Warfield, Systems, G.S. Chandy, and Mathematics Teaching

Above is a picture of John N. Warfield, "the author of two U.S. patents on electronic equipment, and the inventor of Interpretive Structural Modeling, Interactive Management, and General Design Science." Professor Warfield's work has had a significant influence on the contributions of G.S. Chandy to the math-teach@mathforum.org list. In connection with my recent blog entries about "important open questions in mathematics pedagogy," ("Is Mathematics Teaching A Closed Book?"), Mr. Chandy has tried to offer an approach to discussing these and related points of contention on that most contentious of lists that is grounded in the work of Professor Warfield.

I do not profess to know Warfield's work, but I found what Mr. Chandy posted to be sufficiently heuristic to warrant posting what he wrote in its entirety in hopes of generating interest among readers of this blog, as well as those on various lists to which I contribute who many not have ventured into math-teach (probably with good reason).

G.S. Chandy's Post

The 'elements' below are generated from MPG's posting on Aug 22, 2008 8:43 PM at this very thread. (Many of the elements listed below have been generated by GSC - they should be properly articulated by real teachers of K-12 & other school math):

Possible Mission Statement:

M: "To put into place a highly effective way of teaching math (for teachers); a highly effective way of learning math (for students)"

First trigger question:
"What, in your opinion, are some THINGS TO DO to help accomplish Mission M?"
====
Some possible elements, generated in response to above trigger question - these elements largely derived from MPG's posting above-noted:
+++
1. To ensure that the ideas of those who actually teach K-12 are taken into the action planning for future teaching of K-12

2. To learn how to go beyond the lecture-driven, teacher-centered math classes

3. To ensure active student participation in the math class

4. To stimulate the student's genuine interest in math

5. To generate real interest from students

6. To ensure that students become actively engaged in mathematical thinking and math discourse

7. To provide (point to) ways demonstrating HOW students could become truly active in math

8. To ensure that 'grades' are not treated as jokes (by students or by teachers)

9. To use an effective grading system through which both students and teachers could benefit

10. To convince students that it is possible for each of them to become productive and creative in math

11. To convince students that it is inevitable that some would be more productive, others maybe less productive - but that all can be truly productive

12. To help each student 'fill in the gaps' in his/her understanding of math

13. To convince students of the real benefits that a sound understanding of math could bring to them

14. To convince students to put forth enough real effort to become productive and creative in math

15. To convince students of the value of math that goes beyond just 'book-learning' or 'exam-learning'

16. To ensure that students are fairly and effectively tested at various levels

17. To ensure that math-teaching should go beyond vapid 'multiple choice' questions

18. To enable and encourage students to take a real cract at truly challenging math questions (suited to their current level)

19. To ensure that all students always get needed further chances to improve their standings in math

20. To prevent students from attempting to cheat on math - demonstrate that it is really futile to try and cheat in math

21. .... (Etc).

+++
There will obviously be a great many such 'elements'. I suggest that generated such elements should be a continuing task, on which teachers and students could very beneficially spend a few minutes each day.

As we generate such elements, the most useful thing to do with them is to find out just how they may "CONTRIBUTE TO" each other (and to M).

This is done by asking questions like:

"Does, in your opinion,

Element X

CONTRIBUTE TO

Element Y?"
(The strength of the contribution relationship should initially be taken as small, i.e., we initially create the model predicated on the relationship "MAY contribute to". Later, as we become more certain of our understanding of the elements (and of the system in which those elements are active), we could strengthen the relationship to "SHOULD contribute to", and still later to "CONTRIBUTES TO".

Asking and answering those questions about these elements will give participants the opportunity to enhance their understanding of the whole system in which the 'teaching and learning of math' operate. They will also become clear in their minds about each separate element in itself, in its specific context.

(Almost) needless to state, each class would have its own listing, and the lists (and the models generated from them) would see considerable variation. Also, it is entirely possible that there would be considerable disagreement about the contributions or otherwise of various elements. Most of such disagreements can be worked out to the mutual satisfaction of all involved - and it is often possible to work out models representing a satisfactory consensus. Sometimes we cannot come to agreement - I suggest in such cases that the persons involved should create and follow their own models.

Of course, there may be people participating who are, so to speak, more interested in empty argumentation rather than productive resolution of disagreements and issues - just "trolls" - we have a few such in every set of participants. We need to learn how to handle these effectively: not easy at all to do, but it is possible to ensure that trolls do not disturb the flow too much. If we are all actively involved in generated useful elemetns, working on creating productive models of our perceptions about relationships of those elements, then the trolls will disturb working partipants much less. James Wysocki (and others) have suggested practical ways in which such trolls could be dealt with.
+++
Below my signature, I provide a list of some other useful 'trigger questions' about Mission M.

GSC
+++
Useful trigger questions (all are in relation to the Mission M identified above):

Creating the OPMS

Step 1: Identify a desirable Mission M:

MISSION: “______________________________________________”

(In case you have a problem, just put it into ‘Mission’ format. Various examples of ‘Missions’ are available in various examples illustrated).

Step 2: THINGS TO DO (to accomplish Mission):

Generated from “1st Fundamental Trigger Question”:

“What, in your opinion, are the THINGS TO DO to accomplish the above Mission?”

