SUPPLEMENTARY REPORT OF THE MAA ARG

President's Task Force on the NCTM Standards

This supplement to our task force's main report on the NCTM Principles and Standards for School Mathematics contains more specific comments and suggestions made by individual members of the task force.

• CHAPTER 2: Guiding Principles
• CHAPTER 3: Overview of the Standards
• STANDARDS (1 to 10)

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CHAPTER 2: Guiding Principles

[Alan Tucker]

Equity Principle - well done. My only comment is that I would like to see in writing the point that I heard at a meeting from one of the PSSM authors: a higher level of knowledge for all students raises the baseline above which an even higher standard for talented students can be developed. Today, many talented students just get an 'accelerated' program that covers topics at the rate that average students in other countries do. "A rising tides lifts all ships."-- while possibly simplistic, this reasoning has much to recommend it and seems to be a politically savvy way to offer everyone much of what they want.

Mathematics Curriculum Principle. The discussion does not develop the concern about focused instruction adequately-- the second paragraph in this section (page 27) is a good start. The subsection 'Choosing Mathematics that is most important' is meaningless without lots of examples. The important concerns of "coherence" and "comprehensiveness" in the curriculum are discussed too vaguely.

Teaching Principle. The 'Analysis and Reflection' part is much too vague-- no guidance about what is wanted in the classroom. This would be a good place to honestly address the challenge of getting the right balance between drill and exploratory activities.

Learning Principle. This is the section that consistently plays down mastery of skills and says virtually nothing positive about such mastery, e.g., in the first paragraph we find "Students' ability to provide correct answers is not always an indicator of a high level of conceptual understanding." The subsection 'Building on Prior Experiences . . ' is feel-good, e.g.,. "... become members of a mathematical community" but largely useless in guidance for day-to-day teaching. The subsection 'Learning Concepts and Procedures' is the one that really damns skills with faint praise.

Assessment Principle - generally well done. The explicit refusal to give grade-by-grade outcomes appears on line -4 of page 37. Page 39, paragraph 2 is too negative about the value of national tests, like the proposed 8th grade test. Page 40, paragraph 3 is too negative about grades for students. Again, why go needlessly out of one's way to antagonize the mainstream thinking in the country. This should be a consensus building document.

Technology Principle - reasonable and balanced. Because of the politically charged aspects of technology substituting for drill, the authors should be more careful in the paragraph on page 43 about calculators: the fifth sentence about the need for arithmetic skills should come earlier in the paragraph and should be stated more positively.

[Deborah Tepper Haimo]

As for technology, the document, in the introduction as well as in other sections, does mention that students need to become fluent in basic facts, but this always seems to be downplayed. I believe it important to stress that students need to master fundamentals and not turn to technology for simple arithmetic facts that should have become second nature.

[Henry Alder]

There are other places where the Draft is lacking in specific guidance and where teachers will clearly look for it. For example, see the statement on "Calculators" on page 43 where it is stated on line 5 that "Students at all levels should have access to calculators and other technology as they solve problems" and on line 8: "However, access to calculators does not replace the need for students to learn and become fluent with basic arithmetic facts.." No guidance is provided how both of these goals can be achieved simultaneously.

The Draft is completely silent on many other factors which need to be in place for mathematics instruction to be successful. Among the host of other programmatic matters, I would add the importance of uninterrupted classroom instruction, the special importance for students to attend mathematics classes, what constitutes a good lesson (as learned from some of the TIMMS studies), and the importance of parent involvement. I am not suggesting that many pages be devoted to these important factors, but rather that a concise coverage be included emphasizing what is essential for each of these factors.

[Stephen Rodi]

At this point in my reading [Standard 5], I decided that the Principles document would have more chance for positive impact on mathematics instruction in the U.S. if it expanded Chapter Two (pages 21 to 43) beyond generalities to include detailed, specific, hard-nosed direction on course content, course sequencing, hours of instruction, teacher training, teacher certification, use of homework and a host of other pragmatic matters which make the real difference in how much mathematics students learn. (I know. For better or for worse, education is a local matter.)

[Henry Pollak]

Page 37, figure 2.1. I see this as a DIRECTED graph going from N to E to S to W. I am not convinced of the link from W to N.

[Mercedes McGowen]

The authors have stated several of their underlying assumptions, particularly in Chapter 2: Guiding Principles. These are elaborated on and clarified in the following chapters. However, as the document is currently organized, one has to search through several chapters and put the pieces together in order to have a clearer understanding of what these assumptions and hypotheses are. I was expecting that our responses would, at the very least, speak to the beliefs that are explicitly stated in Chapter 2, as they are the foundation for the recommendations in the subsequent chapters. Do we agree with or support these beliefs? As outlined in the draft, they call for some very different instructional practices than are currently in use in many schools.

A basic assumption about learning mathematics is stated in the Learning Principle (p. 34), namely: people do not generally learn concepts by building up pieces of knowledge but rather from grappling with complex and interesting tasks. This assumption about how students learn implicitly challenges the belief that students must learn skills BEFORE they attempt to tackle complex problems. In the past, this has been misconstrued to imply that students do not need to learn skills. The issue many of us wrestle with is not whether to teach skills and algorithms, but WHEN to teach them and HOW to teach them so that students develop meaningful understanding and retain their knowledge in long-term memory. It is at the heart of much of the current debate. It has been argued that students can't use calculators until after they master basic facts; they can't use a symbolic manipulator until after they can do the algebraic manipulations by hand. They can't do word problems until they learn the procedures, etc.

The issue is not whether students use calculators, or symbolic manipulators, but what tasks they are asked to do and for what purpose. Let's focus on what skills, besides knowing basic arithmetic facts, we want students to KNOW upon completion of high school and address the issue of how those skills might differ, in degree, in complexity, in number, etc. for mathematically able students and for students who are not intending to take advanced mathematics. Is it the same set of skills and understandings for both groups? Alan Tucker raised the issue with his question about the two mathematical goals besides the traditional preparation for calculus. Should we not provide the authors of PSSM with some statement regarding these issues?

What UNDERSTANDINGS do we want students to have, in addition to "skills"? We have been quite silent on this point, focusing mostly on the need for basic skills.

[Bob Megginson]

In general, this chapter does a good job of laying out the principles guiding the Standards, although it does so better for some of the standards than others.

Below are some specific comments on the draft.

page 21, lines 4-26: The statement about the need for well-taught mathematics, and the references to the importance of technology and professional development programs, is good.

page 22, lines 1-2, 8-9: The difference between the statements of the Equity and Learning Principles is too subtle. In particular, if "learn" were substituted for "understand and use" in the Learning Principle, then the two would become indistinguishable. The Learning Principle should be reworded to something like this: "Mathematics instructional programs should enable all students to understand mathematics conceptually, not just as a collection of facts and skills, and be able to apply the material in situations not necessarily anticipated during instruction."

