely avoided "carrying." They
hope you won't know or notice that this NYT illustration is
intentionally
deceptive.
- Deception 1: What's the
appeal
of these
constructivist methods? It appears that they have eliminated
carrying.
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
constructivist
methods
look attractive with the simple cases presented, but they quickly
become
labor intensive and unwieldy when you use them to to multiply two
3-digit
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
right
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
cumbersome,
special-case limitations and thus be ready to appreciate the
efficiency,
accuracy, and generality of the standard algorithms. They would
then
look back at the "constructivist" methods as inefficient, preliminary
versions
of the standard algorithms.
Whole
Number Computation in the TERC Fifth (or Sixth) Grade
[TERC instructs the teacher] Write
the following two problems on the board: 253 x 46 701
÷ 27
"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
answer.
Now we're going to try finding the answers to these harder problems
without
a calculator."
U5, Page 128 (Bold in the original indicates TERC's script for
the
teacher.)
The teacher goes on to tell the students
that
they are to solve these problems, but not with standard algorithms.
TERC must state this, since they don't teach these algorithms.
But
some kids have learned them from parents and/or tutors, and all TERC
problems
are trivial for kids who know the standard algorithms and are allowed
to
use them.
We next list the whole number
computation
methods that TERC teaches to fifth graders. You won't find
this list anywhere in the TERC materials. It offers a
distillation
of the ideas found in those materials, but in a much more compact form.
- Remember facts about
landmark
numbers (anchor numbers)
- Example: The student should
remember
that
10 x 25 = 250 and that 20 x 25 = 500
- Note that TERC de-emphasizes
instant
recall
of basic number facts, but they expect kids to remember many other
number
facts.
- Remember or reference (non-basic)
multiplication
facts that have been recorded in "Multiple Towers."
- "The class builds a
Multiple
Tower
on a long strip of adding machine tape, listing multiples of 21 in
order
and looking for patterns in the sequence. They use the patterns
they
find to solve multiplication and division problems involving multiples
of 21." U5, Page 2
- The class builds other Multiple
Towers and
they are expected to use information recorded in these towers to help
them
solve problems. An example involving 32 is given below in this
list.
- Skip-counting, by 5, a
landmark
number,
a large one digit number, or a small two-digit number.
- Addition by "counting on"
from
the
first number, while simultaneously subtracting 1 from the second number.
- 24 + 17 is computed as 24 + 1,
25 +
1, 26
+ 1 ....... 40 +1, with 17 reduced to 16, then 15, then 14, etc..
- Note that students need a method
to
"keep
track."
- Subtraction by "adding on."
- 212 - 98 is calculated by adding
2 +
100 +
12.
- Decompose
numbers
relative
to place value. (TERC says break , not decompose)
- Decompose numbers relative to
landmark
numbers.
(TERC says "break," not decompose)
- 27 = 25 + 2
- 355 = 350 + 5
- Use the distributive law.
(TERC
never
mentions the "distributive law," but they can't avoid using it.)
- 25 x 21 = 25 x (20 + 1) = (25 x
20 )
+ (25
x 1) = 500 + 25 = 525
- Note also the conversion to
landmark numbers.
- Multiplication as repeated
addition,
with the possible efficiency of adding multiples of 10 or 100.
- 27 x 34 is calculated as (10 x
34) +
(10 x
34) + (7 x 34)
- Note the application of the
distributive law
and the remaining difficulty with 7 x 34.
- Multiplication (or division) by
using
related
facts in a "multiplication cluster."
- Example of a multiplication
cluster:
[10 x
32, 20 x 32 , 30 x 32, 5 x 32, 35 x 32] U5, Page 77
- "How can 10 x 32
help
you find 5
x 32? How can 10 x 32 help you to solve 20 x 32? Which of
the
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
teacher.)
- Students are allowed to use
the "Multiple
Tower" for 32 U5, Page 77
- Division as repeated subtraction,
with
the possible efficiency of subtracting multiples of 10 or 100.
- 510 ÷ 24 is calculated by
twice subtracting
240 and then subtracting 24 from 30 to get 21 R6
- Division (or multiplication) by
using
related
facts in a "division cluster
- "The teacher presents
this
cluster:
[12 ÷ 12 120 ÷ 12 132 ÷ 12
133 ÷ 12]." U5,
Page
90
- 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
is
10, and 12
÷ 12 is 1. Put them together and get 132 ÷ 12 is
11.
