************************
From FOCUS [The Newsletter of the Mathematical Association of American], February 2007, Volume 27, Number 2, p. 31. See http://www.maa.org/pubs/focus.html
----------------------------------------
This is the THIRD (and last) of three articles that appeared in the current issue of FOCUS dealing with NCTM's Curriculum Focal Points.
************************
What are the Curriculum Focal Points? And Why Should We Care?
By Barbara Reynolds
What are the Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (National Council of Teachers of Mathematics [NCTM] 2006)? Why should they matter to me? Why should they matter to the mathematics community?
In 2000 the NCTM released the Principles and Standards for School Mathematics, a comprehensive document that built on a bold vision that had been set by An Agenda for Action (NCTM 1980) and expanded in a set of Curriculum and Evaluation Standards for School Mathematics (NCTM 1989). The Principles and Standards sets a challenging comprehensive program, outlining goals that are both broad and deep. Now just six years later, the NCTM has presented us with Curriculum Focal Points. Does this new document represent a change in direction, a strengthening of the guidelines laid out in the Principles and Standards, or (as some might fear) a watering down of the content of the school mathematics curriculum?
When it first came out, I studied the Principles and Standards rather thoroughly, and was excited by its comprehensive vision. If students coming into college and university mathematics courses were educated according to the goals outlined in the Principles and Standards, if they came into college mathematics courses with the depth and breadth of understanding proposed in the Principles and Standards, wouldn't our job as teachers of undergraduate mathematics be so much easier!
My fear as I began to read the Curriculum Focal Points was that I would find a simplified list of competencies, grade-level mastery objectives, and testable outcomes - after all, this is the world of No Child Left Behind (NCLB) - that might represent minimal goals for each grade level of the school mathematics curriculum. My fear was that the Curriculum Focal Points would set minimal goals that would undermine the bold vision set by the Principles and Standards. Would the NCTM sell out to NCLB?
Instead what I see in the Curriculum Focal Points is a balanced set of guidelines that shows one way of developing a prekindergarten through grade 8 mathematics curriculum, guidelines that give focus to one or another area of the objectives set by the Principles and Standards at each grade level, while illustrating ways of making connections to the overall fabric of mathematics throughout the entire prekindergarten through grade 8 mathematics experience of school children. The Curriculum Focal Points are not presented as a curriculum in and of themselves, nor are they presented as a set of lesson plans. Rather they are presented as guidelines for developing an integrated Prekindergarten through Grade 8 mathematics curriculum. These Curriculum Focal Points illustrate one way of making explicit connections both to topics that the child studied in the preceding years, and to content that will be coming in subsequent years. Curriculum developers and textbook writers could use these Curriculum Focal Points as an organizing outline - focal points, actually - to develop mathematics programs that will be integrated and connected across grade levels. Classroom teachers would then be in a position to present mathematics at each grade level that implements the broad vision set by the Principles and Standards, and builds coherently from year to year. The Curriculum Focal Points make it more likely that classroom teachers will have well-designed materials that follow the bold vision set by the Principles and Standards, and that allow them to develop problem solving, reasoning, and critical thinking skills in their students without sacrificing computational skill development.
I began this reflection by asking why these Curriculum Focal Points should matter to me? Why should they matter to those of us who teach undergraduate mathematics? After all, students coming into our undergraduate mathematics classrooms are four or more years beyond Grade 8. What impact might these Curriculum Focal Points have on my own teaching of undergraduate mathematics?
The Curriculum Focal Points could impact my own teaching in two ways: First of all, if more schools adopt school mathematics materials that implement the vision of the Principles and Standards, classroom teachers will need to have a deep and broad understanding of mathematics. Classroom teachers will need an understanding of mathematics that goes well beyond computational fluency and that encompasses deeper mathematical reasoning and problem-solving skills. So we need to think about how to better prepare pre-service teachers, and we need to think about on-going professional development of in-service teachers. Secondly, if the calculus reform movement of the 1980s taught us anything, we must be aware that as school curriculum implements the bold vision expressed in the Principles and Standards - something that will be facilitated by the guidelines set out by the Curriculum Focal Points - we can expect that in four to six years we will be seeing more students whose school mathematics programs implemented this bold vision coming into our own undergraduate mathematics classes. Will we be ready for such students?
