Focus on Focal Points - Part 2

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From FOCUS [The Newsletter of the Mathematical Association of America], February 2007, Volume 27, Number 2, pp. 29-31. See http://www.maa.org/pubs/focus.html
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This is the SECOND of three articles that appeared in the current issue of FOCUS dealing with NCTM's Curriculum Focal Points. The third article will follow shortly.
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Focus on Focal Points

A Commentary on Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, NCTM, Reston, VA, 2006.

By Anthony Ralston

Background

In 1989 and 2000 NCTM (The National Council of Teachers of Mathematics) published two reports, Curriculum and Evaluation Standards for School Mathematics (hereafter the Standards)[1] and Principles and Standards for School Mathematics (hereafter PSSM) [2] on standards for school mathematics. The first of these, at least, was very important in influencing school mathematics curricula but both - although the second less so - have been rubbished by some mathematicians, mostly research mathematicians, many of whom act - and write - as if NCTM were The Great Satan of mathematics education.

But now with the publication under review here there is (almost) universal praise from the most zealous of the traditional (as opposed to reform) Math Warriors (hereafter TMWs) even though the NCTM makes clear that Curriculum Focal Points (hereafter CFP) builds on and is closely tied to PSSM. Phrases such as "an end to the math wars" or "it's about time" for this "role reversal" now roll off the lips of prominent TMWs. How can this be? If the NCTM was as hopeless as it had been portrayed and if CFP only builds on previously denigrated NCTM publications, how can there now be such praise for its latest work? Read on!

Rules of Engagement

If you wish to produce a document on a controversial subject that will be praised or, at least, not damned by any side, the first rule is: Keep it short. If you do, there will be much less text to shoot at. The Standards at 258 pages and PSSM at 402 pages were full of detail that could be castigated by those so inclined.

Avoiding the trap of the Standards and PSSM, the authors of CFP have obeyed this rule admirably. Not counting the boilerplate at the front, CFP is 41 pages short. Of these, 20 pages are an Appendix that relates the Focal Points to PSSM. Nicely color-coded for ease of use, this Appendix may, indeed, be valuable to curriculum developers but it is just what it claims to be: An appendix. The Appendix is followed by one page of references. Ten of the remaining pages are introductory material explaining the motivation of CFP and explaining what Focal Points are, how they should be used, and how they relate generally to PSSM. These do contain the most telling indication that NCTM's approach has not changed:

"[CFP] assumes that the mathematical processes described in Principles and Standards will be implemented in instruction that requires students to discuss and validate their mathematical thinking; create and analyze a variety of representations that illuminate the connections within the mathematics; and apply the mathematics that they are learning in solving problems, judging claims, and making decisions."

The guts of CFP, however, are contained in 10 other pages, one each for the ten grades from Prekindergarten to Grade 8, each of which contains the focal points for that grade.

A second rule is this: Wherever possible, be ambiguous. Then just about everyone can interpret what you have written as supporting his or her perspective. CFP abounds with language subject to whatever interpretation the reader might wish to put on it.

Take, for example, "fluency" (as in "fluency with multidigit addition and subtraction") which appears 25 times whereas "proficiency" appears only once in the entire document and that in the introductory pages. Fluency is the kind of word that can be interpreted as mastery by those who want a back-to-basics approach to mathematics but as meaning only that students can deal with numbers flexibly and efficiently by those who wish to avoid the "drill and kill" instruction of the past. Is this just a quibble? I think not. CFP says in three places that "students should develop fluency with efficient procedures, including the standard algorithm". Clear enough, isn't it? When studying division, this surely impliess that the student should learn the standard algorithm for long division.

Well, not quite. One reader of CFP submitted the following question to the NCTM website for Questions and Answers on CFP (http://www.nctm.org/focalpoints/qa.asp): "Does the Standard Algorithm have to be mastered by all students?" In the answer posted on the website, the response suggests that students should use "efficient procedures, including the standard algorithm - meaning including [italic in original], not exclusively, access to the standard algorithm".

All clear now? And later, "we recognize that use of the standard algorithms may be an issue with some. The key here is the understanding of the algorithm, any algorithm, however it is defined".

Sweet are the uses of ambiguity.

