My Math Education Blog

"There is no one way"

Wednesday, September 21, 2016


In this post, I will outline my approach to this partition problem:

How many ways can you write a positive integer n as a sum of three or fewer positive integers?

Partitions are a standard topic in number theory, but I will limit myself to this specific question. I started trying to figure it out after a conversation about a different (but intimately related) problem, which Marilyn Burns e-mailed me about. Read about the original problem on her blog.

Like Marilyn, I will start by listing some takeaways:
  1. For most people, time pressure is not helpful when trying to work on a challenging problem. I was unable to think about any of this when I felt I should answer Marilyn's questions right away.
  2. Knowing "the answer" does not prevent one from thinking about a problem. It is often a good idea to tell students the answer to a problem, and have them think about why it may be true.
  3. Sometimes, a change of representations is the key to solving a problem. Therefore we are undermining our students' mathematical power if we only teach what we deem to be "the best" way to think about a certain topic.
In any case, according to Wikipedia, the answer to the partition problem is

(Where "round" means "round to the nearest integer.) I found this formula baffling, and decided I needed to find some way to understand how it could possibly be true. The second power seemed plausible, but where did this 12 come from?

The first thing I did was to try to get insight into the problem with small numbers:

To make the rest of this post easier to follow, I suggest you extend this table all the way to n = 9. If you do it correctly, you'll find 12 partitions for 9.

Try to apply the Wikipedia formula to the numbers you found. Sure enough, it does work. But why?

Let's break this down into smaller problems. We have sums of one, two, or three terms, so we will tackle these one at a time. I will use n = 12 as an example, to make this easier to follow.

One term: There is always exactly one way to write n as a "sum" of one term. For example, 12 = 12.

Two terms: Once you choose the first term, the other is forced. For 12: 1+11, 2+10, etc. If the first term is t, the second term is 12-t. However I must stop after 6+6, since the next one would be 7+5, which we already saw as 5+7. So the stopping point is at the half-way mark, when the first term is half of n. There are n/2 possibilities for two terms.

What if n is odd? Look at your calculations for 9. You'll see that there were four possibilities, from 1+8 to 4+5. The next one, 5+4, has already appeared. More generally, in n is odd, there are floor(n/2) possibilities. (Where "floor" means the greatest integer less than or equal to n/2.) In fact, this same formula works for the even case, so that is our answer for two terms.

Three terms: That is what makes the problem difficult, and therefore what makes finding the solution so satisfying. This time, once you choose the first two terms, the third term is forced. Here is a systematic search for 12. The sums are all written with the three terms ranked from least to greatest. Each row stops right before it repeats.

1+1+10, 1+2+9, ..., 1+5+6     5 sums
2+2+8, 2+3+7, ..., 2+5+5   4 sums
3+3+6, 3+4+5   2 sums
4+4+4    1 sum

Note that the last row has a single sum, with all the terms equal to one-third of 12. Making one of those terms greater than one-third would make another less, and therefore would yield a sum we have already listed.

So a total of 12 sums. But how to generalize? Looking at the numbers did not suggest anything to me.

The answer came while I was lying in bed in a bout of insomnia, about ten days after first seeing the problem. Since the third term is forced once the first two have been chosen, we can represent each sum on a Cartesian grid, using the first term as the x-coordinate, and the second as the y. For 12, we get this figure:

Draw a triangle with one side on the y-axis, from (0,0) to (0,6), and the opposite vertex at (4,4):

Note that we could include the one-term and two-term sums on the y-axis, by thinking of those sums as three-term sums, and including 0 terms. But we won't do that: we will stick with the display of the three-term sums.

Shade each unit square that lies "north-west" of one of our dots:

Note that the area of the shaded region is equal to the area of the triangle. This gives us a way to generalize: the base of the triangle is n/2, and its height is n/3. This is not a coincidence, as you will see if you look back at the partitions above, and the accompanying reasoning. So the area of the triangle is n2/12. (Here is our exponent 2, and the 12 in the denominator!) The area of the triangle equals the shaded area, which in turn is equal to the number of three-term partitions.

This does not constitute a full proof, as it relies on the particular case where n is both even and a multiple of 3, but you can see that at least, it gets us close to the final answer. The Wikipedia formula is now much less mysterious.

Indeed, the above argument gives us the following approximate formula by adding our answers for three-, two-, and one-term sums:
...very close to the Wikipedia formula. Accounting for all the cases, (as we did for the two-term sums) should wrap this up. I will leave that project to you.