The Equity Principle

page 23, lines 15-22: The analogy between "mathematics for all" and "English for all" is a very strong and well-put statement about the need for the Equity Principle. It should be kept.

page 25, lines 5-16: If the cited research actually shows that achievement by females has increased to match that of males, rather than that the "gender gap" is closing only because of decreased achievement by males, then this should be said explicitly.

page 25, lines 26-29: It is true that "Schools in middle- and high-income communities are typically able to attract better qualified mathematics teachers than schools in low-income communities." However, a lot of highly dedicated and exceptionally hardworking teachers in low-income school districts, some of whom I have had the pleasure to work with as they have sought to upgrade their mathematical abilities, are quite fed up with hearing that the education in these districts is so bad because the low-paid help is incompetent. The first sentence of this paragraph should be replaced with something like this: "Schools in middle- and high-income communities are typically able to afford a higher-quality educational experience for the students than schools in low-income communities, including a variety of instructional materials and good professional development opportunities for the teachers."

page 26, line 1 through page 27, line 5: The emphasis on research-based analyses of differential performance of students from groups traditionally underrepresented in mathematics is very good, since it addresses the perception voiced by some critics of the current Standards that there is no research basis for modifying or expanding classroom approaches to address the needs of such students. The additional emphasis on holding high expectations of all students, but also making sure that they are provided with the means to satisfy those expectations, is also very good.

The Mathematics Curriculum Principle

The stress throughout this section is that

... the mathematics included in school programs should be sound, significant, and important. Areas of mathematical content and process selected for inclusion need to be arranged into coherent, comprehensive curricula that foster mathematical development over time." (page 27, lines 29-32).

In general, this section conveys this message well.

page 28, lines 13-16 and 32-35; page 29, lines 26-27 and 36-37: It is stated here directly and clearly that rigor, sound arguments, the axiomatic nature of mathematics, and procedural competence are important, as is knowing when to use technology (and therefore when not do so). It is also stated that the emphasis should not be entirely on conceptual knowledge and mathematics that derives from real-world applications. These points are excellent and should be kept.

page 29, lines 23-25: Those who advocate one single national mathematics curriculum for all precollege students could quote the statement that "these standards should consititute the core curriculum for all students" as evidence that the NCTM does not in principle oppose such a notion. I suggest changing "constitute the core curriculum" to "form the basis for the curriculum".

The Teaching Principle

This is a good, solid statement about the need for competent and caring teachers. It is particularly good that it is stated clearly that "Such teaching (to create a classroom environment for thinking and learning) can be accomplished only by teachers who have had strong preparation and who will have ongoing access to effective professional development" (page 30, lines 23-24), and "teachers need to know mathematics well beyond the mathematics they teach" (page 31, lines 17-19).

page 31, lines 31-38: This paragraph is a good statement of the need for a mixture of classroom approaches if a teacher is to reach all students effectively, and it is wise (and correct) to include "direct telling" as an example of one strategy that is often appropriate.

page 32, line 1 through page 33, line 15: These subsections on learning environments and student in-class discourse are more about the core content of the college preparation of every teacher than mathematics standards. Cuts should be made here.

The Learning Principle

This section is more vaguely written and less focused than its predecessors. In particular, the introduction on page 33, line 21 through page 34, line 30 could be tightened up. What it basically says is that (1) in their school years, students should acquire a solid understanding of the conceptual basis underlying school mathematics; (2) such understanding is not the same as the ability to repeat memorized facts and display practiced skills; (3) this conceptual understanding begins with ideas formed about mathematics even before reaching school and should grow as students progress through the grades; and (4) in building on the prior knowledge that students have, teachers must consider both the knowledge base that the class has as a learning community and the differing ones held by students as individuals. It should not take so much space to say this.

page 34, line 32 through page 35, line 21: The entire first paragraph of this three-paragraph subsection on "Learning Concepts and Procedures" could be omitted, and the important points would still be made. This would also remove the reference to "so-called prerequisites" that some will view as inflammatory.

page 35, lines 17-21: The examples given here of student mistakes due to conceptual misunderstandings could just as easily be examples of mistakes due to procedural misunderstandings; students with little understanding of concepts but a solid understanding of procedures would still not make these particular mistakes. Better examples should be sought, or the ones already given should be reworded to show how they reflect conceptual misunderstandings. For example, rather than presenting the miscomputation 100 x 3.45 = 3.4500 as an error a student makes by relying on procedure, it could be stated that a student with a solid conceptual understanding would not be likely to make this error, since the growth of the number 3.45 by a factor of 100 will clearly not result in an answer that is still equal to 3.45.

page 35, lines 22-35: This subsection could be made more concise without substantial loss of content by omitting everything beginning with "They should" on line 28 through "conceptual understanding" on line 32. The material addresses principles that apply to all learning and is not so important to include in a document specifically addressing mathematics learning.

page 35, line 36 through page 36, line 16: The subsection on "Active Engagement with Mathematics" and the concluding summary paragraph for the entire section are fine and well stated.

The Assessment Principle

The writing in this section is direct and nontechnical, and therefore likely to be easily understood by all who are interested in assessment, teachers and nonteachers alike. The section begins with a solid statement (pages 36-37) of the reasons for assessment, making it clear that the aim is assessment of curriculum and progress toward school, regional, and national goals as well as assessment of individual student progress. The rest of the section, which focuses on the structure and process of assessment, is also quite good. I have no suggestions to offer for change. In particular, I really like the concrete example given on page 39, lines 16-25 of assessing student understanding of equivalent fractions beyond the ability to apply an algorithm that constructs them.

