Reply to Critic: Background

Dr. Mulase is a professor of Mathematics at UC Davis. He wrote a vitriolic attack on Algebra: Themes, Tools, Concepts, (ATTC, a textbook I co-authored with Anita Wah,) in response to the decision by the Mathematics Departments in the Davis school district to adopt this book for their algebra classes. A group of Davis parents, including Dr. Mulase, campaigned to remove the book from use in the middle schools in Davis. The District suggested a compromise: parents would be given a choice. As a result, about half the students took Algebra using my book, and the other half used Dolciani, a textbook with the traditional approach favored by Mulase.

The district asked the Mathematics Diagnostic Testing Project, an independent group based in several UC and CSU Math Departments, to compare the results of using the two books. The study was led by Dr. Alfred Manaster of the UC San Diego Mathematics Department. Here is the Summary of Findings for year 1:

This study investigated the performance of students in two Algebra 1 courses taught during 1996-1997 at Davis Junior High School (grades 7-9). The performance measures used in the study were tests of the Mathematics Diagnostic Testing Project. Students in the course called Algebra [using Wah-Picciotto] scored significantly higher on the Geometry Readiness test given in June than students in the course called Alternate Algebra [using Dolciani]. They also scored significantly higher in three of the eight topic areas of this test. Looking at three bands of students in each course with more narrow ranges of Algebra Readiness pre-test scores, the same general pattern occurred in the two lower bands as in the overall student population. For roughly the top third of the students in each of the two courses, the difference in Geometry Readiness test performance was not statistically significant.

Among comparably prepared students, these data suggest that the lower two bands of the students in the Algebra course (in terms of preparation) performed significantly better than their comparison populations in the Alternate Algebra course. There was no significant difference in the performance of the top bands.

In year 2 of the MDTP study, students using my book did better across the bands, but the results were statistically significant only in the lowest band. In other words, the weakest students' algebra skills were helped by my book, at no cost to the middle or top students. Moreover, my book turned out to be very helpful to all on the Golden State Examination in Algebra, as can be seen in the 1998 results:

(For a copy of the Golden State Exam Guide, including sample questions, go to the California Department of Education site.)

I hope these data help Dr. Mulase and others see that math education reform is not automatically for the worse, and that pedagogical approaches they are not familiar with may be worth learning about. What follows is a response to his rant, which I wrote soon after he published it. My point was that the problem-rich and tool-based approach of ATTC, the emphasis on writing and communicating about math, and the connections with geometry and number theory, all can contribute to making algebra more fun, more interesting and more accessible, with no loss in rigor. Some traditionalists disagree, and claim you must master algebra skills before you are allowed to do anything interesting with them. This has a very negative effect on student motivation. A few in the reform movement disagree for different reasons, believing that teaching algebra skills will of necessity make the course inaccessible. Those reformers shortchange their students. I am glad that my book (in spite of its weaknesses) navigates a middle course successfully, at least in the hands of skilled teachers such as the ones in Davis.

[For another debate triggered by the same events, see The Pythagorean Geoboard.

I am sorry that Dr Mulase suffered such pain when reviewing Algebra: Themes, Tools, Concepts (ATTC,) the book I co-authored. Some good may come from his suffering, as I believe discussion among practitioners and teachers of mathematics can only be helpful. Too many of Dr Mulase's colleagues do not pay much attention to what happens at the pre-college level, though I must say that some UC Davis mathematicians (Sherman Stein and G.D. Chakerian) have made magnificent contributions to secondary education.

Of course, I would have preferred a more generous tone in Dr Mulase's criticism, but I have survived the reading of his diatribe, and I may have learned something from it. In this paper, I will respond to his main points, try to explain the approach we used in ATTC and our reasons for it, and sketch how I think a future edition might be different (should I have the opportunity to write it.) Finally, I will try to address Dr Mulase's content objections one by one.