Step 3: Action Planning:

We shall construct an Interpretive Structural Model (ISM) with the responses to above 1st Trigger Question. This ISM (which is technically known as an ‘Intent Structure’) will develop into the ongoing Action Plan for the Mission.

The participants in the Mission should spend about 20-30 minutes each day to develop models relating to progress in their individual parts of this Mission (as seen in their individual One Page Management Systems that should start developing as the organization works towards the global Mission). As a whole, the organization should meet from time to time, as found convenient, to model progress towards the global Mission.



Step 4: Identifying DIFFICULTIES, BARRIERS and THREATS – and overcoming them

These are generated from the “2nd Fundamental Trigger Question”:

“What, in your opinion, are the BARRIERS, DIFFICULTIES & THREATS that may hinder or prevent accomplishment of the chosen Mission?”

Generally, responses to the above trigger question may be:

a) converted into appropriate THINGS TO DO, which would then be integrated into the ongoing Action Plan
b) inserted into a Field Representation – the OPMS software will then help users create Dimension Titles for the Field Representation and also to link up various elements in the Field with appropriate relationships. Various other actions are to be done with Field Representations, which are described in our Workbook.

c) These BARRIERS, DIFFICULTIES & THREATS are also most usefully inserted into a ‘Problematique’ (the governing relationship of which is “aggravates”). It takes a while to learn to develop and use a problematique effectively, but it will be well worth the effort, for the reasons noted in our ‘Basic Presentation’, a copy of which is provided for reference.

The above exercises would help us identify and implement, on a continuing basis, means to overcome BARRIERS, DIFFICULTIES and THREATS discovered during our progress towards our Mission.

Step 5: Identifying STRENGTHS (available/required)

Generated from “3rd Fundamental Trigger Question”:

“What, in your opinion, are the STRENGTHS (available/required) that could help accomplishment of the chosen Mission?”

Responses to the above Trigger Question, if sizable in number, may be inserted into a Field Representation.

The STRENGTHS identified as “Required, not available” would be translated into appropriate THINGS TO DO format and integrated into the ongoing Action Plan to enable users find ways to develop the required
STRENGTHS.

Step 6: Identifying WEAKNESSES – and overcoming them

Generated from “4th Fundamental Trigger Question”:

“What, in your opinion, are the WEAKNESSES that may hinder or prevent accomplishment of the chosen Mission?”

The means of handling WEAKNESSES are exactly the same as we have for handling BARRIERS, etc. That is, responses to the above trigger question would be:

a) converted into appropriate THINGS TO DO, which would then be integrated into the ongoing Action Plan
b) inserted into a Field Representation – the OPMS software will then help users create Dimension Titles for the Field Representation and also to link up various elements in the Field with appropriate relationships. Various other actions are to be done with Field Representations, which are described in our Workbook.

c) These WEAKNESSES are also most usefully inserted into a ‘Problematique’ (the governing relationship of which is “aggravates”). It takes a while to learn to use a problematique effectively, but it will be well worth the effort, for reasons cited in our ‘Basic Presentation’, a copy of which has been provided with our Workbook.

The above exercises help us identify and implement, on a continuing basis, means to overcome WEAKNESSES discovered during our progress towards our Mission.

For individuals as well as for organisations, correct identification of BARRIERS & WEAKNESSES (and defining ways to overcome them) is often a very long and painful task – particularly when organisational Barriers and Weaknesses are derived from an individual’s Weaknesses. It could take months – sometimes people NEVER learn to get over their Weaknesses even when they cause disaster. In fact, most (man-made) disasters are caused precisely because people have not learned how to handle their existing Weaknesses!

Step 7: Identifying OPPORTUNITIES available

Generated from “5th Fundamental Trigger Question”:

“What, in your opinion, are the OPPORTUNITIES available
that may help accomplishment of our chosen Mission? –
and what are the THINGS TO DO to avail the
OPPORTUNITIES discerned?”

Responses to above trigger question are inserted into a Field Representation showing OPPORTUNITIES available – and THINGS TO DO to avail the OPPORTUNITIES identified. The THINGS TO DO to avail the OPPORTUNITIES are also integrated into the Action Plan, to enable us to see how we may work towards availing of the OPPORTUNITIES that arise.



Step 8: Identifying EVENTS & MILESTONES

Generated from “6th Fundamental Trigger Question”:

“What, in your opinion, are the EVENTS/MILESTONES
that may occur during our progress towards our
chosen Mission?”

Responses to above trigger question are inserted into PERT/Gantt Charts (as is done in the conventional ‘Project Management’ software, to show the
status of Milestones during progress towards our Mission.

We observe here that the conventional Project Management software deals only with this single dimension of the OPMS (namely, the EVENTS Dimension). Obviously the name ‘Project Management’ software is a serious misnomer for such software – which may accurately be called ‘Event Management’ software.

The OPMS may be seen to fulfill the role of true Project Management, as it enables users to see all dimensions relating to a specified Mission. However, we prefer to regard the OPMS as an ‘aid to problem-solving and decision making’ (which includes ‘Project Management’, and much else besides).