Between 132 and 133 there's only 1 difference. We're looking for
12 difference, so it's 11 R 1." U5,
Page 90
- Note the hidden use of the
distributive law.
TERC says these are examples of
students'
work at the end of the TERC K-5 program.
- 2015 - 598
- "I found the answer
[1417] by
adding
1000 + 100 + 100 +100 +10 +7. Start with 598. Add 1000 to
get
1598. Add 100 to get to 1698. Keep adding 100's to get to
1998.
Then add 10 to get to 2008. Then add 7 more."
U5, Page 10
- We would mentally add 2 + 1400
+
15.
- 6029 - 4873
- The student wrote 1000 + 100 +
29 +
25 + 2.
Then wrote 1000 + 100 + 25 + 4 + 25 + 2. This was then rewritten
as 1000 + 100 + 50 + 6. Nothing more. U5, Page
35
- Note the conversion to
landmark
numbers.
- We would mentally add 27 +
1100 +
29.
- 26 x 31
- The student wrote 10 x 31
=
310
20 x 31 = 620 25 x 31 = (620 + 155) =
775
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 =
31.
Finally, below these calculations, the student wrote 26 x 31 = (775 +
31)
= 806.
- The details are not given for
5 x
31 = 155
and 775 + 31 = 806. U5, Page 80
- The student used a
multiplication
cluster
for 31 and perhaps a Multiple Tower for 31.
- 767 ÷ 36
- TERC instructs the teacher how
to "lead
students toward different strategies."
U5, Page 82
- "There are ten 36's in
360,
so there
are twenty in 720. In 767, there are twenty-one 36's and a
remainder
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,
so 36
x 20 is
720 (or 720 ÷ 20 = 36). Then 767 - 720 is 47. Take
away
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
involving
small whole numbers.
- Kids need to learn about
carrying
and borrowing.
They need to learn about efficient, accurate, and general methods for
multi-digit
computation. They need methods that work for arbitrary real numbers,
not
just small whole numbers. They can forgot about algebra if they haven't
mastered the standard algorithms. Why? See links.
- They work best with problems that
are
"set
up" to make them look good.
- They're slow and require conscious
analysis
to classify the problem relative to one or more of the relevant
techniques.
- They use non-standard
language.
[Examples:
landmark numbers and familiar fractions.]
- They use non-standard
methods.
[Examples:
multiplication cluster and division cluster.]
- They use tools that are only
available
in
TERC classrooms. [Examples: multiple towers and manipulatives.]
- They can be difficult to master
when
the student
has been denied the necessary orienting framework (the complete facts
and
skills of traditional arithmetic). The TERC techniques are not
difficult
for us who have mastered traditional arithmetic. We use some of them
everyday.
They're easy for us because they actually form a small subset of the
arithmetic
knowledge we have stored in our brains.
TERC's
Hands-On Methods For Fractions, Decimals, and Percents
"The proper study
of
fractions provides a ramp that leads students gently from arithmetic to
algebra. But when the approach to fractions is defective, that ramp
collapses,
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
"This unit does not concentrate on procedures for either
decimal
or fraction computation. Students solve computation problems
using
good number sense, based on their understanding of the quantities and
their
relationships. They carry out addition and subtraction of
fractional
amounts in their own ways and in more than one way, using fractions,
decimals,
or percents, and using any models that make sense to them."
U3, Page I-18
TERC doesn't even attempt to discuss
operations
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.
As for fractions, TERC students learn
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
"models
for fractions," not common denominators.
Shakita and
Tai don't know that 3/48 = 1/16, but they should remember
that
4/8 = 3/6 = 2/4 = 1/2 because these "familiar" fraction facts have been
recorded on their Fraction Equivalent Chart. Shakita
doesn't
know about common denominators. To add 1/4 + 1/5, she
folded
her blue (fourths) and yellow (fifths) fraction strips, put them
together, and compared to the the pink (halves) fraction strip.