Formulas

The question of formulas is a thorny one. One acquaintance of mine, a very sophisticated math teacher who has been fine-tuning a curriculum for 11 years in a high-pressure ultra-academic high school, where most students end up with 5s on the Calculus APs, says "formulas are poison." His point is that students turn off their brain when confronted with one. I have seen this happen more times than I care to remember.

Nevertheless, there is a time and a place for formulas and their memorization, and I will be definitely be rethinking this if I get a chance to work on a revision of ATTC. The key is to decide what is important for a student to know at a given stage, and whether the formula at that point in the student's education serves to hide meaning even as it facilitates computation. Our book is used in grades 7-9, mostly, and in some cases in "remedial" classes. Do those students need to know formulas for arithmetic and geometric sequences and series? I doubt it. Will students who have done the work on sequences in ATTC in grades 7-9 be able to grasp it better in Algebra 2 and learn those formulas then, with understanding? In my experience, yes.

On the other hand, formulas like the distributive law, or the significance of the parameters in the equation y=mx+b should be known and become automatic at about this level, if students are to function effectively in further math and science classes. I must admit that our book is weak on emphasizing "mastery", and that sometimes we went overboard in our warnings against mindless memorization. Still, our efforts at leading students to understand the distributive law (through substituting numbers, through a geometric representation, through comparing graphs) are worth the effort. Similarly, we approach linear functions in many, many representations, hoping to instill a multi-faceted, in-depth understanding, rather than the mechanical parroting of a few phrases.

"Some facts students are led to find are important, such as commutative, associative and distributive laws. But I felt they would simply waste time by the lengthy explanations and explorations. These laws are like 'a red signal light means you have to stop.' Nothing more."

The distributive law, or the slope-intercept form of linear functions is quite a different thing from the arbitrary convention that you stop at a red light. The latter indeed requires only a two-minute explanation. But there is much interesting mathematics encapsulated in the former, and it is definitely worth thinking about for more than two minutes. The approach of telling students to just memorize those formulas with no understanding has of course been tried on a large scale, and has proven to be a failure: many students cannot learn that way, and for those that do, the learning proves to be superficial and the mastery short-lived. How many adults do we each know who got A's and B's in algebra, and yet remember nothing?

To summarize: Dr Mulase is right that some formulas should be memorized in Algebra 1, and that ATTC is not clear enough about that. On the other hand, it is not necessary to memorize all formulas by 9th grade. It is essential, and not easy, to make sure that formulas work to encapsulate understanding the students have, rather than to be a substitute for this understanding. In too many classrooms, formulas are used for the mindless manipulation of ill-understood symbols. Whatever its weaknesses, ATTC incorporates an approach that lays the groundwork for algebra with understanding. It is up to each teacher to make sure that understanding is followed up with the necessary memorization, as appropriate to the class.

"...what Gauss knew when he was about eight years old, namely, the summation formula of the arithmetic progression."

In the famous story about the young Gauss, my understanding is not that he applied a formula he knew, but that he used precisely the algorithm we lead students to discover in ATTC. And at any rate, if Gauss was one of my students, I would not subject him to either ATTC or to any of the currently available textbooks.

Alas, most Algebra 1 students are not Gauss, and I don't think that math reform can be blamed for that.

Proofs

It is difficult to take Dr Mulase's complaint about the lack of proofs seriously. ATTC is used primarily in grades 7-10, and frankly, proof is difficult to introduce to our best students in 10th grade. Few people involved in teaching Algebra 1 make proof a priority. Nevertheless, I maintain that ATTC does a reasonably good job of preparing students for mathematical proof.

Take one of the examples that were mentioned by Dr Mulase: the sum of a sequence of consecutive natural numbers. In the lesson Dr Mulase complains about, students are guided through two approaches (with specific numbers) which would (with variables) constitute a proof of the result. One method is geometric (making a rectangle from two triangular numbers), and the other is algebraic (adding the numbers forward and back, and recognizing that twice the sum is n(n+1) ). Understanding the argument with numbers is an excellent way to prepare students for understanding the proof later, perhaps in Algebra 2. Moreover, the argument / embryonic proof generalizes much more readily than the formula to the case of a general arithmetic sequence.

