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

• The TERC Program: A Summary View
• The TERC Philosophy of Elementary Math Education
• Searching For "More and Deeper Mathematics"
• TERC Calls It Data:  Statistics Without The Arithmetic Mean
• Probability Without Fractions:  It's Guess and Check
• TERC Calls it Space: Volume Without Formulas
• TERC's "Mathematics of Change" >>> Moving Towards Calculus
• The Math Ideas Found in TERC's Book For Teachers
• TERC Calls it Number:  Arithmetic Without Standard Methods
• There's Trouble in TERC City:  Kim is Using a Standard Algorithm!
• Whole Number Computation in the TERC Fifth (or Sixth) Grade
• TERC's Hands-On Methods For Fractions, Decimals, and Percents

The TERC Program: A Summary View

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.

1. 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.
2. 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.
3. 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."
4. 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.
5. TERC omits standard formulas.
6. TERC emphasizes estimation and many right answers.
• They suppress the concepts of precision and accuracy.
7. 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.
8. 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 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 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.

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

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 (2x²  + 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.]

• 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

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
• 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
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
• 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
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
• 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
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
• 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?

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.

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 12¹/₂ , both 12 and 13 are likely predictions, but be prepared for a variety of responses."   U4, Page 19
• TERC introduces their concept of 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

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)

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
• 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.
2. 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.
3. 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.
4. Enter plastic "centimeter cubes."
• TERC instructs the teacher to "show a centimeter cube as you introduce the first activity."  U8, Page 40
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)
• 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.
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
• 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
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
• 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
9. 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.
10. 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
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
• "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?

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.
• Wait, there is a way!  "See-through graduated prisms" are always conveniently available these days.
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

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

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 ³/₄  cups."  BA, Page 6
• Agreed, but now I have ²/₅ of a cup and want 1¹/₄ 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?"
2. 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.
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
• 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
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
• "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."
7. 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]
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
• "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
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
• 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.
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
• Agreed.  Most of us would quickly figure 15,  without using long division.  [Recall related discussion above.]
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
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.
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
• 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.
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
• 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.
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
• What was the mother thinking?
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!

"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

• "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.

• "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

1. 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.
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.
• 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.
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.
• Remembered, but never memorized.  TERC wants that to be clear.
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.
• 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.

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.

Leading Mathematicians Explain the Educational Importance of the Standard Algorithms

1. The Math Wars, Conceptual Thinking, and Traditional Algorithms  by David Ross
2. Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education by H. Wu
3. Mathematicians Respond to the NCTM's Request For Information About Algorithms
4. Long Division by David Klein and R. James Milgram
5. Algorithms, Algebra, and Access by Stanley Ocken
   

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.)

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)
• 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.
2. 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.
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.
• 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."
5. Subtraction by "adding on."
• 212 - 98 is calculated by adding 2 + 100 + 12.
6. Decompose numbers relative to place value.  (TERC says break , not decompose)
• 344  = 300 + 40 + 4
7. Decompose numbers relative to landmark numbers.  (TERC says "break," not decompose)
• 27 = 25 + 2
• 355 = 350 + 5
8. 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.
9. 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.
10. 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
11. 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
12. 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.

1. 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.
2. 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.
3. 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.
4. 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?

1. 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.
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

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.

1. 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
2. "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."
3. 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.
4. TERC heavily promotes estimation, rather  than accurate answers.  9/20 is about 1/2 in TERC-think.

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.
 

Leading Mathematicians Explain the Educational Importance of Computing with Fractions

1. Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education by H. Wu
2. Knowing and Teaching Elementary Mathematics by Richard Askey
   

References     (See TERC Investigations Price List at scottsforesman.com)

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)

Next?

• The Anti-Content Mindset: The Root Cause of the "Math Wars"
• Memorize Multiplication Facts?  Cheney, Yes.  Romberg, Abstain.
• Understanding the Revised NCTM Standards: Arithmetic is Still Missing
  
• Understanding the NCTM's Math "Standards" and NCTM "Math Reform"  (Home)