Step 9, et seq: ‘System’ Dimensions of the OPMS

When sufficient numbers of elements have been generated in response to the Six Fundamental Trigger Questions as described above (and, further, those elements have been appropriately inserted into models AND linked up), the users would find it necessary and useful to start working on the ‘System Dimensions’ of the OPMS:

• PLANNING SYSTEM(S)
• INFORMATION SYSTEM(S)
• MARKETING SYSTEM
• PRODUCTION SYSTEM (for manufacturing organizations. For educational institutions, research institutions, etc., the title of this dimension may be modified appropriately).
• PROBLEM SOLVING SYSTEM AND LEARNING SYSTEM: The OPMS itself defines these two systems, and no work is required to be done by users in regard to these two closely coupled systems)
• MONITORING & EVALUATION SYSTEMS
• FINANCE CONTROL SYSTEM
• OTHER(s) (as required)

The OPMS would help users develop all above dimensions.

In general, the models in these ‘system’ dimensions of the OPMS are small, but powerful, ‘meta models’ derived from the larger models appearing above the System Tie Line. It is generally found that these models start developing effectively, in a very natural way after a certain ‘richness of connection’ has been established between the fundamental models above the System Tie Line.

The models within the OPMS are models of human perceptions relating to the chosen Mission. The purpose of creating such models is primarily to show a simple ‘action path’ to each person involved in accomplishment of the Mission. That is, these models show what each person and each group involved should do each day in pursuit of the Mission. The THINGS TO DO identified in the Action Planning structure as ‘FOCUS elements’ are the crucial activities at any point of time – these would be just a few (typically, 3 to 5 each day). These focus elements will naturally change from time to time as we progress towards accomplishment of the Mission.

More Problems of Remediation and Assessment

In response to one of my previous posts on the issue of meaningful open questions about mathematics pedagogy, an anonymous contributor wrote in part: "How can I teach a student at level N who hasn't mastered all the work of level N-1 (or maybe even N-2 for some things)? What can be done to help with the learning of the current content while remediating gaps in learning?"

This is a particularly nagging problem for most mathematics teachers in K-12 and beyond, though at the college level it is easier perhaps for instructors to be blithely condescending and dismissive of the problems of students who come to class ill-prepared for the level of coursework expected of them. College is, after all, a choice, and professors are not obligated to provide remediation, although there are generally non-credit classes offered that do just that, or at least purport to do so.

I wrote on math-teach@mathforum.org in response to the above question:

The first question is nearly ubiquitous in mathematics education and of course there is no simple answer to it (not that we won't likely here some in this venue from non-teachers).

I suspect that we'll never see a time where teachers don't have to ask that question for the simple reason that there will never be a "system" that eliminates differential intellectual development, readiness, or individual variability in motivation for learning any given subject. It seems inevitable that kids will not be equally ready in all sorts of ways for any given mathematical topic at a given age/grade level.

As long as we try to mass educate under the current model, with bizarre expectations that we can legislate kids to level N or intimidate schools, parents, teachers, or kids to some pull rabbits out of hats in an attempt to pretend that all kids are at level N at the appointed age/grade level, we'll miss the boat. A saner approach is to do the best we can to get kids ready for school and then teach them where they actually are, using differentiated instruction and methods to brings them as far as we can given where they start. That requires more flexible ideas about content and sequence, about instruction and tools, about the nature of classrooms, and about meaningful, useful assessment than we can reasonably expect to see given conservative/ reactionary opposition to sensible approaches to public school in particular, and education in general.

Richard Strausz replied:

Michael, I agree with what you say. Some critics use such real-world observations as more ammunition for their crusade against public education.

However, in talking with math teachers in Catholic and Jewish high schools, I hear similar situations in their classrooms.

And I commented:

I'm not surprised in the least. Didn't mean to suggest there was something unique going on in secular schools. I suspect we'd hear similar issues and concerns from private, non-sectarian classrooms.

Of course, the usual suspects in mathematics education, right-wing ideologues who pretend that all was once well in math teaching back in some imagined day, and/or that it will all be well again if only we had "real standards," (who decides and why they are qualified to do so always turns out to be those same ideologues and their like-minded brethren), "real accountability" (to whom is never quite clear, but it generally means to people who are at best marginally involved in actual teaching), "real textbooks" (and of course, nothing BUT textbooks, generally of the skill-based, routinized kind best exemplified by "teacher-proof," learning-proof Saxon Mathematics products), "real methods of instruction" (direct instruction with the teacher firmly in the middle, doing most of the talking and most of the mathematics), then all would be well.

Wayne Bishop Chimes In

In that light, the following predictable nonsense was offered up by the reliable anti-reformer, Wayne Bishop:

You are right about nonpublic schools but you seem to have misinterpreted the real problem with your identification of this problem being "more ammunition for their crusade against public education". It's ammunition for "their" crusade against colleges of education, the problem so well identified by Reid Lyon. The aversion against standards-based education (prior to the collegiate level where, at least in mathematics, standards tend to be used and accepted) is a direct consequence of the teaching of schools/colleges of education. For inexplicable reasons, almost religious in appearance, the industry prefers to place students by age-level even when it is obvious that students are being placed into situations where they are doomed to fail if honest performance standards are maintained.