Dr Mulase takes a substantial amount of space discussing the so-called "stamp" problem:

"Let a and b be two relatively prime positive integers. Find the largest positive integer that cannot be expressed as ma + nb, where m and n are non-negative integers."

The answer is that it is the product of the numbers, minus their sum. Dr Mulase thinks the problem belongs in graduate school. Perhaps. Because this may be the view of many, the lesson is clearly marked as optional. However our best algebra students should be able to go well beyond the "pattern recognition" we are charged with promoting. After an initial exploration, we show the concepts underlying a proof. Indeed, for the case a=5 and b=6, we suggest students write numbers in an array:

 
 0  1  2  3  4

 5  6  7  8  9

10 11 12 13 14

15 16 17 18 19

20 21 22 23 24

etc.

Then we suggest circling the multiples of 5, the numbers that are equal to 6 + a multiple of 5, 12 + a multiple of 5, 18 plus a multiple of 5, 24 + a multiple of 5. All these numbers can be expressed in the required form. Because of the layout of the numbers, it is clear that 24 and all numbers greater than 24 can be expressed that way. The greatest impossible number is the one above 24 in the chart, in other words 24-5=19.

This algorithm would work for any pair of relatively prime numbers, and is accessible to most students at this level. Understanding a formula is more difficult, as it requires us to generalize, but the essential foundation of the proof has been laid. We would use a columns, and the answer would be

b(a-1)-a=ba-b-a.

This argument is outlined in the book. It probably could have been outlined better. (In fact there is a typo there.) The full proof is probably too difficult for most students, and teachers have to decide how far to take the lesson, but the introductory part is not an argument that needs to wait till graduate school, nor is it mere "pattern recognition" as it enhances students' insights about numbers, multiples, and the relationship between addition and multiplication.

Explanations

More interesting than Dr Mulase's view on proofs is his criticism of the fact that while students solve many good problems in ATTC and discover many patterns, they are not "rewarded" by a mathematical explanation of the patterns. I will not bother responding to the curious objection to the search for patterns ("Pattern recognition is for chimpanzees and seals in a circus, but definitely not for our children!") Of course, discovering patterns is at best only the beginning of understanding. But it is a useful beginning.

That beginning should indeed be followed by an explanation. However, the reality is that such explanations in math books are rarely read and almost never understood by students. Worse, they often sabotage the students' own engagement with the material, as they get the idea that math is about reading how somebody else summarizes something that they feel they themselves could never aspire to understand. Still, I think our critic has a point, and judicious inclusion of some explanations would have strengthened the book, particularly in the hands of a teacher (or parent) whose math background is shaky.

Still, it is far more rewarding and educational to students to write a good explanation that to read one. Of course, it is not easy to do, and on important questions it may be helpful to both write and read an explanation.

I would be interested in hearing from ATTC users how they feel about these questions. Do we need more explanations in the book? If so, for which topics? Which formulas should we highlight, and expect our students to memorize (as well as understand)? And more importantly, how can we do it without taking away from the strengths of the book, which lie in the presentation of worthwhile problems?

Content Questions

Here are some comments in no particular order on some of the content points raised by Dr Mulase.

Integers, lattices:

Dr Mulase feels we do too much with integers. Actually, doing a lot with integers is developmentally appropriate, as it is a zone where students are fairly comfortable and challenging problems can be posed that in some cases serve as previews of the same problems in continuous form.

Moreover, we were trying to broaden the content of the course beyond its traditional limits, which are not the result of careful thought, but of historical accident. Some of the digressions (e.g. the "McNuggets" problem discussed by Dr Mulase as the hardest in the book) do not immediately contribute to traditional Algebra 1 / Geometry topics, and those are clearly marked as optional. Others do. For example, the (discrete) work on geoboards leads to an interesting approach to the Pythagorean Theorem, an approach we offer not as a substitute to the familiar geometric approaches, but as a useful complement to and preview of those approaches. What makes it effective is the fact that students are able to discover their own strategies for area, which they understand well, and that those lead to a way to find distance on a Cartesian plane. A proof of the Pythagorean Theorem is an easy step away from that strategy.