Throughout CFP the language has been very carefully chosen so as not to upset any of the most vocal traditionalists, mostly research mathematicians, who were so critical of the Standards and PSSM. The result is a document so spineless that traditionalists have praised it and reformers will only be mildly dismayed. (Full disclosure: I have been a peripheral warrior in the Math Wars but hardly a neutral one. I am totally unsympathetic to the traditionalists whom, I believe, have utterly failed to grasp how mathematics education needs to adapt to a world where calculators and computers are ubiquitous. But neither am I a fan of the so-called reform curricula that are generally much too timid in proposing changes in school mathematics.)

The C-word

Well, there I've done it by using just the word the TMWs feel so strongly about, namely "calculators". It is particularly noteworthy that the word "calculators(s)" appears nowhere - I repeat, nowhere - in CFP . How can this be at a time when whether or how much students use calculators in elementary school arithmetic is one of the most, perhaps the most controversial issue in elementary school mathematics education? The authors of CFP would, I think, answer this question by pointing to the section in CFP on "How Should Curriculum Focal Points Be Used" where they say "Its [CFP's] presentations of the focal points include neither suggestions for tools to use in teaching nor recommendations for professional development in content or pedagogy."

Thus, we won't discuss calculators because we aren't giving any "suggestions for tools". On the other hand there are five references to those other tools, "pencil-and-paper". The fact is that one just cannot speak or write persuasively about the elementary school mathematics curriculum in the 21st century without dealing with the issue of whether or, if so, when calculators should be used in teaching that curriculum.

CFP does, at least, pay lip service to the benefits of mental calculation which is mentioned four times. Three of these are in the context of estimating sums, differences, products or quotients or calculating them mentally "depending on the context and the numbers involved", making it clear that little more than knowledge of the addition and multiplication tables is expected here. (Am I wrong? Perhaps but, if so, this is another example of purposeful ambiguity.) The fourth instance advocates the building of "facility with mental computation" to do "addition and subtraction in special cases such as 2,500 + 6,000 and 9,000 - 5,000". While I applaud any mention of mental arithmetic, CFP essentially trivializes what students may accomplish in this domain.

The absence of even the word "calculator" is the most important reason why CFP has been so widely praised by NCTM's heretofore opponents. But the refusal even to discuss the crucial issue of calculators just panders to the anti-calculator brigade; it means that anyone using CFP for curriculum development has no guidance whatever on how much or how little use of calculators to build into a curriculum.

Fuzzy Math

The paeans of joy in the American press (e.g., the Wall Street Journal [3], the New York Times [4], the New York Sun [5], the Washington Times [6], the Jewish World Review [7], and probably elsewhere also) all praised the retreat of NCTM from the "fuzzy math" of the Standards and PSSM. What is "fuzzy math"? It is, as I have written elsewhere [8], "a fuzzy concept meaning whatever the critics of new [i.e., reform] curricula want it to mean at a given time". Sometimes it refers to those [mythical] people who wish to favor inexact rather than exact answers. At other times it refers to anyone who favors "constructivist math" [5] (whatever that may be).

In fact, the canard that NCTM ever favored fuzzy math, however you might define it, has never been true; it is, indeed, a lie which is repeated endlessly without any evidence whatsoever in the hope that if you repeat a lie often enough, people will believe it. I know of no one in NCTM or the math education community generally who has ever espoused teaching children that exact answers are not important and always to be desired, when they can be obtained with reasonable effort. Nor does any math educator doubt that instant recall of the addition and multiplication tables is necessary for anyone studying arithmetic.

At least, CFP, like the Standards and PSSM before it, does stress the value of being able to estimate, not as a way to avoid calculating exact answers but rather when an estimate is all that is required or to enable checking the reasonableness of answers on a calculator. Indeed, any good contemporary elementary school mathematics curriculum must emphasize the value of being able to estimate answers.

The Math Wars

Various comments about CFP ([3], [9]) have expressed the belief or hope that its publication would bring an end to the Math Wars that for years now have roiled the US math education scene. On one side have been the TMWs, the most vocal of whom are research mathematicians but also including parents, business groups and some teachers. On the other side are the RMWs (reform math warriors) whose leaders are mainly math educators but with significant support from parents and teachers.