It's always fascinating to hear Dr. Bishop cite, directly or otherwise, the execrable Reid Lyon, a former Bush education "expert" who was in part responsible to the multi-million dollar boondoggle known as READING FIRST, a project that appears to have been both rife with corruption and utterly ineffective, neither of which comes as a shock to Lyon's critics or fair-minded observers of how the current administration has lowered to every opportunity to help those most in need of its assistance. It was of course Mr. Lyon who said "You, know, if there was any piece of legislation that I could pass, it would be to blow up colleges of education" (McCracken, Nancy. "Surviving Shock and Awe: NCLB vs. Colleges of Education." English Education, January 2004, 104-118.) Scratch a right-wing ideologue these days, you may just turn up an anti-democratic terrorist with his hands in the public trough up to his shoulders.

My Proposal

In any case, here is my response to Dr. Bishop's latest screed:

For those of us who actually teach K-12, the issue that is far more onerous is not what book was used, what teaching methods were most prominent (well, that's a concern if one is using eclectic instructional approaches with students who've been taught in totally lecture-driven, teacher-centered classrooms in which student passivity is so ingrained that ANY requirement that they become actively engaged in mathematical thinking and mathematical discourse is useless until the teacher shows students HOW to become active in mathematics and convinces them that it is necessary and productive to do so), or even to some extent whether the previous class covers every topic that s/he would have. Rather, it is the passing of students from previous classes with grades of D and, generally, C. Frankly, I'd say that in my experience, unless we're talking about an honors course of some sort or an extraordinary school, anyone coming into Algebra II with a grade less than a solid B in Algebra I is going to have more mathematical holes in his/her head than it takes to fill the Albert Hall or are found in the typical argument from members of Mathematically Correct and NYC-HOLD about any aspect of mathematics education. And that's a lot of holes, let me tell you.

The obvious solution is to stop making grades the joke they are and instead go to minimum exit/entrance exams with multiple sorts of assessment employed to determine whether students are ready to move on. I suggest both exit and entrance exams so that there is a double check in place: one on leaving level N-1 and one on beginning level N. And also so that those who marginally fail to gain exit at level N-1 can get another shot at the beginning of the following year. Should they have gotten themselves up to speed by then, they should not be denied a chance to prove themselves ready to proceed.

Naturally, I'm not calling for a nationally-normed testing instrument here, especially if it's strictly some vapid multiple-choice test that values only one narrow set of skills. But within each state's standards, if there is a real attempt at meaningful formative assessment that leads directly to re-teaching areas of weakness before allowing students to take a crack at more challenging mathematics that builds directly on previous knowledge, then such a system would potentially reduce the number of students who are simply pushed further down the rabbit hole with nothing to hold onto.

For such a system to work, however, it can't be linked to a bunch of politicized punishments intended to help ideologues and politicians use teachers, schools, and kids as footballs for making propaganda.

Minimally, that would mean that people like Wayne Bishop would need to be keep as far as possible from influencing the assessment process. Instead, knowledgeable assessment experts who actually understand and stick to psychometric principles must have major input into each state and district's development of assessment tools. Close attention must be paid to some of the issues raised in two papers by SUNY @ Stony Brook mathematician Alan Tucker about cut scores, the theory of performance standards, and related issues. A balanced approach to assessment, regardless of the religious objections of the nay-sayers, must be used, and any structure that has been proven to promote wide-scale cheating due to absurd political pressure on educators, parents, and students, must be closely examined and, if found incapable of improvement, abandoned.

Is authentic, meaningful assessment expensive? You bet. But then, finding out what's really going on in a complex world is always dearer than looking for surface data that tells you what you already believed to be the case beforehand. And crafting real solutions rather than destroying public education so that the rich can get richer is always unpopular. . . among the powerful and their lackeys and mouthpieces.

Keith Devlin Continues on Multiplication

The multiplication issue continues to live on in various quarters, and now Keith Devlin, who started the current round of debates, discussions, and arguments, has posted a third column on the subject, "Multiplication and Those Pesky English Spellings." I'm still chewing over what he says there, but I can report with pleasure that this blog, along with three others, is mentioned favorably in his new column:

Of the blogs I looked at, which had threads devoted to my "repeated addition" columns, the following all had some good, thoughtful comments by their owners and by some of their contributors - by no means all agreeing with me - though the discussion in "Let's Play Math" soon descended to uninformed and repetitive name calling, and the owner eventually closed the thread, which unfortunately soon reappeared elsewhere.

* http://letsplaymath.wordpress.com
* http://www.textsavvyblog.net
* http://rationalmathed.blogspot.com
* http://homeschoolmath.blogspot.com/

Those blogs each have interesting and useful things to say on other math ed topics as well. Most of the others I saw seem to be little more than sounding boards for people who are so convinced that the overwhelming mass of evidence must all be wrong - since it runs counter to their beliefs - they don't even bother to read it. Those (other) blogs are not information exchanges from which you can learn anything, but platforms for people to espouse their own particular, unsubstantiated and often wildly wrong beliefs. Mathematicians who care about our subject and who like to think that the students who pass through our classrooms emerge with a good understanding of the mathematics we taught them, should be advised that they venture into any other mathematics education blog at their own risk.