Friday, September 9, 2016

Assessment Postscript

A few months ago, I wrote a series of posts on the subject of assessment. (It starts here.) Even though the series extended to eight posts, I didn't manage to include everything I had wanted to say. Here are a few thoughts that didn't make it into the series.

1. Obedience

For some of us, assessment policies reward memory and docility more than understanding. I kid you not: some teachers take points off for a staple in the wrong location. Many will penalize students irrelevantly by having their attendance or punctuality affect their grade. Yes, there's a place for that, but it's not why we became math teachers. Our job is to teach math, not obedience to authority figures. Moreover, there are all sorts of biases built into this, because students from different backgrounds (and in fact, different genders) often have different relationships to authority, and that has little to do with their ability to do math.

2. Points

One thing that reveals the subjectivity of grades is the fact that the points that are its ingredients get added up even though they represent incommensurable things. x points for class participation, y points for homework, z points for quizzes, etc. It's like adding a student's height, weight, and temperature, in the hope of getting a meaningful sum.

3. Alternative assessments

My post on Assessment Tools and Strategies reflected my own practices. However, there are other options. I will not say a lot about those, as I am not an expert, but here are a few ideas:
  • Group tests, with the score determined by a random drawing among each group's papers.
  • Participation quizzes, where you watch the class work and make notes on students' work habits.
  • Observing and evaluating students' class work.
  • Notebook checks, which give you a different window on student understanding.
  • Holistic scoring of student written work (a lot faster than rubrics.)
  • Quick Yes/No rubrics (see Algebra: Themes, Tools, Concepts, Teachers' Edition p. 560.)
  • Portfolios: a student-compiled folder containing the student's best work.
Student self-evaluations in journals or other forms can help round out the picture.

Anita Wah and I elaborated on many of these ideas in ATTC TE, pp. 552-555.

4. Grade Grubbing

Teachers like to complain about the grade grubbing culture at their school. We like to imagine a world where all students are strictly motivated by their interest in what we are teaching. I sympathize, but I don't blame the students: they reflect the broader culture, and especially the culture and structures of our own school, and our own complicity in those. If you want to reduce grade grubbing, figure out ways to de-emphasize grades.

I hope to combine the original series of posts into an article for my Web site. When I do, I'll insert the above bits in the appropriate places.


Links to the original series:
     Legitimate Uses of Assessment
     Problematic Uses of Assessment
     The Meaning of Grades
     De-emphasizing Grades
     Grades: the Research
     The Perils of Backward Design
     Assessment Tools and Strategies
     Forward Design

Friday, September 2, 2016

Big Dodecagon

A classic activity is to cover a 1-inch-side dodecagon with pattern blocks. This provides a great context to discuss symmetry (see Geometry Labs 5.6.)  Here is one way to do it:

See many others, found by Simon Gregg's students.

In the past few days, I've had fun making a double-size, quadruple-area dodecagon:


You too, and your students, can play this game! Download the big dodecagon, and get started!


PS: Visit my Pattern Blocks home page!
...and my new Dodecagons page! (More images, and some math questions for your students to ponder.)

Wednesday, August 31, 2016

Fads and Memes

My defense of eclecticism in teaching generated a strong positive response from teachers, perhaps because I articulated a widely held resentment about the fads that blow through the educational landscape. But interesting questions were raised about what I wrote. In my last post, I tried to clarify my views on math education research. Today, I continue thinking about other points that were raised in relation to my original post.

Patrick Honner wrote: "Oddly, this seems a bit like a defense of edu-faddism. We can always learn something from fads, so keep them coming!" I certainly didn't intend to say that, but it is absolutely true that I have learned something from almost every one of the fads I listed in the post, and from some I did not list. So yes, I have learned to coexist with the fads. None of them have all the truth, but most have some piece of it, and I'm open to that. Like I said, I'm eclectic, and I reject the defensive and cynical stance that rejects any and all new ideas. Keep'em coming! I trust myself and my colleagues to separate the useful from the ridiculous.

The reality is that fads will always be with us. Consider their life cycle: researchers uncover an important idea about how children learn. They care about this, and find ways to spread the word about their discovery. Sometimes their discoveries are the result of questionable studies, but even if the results are valid, they can be misinterpreted or overgeneralized. In any case, they gain further traction as administrators want to spread them into their schools and districts, usually with the best intentions. Some consultants tap into this phenomenon, and become the carriers of that gospel, until the next fad comes along. I see no way of stopping this. Many people want to help teachers, without being teachers themselves, so they contribute to one or another link in that chain. I appreciate their efforts, even if I don't buy their often naive claims.