The Technology Principle

page 40, lines 30-40: The introduction to this section, with its emphasis on the responsible use of technology in the classroom, is good.

page 41, lines 1-12: I disagree that technology should be portrayed as an effective tool for changing students' perception of mathematics as a dead discipline. As mentioned, technology can be used effectively for geometric visualization and the exploration of large data sets, but the students will almost certainly recognize that these are applications of technology to the better understanding of old knowledge. The portions of cryptography that they can explore with understanding at the precollege level, even with the use of technology, still seem to be rather classical, and the presentations of fractals and chaos that I have seen given at the precollege level seem to be quite descriptive, with technology used to explore particular fractals in detail rather than to convey any real understanding that there is new mathematics involved in analyzing what is seen on the screen. I am not familiar with all of the materials used to present these growing fields to precollege students, so there may be much here about which I am not aware, but I remain wary of touting technology as a device for convincing precollege students that mathematics is a rapidly growing subject.

page 41, line 13 through page 42, line 41: This subsection discusses the real value of technology in the mathematics classroom, namely, to aid mathematical explorations that would be overly tedious or otherwise impractical without the technology. The example of a young student exploring which numbers one can count by to arrive exactly at 100, and then (we hope) exploring some of the mathematical questions that arise from this, is good.

page 41, lines 37-38: It is not clear why "Children can study graphs and tables at an earlier age when they have technological help to generate them," though technology can certainly help with the exploration of complicated graphs and large tables at any age.

page 42: This entire page is a solid, well written statement of the importance of technology in the modern mathematics classroom. The caveats in the last paragraph, particularly the strong statement ending this subsection that "The technology should be used to support conjecture, but teachers need to provide an emphasis on the importance of proof," are quite appropriate and welcome.

page 43: This page is the document's statement about the use of calculators and other technology. Due to the controversy surrounding this issue, this must be said very carefully. As it now stands, the statement that "Students at all levels should have access to calculators and other technology to use as they solve problems" (lines 5-6), and later statements about giving students "routine access" to technology so that it is "available when needed," are open to misunderstanding and quotation out of context. The document seems to imply that there are situations in which it is appropriate not to allow students to use technology, and that "routine access" to technology need not be given when the students are tested on their mastery of basic skills, but it never flatly says so. It should flatly say so. After all, students also use books extensively in their everyday lives and will be expected to be able to learn from them in the workplace, but this does not mean that all tests must be open-book.

CHAPTER 3: Overview of the Standards

[Henry Pollak]

The four societal needs on page 46 should really be five. What's missing explicitly is the mathematics of intelligent citizenship. Number 2 is called "Cultural Literacy", and says what "our citizens" should learn. But what is being learned here is cultural literacy. Then in Number 3, it says "Just as the level of mathematics needed for intelligent citizenship has increased..." - as if that's what the previous paragraph had talked about! It seems to me that mathematical literacy is number 1, cultural literacy is number 2, mathematics for intelligent citizenship is number 3, mathematics for the workplace is number 4, and mathematics for client disciplines is number 5. If the list of societal needs must be kept down to four, then the workplace and the science and engineering and other users really go together better than submerging intelligent citizenship between the second and third.

In the parts that I read, the word "model" often means a mathematical idealization of a phenomenon outside of mathematics (e.g. page 50, line 15 or page 307, lines 16 and 17). But sometimes the word "model" may mean a mathematical reformulation, a different representation, of something elsewhere in mathematics. That's the way I would read page 48, line 12 and line 15. "Model" IS used in several ways in mathematics, and those are two of them. They are different; this illustrates the value of a glossary.

It is asserted on page 46, line 22 that "the main topics of discrete mathematics are included, but are distributed across the standards". Are we sure? There is research by Harold F. Bailey, "The Status of Discrete Mathematics in the High Schools", DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 36, "Discrete Mathematics in the Schools", AMS and NCTM, 1997, pp. 311-316. The six topics of highest frequency reported in that survey were, in order, set theory, combinatorics, graph theory, functions and relations, trees, and probability. I have seen the graph theory and the probability in my reading so far; are the others covered?

[Deborah Tepper Haimo]

1. There is one overall difficulty that disturbs me. It is the fact that all results of mathematics are based on hypotheses whether these are explicitly given at the start of a problem or are implicit and thus are assumed to hold if any reasonable person would not dispute their existence. In early grades, there is generally no problem, since what is given or known is largely obvious so that no difficulties arise. As students move ahead in mathematics, however, the fact that hypotheses are often not explicitly given can be a problem both for students and their teachers. For example, this turned out to be serious and was apparent in some of the earlier NCTM materials where unnecessary mistakes were made. By ignoring hypotheses, questionable results were obtained for problems highlighted as vignettes. Such errors can and should be avoided by a clear statement to the effect that a given set of hypotheses leads to a single correct answer (even if there are various ways to arrive at a solution -- a fact that is often stressed in these pages as it should be).

2. Page 50, lines 33-34: I wonder about that last sentence. Do students who have number sense really "NATURALLY decompose numbers, etc."? I rather doubt this from my experience.

3. Page 51, line 3 of last sentence: I would prefer deletion of the last five words which seem to me unwarranted, and their replacement by "with numbers are ready to advance in their study of mathematics."

4. Page 51, lines 6-7: I would delete the words " the single digit", "combinations and the counterparts for". I would start the next sentence with "Understanding" and follow "skill" with "of such facts", deleting "Understandings and skills can also be developed" at the beginning of the next sentence which I would combine with the first by introducing the words "as well as".

5. Page 51, line 10: It might be preferable to have some parallel structure by introducing the word "at" before "school".

6. Page 51, lines 14-15: I suggest changing the end of the sentence to "and in successfully studying many other areas of mathematics."

7. There is much repetition of ideas, phrases, and words throughout the document extending it to well over 300 pages! Will it really be read by anyone? I guess the authors realized this, but they nevertheless did not seem to restrict themselves, perhaps because of lack of time.

8. Perhaps on page 51, line 16, the word "use" could be changed to "will turn to".

9. Page 51, lines 20-21: Replace the words "and that" with "to".

10. Page 51, line 22: A plural is needed for the verb "help".

11. Page 51, line 23: All that follows the second "and" would be questionable. At least the word "will" should be replaced by "may".

12. Page 51, lines 26-27: The words "and the inclination" might better be replaced by "making sure", and the further repetition of "and inclination" in the next line should be deleted. .

13. Page 51, line 34: Why is it necessary to distinguish between "estimation and approximation"? Are these terms clearly defined and universally accepted by educators? I have asked several mathematicians and they could not differentiate meaningfully between those terms.

14. Page 51, last paragraph: This paragraph fails to stress how important it is to know the basic arithmetic facts as well as those in, say, the calculus. The first sentence might be completed by including the statement "but all students must know thoroughly all the basic facts of arithmetic without relying on calculators for such information." This might be followed by "Only after they have mastered these facts is the time appropriate to turn to calculators as computational tools to deal with cumbersome computations necessary for solving some problems." The rest of the verbiage is poor justification to introduce calculators promiscuously into the mathematics classroom and is better omitted entirely.

15. Page 52, line 5: After "aided" add "when necessary".

16. Page 52, line 6: Replace the first five words with "manipulatives".

17. Page 52, lines 11-13: Change "collected" to "gathered"; combine the following sentences to read instead, "It may be that, in the early grades for example, "24" is seen as ... twelve".