Hard not to feel pleasure at getting that kind of comment from someone of Prof. Devlin's stature, and I like the other blogs he mentions as well, so to be put in their company is doubly flattering.
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Meanwhile, as I try to digest Devlin's latest column, I thought I'd let readers of this blog know that I recently made a very fortuitous discovery at a used book sale while visiting Fenton, MI last Saturday: about 50 issues of THE MATHEMATICS TEACHER from 1965 to 1973. There are so many remarkable things in these magazines that are worth considering in light of the last 20 years or so that I may well have material for several months' worth of blog posts just from what is in them.

As a teaser, the following is from an Educational Testing Service (Cooperative Test Division) advertisement in the February 1967 issue:

Cooperative Mathematics Tests

This series reflects the ferment and change in the mathematics curricula. At the same time, content has been carefully selected to ensure appropriateness of the tests for most students.

Ability to apply understanding of mathematical ideas to new situations and to reason with insight is emphasized. Factual recall and computation are minimized.

No, I didn't make that up. No, I'm not mistyping the date. That is 1967, not 1997 or 2007. It would be fascinating to see these testing instruments, and if anyone knows of them, please let me hear from you in the comments section or directly by e-mail

Comics, Closure, and Mathematics Education

While some ideologues are busy denying that there are any open questions in mathematics pedagogy, lots of bright folks are actively exploring many such meaningful questions. Only those who do not teach (and some who do) could believe that there are neither meaningful questions about how to more effectively teach mathematics nor people actively engaged in pursuing answers to them. But such is the nature of the Math Wars and the current American adaptation of The Big Lie strategy: small cadres of dedicated and destructive individuals would sooner invest in trying to undo the good work of others than to do anything original and creative themselves. As Kandinisky said so well in "On The Problem of Form,"

This evolution, this movement forward and upward, is only possible if the path in the material world is clear, that is, if no barriers stand in the way. This is the external condition. Then the Abstract Spirit moves the Human Spirit forward and upward on this clear path, which must naturally ring out and be able to be heard within the individual; a summoning must be possible. That is the internal condition.

To destroy both of these conditions is the intent of the black hand against evolution. The tools for it to do so are:
(1) fear of the clear path;
(2) fear of freedom (which is Philistinism); and
(3) deafness to the Spirit (which is dull Materialism).
Therefore, such people regard each new value with hostility; indeed they seek to fight it with ridicule and slander. The human being who carries this new value is pictured as ridiculous and dishonest. The new value is laughed at as absurd. That is the misery of life.

I was brought to Kandinsky's remarkable essay via a surprising source: Scott McCloud's brilliant and heuristic book, UNDERSTANDING COMICS: The Invisible Art. I've been thinking a lot about the medium of comic books as a way to teach mathematics (not an original thought, and one most recently planted in my head by Fred Goodman), who is also responsible for my looking at McCloud's work.

One point that struck me intensely that McCloud spends an entire chapter looking at is that of "closure" and the role of the gutter in comics (I will refrain from the more specific "comic books" or "comic strips" to keep things as open as possible). For those unfamiliar with the term, McCloud defines the gutter as the space between borders. He calls the gutter one of the most important narrative tools in comics, invoking as it does the procedure he defines as closure.

What, then, is "closure," and what is its importance for mathematics education, if any? In UNDERSTANDING COMICS, McCloud says, "This phenomenon of observing the parts but perceiving the whole has a name. It's called CLOSURE." But I'm not sure that particular definition quite does justice to how McCloud develops this notion in the book. What comes through is that between any two panels there is a gap in space/time, and into that gap, represented literally by the empty space of the gutter, each reader pours his/her imagination to create closure, thereby determining their own connections between separate moments in the sequential visual narrative that is comics. No two readers can conceivably do this identically for a host of reasons not unlike what undergirds constructivist learning theory.

While I'm not claiming that this is somehow unique to comics, McCloud makes a persuasive argument for it being sharply defined in this medium in ways that offer no choice for readers but to engage in the closure process dozens of times. And it is that notion that grabbed my attention as I read his book, not just because I come from a literature background with a deep interest in narrative and in the relationship between author/text/reader, director/movie/viewer, etc., but because I am a committed mathematics educator with an abiding passion for how teachers craft lessons and how students engage (or fail to engage) in them.

And so I found myself thinking about the implications for mathematics education in McCloud's notion of closure. In particular, I have thought about some of the best teaching I've witnessed or experienced, what to me made the lesson and the execution of it compellingly successful and remarkable, in the radical sense of that word. In all cases, there was a delicate hand at work: in the choice of problems and examples, in the mathematical questions students were asked to engage in, in the scaffolding process, in the conducting of discourse between teacher and students, student and student, and between individual students and the whole class. And always there were gaps, spaces that appeared to be intentionally left blank into which students were asked to engage their thinking, employ their prior knowledge, strategies, methods, and understanding, in order to make connections and move ahead towards solving a problem, deepening their understanding of a previously-considered concept, etc.