Viewed by itself, it is easy to see why CFP appears to signal an end the Math Wars. It has so little content and that which it has is expressed in such anodyne or ambiguous form that no one is likely to get very agitated about anything it says. Thus, it is possible for TMWs to use CFP to declare victory in the Math Wars while RMWs will view such a claim as ridiculous.

Indeed, viewed as the authors of CFP state they intended, namely as an extension of the Standards and PSSM, CFP resolves none of the issues in the Math Wars. Moreover, despite the response of TMWs to CFP and the prior publication of the Common Ground report [10], none of the really significant issues in the Math Wars have been resolved nor can they be in any foreseeable future.

Briefly stated, at the elementary school level, these issues revolve around the question of whether arithmetic should focus on attaining proficiency with the classical pencil-and-paper algorithms for the four arithmetic functions or whether the elementary school curriculum should embrace the wide use of calculators in teaching arithmetic to achieve sound understanding of arithmetic itself as well as to prepare students as well as possible for the further study of mathematics. There are similar issues with respect to middle school and secondary school mathematics but most of the heat in the Math Wars has been focused on the elementary school curriculum.

These issues are nowhere near being resolved. While we should all applaud any attempt to achieve a debate more civilized than at some times in the past, publications like the Common Ground report and now CFP merely fudge the important issues. But resolution of the arguments in the Math Wars, not fudging, is crucial to the future of American - but not just American - school mathematics. My own view is that the main controversies in the Math Wars will not be definitively settled for many years until, at least, the main protagonists have long since left the field of battle. In the meantime, it is important that those who feel strongly about the reform mathematics agenda fight for their beliefs with undiminished intensity and without propitiation of their antagonists.

The Real Issue

Sadly, however, despite my strong belief in the need to reform American school mathematics, neither the success nor failure of this reform will have much effect on American mathematics education for the foreseeable future. The real tragedy of mathematics education in American schools is the declining number of first-class mathematics teachers (and the growing number of uncredentialled teachers) in secondary schools and the growing number of mathematics-averse teachers in elementary schools. Nothing in the No Child Left Behind Act (NCLB) will reverse this trend. Indeed, the opposite is much more likely with NCLB already beginning to prove that act most destructive of good education ever passed by the United States Congress.


This is not the place to discuss why teaching, particularly mathematics teaching, is failing to attract the best and the brightest that we need in American schools (but see [11]). Nor is it the place to discuss the disaster that the testing regimen in NCLB is wreaking on American schools. But until the teaching profession does start to attract large numbers of the best and the brightest, a publication like CFP, whatever you think of it, cannot possibly contribute much to improve the state of American school mathematics education.

References

1. National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics, NCTM, Reston, VA, 1989.
2. National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, NCTM, Reston, VA, 2000.
3. John Hechinger, New Report Urges Return to Basics in Teaching Math, Wall Street Journal, 12 September 2006.
4. Teaching Math, Singapore Style, Editorial, The New York Times, 18 September 2006.
5. Andrew Wolf, Turnaround in the Math Wars, The New York Sun, 15 September 2006.
6. Phyllis Schafly, Parents Right, Experts Wrong, Washington Times, 27 September 2006.
7. Debra J. Saunders, Fuzzy Memory on Fuzzy Math, The Jewish World Review (http://jewishworldreview.com/0906/saunders091406.php3), 14 September 2006.
8. Anthony Ralston, Research Mathematicians and Mathematics Education: A Critique, Notices of the American Mathematical Society, Vol. 51, 2004, p408.
9. R. James Milgram as quoted in Education Week, 12 September 2006.
10. Ball, D. L., Ferrini-Mundy, J., Milgram, R. J., Schmid, W., Schaar, R., Reaching for Common Ground in K-12 Mathematics Education, http://www.maa.org/common-ground, also in Notices of the AMS, Vol. 52, pp1055-1058. (See also: A. Ralston, K-12 Mathematics Education: How Much Common Ground Is There?, FOCUS, January, 2006, pp14-15.)
11. Anthony Ralston, The Real Scandal in American School Mathematics, Education Week, 27 April 2005 (also: http://www.doc.ic.ac.uk/~ar9/TeacherQual.html)