18. Page 52, lines 29-39: What about more complex fractions such as 2/7 or 4/9? When are students exposed to these and their operations? Is that included in the grades 6-8 material mentioned on the next page, at the beginning of the first full paragraph?

19. Page 53, line 31: Is the phrase "missing-addend situations" conventional?

20. Page 54, line 5: Add, after "identities", the words " for addition and multiplication, respectively".

21. Page 54, line 16: At some point, there was discussion of division's relationship to subtraction. What is regarded as "inverse" here?

22. Page 54, line 37: Who is the judge of "tedious"? Might not this word be omitted?

23. Page 55, line 16: Why not say "have memorized" rather than "be able to recall"? Politically, memorization is out, but why?

24. Page 55, line 31: Here also, "develop recall of" might be changed to "memorize".

25. Page 56, lines 19-24: I find it confusing to have patterns and functions, which are generally regarded as parts of algebra and play a role in its study, separated out as topics here. I would suggest that the whole first paragraph be omitted.

26. Page 56, line 26: Add, after "including", the words "the search for patterns, the study of functions, ... notations."

27. Page 57, line 5: Young children should be alerted to the fact that when they are given a small number of terms their finding of a next term or formula that generates what has already been given is only one possible answer if nothing further is mentioned.

28. Page 57, line 16: After the word "case", add "very likely".

29. Page 57, line 21: After the word "generated" there might be a comma and the continuation of the sentence to read "but should be aware that if a general rule to describe the pattern is not given, they may find a variety of possible generalizations that produce the same few given terms at the start".

30. Page 57, line 31: The wording can be considerably abbreviated here, as elsewhere, without changing any meaning. Instead of the many repetitions, after the sentence ending with "study", add a new one that reads "As students continue in grades 9-12 and systematically consider other families of functions, they use algebraic ideas to build on earlier school experiences, and extend their knowledge by including polynomial, rational, exponential, and trigonometric functions".

31. Page 58, line 13: It would be wise, it seems to me, to avoid pompous declarations, and the overused word "power" is a case in point. I would thus recommend that the first four words of the second sentence here be replaced by "It" and that the last four words be replaced by "finding a solution".

32. Page 58, line 15 : Aside from the word "power," which I again would delete in favor of, say, "learning", I am concerned about the emphasis on mathematical ideas encountering "conceptual obstacles" or being "complex". It seems to me that the overall picture is of a discipline too hard to learn. Why instill such negative thoughts in those who should give every indication that they are involved with a dynamic, exciting, and vibrant area of study?

33. Page 58-60, lines 15-23: There is too much negativism here and far too much detail. The entire section can be considerably shortened!

34. Page 60, line 25: The entire paragraph here might read "Mathematical modeling of phenomena provides an important application, symbolic notation being central here with algebra implicit throughout." Again, the rest of the section can be abbreviated considerably.

35. Page 61, lines 11-13: The words "change" and "multiple" are too vague here and need to be replaced.

36. Page 76, line 5: An important part of understanding mathematics might be to change the phrasing of the second bullet to read "develop the ability to formulate problems by identifying hypotheses and conclusions, to represent them in various ways, and to use algebraic notations to abstract and generalize these". Why specify "outside mathematics" since all examples seemingly outside the discipline will be dealt with by mathematical tools?

37. It is important that problem-solving not be confined to a set of isolated situations that will fragment the discipline. For this reason, it is vital that mathematical concepts be carefully considered and mastered so that students have a sense of a well thought-out discipline culminating in important results. Indeed, the same mathematical tools, given different interpretations, can be applied to seemingly unrelated disciplines to solve a highly varied array of problems in many different areas if their essence is recognized.

38. Despite the phrasing of needing to "develop a disposition," the mere introduction of that noun furthers the prevalent notion that mathematical talent is somehow innate and those without it cannot learn. I would by far prefer to change the word "disposition" to "ability"!

39. Page 76, lines 11-21: The paragraph might be changed to read "Developing the ability to solve problems is important and should be a well-integrated part of the curriculum. It requires finding a solution method not known in advance, and consequently strengthens students' use of the strategies they have learned. Students must be given ample opportunity to deal with complex problems that require significant effort to solve."

40. Page 76, lines 22-37: The introductory questions are repetitive of earlier discussion and should be omitted. The example for middle school to provide some understanding of proportion seems confusing, particularly if recipes have more ingredients than the same juice and water. (Is that intended to be dealt with in what is in the parentheses?) The problem as stated is far too vague to be considered a good mathematical illustration, and much too much time seems devoted to getting the notion of proportion across by this example.

41. Page 77, lines 1-11: The paragraph might read as follows: "When students learn a given concept such as the operations of arithmetic in the lower grades, their knowledge should be reinforced with good problems. For example, a teacher may say 'I have pennies, dimes, and nickels in my pocket. If I take three coins out of my pocket, how much money could I have taken?' (adapted from NCTM 1989, p. 24) The knowledge needed to solve this problem is the value of the coins in question as well as the operation of addition. Students must have content knowledge to deal with problems, but they can determine the information they lack and need if they have acquired real understanding of the concepts involved."

42. Page 77, lines 14-16: Delete this material.

43. Page 77, lines 26-29: Replace the first sentence with "Individuals who have mastered mathematical concepts tend to abstract and generalize results. In the second sentence, after "individuals" add the word "generally" and then delete "making ... and ".

44. Page 77, line 31: At the end of the line, add "with equal horizontal and vertical sides".

45. Page 77, lines 34-35: The last sentence of the paragraph might read: "These children have learned that there may be many ways to look at a problem and finding one approach may not necessarily be a signal of being finished. One may well seek another answer which may be shorter or more elegant, highly prized in mathematics."

46. Page 78, lines 4-16: Why are both examples for high schools?

47. Page 78, lines 24-31: Most of the material in the paragraph has been repeated many times and the paragraph should be deleted.

48. Page 79, lines 7-8: Delete the entire second sentence. Add, at the end of the third "and cannot be generalized."

49. Page 79, lines 9-15: The last sentence there might read "Clearly different strategies are accessible at different ages, including, for example, the very specific and mathematically powerful proof by contradiction, ... strategies."

50. Page 79, lines 15-17: Delete the sentence there, and have as the last sentence, " Strategies are learned over time, becoming increasingly complex to deal with more involved problem situations."

51. Page 79, lines 21-22: Add after "they" the words "should have mastered". Delete the next sentence.

52. Page 79, lines 22-24: After "They... problem" introduce a comma and continue with "reading it carefully if it is written down and asking questions until they fully understand it, if it is told to them orally." Delete the sentence that starts with "Effective".