While on the subject of the Pythagorean theorem, the formulation "leg² + leg² = hyp²", while it carries more content than the traditional a²+b²=c² does have the problem pointed out by Dr Mulase, and would be improved by being written "leg² + LEG² = hyp²".

Similarly Dr Mulase objects to combinatorics and group theory. He is entitled to his prejudices, but the question is not so much whether there is too much with integers, as number theory is an immensely profound and interesting part of mathematics, and likewise for the other parts of discrete math we touch on. The question is more whether there is enough coverage of the topics we expect to cover in Algebra 1. The tables in the back of the Teacher's Edition (p 534) makes clear that the most important topics are treated in depth.

The late appearance of the quadratic formula stems from our belief that this (very important) topic is best treated in Algebra 2, or in a very strong Honors Algebra 1. By the way, my work on completing the square, and the connections between the Cartesian representation and the Lab Gear representation of quadratics is better developed in Lab Gear Activities for Algebra 1 (title chosen by the publisher, over my objections.) This approach is very effective (at least it is in Algebra 2), and it makes a topic that was previously meaningful to a tiny number of students accessible to many more. This gives the quadratic formula a grounding that was previously not available. Far from trying to keep this classic result from students, ATTC and the Lab Gear are a way to bring it to more students than ever.

The Lab Gear:

Dr Mulase has a number of objections to the Lab Gear, the "toy" that is equivalent to counting on one's fingers. The first is that the number of dimensions is not necessarily what the degree of a polynomial refers to. Well, of course not. There is no shortage of examples, say in physics, where the square of a quantity does not refer to area. Does Dr Mulase conclude that we should not use any context at all, for fear students may conclude any context we use is universal? That would make for a rather dry course! Our book uses many, many other ways to represent polynomials, far more than our critic believes necessary, apparently. But in that galaxy of models, the dimensional one has an important role. (I will not get into the historical reasons to mention this model, as children's development need not parallel the history of math.)

Dr. Mulase may never have seen student mistakes like x + x² = x³, or (x+5)² = x² + 25. Or perhaps he is only interested in those students who don't make those mistakes. The geometric approach he complains about helps students make sense of the variables, and reduce the number of errors of this type. And when students do make those mistakes, instead of telling them for the n^th time they're wrong, and how it's done, the teacher who has used the geometric and manipulative approach can lead the student to correct their own mistakes by reference to the dimensional model. Students who understand this geometric connection are better equipped than those who just got the "two minute explanation" promoted by our critic.

By the way, Dr Mulase is wrong when he says that we force students to use the Lab Gear. When a student is ready to take off and stop using it, there is no reason to discourage him or her. In fact, we propose a strategy to go from blocks to symbols for each of the main uses of the Lab Gear. Note that it is important to distinguish the student who dislikes the Lab Gear because of not understanding the geometry, from the student who dislikes it because of having a full understanding of it, and therefore finding it unnecessary. The Lab Gear is a pedagogical tool to highlight the geometric connection, it is not an end in itself.

One good point made by Dr Mulase is the criticism of the Lab Gear as a way to clarify -(-x)=x. That is not its strongest use of the tool, and it is certainly arguable that in some classes, it is best to limit the Lab Gear to throwing light on the distributive law and factoring. In fact, the main purpose of the Lab Gear is to promote discourse in the algebra class, rather than to serve as a "do-as-I-do" algorithmic model. For example, teachers need to build upon the questions students often raise about the limits of the model, about the connections between blocks and symbols, or about the various meanings of "variable". And if students raise no questions, then it is definitely the teacher's responsibility to raise them.