---------------------------------
Barbara Reynolds, SDS, is professor of Mathematics and Computer Science at Cardinal Stritch University in Milwaukee, WI. She has a passion for teaching for understanding.
----------------------------------
THIS IS THE THIRD AND LAST POSTING ABOUT THE Curriculum Focal Points.
Tuesday, March 13, 2007
Focus on Focal Points - Part 2
************************
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
----------------------------------------
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.
************************
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)
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
----------------------------------------
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.
************************
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)
In FOCUS: Curriculum Focal Points - Part 1
************************
From FOCUS [The Newsletter of the Mathematical Association of America], February 2007, Volume 27, Number 2, pp. 27-28. See http://www.maa.org/pubs/focus.html . The articles are posted with the approval of the Editor of FOCUS.
----------------------------------------
This is the FIRST of three articles that appeared in the current issue of FOCUS dealing with NCTM's Curriculum Focal Points. The other two articles will follow shortly.
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In FOCUS: Curriculum Focal Points
As noted in our November issue, the National Council of Teachers of Mathematics (NCTM) recently released a document entitled Curriculum Focal Points. The document is available online at http://www.nctm.org/focalpoints/downloads.asp, where one can also find a lot more information and discussion, including questions and answers on the document. NCTM says the document extends the Council's leadership of more than twenty-five years by describing an approach to curriculum development that focuses on areas of emphasis within each grade from prekindergarten through grade 8. Curriculum Focal Points, widely reported in the news media, was hailed by some as a retreat from NCTM's previous positions. This misreading is addressed in each of the three articles that follow. We hope that they will shed some light on the goals and content of Curriculum Focal Points. The articles below are the opinions of the authors and do not reflect a position or stance of the MAA.
NCTM's Curriculum Focal Points
By Francis (Skip) Fennell
The publication by the National Council of Teachers of Mathematics (NCTM) of Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence marks the Council's next step in implementing the vision set forth in the Principles and Standards for School Mathematics (NCTM, 2000), with a particular emphasis on curricular expectations.
The genesis of the Curriculum Focal Points, released on September 12, 2006, was a conference at the Park City Mathematics Institute in 2004 organized by NCTM with the Association of State Supervisors of Mathematics (ASSM). It brought together mathematicians, supervisors of mathematics, and mathematics educators with the intent to examine the K-12 mathematics standards of each of the states and how they were influenced by Principles and Standards for School Mathematics. The expectation was that the consistency of the findings would guide a discussion that could begin to lead to a meeting of the minds about the important mathematics that should be taught at various grades. Instead, the outcome was the discovery that there was little consistency between standards and expectations or of what content fell in what grade, and perhaps most troubling was the consequent realization that this lack of consistency was inevitably detrimental to the teaching and learning of K-12 mathematics nationwide.
An analysis of the conference findings resulted in the publication entitled "Standards and Curriculum: A View from the Nation," (NCTM, 2005). This joint report by NCTM and ASSM provides insights into where we appear to be headed in our expectations for students' mathematics learning. The report is an initial attempt to examine across states the impact of Principles and Standards for School Mathematics on curriculum reform, discern how state educational agencies approached the task of developing state standards, and bring to light areas of commonality and difference. The findings from this endeavor laid a foundation for discussions about the future direction of local, state, and national mathematics curricula. From this were born the idea and concept of what became the Curriculum Focal Points.
Extensive, thoughtful, and at times intensely debated discussions among a group of nine writers generated the concept and early drafts of the curriculum focal points. The written draft of the writers, who represented expertise in mathematics and mathematics education as well as classroom experience from prekindergarten through grade 8, was reviewed and commented upon by some 70 reviewers, including mathematicians, mathematics educators, and policy makers. After extensive further revisions, the publication was presented to the NCTM Board of Directors, which on April 24, 2006 approved it for publication.