53. Page 79, line 31: Delete the nine words beginning with "in" and ending with "mind".

54. Page 79, line 36: Add after "problems" the words "if these are not isolated but clarify and enhance important mathematical concepts" with appropriate commas.

55. Page 79, lines 38-39: After the fourth word of line 38, "that", add "stress the importance of developing such ability, consider a variety of mastered strategies, do not hesitate to make adjustments and change course if need be" to compete the sentence.

[Susanna Epp]

The subsection Standard 7, discussed on pages 80-85, is generally good to excellent. My main reservation concerns the use of the words "convincing" and "persuasive" to describe arguments, and this reservation applies to all the many sections in PSSM where this language is used. The quote from John Mason (page 81), "Convince yourself; convince a friend; convince an enemy," sounds great to a mathematician, but in the real world, rational thought processes play only a small role in convincing or persuading anyone of anything. In particular, they essentially never work with one's enemies, and polite friends normally downplay disagreements of all kinds. That is why it is difficult to get students to criticize each others' arguments in a mathematics class.

[Mercedes McGowen]

As Deborah Haimo pointed out: "the fact that hypotheses are often not explicitly given can be a problem both for students and their teachers." I believe that we, along with the authors of PSSM, should also explicitly state hypotheses AND underlying assumptions about teaching and learning mathematics.

STANDARDS

STANDARD 1: Number and Operation

[Alan Tucker]

• K-2 The Focus Areas are good. The discussion is good except on the lower half of page 114 and on page 115 where mastery of addition facts is downplayed too much.

• 3-5 The Focus Areas and discussion are good. Informative examples.

• 6-8 The Focus Areas and discussion are generally okay. The second bullet "develop meaning for integers . ." in the 'Understand Number. .' Focus Area seems to be a primary grade topic. The 2nd paragraph on page 218 and the 3rd paragraph on page 219 seem to be discussing material that should have been covered fully in earlier grades -- there appears to be little new here.

• 9-12 The Focus Areas and discussion are good. I like the emphasis on matrices and complex numbers.

STANDARD 2: Patterns, Functions, and Algebra

[Alan Tucker]

• K-2 The Focus Areas and discussion are reasonable (not much going on for this standard at this level).

• 3-5 The Focus Areas and discussion are good. Reasonable examples.

• 6-8 The Focus Areas and discussion are good. Nice examples.

• 9-12 The Focus Areas and discussion are good. However, on page 281, paragraph 3, I'd like to see an explicit, positive statement about the importance of mastery of symbolic manipulation skills.

As noted earlier, one curriculum for all high school students seems wrong to me. Calculus in high school should be discussed. There are important points to be made here, e.g., require students in AP calculus classes to take the AP exam, rather than treat the course as a warm up for calculus in college.

[Mercedes McGowen]

The discussions about representations and, in particular, about symbolic notation, are found throughout the document in Standards 2 and 10. The underlying assumptions on which the recommended changes in practice are based are not stated directly and concisely in one place, but dispersed throughout the document. These discussions recommend changes in traditional teaching practices which, if widely adopted, would impact bench marking and expectations about what students should be able to do at certain specified times. The changes have not been noted or commented on in any of the task force discussions to date.

In the content standards 1 and 2, the authors explicitly recommend that "conventional symbolic representation and manipulation begins to be an important part of the childs' mathematical activity toward the end of this grade band." (p. 120, lines 1,2). They argue eloquently for allowing children flexibility in their choice of representations and state that teachers "should provide as many different representations as necessary" in order that children might develop understanding of mathematical concepts and procedures. (page 148, lines 4-5). The authors suggest that it would be useful "to consider not only how conventional representations might be better taught to students, but also to consider the role that other, less conventional, representations might play in the mathematics classroom." They support their recommendation on p. 95 and 95 (Chapter 3).

In the Grade 3-5 band, flexibility in choosing and creating representations with discussion of the advantages and limitations of a particular representation are discussed. Expecations are not stated clearly for what skills and understandings students who complete 5th grade should be able to demonstrate with respect to symbolic notation.

However, the discussion about structure and symbolic notation in the Standards for 6-8 are quite specific (pages 224-226). A caveat is also stated: "Learning to interpret and write symbolic expressions and to solve symbolic algebraic equations is important in the middle grades, but for most students extensive work with symbols should come only after they have become fluent with verbal, tabular, and graphical representations of relationships that symbols represent."

Once again, the PSSM recommendations explicitly challenge the traditional practices of when symbolic notation is introduced and how it is introduced. They also raise the question about what the 9th grade formal algebra course should consist of. What is envisioned as content for middle grades students is much of the foundation many students lack AFTER completing two years of high school algebra as currently taught.

STANDARD 3: Geometry and Spatial Sense

[David Kullman]

I have no problems with the four points of the standard itself, nor with most of the Focus Areas for specific grade bands. However, it isn't clear to me what is meant by "use spatial orientation to navigate to a point" (page 171, line 13) or whether "geometric relationships among two-dimensional and three-dimensional figures" (page 227, line 18) refers to each dimension independently or to some connections between them.

As I noted elsewhere, I think that the detailed examples are overdone. On the other hand, the Focus Areas are vague enough that their implementation may be accomplished in various ways, not all of which will meet OUR interpretations of what the standards say.

[Bob Megginson]

General Discussion of the Standard

It is good that mention is made right up front about the role of geometry in developing students' ability to "use formal reasoning and proof in their study" (page 61, lines 34-35). This point is repeated at appropriate places throughout the discussion of this Standard, for example, on page 62, lines 21-26 and 34-36 and page 291, lines 20-22. Appropriate emphasis is also placed on the use of technology to aid geometric visualization and investigate geometric conjectures, but the eventual need for rigorous justification of these conjectures is still emphasized.

The overall discussion of this Standard makes good use of specific examples, both in the general discussion and in the specific descriptions of the implementation at differing grade levels. The emphasis on using coordinate geometry as a "bridge for linking the worlds of algebra and geometry" (page 63, lines 3-4) falls right in line with one of our ARG's recommendations for the treatment of geometry.

The subsections on transformations and spatial reasoning (pages 63-65) are well written. In short, I really do not have any suggested changes for the presentation of this Standard in Chapter 3.

Chapter 4: Standards for Grades Pre-K-2

The organization of these gradewise discussions is very good, with specific bulleted descriptions of implementation items followed by discussions that are loaded with examples. This organization should be kept.

page 123, lines 23-24: It seems far-fetched to claim that students are developing the concept of a number line when using colored connecting cubes. I recommend deleting this example, and instead bringing up the modeling of number lines in the discussion of rulers on page 129.

