TERC Hands-On Math: The Truth is in the Details

An Analysis of The First Edition of TERC's Investigations

by Bill Quirk  ( wgquirk@wgquirk.com)

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

  1. 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.
  2. 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
  1. TERC insists on the ongoing use of hands-on tools (manipulatives, "models," and calculators).
  2. TERC rejects the need for memorization and practice.
  3. TERC fails to clearly define terms.
  4. TERC emphasizes "familiar numbers."
  5. TERC omits standard formulas.
  6. TERC emphasizes estimation and many right answers.
  7. TERC proudly rejects standard computational methods.
  8. TERC attempts to directly teach their shallow and misleading content.

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:

TERC Says Manipulatives and Calculators Are Essential as Students Work Collaboratively

TERC Says Their "Constructivist Math" Approach:

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:

Note:  TERC claims that their fifth grade materials are "Also appropriate for grade 6."

One-Half is a Big Idea for TERC

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 102 + 4 x 10 + 7.  Eventually we want them to think: 8,247.942 = 8 x 103 + 2 x 102 + 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 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 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 + 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. 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.

  1. The teacher tells the students:  "at USA Today, the goal is to sell ads to fill two-fifths of the paper."  U9, Page 70
  2. 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
  3. 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
  4. 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.
  5. 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
  6. 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
  7. 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
  8. 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

Significant Statistics At The End of the Fifth Grade?

  1. Eyeball estimates to associate a "familiar" fraction with the (combined area) of the ads on a page.
  2. The central role of manipulatives (Recording Strips and Data Strips)
  3. Fifth graders who can't add familiar fractions and need a calculator to divide 48 by 3.
  4. 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.

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
  1. 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
  2. TERC then changes the focus to box relationships and  links to the big idea of one-half.  U8, Pages 16-23
  3. Next it's filling boxes with identical packages of 2 or more cubes.  U8, Pages 24-37
  4. Enter plastic "centimeter cubes."
  5. 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)
  6. 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
  7. Students measure "the space in the classroom."  U8, Pages 38-60
  8. 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
  9. Next it's "Comparing Volumes of Related Shapes."  U8, Pages 66-74
  10. Enter "a special measuring tool - a see-through graduated prism."  U8, Pages 74-78
  11. "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

Significant Geometry at the End of the Fifth Grade?

  1. Discovering exact relationships by pouring sand.  That is significant!  Let's call the New York Times.
  2. More kindergarten activities: cutting, folding, taping, pouring,  and building a paper robot.
  3. 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.
  4. 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.
  1. "I have 1/2 cup of flour and need 11/4 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 (11/4  -  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/cups."  BA, Page 6
  2. Add 58 + 57 by first adding 60 + 60,  and then subtracting 5 from 120.   BA, Page 6
  3. 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
  4. Challenge in a second grade classroom: "Can you land exactly on 100 if you count by fives?"  BA, Page 14
  5. 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
  6. 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
  7. Challenge in kindergarten: "How many eyes are there in our classroom?"  BA, Page 28
  8. 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
  9. A first grader indicated "how she used groups of 5 to count her set of objects."  BA, Page 50
  10. Martin Luther King was born in 1929. Hold old would he have been be in 1996?   BA, Pages 68-71
  11. "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
  12. "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 1
    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
  13. 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
  14. "Multiplying 1346 x 231 is problem that is best solved with a calculator."  BA, Page 76
  15. "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
  16. "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
  17. "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 TERC and The Standard Algorithms: As Stated in The Algorithm Issue  (Essay at the CESAME Website)

The Truth About the Standard Algorithms

  1. TERC doesn't teach the standard algorithms, but they can't hide them because parents and tutors teach them.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
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.

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.

  1. Remember facts about landmark numbers (anchor numbers)
  2. Remember or reference (non-basic) multiplication facts that have been recorded in "Multiple Towers."
  3. Skip-counting, by 5, a landmark number, a large one digit number, or a small two-digit number.
  4. Addition by "counting on" from the first number, while simultaneously subtracting 1 from the second number.
  5. Subtraction by "adding on."
  6. Decompose numbers relative to place value.  (TERC says break , not decompose)
  7. Decompose numbers relative to landmark numbers.  (TERC says "break," not decompose)
  8. Use the distributive law.  (TERC never mentions the "distributive law," but they can't avoid using it.)
  9. Multiplication as repeated addition, with the possible efficiency of adding multiples of 10 or 100.
  10. Multiplication (or division) by using related facts in a "multiplication cluster."
  11. Division as repeated subtraction, with the possible efficiency of subtracting multiples of 10 or 100.
  12. Division (or multiplication) by using related facts in a "division cluster
TERC says these are examples of students' work at the end of the TERC K-5 program.
  1. 2015 -  598
  2. 6029 - 4873
  3. 26 x 31
  4. 767 36

What's Wrong With TERC's Methods for Whole Number Calculations?

  1. They're limited to simple problems involving small whole numbers.
  2. They work best with problems that are "set up" to make them look good.
  3. They're slow and require conscious analysis to classify the problem relative to one or more of the relevant techniques.
  4. They use non-standard language.  [Examples: landmark numbers and familiar fractions.]
  5. They use non-standard methods.  [Examples: multiplication cluster and division cluster.]
  6. They use tools that are only available in TERC classrooms.  [Examples: multiple towers and manipulatives.]
  7. 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:
  1. TERC insists that children use estimation strategies (only) to make their colored fraction strips.
  2. "Fraction on Clocks" and "Fractions Strips" are the only "Models for Fractions" available for adding and subtracting fractions.
  3. The problem 1/4 + 1/5 actually occurs as a "more difficult problem" in the TERC materials (U3, Page 99).
  4. 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. 

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.