Curriculum focal points are important mathematical topics for each grade level. They are the related ideas, concepts, skills, and procedures that form the foundation for understanding, lasting learning, and success in higher level mathematics, beginning with algebra. NCTM views the Curriculum Focal Points as a framework for developing mathematics curriculum at the state and school district level. The focal points are intended to frame discussions that will eventually inform the decisions of textbook publishers and assessment developers, as well.
The Curriculum Focal Points address curriculum, or what is taught, rather than instruction, or how it is taught. By design, there is no mention of instructional strategies, instructional materials, technological tools (e.g. the calculator), or manipulative materials. This was the intent of the writers of the focal points-to provide a publication that would foster discussion, dialogue, and decision-making relative to the important mathematics for prekindergarten through grade 8. The ultimate goal would be for these suggestions, the focal points, to lead to the development of mathematics curriculum goals that are more cohesive from grade to grade and from school to school. Through its Connections, the new publication also shows additional ways in which the focal points connect to Principles and Standards.
The Curriculum Focal Points provides an example, a critical foundation, for the next generation of curricula and related assessments. Curriculum developers can place Curriculum Focal Points and a local or state curriculum side by side when refining their current curricula. Curriculum developers can determine how much time to devote to the focal points as the mathematical core for a particular grade level, and then build other mathematics topics around these important areas of focus.
Media Coverage
The Curriculum Focal Points release was widely reported in the news media and generated considerable discussion within the mathematics and education communities. Some inaccurate coverage raised questions among many who were asking "Is what the Wall Street Journal and New York Times reported true? Is NCTM really going back to basics?"
In a letter sent to the Wall Street Journal and published on September 26, I wrote, "Contrary to the impression left in your article, learning the basics is certainly not 'new marching orders' from the NCTM, which has always considered the basic computation facts and related work with operations to be important. Nor is the new focal-points approach to curriculum development a 'remarkable reversal' for NCTM. As stated in NCTM's 1989 and 2000 standards, conceptual understanding and problem solving are absolutely fundamental to learning mathematics. The council has never promoted estimation 'rather than precise answers.' Estimation is a critical component to the overall understanding and use of numbers."
A letter to the editor of the New York Times published on September 24, stated, "What some refer to as basic skills (for example, multiplication facts, and fluency with the addition, subtraction, multiplication and division of whole numbers) have always been a fundamental core of elementary school mathematics. Always. But we want more. We want children to understand the mathematics they are learning and we want them to be able to solve problems, which is, in the long run, why we do mathematics."
The Curriculum Focal Points are in no way a reversal of the Council's long-standing position on teaching students to learn critical foundational topics (e.g. multiplication) with conceptual understanding, and they are not a retreat from Principles and Standards for School Mathematics. Rather, the Curriculum Focal Points are the next step in implementing the Standards. The appendix in Curriculum Focal Points directly links the focal points to virtually all the expectations in Principles and Standards.
One of the questions asked most frequently is about the standard algorithm and whether the Curriculum Focal Points expressly states that all students must learn the standard algorithm. The grade 2 focal point suggests efficient procedures, including the standard algorithm-including, not exclusively, access to the standard algorithm. Students should use what they can do efficiently and accurately. Most important, they should not use any algorithm until it is understood.
Similarly, for grade 4 the quick recall of multiplication facts and fluency with efficient procedures, including the standard algorithm, is a focus. Again, and importantly, fluency emerges through deep understanding of the multiplication process-how multiplication is represented and how properties, particularly the distributive property, are used when multiplying. Students become fluent through their understanding of how and why procedures work - with a focus on place value and properties of operations.
The Purpose
Today's mathematics curricula tend to be dominated by long lists of very specific goals, standards, objectives, or learning expectations. By contrast, Curriculum Focal Points describes significant mathematical concepts and skills for each grade level. They are a way to organize and connect critical mathematics topics from grade to grade. Organizing a curriculum around the focal points can provide students with a more coherent ever expanding body of mathematical knowledge.