Chapter 5: Standards for Grades 3-5

page 172, lines 6-11: This example is not well chosen, since the exploration would not seem to require any assistance by computer. It can be done very well visually.

page 172, line 30 through page 173, line 3: The emphasis on the development of both two-and three-dimensional geometric intuition is good.

page 172, lines 4-12: Another very good feature is the emphasis on correct use of standard mathematical terminology.

Chapter 6: Standards for Grades 6-8

page 227, lines 4-6: Students SHOULD engage in formal validations using proof in high school. The word "may" should be changed to "should" here.

page 228, line 26 through page 229, line 12: This is an excellent exposition of the importance of exploration, conjecture, and justification for this grade band. It should be kept as is. The same comment applies to the discussion of the role of the Pythagorean theorem given on page 230, lines 4-20.

page 230, line 23: The use of acronyms whose meanings are not widely known outside the mathematics education community, such as MIRA that appears here (about whose meaning I am clueless), should be avoided.

Chapter 7: Standards for Grades 9-12

page 294, line 15 through page 295, line 6: I am not happy with the way this problem is stated, specifically, that the line segment is "randomly divided into three lengths" and the supposition that "x and y represent two of the three lengths." I suppose that this means something such as the segment is "divided into three subsegments by two points chosen at random along its length" and that "x and y represent the lengths of the two noncentral subsegments," which then justifies the analysis given. An alert student might (and should!) worry that some other interpretation might be assigned to the notion of randomly dividing the segment into three "lengths" and that the answer will then be different.

page 298, line 31: Here is another undefined acronym, ARISE.

STANDARD 4: Measurement

[David Kullman]

I like the overall approach to this standard, starting with "the process of measuring" in grades Pre-K-2 (page 128, line 6) and eventually characterizing measurement as "the process of assigning one or more numbers to the properties of an object" (page 300, line 7).

[Henry Pollak]

In Standard 4, both overall and in the individual levels, there are several items which seem to be delayed too long or missing. One is the fact that in the context of measurement, 3 and 3.0 are not the same thing. 3 inches means that the actual length is between 2.5 and 3.5 inches; 3.0 means that it is between 2.95 and 3.05. I didn't see this anywhere in the measurement standard; it might be in the number standards somewhere -- I haven't read all of those. There is good discussion in various places about measurement precision and errors (e.g. pages 177 and 236), but not of the implication for notation. Next, the number of places to keep in a computation is nicely done on page 304, but that's the Standard for grades 9-12! Shouldn't this come up sooner -- like grades 3-5? Third, there is an issue which I didn't see mentioned at all, but which I think belongs there. That is the implied accuracy of numbers in everyday speech. When we say that A is 82% of B, this usually means that the computation gives a number between 81.5 and 82.5. If we say 80%, it often means between 75 and 85; 50% often means that it's not close to either 0 or 100, and no more than that. Another example of the same phenomenon: Conversions between metric and English units when the original number is much rounded often gives ridiculous-looking results in the newspapers.

Page 303, lines 7-15. The write-up makes no point of the fact that the coefficient of log x on line 8 is 1.500 to three decimal places. Historically, that was enormously significant - Kepler's Third Law! Does the omission of this send a message that we want sent? Thoughts from Standard 6 are found elsewhere in the discussion of Standard 4 - especially in Grades 9 - 12. Why not here?

[Stephen Rodi]

It is indeed the better part of common sense to include, as the document does, both metric and "customary" measurement systems in the curriculum (page 67, lines 36-37).

I thought the sequencing of the measurement material from Pre-K through Grade 12 was well laid out. In particular, I was impressed with the frequent use of short examples or model problems to clarify ideas. I thought this technique of exposition especially effective for teachers in the lower grades (prior to Grade 9) who might wonder exactly what or what level of sophistication was intended.

The emphasis on estimation and approximation in Grades 3 - 5 and again in Grades 6 - 8 is excellent.

I suggest that the word "emerge" on page 178, line 34 (as in, formulas for measuring perimeter, area and volume should "emerge" from student recording of information and looking at patterns) may come back to haunt the document. There is a lot of potential pedagogical misinterpretation in that verb. Maybe something like, "In their work with measurement, students should begin to recognize the standard formulas for....."

In Grades 6-8, the presentation of measurement had a very strong geometric flavor, which I thought appropriate. As I read this material, the following thought struck me. One way of resolving the dilemma "Should algebra be studied in 8th grade?" would be to start the formal study of algebra in 9th grade but to substantially beef up the study of geometry in Grades 6-8 with some deductive geometry and a little informal algebra embedded. This might better prepare students to deal with the abstract symbolism of algebra which seems to cause so many students trouble these days. In fact, from what I could tell, Principles was recommending a course similar to this.

I agree with the statement directed to Grades 6-8 on page 232, line 21, that "measurement should be emphasized throughout the school year, rather than treated as a separate unit of study." However, I think at these grade levels some specific instructional units might concentrate directly on measurement as a transition from the intuitive, object-based experiences in earlier grades to a more formal and abstract understanding . This should not be overdone, but also should not be lost in a more "spread out" instruction on measurement.

I would like to see a comment (similar to the one above on page 232, line 21) included in the material for Grades 9-12. Here it should have the additional emphasis that in Grades 9-12 students are in science classes where measurement is critical. Measurement ideas have to be drilled, repeated, and "exampled" constantly with students so that they become second nature.

I was surprised in the Grade 9-12 material to find very little (in fact, I think, no) linkage between measurement in mathematics classes and in science classes, even though the major example in this section was a physics problem! This appeared inconsistent with the emphasis in the Pre-K-2 and Grade 3-5 presentations of the importance of physical experience in learning about measurement. It seemed the draft writers were avoiding reference to the "real world" precisely when one got close to the "real" study of the real world in science! Is the orientation for "all students" affecting this presentation?

At the other extreme, the "log-log" example in the Grade 9-12 material seemed overly sophisticated for the point being made in this expository document. The document may need a more centrist example here.

I suggest eliminating the word "some" in line 21, page 304. A goal of "some understanding" is too low, even for "all" students. At the Grade 9-12 level, it is not too much to expect all students to have a "full" understanding of "basic formulas for surface area and volume."

As a final comment, I observe that my biggest trouble in teaching calculus in college, especially to business and social science students, but even to science students, is their almost non-existent sense of unit and measurement. Even more important than coming to me with algebra skills (which they do not) would be their arrival with a clear idea of how rates are measured and what such measurements mean: dollars per year, people per day, milliliters per minute. The actual meaning of the derivative, as opposed to grinding out numeric answers, remains an eternal mystery to them because they do not understand these basic measurement concepts and how to express them. Their level of ignorance reveals itself daily when asked to put a measurement label on a numeric answer.

