Changing Order of Topics: An Example From Practice

Some day, my princess will come. And she'll be passionately interested in discussing questions of actual teaching practice in mathematics classrooms. Actually, of course, many such exist, but getting them together on this blog or in open, active public forums, having the time to participate actively, etc., isn't always a trivial task. For full-time teachers, time is always at a premium, and while there are many good resources on the Internet, tracking down what would really be productive for one's own work can be difficult.

From my perspective, one of the shortcomings of even the best conversations about practice, however, is the lack of access to specific classroom data, particularly video with supporting documentation (teacher journals, samples of student work, assessment instruments, observer notes, etc.) that would allow other practitioners to engage in deep analysis of what goes on in mathematics teaching based on common access to the same materials. As a former literature doctoral student, I am reminded of the countless frustrating conversations I had with undergraduate students of mine (and occasional fellow graduate students) about the absurdity of discussing literature without making much or even any direct reference to things in the text(s) under examination. Hearing someone offer up "personal reactions" to Dylan Thomas' "And Death Shall Have No Dominion," or Marlowe's DR. FAUSTUS (e.g., "I could really relate to that Mephostophilis character. He reminded me of my best friend in high school") just doesn't quite measure up to the level of discourse one hopes for in serious literary analysis. I kept wanting students to show me where IN THE TEXT they were getting their ideas and HOW what they said was going on actually worked. Generally, I was very disappointed even when I gave meaty selections from the texts that we'd dealt with in class on exams and asked students to offer some sort of analysis that stayed primarily "on the page."

To expect deep conversations about teaching, one must have similarly detailed examples of practice about which to converse. There's really no substitute for specifics and no specifics better than seeing what teachers do, with supporting documentation to flesh out and give depth to one's analysis. But as of this writing, there seems to be no readily-accessible, free web resource that could be used by groups of teachers, parents, researchers, and other interested stake-holders to discuss actual mathematics practice grounded in the sort of data needed for "close textual analysis." (Of course, perhaps I'm in error: if so, I hope to be flooded with information about just such resources and will, you can be sure, flock to them post haste).

While waiting for Godot (or, in fact, trying to figure out how to create just such a resource), I will resort to the less-satisfactory standby of anecdotal evidence, offered strictly in writing and through the admittedly-flawed lens of my own memory for something that's now about five or six years in the past. Any inaccuracies in the recounting are, I hope, due merely to incipient senility rather than purposeful misrepresentation of fact.

A Fortuitous Decision

My focus is the decision I made in the winter of the 2002-2003 academic year to change the order in which I taught two topics in an intermediate algebra course I'd been teaching with various materials for 2 1/2 years at a public charter high school in Ann Arbor. The goal of the course was to prepare high school sophomores (for the most part) to take the same course subsequently with college instructors for dual-enrollment credit towards both high school graduation and credit at the community college that sponsored and hosted our school. My course was intended to help bridge the gap between taking first year algebra over the period of one year at the typical pace of high school and taking a second algebra course at a college pace in one semester.

I had been using various books for the course, including the same text mandated for the college class, an indifferent text to my mind written by past and current members of the math department, designed for use both in regular classes and in self-paced courses. As a result, it was similar to programmed instruction texts from the 1960s to some extent. I had also used COMAP's Mathematics: Modeling Our World, a more progressive text that was heavily grounded in applications and mathematical modeling. While each book worked well with a subset of my students (and, frankly, with my own development as a teacher), neither was without flaws and neither reached a core group of students I had each semester who were low achievers and/or what might politely be called slackers when it came to school in general and mathematics in particular.

Although I eventually wound up using a third text (ALGEBRA 1: A PROCESS APPROACH, from the University of Hawaii Curriculum Research and Development Group) to reach that group in a shortened spring semester, when other students had gone on to the college course, I'm not sure it or any book I've seen would have been right for all my students or been fair to all these groups of kids I wanted to reach in ONE semester.

An example of a difficulty none of the books seemed to address adequately was teaching absolute value inequalities and their graphs, as well as the broader question of helping students see the connections between linear and quadratic equations and their respective graphs. But merely solving absolute value inequalities was always a sticking point for all but my best students. For reasons I had not yet determined, this topic always came after we had covered quadradic equations and their graphs (though not immediately thereafter). And even for students who did reasonably well with the pure algebraic manipulation and solving required in the units on quadratics, understanding how changes in the coefficients impacted their graphs and vice versa proved more difficult. So I had two stubborn but seemingly unrelated problems to deal with in these units.