Mathematics leaders should use the Curriculum Focal Points to launch discussions about the next generation of curriculum standards, textbooks, and tests. Such dialogue, discussion, and debate is critical and can lead to the development of new models for curriculum, instruction, materials, and assessments. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics represents an important, initial step in advancing collaborative discussions about what mathematics students should know and be able to do.
From FOCUS [The Newsletter of the Mathematical Association of America], February 2007, Volume 27, Number 2, pp. 27-28. See http://www.maa.org/pubs/focus.html . The articles are posted with the approval of the Editor of FOCUS.
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This is the FIRST of three articles that appeared in the current issue of FOCUS dealing with NCTM's Curriculum Focal Points. The other two articles will follow shortly.
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In FOCUS: Curriculum Focal Points
As noted in our November issue, the National Council of Teachers of Mathematics (NCTM) recently released a document entitled Curriculum Focal Points. The document is available online at http://www.nctm.org/focalpoints/downloads.asp, where one can also find a lot more information and discussion, including questions and answers on the document. NCTM says the document extends the Council's leadership of more than twenty-five years by describing an approach to curriculum development that focuses on areas of emphasis within each grade from prekindergarten through grade 8. Curriculum Focal Points, widely reported in the news media, was hailed by some as a retreat from NCTM's previous positions. This misreading is addressed in each of the three articles that follow. We hope that they will shed some light on the goals and content of Curriculum Focal Points. The articles below are the opinions of the authors and do not reflect a position or stance of the MAA.
NCTM's Curriculum Focal Points
By Francis (Skip) Fennell
The publication by the National Council of Teachers of Mathematics (NCTM) of Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence marks the Council's next step in implementing the vision set forth in the Principles and Standards for School Mathematics (NCTM, 2000), with a particular emphasis on curricular expectations.
The genesis of the Curriculum Focal Points, released on September 12, 2006, was a conference at the Park City Mathematics Institute in 2004 organized by NCTM with the Association of State Supervisors of Mathematics (ASSM). It brought together mathematicians, supervisors of mathematics, and mathematics educators with the intent to examine the K-12 mathematics standards of each of the states and how they were influenced by Principles and Standards for School Mathematics. The expectation was that the consistency of the findings would guide a discussion that could begin to lead to a meeting of the minds about the important mathematics that should be taught at various grades. Instead, the outcome was the discovery that there was little consistency between standards and expectations or of what content fell in what grade, and perhaps most troubling was the consequent realization that this lack of consistency was inevitably detrimental to the teaching and learning of K-12 mathematics nationwide.
An analysis of the conference findings resulted in the publication entitled "Standards and Curriculum: A View from the Nation," (NCTM, 2005). This joint report by NCTM and ASSM provides insights into where we appear to be headed in our expectations for students' mathematics learning. The report is an initial attempt to examine across states the impact of Principles and Standards for School Mathematics on curriculum reform, discern how state educational agencies approached the task of developing state standards, and bring to light areas of commonality and difference. The findings from this endeavor laid a foundation for discussions about the future direction of local, state, and national mathematics curricula. From this were born the idea and concept of what became the Curriculum Focal Points.
Extensive, thoughtful, and at times intensely debated discussions among a group of nine writers generated the concept and early drafts of the curriculum focal points. The written draft of the writers, who represented expertise in mathematics and mathematics education as well as classroom experience from prekindergarten through grade 8, was reviewed and commented upon by some 70 reviewers, including mathematicians, mathematics educators, and policy makers. After extensive further revisions, the publication was presented to the NCTM Board of Directors, which on April 24, 2006 approved it for publication.
Curriculum focal points are important mathematical topics for each grade level. They are the related ideas, concepts, skills, and procedures that form the foundation for understanding, lasting learning, and success in higher level mathematics, beginning with algebra. NCTM views the Curriculum Focal Points as a framework for developing mathematics curriculum at the state and school district level. The focal points are intended to frame discussions that will eventually inform the decisions of textbook publishers and assessment developers, as well.