In addition to these grand fads, we also have teacher-initiated memes, which I unfortunately lumped together with the fads in my post. Those are often more useful and less pretentious, as long as we avoid the temptation to see them as more than what they are. You can get acquainted with some of those on the excellent #MTBoS homepage, though it does not include some of the most successful memes, such as Dan Meyer's "three act" lesson format.

As it turns out, Dan Meyer was the attendee who had criticized my workshop for lacking an overall pedagogical framework. After my post, he elaborated on Twitter: "Loved the workshop. I wasn't asking for more fads, though. The opposite. The Internet is awash is interesting math lessons. I'm interested in the bigger ideas about learning that undergird them, that survive fads, that generate more lessons." I too am interested in such ideas, but I stand by my call for eclecticism. Teachers have little time for theory and are happy to collect interesting lessons from the Internet, from conferences, or from colleagues. I am happy to contribute such lessons on the Web, in my books, and in my workshops. I have been prolific, and my curriculum creations over the decades have varied widely. I am certain they cannot possibly all fit within a single framework.  Nor should they: trying to stay within such a framework would have paralyzed me.

Still, I have managed to write about pedagogy. My most coherent contribution at that more general level may be my idea of a tool-rich pedagogy, which incorporates manipulative, technological, and conceptual tools under one theoretical umbrella. (See: For a Tool-Rich Pedagogy, Math: Visual and Interactive!, A New Algebra, What Are Themes, Tools, and Concepts? plus who knows how many articles about specific tools, such as the Lab Gear, function diagrams, graphing technology, the interactive whiteboard, etc...) 

The idea of a tool-rich pedagogy is a fertile foundation for curriculum creation, more so than the more specialized memes and fads mentioned above, because it covers so much ground. It does frame much of my output as a curriculum developer, but it does not and cannot encompass all of it. By staying in the classroom for 42 years, and by teaching just about everything from counting to calculus, I have come across widely different teaching challenges, which have led to widely different pedagogical responses. This is true of all teachers: we don't have the luxury of constraining ourselves to a single theory, because our work does not allow it. We get ideas from many sources, and evaluate them by using them. The best of those ideas end up in our repertoire. We have no choice but to be eclectic.

I'll let others theorize, and I'm sure I will learn something from what they come up with!


Tuesday, August 23, 2016

Math Education Research

In my last post, I argued in favor of eclecticism in teaching. The response I got was unusually enthusiastic in terms of numbers of visitors, retweets, and comments on Twitter. Unusually enthusiastic compared to what I'm accustomed to: I'm far from being an Internet celebrity. Still, it felt like I said something that resonated.

However there were three substantial disagreements / questions aired in the Twitterverse.
1. Not all ed research is flawed
2. Asking for the big ideas underlying good lessons is not asking for fads
3. One should oppose faddism, not coexist with it

I'll respond to #1 today, and to the other two in a future post.

Aran Glancy says: "characterizing all educational research as flawed is unfair." Of course, he is right. The point I was trying to make was that even valid research that is used to support the unrealistic claims at the core of various fads should not be generalized beyond reason. As Aran puts it: "the issue is much, much less about the quality of the research and much more about how it is misused to create the fad". Fair enough. We agree on that.

As evidence that I don't think all research is flawed, here are links to past blog posts in which I praise specific bits of ed research:
and a guest post:
So far, so good. But Aran took exception to my statement "I like research that confirms my beliefs, and appreciate the work that goes into it." I am not retreating from that. Much research claims to prove things I know are false, or at any rate not useful to me. I have neither the time nor inclination to analyze those papers.

For example, I was told that short periods are better than long periods for teaching math, because someone found a correlation between higher test scores and shorter periods. (If I remember right, this argument was used by administrators at a school that had a very strong math program, which was catastrophically undermined when moving to short periods.) While I am sure some good teaching can happen in shorter periods, I believe it's absurd to make a blanket claim that those are superior to longer periods. (See Math in the Long Period and Teaching in the Long Period for an explanation of my views on this.) If there is such a correlation, it could be due to the fact that it is schools that were less successful on tests that felt a need to try block schedules. Or it could be that the tests measured something other than depth of understanding. Or it could be something else. I don't have time to look into it. I hope and assume that someone else's research sooner or later will establish the opposite result.