[Bob Megginson]

Chapter 3: General Discussion of the Standard

This discussion lays out well the general intent of this standard. The progression through the grades seems reasonable. Appropriate emphasis is given to connections with other areas of mathematics, such as geometric notions of congruence and similarity (page 68, lines 6-17) and the uses of under- and overestimation in calculus (page 68, line 36 through page 69, line 5).

page 69, lines 17-32: I agree strongly with the statement that "Strong connections need to be made between the formula and the actual object." However, I am worried about the example involving the formula for the area of a circle. Using this plausibility argument to convey intuition about the area of a circle (1) ultimately relies on another geometric formula for which the students may not have any better intuition than for the area formula, namely, the formula for the circumference of a circle; (2) does not convey the important point that the real idea behind the formula is that the areas of similar figures are in proportion to the squares of corresponding linear dimensions; and (3) does not in any case seem intuitive enough to come to mind quickly when students are using the formula. This plausibility argument can be shown to students as a way to relate the constants of proportionality in the formulas for the circumference and area of a circle, but the primary intuition for the formulas should come from their relationship to standard facts about proportionality for similar figures.

Chapter 4: Standards for Grades Pre-K-2

page 129: The use of rulers is mentioned several times. This is an appropriate place to discuss the development of the notion of a number line in the minds of students in this grade band, rather than through the example on page 123, lines 23-24, that uses colored connecting cubes.

Chapter 5: Standards for Grades 3-5

This section is particularly well stocked with examples to illustrate the ideas.

page 179, lines 9-15: Although the instructions for the ball drop experiment say to drop the balls from different heights, the table data show that all balls were dropped from the same height, 100 cm. This minor discrepancy should be eliminated.

page 179, line 21 through page 180, line 6: It is not clear that the students will see that these measurements, such as the amount of water consumed daily at the school water fountain, are being "carried out for a purpose." Some of the examples should be replaced by ones for which the real-world need is more apparent.

Chapter 6: Standards for Grades 6-8

page 233, lines 25-26: It is stated here that "there is no need for students to memorize metric and customary measurement conversion equivalencies," but on page 67, lines 39-40, it says that "Students will find it helpful to know a few English-metric equivalents." I would strongly urge sticking with the original statement from page 67. It really is helpful for students to know, for example, that there are 2.54 centimeters in one inch.

Chapter 7: Standards for Grades 9-12

This presentation seems fine. I have no suggestions to make.

STANDARD 5: Data Analysis, Statistics, and Probability

For more general concerns about the impact of adding statistics to the mathematics program, see earlier comments by Henry Alder and Stephen Rodi in CONCERN #5 of our main report.

[Stephen Rodi]

Perhaps more discussion could occur in the Pre-K-2 section on using play as a source of informal introduction to these ideas. As the document only partly recognizes, in these grades children are very young and intellectually very non-abstract. Inclusion of these topics in the curriculum, it seems to me, would have to be very informal at this level.

For Grades 3-5, I thought some of the goals (e.g., page 182, line 1-7) a bit sophisticated, especially for Grades 3 and 4. The level of abstraction of the actual classroom presentation would make a difference. But, if presentation is as important an issue as I think it will be, Principles should give more guidance, rather than lumping all of these grades together. My parenting experience tells me that third graders and fifth graders are very different creatures as regards readiness to deal with some of these ideas.

I very much like the statement in line 28, page 237: "Students' work with data analysis and statistics in grades 6 -8 draws on and integrates their knowledge of ratios, fractions, decimals, percent, graphs, and measurement." This sort of integration, in my opinion, is exactly what should be occurring in the middle school grades, rather than a rush to algebra for all students. As I have said above, I think geometry should be the other main support of the middle school years.

I don't know what is intended by the statement on page 310, line 14 that "probability is counter-intuitive for most people." I find the basic idea of probability very intuitive for most students. In a world of lottery pay off, Las Vegas vacations, and weather reports, the intuitive idea of measuring "chance" is one of the few ideas I can depend on students having when they show up in class. The formalizing of that intuition is what gives trouble.

The breadth, depth, and specialization of the Grade 9-12 material reminds me that a vast part of the secondary school teacher corps will be unable to deal with this material in an insightful way, if it is dispersed throughout four years in an integrated mathematics curriculum. Will this stark reality in effect move these topics into separate courses taught by specialists and effectively undermine the goal of all students seeing these topics? If not, how will the ability of the teacher corps to teach these topics be managed?

[Henry Pollak]

In the overall discussion of probability, conditional probability is mentioned in the Chapter 3 overview (page 75, line 28). The experimental evidence is certainly that this is difficult for students, and the only place it occurs is in the 9-12 standard on page 310. At this point, one might mention that it is a good application of rational functions. All the way through, it might be worth saying that probability is not only an excellent application of rational numbers, but can also be used to motivate a variety of operations on rational numbers.

In the discussion of data analysis at all levels, there is frequent mention of students taking their own data and then analyzing them. I am delighted with that, but I notice that there is never any mention of data presented in the textbook which students then work with. It seems to me there is a place for this also.

Page 183, Figure 5.14. The numbers on the abscissa are meant to represent intervals, not coordinate points, but that isn't clear.

Page 240, Figure 6.8. Brand 1 has two entries of 1 pepperoni each in the bar graph, but these have disappeared from the box-and-whiskers plot, which has 2 pepperoni as the minimum value. Why? Are they outliers more so than the 13's? There is more than one accepted definition of whiskers -- which are you using?

Page 309, Figure 7.18. People might see a logistic curve from 1790 to 1940, which then got a new "kicker" by the time 1950 came around. Should something be said about this possibility?

STANDARD 6: Problem Solving

[Henry Pollak]

Page 136, lines 18 - 20. I am sorry, but I don't understand that sentence at all.

Page 187, lines 23 - 29. Changing the fractions into decimals and then comparing is not explicitly listed as a strategy. Didn't anyone try this, for example on a calculator? Some people might take the omission as a judgment.

Page 189, line 27 through page 190. There is no mention of handing out 8 brownies, one to each person, and only dividing the 9th. Why not? Again, some people might read something into this omission.

Page 244, up to line 21. The statement of that problem does not require the "figure" to be connected, only that each square must have at least one edge-adjacent neighbor. Is the union of two disjoint rectangles, 1x4 and 2x5, considered legal? Do you expect that question to be raised? Is part of the learning what the problem does NOT say?

STANDARD 7: Reasoning and Proof

[David Kullman]

For each grade band, two common questions are addressed. (1) What does reasoning (or reasoning and proof) look like in these grades? and (2) What is the teacher's role in developing reasoning and proof? I think this is a useful way to consider the standard.