I contrast this with some much of the more mundane mathematics teachings I've seen, experienced, and been responsible for in my own practice. And what inevitably is lacking is the sort of things that persuade students to engage as deeply as seems necessary for students to carry away meaningful mathematical residue. How many times have teachers presented what at least to another reasonably knowledgeable math person would appear to be a clear, logical, organized lesson on some standard topic in the curriculum - an arithmetic operation on the integers, comparing fractions, converting from decimals to percents, long division, graphing linear equations, etc., only to discover shortly thereafter that a sizable number of the students are clueless during, immediately at the end of, or on the day following the lesson (if not throughout all three)? What has gone wrong? Was there something wrong with the teacher's explanation? Were the examples unclear, poorly chosen, badly explicated? Were the practice problems too hard, too dull, too disconnected from the lesson?

Of course, it's a commonplace amongst far too many teachers to conclude that the fault likes not with ourselves and our lessons but with our students (and of course, the poor job LAST year's teachers did in getting the students up to speed, though we wouldn't repeat that to the teachers from the previous grade, at least not to their faces). But let's pretend just for a moment that most students might just learn mathematics more effectively if our lessons were better. What would have to be the nature of those lessons? What would be necessary in the lesson content, structure, and presentation for it to be a rarity for a student to, barely seconds after "experiencing" the lesson, or barely seconds after having gotten some help with a particular problem related to that lesson, come right back to ask for help on essentially the same idea, concept, procedure, etc. in a nearly identical problem?

And it is here that I believe McCloud's idea about closure can tell us a great deal. Because it is my belief that most of our students who are doing poorly in mathematics and who evince deadly passivity as they sleepwalk from math course to math course, are not engaging in any sort of meaningful closure during lessons. They are operating as if math class were television, or some other medium that does not invite closure, rather than comics, which demands it. And their passivity does not lead to the sort of mathematical learning most teachers would like for students.

I suggest that a key question we need to be asking in mathematics education is how to build lessons that effectively get students to engage in the closure process. How do we craft lessons that leave a reasonable number of reasonably-sized "gutters" that will get students to engage in the closure-process, filling in the gaps with their best ideas, actively making connections rather than sitting back waiting for someone to magically implant understanding and mastery in their heads. Who knows? Maybe the very medium of comics itself is part of the answer, something I'm increasingly thinking about. One thing I'm sure of: the main methods we've been using in US mathematics classrooms aren't cutting it for a huge percentage of our students, and if we want to change that, it seems worth thinking about the nature of the media we're using to present and deliver instruction.

Is Mathematics Teaching A Closed Book?

Hans Freudenthal

A. Dean Hendrickson

It's never a good idea to get involved in a fight with an Internet "ghost." But if you play in the sty formerly known first as nctm-l@mathteach.org and now called, ironically, math-teach@mathforum.org, you can't avoid it. One of the more prolific voices there is a fellow who posts under the name of "Haim Pipik" (if you don't "get" the Yiddishism, you're not missing much). However, a few of us who've been around the Math Wars for more than a decade know him better as Edmond David (6th from left standing), a resident of Brooklyn, erstwhile member of NYC-HOLD, self-identified "NYC parent," and a fellow who doesn't mind slinging mud, pushing his quasi-libertarian, transparently right wing and vehemently anti-liberal agenda behind his pseudonym. All in all, not the most courageous guy in the Math Wars or any other battle: by their lack of courage shall ye know them, I suppose.

Ordinarily, I wouldn't bring the empty slings and dull arrows of this outrageous fellow to my blog, but the fellow has been most insistently issuing a phony challenge on math-teach that is worth mentioning, though not necessarily getting involved with (I am, for my own purposes, but I don't suggest anyone reading this who isn't already wasting time on that list head there to see Ed/Haim in action: it's not really worth your time.

The nature of "Haim's Challenge" is his claim that "there are no open questions of mathematics pedagogy." He has his own reasons for making this claim that ostensibly have to do with the discourse on math-teach and his own agenda which primarily seems to be to assail public education at every turn, call for privatization, vouchers, dismantling public schools (if I understand him correctly, anyway), and blaming all our educational woes on unions, education professors, progressive educators, bad policy makers, etc., all of whom he lumps together alternately as "the Education Mafia," "Educational Mullahs," and similar witticisms. Our Haim is short on proof that any such entities exist, of course, but when you're arguing by name-calling, what need you evidence to offer?

Regardless, I am curious as to how readers of this blog feel about his fundamental claim, which he extends to the broader one that clearly he is right because "no one is interested in discussing math teaching; no active conversations are in evidence;" and so on.

Of course, I think this is arrant nonsense, but maybe I have a distorted understanding of what one is to make of the literature in mathematics education or the lists, web sites and blogs I frequent where concerns about pedgogy and the relationship between content and how to effectively teach it (be it to oneself, home schooled kids, or students in more traditional school settings) so as to improve learning are very much in evidence. Add to that active research programs by mathematics educators at universities and other institutions throughout the world and it's hard to imagine how anyone could offer a less accurate, more patently false claim.

But what do YOU think? Is mathematics teaching a closed book? Do we know all we need know about how to teach math? Is Haim/Ed right? And if not, what are some important open questions about mathematics pedagogy you are pursuing or would like to discuss with others?