Here's another example. I'm a big fan of the "growth mindset" fad, and in fact have been making some of those points for years. Having some research to back this up is excellent. But frankly, some of the research is hard for me to take seriously. The claim that being exposed to this concept in a slide show is sufficient to change a student's mindset is hard to believe, even if some short term effects can be observed. What may have a lasting effect on student mindset is restructuring one's teaching to make clear in practice, not just in words, that students can get better at math. This would include such policies as extending exposure, lagging homework, de-emphasizing grades, valuing test corrections, making explicit every day that getting it wrong is often a necessary stage on the way to getting it right, and so on. In other words, a classroom culture that challenges the dominant culture. I would love to be able to point to research that supports those things, so if someone reading this post can do a study comparing such practices with the single slide show approach, I'd love to know how it turns out.

The fact is that research is influenced and framed by the researcher's values. As teachers, we don't have time, and most of us don't have the interest, to survey the literature, evaluate papers, and impartially reach conclusions. We need to find math ed researchers who share our values, and use their results to refine our practice, and to dialogue with administrators. For example, if equity is an important goal for you, find the researchers who share that concern. Their work is likely to be useful. If you prioritize understanding over memorization, or collaboration over competition, find the research that is about that. And so on. But to be honest, it will often be more fruitful and practical to get your ideas from fellow teachers.

Some years ago, at a conference, I saw a math ed prof who was an expert on the learning of geometry. I had recently read one of his papers, and I said to him: "Your research confirms my beliefs!" Without a moment's hesitation, he replied, with a big smile: "That's what it's for!" Was he joking? I don't think so. That is indeed what it is for.


(To be continued!)

Wednesday, August 17, 2016


In between June 27 and August 4, 2016, I presented seven to ten workshops (depending on how you count) ranging from a couple of hours to four days. I share most of the handouts, resources, and slides on my Summer Workshops site. (See below my signature for more details on what's there.)

The site will remain available until some time in December or January. At that point, if I am to offer workshops in Summer 2017, I'll probably shut it down, and start setting up next summer's site. (Whether I offer workshops next year depends on whether someone is willing to host them!)

These workshops are exhausting, as I pack a lot into each one, and go all day with no prep periods. One of the reasons I keep doing them is that participants seem to find them useful. I get nearly unanimously rave reviews in the workshop evaluations. However, one of the participants in my Making Sense in Algebra 2 workshop had an interesting criticism. That anonymous participant pointed out that I presented no coherent pedagogical framework for the activities I shared. Good point! I did not present a coherent framework because, well, I do not have one to present.

Certainly, the title of the workshop captured my goal: the activities I shared were intended to help students make sense of Algebra 2 content. But that is not a pedagogical framework. In fact, I think every activity was different: I used labs, spreadsheets, kinesthetic strategies, measurement, graphing technology, games, slides/lecture (yes!), big anchor problems, and perhaps other formats.

According to Merriam-Webster:

    Eclectic: selecting what appears to be best in various doctrines, methods, or styles

That pretty much describes my stance as an educator.

During my four-plus decades in the classroom, I've seen many math edu-fads come and go: new math, individualization, manipulatives, problem-solving, group work, constructivism, constructionism (yes, that's a thing), portfolios, complex instruction, differentiation, interdisciplinary-ism, backward design, coding, rubrics, problem-based instruction, technology, Khan Academy, standards-based grading, making, three acts, flipping, inquiry learning, notice-wonder, growth mindset... not to mention various generations of standards.

It doesn't take long for a conversation between teachers to include something sarcastic about the fad du jour. By being sarcastic, we put up an umbrella to try protect our sanity from the ideas raining on us from administrators, academics, and yes, even colleagues. I will go further, and boldly say to the proponents of the current pedagogical panacea: I'm sorry, but whatever "evidence-based" product you're selling today, I'm not buying. The research it is based on is flawed. The anecdotes that support it only apply to specific circumstances which are not easy to replicate. In short, as I have written before: nothing works.

I guess that sounds cynical.

But I am the opposite of cynical! Nothing works for every student, every class, every period, every day, every teacher, every department, every school, every district... That is just a fact. There is no one way. But this is what makes our job interesting! We need to be eclectic, and select "what appears to be best in various doctrines, methods, or styles." Instead of rejecting the fads wholesale, we need to consider each one as it comes along, as all (or almost all) have some validity. Instead of shutting our classroom door and continuing business as usual, we should keep it wide open. Without becoming a dogmatic across-the-board adopter of each pedagogical scheme, we need to learn what we can from it, and incorporate that bit into our repertoire. This is how we get the sort of flexibility that makes for good teaching. If we do that, our lessons will not fit a standard mold. Quite the opposite: they will depend on the myriad variables that make teaching such a complex endeavor.