Functions and function diagrams:

ATTC emphasizes functions throughout. Many teachers find that with ATTC, they are successfully introducing key ideas about functions in Algebra 1, that are difficult to get across in Algebra 2 or Precalculus to students who have not been exposed to this approach. The approach of putting functions at the center is fairly widely supported by the "multiple representations" current in algebra and calculus reform, and I am neither the first nor the last to point out the pedagogical and mathematical power of this, especially in conjunction with intelligent use of technology.

My contribution to this is mostly the use of the function diagram (parallel x and y axes) as a way to get additional insight into linear functions and into concepts such as definition of function, inverse function, composition of functions, and iteration of functions. All of this is far, far clearer in the function diagram than in the cartesian graph, and I believe that Dr Mulase would agree if he gave it some more thought.

As for iteration of functions, Dr Mulase is correct that we use the linear case as an introduction to dynamical systems and chaos theory. We merely introduce, because this is only Algebra 1. Should I get the opportunity to write an Algebra 2 or Precalculus text, you can be sure that I would follow up on this.

In addition, the function diagram can be used to illustrate rate of change and therefore the concept of the derivative. In fact, I have been told that his illustrious colleague Dr. Sherman Stein uses function diagrams in a calculus textbook for precisely that reason. At any rate, to say function diagrams are nonsense, as our critic does, is absurd.

Of course, Cartesian graphs remain essential, and in the long run far more important than the function diagram, which is merely a pedagogical tool. This is in part because of the reasons enumerated by Dr Mulase, which I thoroughly agree with. This is why graphs appear in 156 pages of ATTC, while function diagrams, cool and useful as they are, only appear in 32 pages.

"Real world" problems:

We are charged with using "unrealistic real world problems". Let's be real. The world is entirely too complicated to be analyzed as is by Algebra 1 students. This is why "real world" problems at this level are invariably based on somewhat oversimplified models, no matter what the textbook. Let's look at the examples one by one:

- The Algebank problem is silly because of the 100% rate and $100 fee

Sillier than the age problems or train problems of old? Give me a break. This is an extremely engaging lesson, which has generated intense involvement from students and teachers alike. Sure, we could have used a 1% yearly interest rate and a service charge of $5, but how different would that be? Lighten up! Fortunately, most students and teachers are able to see that the structure of the problem is what matters, not which numbers we use to illustrate it. In fact we suggest that the students use what they learn in this case and apply it to real bank policies. Interestingly, when they do, they find that the amounts children put in savings account are generally below the fixed point for those accounts.

- Asking for the difference between Earth-Moon and Earth-Saturn distances is worse than a problem asking students to add up the temperatures of stars

Huh??? Not if you are interested in comparing the time a space probe would take to get to the Moon with the time it would take to get to Saturn. This difference would at least get us started in thinking about that. The fact that the number is very large, essentially equal to the Earth-Saturn distance, is surprising to students, and a great opportunity to discuss order of magnitude (which is what the Teacher's Guide suggested). Moreover, the question asked was to calculate the ratio of the distances as well as the differences. Clearly, in this context, it is the ratio that is the more interesting number. Asking for both makes that crystal clear.

- How many pages does a book have, if it has 1992 digits in its page numbers

This is just a harmless puzzle, with the bonus that it forces some thinking in how to combine multiplication and addition. This is the sort of problem that people who make up math competitions use all the time, for a good reason: they reveal how well the solver understands numbers and operations.

- The window problem where the price is calculated as a linear combination of the area of the panes and the length of the wooden bars that frame them

The main problem with this one is that it is far too difficult for most classes, and I foolishly discouraged hints. In fact, a good hint is that the result involves the areas of the panes and the total length of the wooden bars, not so far-fetched an idea, given that the length of the wooden bar may be a way to approximate the labor involved in making the window. The terminology "perimeter" to describe it was a bad choice on the part of the person who wrote the solutions.

- The widget problem where if the price drops by $1 there are 10 more customers.