More on Devlin and the Nature of Multiplication

Mark Chu-Carroll

My previous post, "Devlin On Multiplication (or What is the meaning of "is"?)" engendered a great deal of discussion on several lists and blogs. However, primary credit for both serious and heated conversations about whether it's sacrilege to teach kids that "multiplication IS repeated addition," "repeated addition is a valid model for learning multiplication," "multiplication of integers can be effectively thought of as repeated addition," and many variations on this theme clearly goes to Professor Devlin himself. 

One particularly interesting conversation has arisen on a blog I enjoy reading on occasion, GOOD MATH, BAD MATH by Mark Chu-Carroll, a PhD in computer science. His July 25th, 2008 post, "Teaching Multiplication: Is it repeated addition?" has resulted in over 65  responses as of this writing and seems likely to continue to give rise to interesting comments about mathematics, its teaching, and much else. While I don't always agree with what Chu-Carroll has to say about mathematics education (not his main focus in any case), he's worth reading, and for the most part those who comment on his blog manage to refrain from the sort of sniping that dominates more politicized venues (e.g., math-teach@mathforum.org). 

Devlin On Multiplication (or What is the meaning of "is"?)

I generally like to be pretty sure I'm right about something before I blog about it here (or at least that I'm not absolutely and irredeemably wrong). Some topics, of course, are relatively safe in that it's pretty hard to determine a definitively correct answer (though perhaps that doesn't prevent some answers from being almost certainly wrong nonetheless): matters of just how to teach something in mathematics or other subjects rarely avail themselves of a single right answer that serves all students equally well for all times and places (regardless of what my antagonists at Mathematically Correct, NYC-HOLD, and similar nests of absolutism may believe or claim). On the other hand, it's easy to be shown to be entirely wrong about specific pieces of mathematics, especially if one is not a mathematician, something I neither am nor have ever claimed to be. Not that members of that priesthood never err or that we laypeople are invariably wrong when we disagree with one of the elect on matters mathematical, of course. But a betting person would do well to give the odds and go with the mathematician when such disputes arise about mathematical particulars. I've known of errors in mathematics problems published by the Educational Testing Service, too, but you'd clean up betting on them against the likelihood they've erred. 

Imagine then my trepidation at taking on someone like Keith Devlin, the popular mathematician who writes regularly for the general public and the profession about mathematics in books, in a monthly column for the Mathematics Association of America, and on National Public Radio. His June 2008 column ("It Ain't No Repeated Addition") came to my attention this morning because several people who blog about home school mathematics teaching (Maria Miller and Denise,  an Illinois-based homeschooling mom who blogs under her first name only at Let's Play Math) were puzzled by what Devlin had to say.

In particular, as his title suggests, Devlin argues quite vehemently that "(m)ultiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not." It's hard to be more unequivocal than that. Instead, he argues, we need to teach students from the beginning that addition and multiplication are two basic things we can do with numbers that share some similarities in places but are not the same thing (I insert the similarity issue that he mentions in a way that likely gives it more attention than he would like, so responsibility for where I plan to go with that point is mine alone).

By his own admission, Devlin's not a K-12 mathematics teacher and he is perhaps a tad disingenuous when he mentions this at the end of his article, stating, "I should end by noting that I have not tried to prescribe how teachers should teach arithmetic. I am not a trained K-12 teacher, nor do I have any first-hand experience to draw on." But considering the respect both K-12 teachers and the general public have for professional mathematicians, it's hard not to think that his intent and deep hope is that his position might carry enough weight to have a great deal of influence in how arithmetic is taught. This isn't, after all, a major league baseball player endorsing a presidential candidate, something that would carry zero weight with most of us; it's a world-famous mathematician with a good deal of media exposure (more than the vast majority of his colleagues) making a claim about what multiplication is and is not. That would likely influence more than a couple of elementary school teachers, school administrators, and other interested and concerned stake-holders. Or so I would expect.

So now I need to go out on a limb and suggest that while Devlin isn't wrong, he also isn't quite right, and the problem lies with the nature of school mathematics and its teaching, as well as issues of mathematical maturity. His argument reminds me of a similar one I have when I do professional development with K-5 teachers regarding things they tell students that simply aren't true. Well, that's actually not always the case, either. The problem is that so much depends on what set of numbers one is working with. Teachers state without hesitation to lower elementary students that "you can't subtract a larger number from a smaller number" and "you can't divide a bigger number into a smaller number" (alternately on this last one, we hear "six doesn't go into three, so. . . "). And if one is working in the natural or whole numbers, both statements are true. But in the integers, the first statement is false, and in the rational and real numbers, both statements are false. (And one could go on to look at other sets of numbers, but for K-5, this suffices to make the point). So are teachers wrong to make the statements they do?

If you say "No," unequivocally, you're suggesting that you likely agree with the hypothetical teachers Devlin cites who believe that kids simply aren't adequately sophisticated at a given age (or individual point of development not directly attributable to age alone) to deal with more advanced or sophisticated mathematical ideas. It's tough enough to understand subtraction and division of whole numbers without throwing negatives or rationals (let alone irrationals) into the mix. We must walk before we can run, after all.

If you say "Yes," unequivocally, you likely find the reasoning above objectionable, as does Devlin, and conclude that the solution is to eschew making any untrue mathematical statements to kids at any stage of the game (assuming one knows all the requisite mathematics, something perhaps much to be wished for, but not so easy to guarantee in practice, even if we raised not just the bar, but the actual minimal mathematical knowledge of anyone teaching K-5 mathematics in this country in public education). 