Thus, when I share the materials from my workshops, I am not saying they are sure to work for you just as they are. Au contraire! Take from it what you want, and adapt it to your own classes, your own personality, your own math background, your own school schedule, you own beliefs. And let me know how it goes!


PS: The workshops whose materials I share on my Summer Workshops site are: Visual Algebra, Making Sense in Algebra 2, Transformational Geometry, No Limits, and Hands-On Geometry. Not there, but relevant to other summer workshops I offered: Abstract Algebra and Common Core: A Closer Look.

Wednesday, August 10, 2016

Math in the Long Period

In this post, I will expand on some of the ideas from a previous article about Teaching in the Long Period. (Read that article first!) I will also add some thoughts that were missing in the original, and try to answer some specific concerns often expressed by math teachers whose schools are adopting block schedules.

Do students have the necessary attention span to remain engaged during a long period?

Yes, they do, if you break the period into two or more chunks. Few activities  will work well for the duration of a long period. In your planning, get used to breaking it up. Here are two possible templates, and a blank one if you don’t like either of them:

(These templates are intended for a 70-minute period. Adjust as necessary.)

The opener can be a transitional piece into the day’s work. What I did was have students go over the homework in their groups. Another possibility would be a well-chosen warm-up or "do now" which will help set up the main topic of the next segment.

The One Main Activity template is good for a lab or collaborative project, the sort of thing one doesn't have time for in a shorter period.

The Basic Routine template is one way to set up a daily rhythm. The central segment is where you’re forging ahead with new material. The last segment is where you might review some past work, apply new ideas, or preview some future topics. It is not to do homework or have a study hall, and should be planned just as seriously as anything else.

Breaking up the period is only partly about content. The main point is to use different formats:
◊ whole class / groups / pairs / individual
◊ paper-pencil / verbal / hands-on / technology
◊ formal / informal
◊ watching / reading / writing / making
Format-switching helps students stay focused, and in any case is better teaching, as it increases engagement, motivation, and understanding. (See my Art of Teaching worksheet to help you think about broadening your pedagogical horizons.)

Use this worksheet to draft possible long periods:
◊ generic for your department or all your classes
◊ special versions for different grade levels and courses
◊ actual implementation for a particular day
...or all three. Whatever would be most useful.

Can students remember what they learn if they don't have math every day?

They will if you rely on understanding rather than memorization. One way to do that is to teach the most important topics in multiple representations, and/or with the help of technological or manipulative tools. Take advantage of the long period to diversify your toolbox, and use your new tools strategically on the key topics. Longer periods make it possible to build interesting review into your course. (Teaching the same thing the same way is the worst possible kind of review, as it is boring for students who got it the first time, and usually unhelpful to the ones who didn't.)

I discussed other strategies to strengthen retention in these posts:
- lagging homework
- separating related topics
- and generally extending exposure
(Practical advice on implementing these ideas.)

Will I be able to cover as much material in a block schedule?
Probably not, because even if you see students for the same number of minutes, there's only so much they can learn in each school day. If your goal is to teach for understanding, you will need to prioritize the most important topics, as suggested above. To do that will require eliminating or giving less time to other, less crucial topics. See my guidelines for pruning the curriculum.

But even then, you still run a risk of not covering even a pruned curriculum if you allow your classes to be too leisurely. Do not let long periods lull you into a false sense that there is plenty of time. There isn't. As suggested above, I do believe in eternal review, if it is done well, but the other side of that coin is constant forward motion.

Constant forward motion is helped by lagging homework and the other strategies listed above. Another way to keep moving forward is to pursue two units at a time. I realize that is truly countercultural, but my department has done this for years, and it's worked very well for us. For one thing, if you're pursuing two units, you can use that to break up your period into chunks as suggested above. Also, if things bog down in one of them you can switch the emphasis to the other one while you figure out what to do. Finally, pursuing two units forces students to be alert and not turn into automatons.

Ideally, the units would be unrelated, and as different from each other as possible as to their "feel". For example, a unit on the properties of special quadrilaterals can be run concurrently with one on right triangle trigonometry. Or, a unit on systems of equations along with one on the Pythagorean theorem. And so on.

Will I be able to change my habits?

I cannot answer that for you. I'll just say that the long period is unforgiving: if you don't heed at least some of the above suggestions, it will feel endless, and you'll conclude that "it doesn't work". On the other hand, the long period can be a motivation and an opportunity for ongoing professional growth. In my own career, it has both required, and facilitated my becoming a better teacher. If you're up for the challenge, the long period is a wonderful gift to you and your students.