In Chapter 5, the question, "What do students learn about mathematical reasoning in grades 3-5?" seems redundant and should be subsumed under question 1.

On the other hand, in Chapter 7 the question, "How does technology affect the role of reasoning and proof in grades 9-12?" is well put. However, I do not believe that it is adequately answered. In particular, I would like to see a stronger, more direct statement to the effect that, while one counterexample is enough to show that a general assertion is false, the truth of such an assertion in several (or even many) cases does not allow the conclusion that it is true in all cases.

I was disappointed that nowhere, in the discussion of reasoning and proof for grades 9-12, is the phrase "deductive reasoning" used. There is also no mention of the principle of mathematical induction nor of a deductive system. (The latter IS included as a Focus Area in the 9-12 Geometry standard.)

[Henry Alder]

I think Standards 7 (reasoning) is very well done. I like the early introduction of mathematical reasoning, particularly stressing it in the middle school grades, and its strong emphasis in the high school grades. This is something I know is badly needed. I teach only upper division mathematics courses at the present time, but I observe even at that level how much the students' mathematical reasoning abilities have declined, especially in the last few years. Questions are raised by students on simple logic which I had never encountered until two or three years ago.

My criticisms of the treatment of this Standard are minor and are equally applicable to other parts of the Draft such as:

1. The use of meaningless trendy terms such as "powerful reasoning" (see page 139, line 21).

2. Statements which are unclear such as (page 191, lines 13 and 14): "A focus in these grades (3-5) is for students to create mathematical ideas themselves and to articulate, examine, and evaluate these ideas". It is not clear to me how a third grader would do this.

3. Use appropriate mathematical terminology wherever possible, for example, "proof" instead of "convincing mathematical argument" (see page 316, line 13).

[Susanna Epp]

By and large, the discussion of reasoning and proof in the grade-band sections was disappointing. If students are to succeed in proving and disproving basic statements about numbers and geometry in high school, the discussion in the preceding sections of PSSM should indicate how the thought processes necessary to achieve this goal are to be built up over the years. What experiences will lead students to understand that a few examples do not prove a general statement but that one counterexample will disprove it, that the truth of "if A then B" implies the truth of "if not B then not A" but not necessarily of "if B then A," and that to show a statement P is true, it suffices to show that not P is false? More generally, how will students learn to distinguish valid from invalid reasoning? The draft PSSM hardly addresses these crucial issues at all.

In particular, there are no references to the use of logical puzzles, which are effective in giving students a sense both for chains of inference and for proof by contradiction. Nor is the concept of "generic example" discussed, where one works through an example illustrating a general property with one set of numbers and then points out how each step in the solution would be equally valid if any other numbers were substituted. Generic examples provide a transitional stage in the development of the concept of proof by making it possible to engage in deductive reasoning about "any" things with students not yet mature enough to deal with sophisticated mathematical notation. For example, a proof that the decimal representation of any rational number is terminating or repeating would involve extensive use of subscripted variables but the idea can be effectively conveyed through discussion of a generic example.

In general, these grade-band sections do not go far enough in suggesting ways that teachers can help students learn to distinguish an argument justifying a statement from evidence establishing its plausibility. Justification might be sketchy, contain gaps, and neglect special cases, but suitably enlarged, a justification for a statement should be capable of being expanded into an actual proof. On the other hand, noticing a pattern in a certain finite set of data or observing an analogy between two mathematical structures merely establishes plausibility for a general property by providing supporting evidence that it might hold. But such evidence does not constitute valid justification that the property holds in general.

STANDARD 10: Representation

[Stephen Rodi]

I was delighted to read the material in pages 94-100. The nexus between mathematics and language has been under-used in teacher preparation and in instruction. Students, and probably too many of their teachers, see mathematics as a great puzzle to which one needs to learn the tricks and keys. They don't appreciate mathematics as a way of expressing knowledge and understanding, in part because they do not understand language either. Pages 94-100 address this phenomenon head-on. It should be a core unit in every university mathematics education curriculum.

The discussion of "idiosyncratic" representations (page 96, line 14-28) offers an area wherein the Principles document opens itself for criticism and wherein it should choose its words carefully. I would suggest one change in lines 25-26. Drop the phrase "when appropriate" at the end of the sentence "teachers can build bridges from students' personal representations to more conventional ones, when appropriate." There are those who believe such a bridge always is appropriate, but by omitting the phrase the issue is left more open for individual interpretation.

I found the "level" of material discussed under this Standard for Grades 3-5 distinctly less sophisticated than the Grades 3-5 material for Standard Five. This only re-enforced my instinct, alluded to above, that the presentation (or perhaps actual level) of the Grade 3-5 material for Standard Five needs to be re-visited.

The "algebra paragraph" in this "Representation" Standard for Grades 6-8 (page 265, lines 3 -12) seems pitched at a more sophisticated level than what is proposed for algebra under Standard Two (page 222, line 1-10) for the same grade level.

On page 267, line 25, "relatively large scale, motivating, and significant problems" are proposed as "important." I suggest omitting the word "relatively." I also note that some people (at least me) have an instinctually negative reaction to what I call "semi-emotive" descriptive words in mathematics education documents like "powerful" mathematics or "meaningful" mathematics or "robust" understanding (page 268, line 4). (My dictionary gives the fifth definition of "robust" as "marked by fullness or richness" and applies it only to wine.) Perhaps three such adjectives in the same sentence on page 267 is overkill. Finally, my question again arises on how the rest of the curriculum will be carried out while such "relatively large" problems are being solved? Teachers and others will need direction on this.

The issue of idiosyncratic representations arises again on page 268, lines 6-7. I would suggest that the mandate given there that "teachers need to be careful about trying to move students too quickly towards conventional representations" needs discussion and perhaps re-phrasing.

In a similar vain, I would recommend omitting the word "eventually" in the sentence (page 268, line 20) "Eventually students must learn to use conventional notation so that they can communicate effectively with others." "Eventually" sounds like this learning could be put off for an awfully long time!

I also suggest that conventional notation does more for the student than allow communication with others. Notations provides the very epistemological "words" and concepts needed for a fuller understanding than the student's home-made "representational vocabulary" allows. In this light, one should be moving students toward conventional representations more, rather than less, rapidly. Much lack of understanding can be disguised in home-made representation.

The caveats (page 268, line 38-44) on testing student understanding of personal representational presentations to verify if these reflect real understanding are excellent. Such "testing" goes on daily in my business calculus classes, long after students leave grade six!

I applaud the sentence of page 330, lines 19-21: "As they progress through high school, students must continually develop their ability to meaningfully interpret and use conventional, and increasingly abstract, forms of representation fluently and flexibly."

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