Naturally, I wouldn't be writing this if I thought Devlin's analysis was sufficient and proposed solution ("Don't tell kids that multiplication is repeated addition"), such that he's made one , adequate. One thing I find lacking in his piece is a solid example that would communicate well and clearly to K-5 mathematics teachers (based on the ones I've known and worked with) how multiplication differs in some deep way from addition. I does not suffice merely to assert that the two are, for the most part, not the same. Teachers at the K-5 level may be intimidated into accepting an order not to claim otherwise, at least when they're being watched, but if in their hearts they believe otherwise, that belief will surely inform their teaching and the idea is going to be communicated to their students. Similarly, research indicates that college mathematics students struggle to accept fully such concepts as the equality of 0.9999... and 1 even when they can be taught to repeat that this is factually true. This gap between surface knowledge and deep understanding and concomitant belief should not be dismissed.  

But even more importantly, what Devlin proposes presumes that there is one correct model or metaphor for understanding multiplication. This assumption runs counter to vast teacher experience in K-5 mathematics and beyond. The multiplication-is-repeated-addition metaphor fits a number of physical models of multiplication grounded in even more fundamental and deep metaphors of arithmetic discussed extensively by Lakoff and Nunez in WHERE MATHEMATICS COMES FROM. Further, it is clear from the perspective of how arithmetic developed historically that the use of multiplication from a practical perspective can readily be called "fast addition," just as exponentiation can be called "fast multiplication." The associated algorithms that emerged for doing whole number and integer multiplication compress some of the individual steps through the clever use of place value in ways that are sometimes confusing to many students, and it is often helpful to show students how this viewpoint is valid by "unpacking" the algorithms and recreating the "repeated addition" that could be done to obtain the same results. Of course, at the same time, we want students to understand why using multiplication algorithms to find the product of even moderately large numbers is preferable. 

The modern view of multiplication Devlin advocates as "one of the things we can do with numbers" that is different from addition is not necessary for students who are learning arithmetic with any set of numbers up to and including the integers. Therefore, it is questionable that students would be advantaged by being given that more sophisticated idea immediately. Further, Devlin's claim that they are actually disadvantaged by being told something else is something he would need to provide proof or at least strong research evidence for, something that is absent in his article. I'm reminded in part of those who argue that failure to stress or thoroughly teach the long division algorithm by, say, fourth grade, will make students unable to learn and do synthetic division in algebra class. This argument sounds plausible to some people (generally those who know little or nothing about teaching K-12 mathematics), but I have yet to see anyone provide even a shred of evidence to support the claim.

The "obvious" solution to this seeming quandary (lie to students about what multiplication "is," or run the risk of over complicating things for them unnecessarily and losing one of the more effective and understandable metaphors/models for it) is, on my view, the same one I have always suggested would solve the above-mentioned issue of lying unnecessarily to students about the nature of subtraction and division: make clear to students that how we define specific arithmetic and other mathematical operations depends upon what set of numbers we're speaking of and that while they may currently be dealing with numbers that do not allow us to answer questions like 7 - 9 = ? or 3 divided by 5 equals ?, the fact is that there ARE numbers that they will learn about in a few years (if they don't know about them already!) that will allow us to answer such questions. It's not that multiplication "is" repeated addition, but rather that there are contexts in which that is one reasonable way to think about it. However, like all the arithmetic operations, it also "is" many other things, each of which will prove valuable to consider in turn.

In higher grades, teachers constantly tell students that "We can't take the square root of a negative number," but the fact is that this is true only if we are restricted to the set of real numbers or one of its subsets. Using Devlin's argument, that practice should be outlawed because we might be making it "too confusing" for these students when they encounter complex numbers or any other new set of numbers (e.g., quaternions). We shouldn't tell them that multiplication is commutative, because they will by high school encounter matrix methods for solving systems of linear equations and be taught that matrix multiplication with elements drawn from the real numbers is NOT commutative. Surely THAT should be a mind-blowing experience that will destroy students' abilities to progress in mathematics! 

Of course, nothing of the sort happens. And to the extent that it might with some pupils, the same approach I've mentioned previously would smooth the way: simply be honest with students about what restrictions apply to broad generalizations we ask them to accept (or even prove!) in class. This practice would actually raise the level of sophistication students bring to the mathematics they are doing, making them more careful about simply accepting any given statement as true for "all numbers" and any given operation. Instead, we will have impressed upon them the necessity of considering what set of numbers is under consideration and whether statements made about a particular operation are restricted to a given set and situation. Multiplication is NOT always commutative. It also isn't always repeated or fast addition. But it surely is commutative in many circumstances and it is important for students to be taught this and to see that it makes sense. Similarly, it is valuable for students to explore the notion of multiplication as repeated or fast addition, as long as they are given the tools to understand that this might not always make sense as they consider other, more abstract sorts of numbers or applications of the multiplication concept. Students so taught might well develop the valuable mathematical habit of mind of questioning how broadly a particular mathematical idea applies. And that is one of the things upon which they will be able to build a deeper and more mature knowledge of higher mathematics.