Dr Mulase points out that if the price is 0, then there would only be 240 customers. Clearly this cheap economic model breaks down outside a a specific range. This was an attempt to use a quadratic function in a non-geometric, non-physics context. I heard this problem at a conference, and uncritically incorporated it into the book. Perhaps I should not have. I am not particularly attached to it, nor do I think it is harmful, since all modeling in the social sciences depends on discussing assumptions about domain and range. A good question to think about would be the one raised by Dr Mulase. With his permission, I would like to include it in the next edition.

- Chain letters

The only objection raised by Dr Mulase is that he hates the problem. On the other hand, many students find it interesting. A little calculating certainly helps one think about chain letters, pyramid schemes, and related topics, and it is not crucial that Dr Mulase enjoy the problem to make it worthwhile.

- Polyominoes

The opening pages of ATTC are centered around the problem of the relationship between the area and the perimeter of a polyomino. This is one of the great problems in this or any textbook, if I say so myself, even though Dr Mulase finds it unrealistic and frustrating. What is great about it is that it engages students and teachers at many levels. It is a problem I have used in a huge variety of settings, with students of many ages, with parents, with teachers. Invariably, the level of engagement is high. Everyone understands the problem, almost everyone can find a formula for the maximum perimeter for a given area, and many are intrigued by the minimum perimeter problem. However, the latter is much tougher to crack and the search is quite fascinating, with interesting false starts and partial results along the way. While it will not accomplish miracles in teaching any particular topic, it is a great introduction to what doing math should be like. Students are introduced to a manageable problem unfiltered by oversimplification, and they think about it geometrically, numerically, and on a cartesian graph. And they enjoy it! What's so bad about that?

Conclusion

I am usually interested in feedback from professors of math. Many are beyond naive about pedagogical matters -- one math prof at UCB wrote that variables are just like numbers, and all you have to do is explain that to students -- but we share common goals, at least in theory. Besides, they have to teach the students we send them.

It would be nice if the exchanges were civil, and based on the not unreasonable assumption that we all want to improve math education, but I'm afraid not everyone can be mature.

Dr Mulase complains that some of the discoveries students are led to in our book belong in graduate school, not Algebra 1. Those are few and far between, frankly, and they are often very interesting to students. But I am sure most users of the book skip them. They are clearly marked as "optional". Other topics he deems unimportant, and he is entitled to his opinion. But a look at the index on page 534 of the Teachers' Edition shows that the topics that get the most attention by far are Cartesian graphing, linear equations and functions, the distributive law and factoring, fractions, opposites and reciprocals, the correct use of minus, ratio and proportion, slope and rate of change, square roots, etc. Not a particularly controversial list.

"Now I understand the source of my feeling. The book is designed to destroy the mathematical sensibility of the students."

This is exactly what can be said about the traditional algebra texts, which have brilliantly succeeded in reserving algebra for a tiny elite, a handful of which become professional mathematicians. This is well documented, and I need not go over it here. Our book has flaws, but one thing it does (in the hands of an adequately prepared teacher) is turn kids on to thinking and problem solving, which is really a prerequisite to enjoying math. The plug-and-chug disaster that our critic is waxing nostalgic about has been tried widely, on a scale many orders of magnitude greater than my commercially insignificant book, and the results are plain to see: most people who got A's and B's in the traditinal algebra one course remember nothing of it as adults.

I realize that the failure of the traditional course does not absolve me from being accountable for the weaknesses in our book. There are weaknesses, especially in the hands of a teacher who does not have a strong math background, or in the hands of a mathematician who does not have a sense of how actual students actually learn. I hope to have a chance to do better in the next edition should there be one, though I am rather proud of the striking successes of this book in some places where other approaches have failed.

I would encourage Dr Mulase to get more involved in the teaching of algebra to young people. As they say in French, criticism is easy, art is difficult. If you can write a better algebra book, please do it. What is known is that old ways have not worked. I have no illusions that our book, or the NCTM Standards, or any specific remedy is a panacea. It is but an attempt to take up a difficult challenge. Anyone willing to lend a hand is more than welcome.