For reasons that have become lost in the mists of time (I wish I could say that I had some sort of brilliant insight following much mediation or had been led to an epiphany through conversations with colleagues, but if either was the case, I have no recollection to report), I decided to try teaching the absolute value inequality unit before we did quadratics. It may be that I just wanted to get it out of the way and move on, or that I wanted to weed out a few students who were annoying me. But most likely it was nothing quite so insidious or inspired.

To what I recall as my great surprise and greater pleasure, the students on the whole seemed to do much better on both units, including how they approached solving the absolute value inequalities. But the biggest payoff, based both on the sorts of things they said in class and how they performed on assessments was in their understanding of the graphs and their connections with their symbolic representations. In particular, ideas about horizontal shifts, which heretofore always seemed extremely elusive for the majority of my students, now seemed to make sense to a lot more of them. And when we revisited these and related issues in a subsequent unit on quadratic equations (and even later on quadratic inequalities), they were to a much greater degree than in previous semesters I'd taught able to grapple successfully with what was going on with the graphs as we fiddled with the parameters.

To what do I attribute the apparent changes? Naturally, I wish I could say it was a vast improvement in the quality of my instruction, and perhaps that was true in one particular way. Because as I taught the transformations of the graphs of the absolute value inequalities, it hit me for perhaps the first time (or at least for the first time so overwhelmingly clearly) that there were a host of parallels between these changes and those associated with quadratics. In fact, to my mind, it seemed that as we covered the topic in the unit on absolute value inequalities, it was somehow easier to communicate what was happening with the graphs as we changed the values of the coefficients.

What occurred to me, though I only had scattered informal feedback from some students to confirm this (and it was in retrospect, when we treated the quadratics unit later), is that absolute value graphs are extremely simple because they consist only of two straight lines. Students are comfortable with straight lines compared with curves, and of course a linear relationship is easier to plot and to draw the graph for with complete confidence compared with graphing any curve, which always seems imprecise by comparison when drawn by hand or even with tools to help aid in precision on graph paper. Students quite quickly "get" the symmetry in the absolute value graphs, and of course the fact that there are two predictable (and opposite-signed) slopes operating on the two "legs" of the graph makes hand-graphing a snap.

While it may seem like a minor point, contrast that with graphing parabolas. Although the symmetry is there, of course (in fact, ALL the same properties are there EXCEPT for the straightness), it's harder to graph individual points except for lattice points with complete confidence (it can be argued that the same is true with absolute value graphs, too, but consider how the issue of slope comes into play and it is likely that one will see how relatively easy these are compared with any graphs involving curves). Moreover, and this may be the crucial point, it's easier to picture mentally and to anticipate what's happening on a point-by-point basis with the absolute value graphs, which in their simplest forms (with "leading coefficient" of one) are, after all, pieces of y = x and y = -x, possible shifted vertically and/or horizontally.

And it's particularly those horizontal shifts that are difficult for students to grasp. The idea that adding a positive quantity to the independent variable "inside" the function results in a left shift and adding a negative quantity produces a right shift is counter-intuitive at first blush. Without having to deal with the complication of squaring, many students seemed more able to just think about the impact of this one change on the resulting y-values. Later, when the same issue arose with quadratics, most students seemed able to transfer what they'd already learned from their previous experiences with the absolute value graphs.

Research "Inside" Education?

Is the above "proof" that what I observed is definitely the case and that all teachers should follow suit? Of course not. It's simply food for thought and, I hope, someone's formal research. But this is one example of where classroom practice could inform research and that the subsequent research could later influence broad-scale practice. If someone decides to set up and conduct a sound comparative experiment, it is conceivable that they could do more deeply the kind of analysis that Deborah Ball and Francesca Forzani call research "inside" education rather than about it ("What Makes Education Research 'Educational'?", looking at the interplay between content, teacher, and student on this and related issues. Thus, I suggest that while what I have presented here is not a substitute for either formal research or the kind of deeper and broader data-base I mention in the beginning of this piece as a source for practitioners and others looking to improve classroom practice, what I've offered is a meaningful contribution to research and practice.

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?