She
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
surprised,
but convinced after checking Tai's work. She then suggested that both
discoveries
should be recorded on the Fraction Equivalent Chart. Tai wasn't
sure.
The Fraction Equivalent Chart didn't currently show fraction sums, just
equivalent fractions. But surely these facts were acceptable.
After
all, they were about the big idea of one-half!
Later that day, Shakita shared their
discovery
with Sarah, her home-schooled friend. Sarah said, if that's true,
1/4 = 1/5. Now Shakita was excited. This new equivalent
fraction
fact definitely belonged on the Fraction Equivalent Chart. But
Shakita
needed to be sure, so she asked Sarah to explain. "It's
easy,
if 1/4 + 1/5 = 1/2 and 1/4 + 1/4 = 1/2, then 1/4 + 1/5 = 1/4 +
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
prove
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.
OK, this story isn't found in the TERC
teaching
materials, but parents in New York City aren't laughing. It's too
close to the truth of their "Saturday morning live" experience ("mom,
that's
not the way I'm supposed to do it"). The story contains no
distortions
of fact and is quite plausible, especially when you consider:
- TERC insists that children use
estimation
strategies (only) to make their colored fraction strips.
- TERC instructs the
teacher: "List
the five colors, each paired with a fraction, on the board or chart
paper.
Include a model showing students how to write the corresponding
fraction
name at the bottom of each sheet For example: pink -
halves
green - thirds blue - fourths yellow -
fifths
white - sixths." U3, Page 42
- "The students' challenge
is
to partition
the strips accurately to show different fractions."
U3, Page 42
- "When marking their
strips,
students
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
the
marks until they think they are quite accurate."
U3, Page 42
- "Fraction on Clocks" and
"Fractions
Strips"
are the only "Models for Fractions" available for adding and
subtracting
fractions.
- TERC teaches kids to solve
problems
using
the available models. It's exclusively "hands on."
- The problem 1/4 + 1/5 actually
occurs
as a
"more difficult problem" in the TERC materials (U3, Page 99).
- Children would naturally expect
that
the methods
that they have been taught are adequate for solving the problems given
to them, unless they've been told otherwise. But TERC never instructs
teachers
to tell kids that "strip addition" doesn't "work" for many pairs of
familiar
fractions. The familiar fractions aren't closed under
addition.
That is, the sum of two familiar fractions may not be familiar.
Certainly,
9/20 (1/4 + 1/5) is very unfamiliar to TERC fifth graders.
- TERC heavily promotes estimation,
rather
than accurate answers. 9/20 is about 1/2 in TERC-think.
TERC does want children to learn how to
compare
fractions and understand fraction "relationships." But once again they
want to avoid the complexity of "common denominators."
Solution?
Students work with models and play games until they remember (some of
the)
the relationships. [4/5 is less than 7/8. Isn't
it?
Let's check the chart.]
Finally, TERC wants kids to remember
"familiar
fact equivalents." For familiar
fractions
only, students complete strips, charts and tables for "Fraction
and
Percent Equivalents," "Fraction to Decimal Division," and
"Fraction,
Decimal, Percent Equivalents." Students are eventually given
paper
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
fractions,
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
BA: Beyond
Arithmetic:
Changing Mathematics in the Elementary Classroom
IM: Implementing the
Investigations in Number, Data, and Space
TERC
Investigations Grade 5 Teacher Books (Units 1 - 9)
U1: Mathematical
Thinking
at Grade 5 (Introduction and Landmarks in the Number System)
U2: Picturing
Polygons
(2-D Geometry)
U3: Name That
Portion
(Fractions, Percents, and Decimals)
U4: Between Never
and Always (Probability)
U5: Building On
Numbers
You Know (Computation and Estimation Strategies)
U6: Measurement
Benchmarks
(Estimating and Measuring)
U7: Patterns of
Change
(Tables and Graphs)
U8: Containers and
Cubes (3-D Geometry: Volume)
U9: Data: Kids,
Cats,
and Ads (Statistics)
Click for link to
TERC
CESAME Essays
AI: CESAME:
The Algorithm Issue
PV: CESAME:
Place Value
WU: How
to Prepare Students For Algebra by H. Wu, (American Educator,
Summer 2001)
Copyright
2002-2011 William G. Quirk, Ph.D.