The Curriculum Focal Points address curriculum, or what is taught, rather than instruction, or how it is taught. By design, there is no mention of instructional strategies, instructional materials, technological tools (e.g. the calculator), or manipulative materials. This was the intent of the writers of the focal points-to provide a publication that would foster discussion, dialogue, and decision-making relative to the important mathematics for prekindergarten through grade 8. The ultimate goal would be for these suggestions, the focal points, to lead to the development of mathematics curriculum goals that are more cohesive from grade to grade and from school to school. Through its Connections, the new publication also shows additional ways in which the focal points connect to Principles and Standards.
The Curriculum Focal Points provides an example, a critical foundation, for the next generation of curricula and related assessments. Curriculum developers can place Curriculum Focal Points and a local or state curriculum side by side when refining their current curricula. Curriculum developers can determine how much time to devote to the focal points as the mathematical core for a particular grade level, and then build other mathematics topics around these important areas of focus.
Media Coverage
The Curriculum Focal Points release was widely reported in the news media and generated considerable discussion within the mathematics and education communities. Some inaccurate coverage raised questions among many who were asking "Is what the Wall Street Journal and New York Times reported true? Is NCTM really going back to basics?"
In a letter sent to the Wall Street Journal and published on September 26, I wrote, "Contrary to the impression left in your article, learning the basics is certainly not 'new marching orders' from the NCTM, which has always considered the basic computation facts and related work with operations to be important. Nor is the new focal-points approach to curriculum development a 'remarkable reversal' for NCTM. As stated in NCTM's 1989 and 2000 standards, conceptual understanding and problem solving are absolutely fundamental to learning mathematics. The council has never promoted estimation 'rather than precise answers.' Estimation is a critical component to the overall understanding and use of numbers."
A letter to the editor of the New York Times published on September 24, stated, "What some refer to as basic skills (for example, multiplication facts, and fluency with the addition, subtraction, multiplication and division of whole numbers) have always been a fundamental core of elementary school mathematics. Always. But we want more. We want children to understand the mathematics they are learning and we want them to be able to solve problems, which is, in the long run, why we do mathematics."
The Curriculum Focal Points are in no way a reversal of the Council's long-standing position on teaching students to learn critical foundational topics (e.g. multiplication) with conceptual understanding, and they are not a retreat from Principles and Standards for School Mathematics. Rather, the Curriculum Focal Points are the next step in implementing the Standards. The appendix in Curriculum Focal Points directly links the focal points to virtually all the expectations in Principles and Standards.
One of the questions asked most frequently is about the standard algorithm and whether the Curriculum Focal Points expressly states that all students must learn the standard algorithm. The grade 2 focal point suggests efficient procedures, including the standard algorithm-including, not exclusively, access to the standard algorithm. Students should use what they can do efficiently and accurately. Most important, they should not use any algorithm until it is understood.
Similarly, for grade 4 the quick recall of multiplication facts and fluency with efficient procedures, including the standard algorithm, is a focus. Again, and importantly, fluency emerges through deep understanding of the multiplication process-how multiplication is represented and how properties, particularly the distributive property, are used when multiplying. Students become fluent through their understanding of how and why procedures work - with a focus on place value and properties of operations.
The Purpose
Today's mathematics curricula tend to be dominated by long lists of very specific goals, standards, objectives, or learning expectations. By contrast, Curriculum Focal Points describes significant mathematical concepts and skills for each grade level. They are a way to organize and connect critical mathematics topics from grade to grade. Organizing a curriculum around the focal points can provide students with a more coherent ever expanding body of mathematical knowledge.
Mathematics leaders should use the Curriculum Focal Points to launch discussions about the next generation of curriculum standards, textbooks, and tests. Such dialogue, discussion, and debate is critical and can lead to the development of new models for curriculum, instruction, materials, and assessments. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics represents an important, initial step in advancing collaborative discussions about what mathematics students should know and be able to do.
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