My Math Education Blog

"There is no one way"

Monday, October 3, 2016

Reading Algebra

Symbol sense is an essential part of mathematical literacy. It is the understanding that undergirds effective symbol manipulation, and perhaps more basically the ability to interpret and create algebraic expressions. Symbol sense, like number sense and operation sense, is not learned so much through listening to a teacher. Rather, it grows as one gets practice generalizing numerical relationships, translating functional relationships into formulas, and reading algebra. The latter is the topic of this post.

Successfully reading algebra does require some clarity on the conventions that are accepted in the international institution of mathematics. Because these are conventions, they can only be transmitted by explicit instruction. (This is unlike, for example, the properties of proportional relationships, which can be explored and discussed in different contexts. Students can "discover" some of the properties, as well as be introduced to them by the teacher.)

Here are some examples of conventions in the writing of algebraic expressions, and therefore in reading them.

1. The Three Meanings of Minus

In front of a number, minus means negative. For example, -2.

In front of any expression, minus means the opposite of. For example:
-(y + 1)
In between two expressions, minus means subtract. For example:
2 - 3
2 - (y + 1)
Of course, for this to be meaningful, students must understand the concepts expressed by the symbol. (For example, that the opposite of a quantity is that which you add to it to get zero.) In other words, it is pointless to try to have students memorize these ideas if they don't have a need for them.

A good question to discuss is: "Is -x negative?"

 On some calculators, there are different keys for minus as "opposite" and minus as "subtract". That is inconvenient, perhaps, but it does provide an opportunity to have this discussion.

On the meanings of minus, see my Algebra Lab Gear: Basic Algebra, Lesson 3.

2. Sum or Product?

What is the result of this calculation: 2 + 3 · 5? If you just read from left to right, you get 25. If you start with the multiplication, you get 17. Discussing examples of this type shows that a convention ius needed. And, of course, the convention is to multiply before you add, so the "correct" answer is 17. Nix the Tricks, an excellent booklet by Tina Cardone and others, has a good discussion of this and recommends GEMA (grouping, exponentiation, multiplication, addition) instead of PEMDAS. Including division as part of multiplication, and subtraction as part of addition makes a lot of sense, and deserves a conversation with your students.

Once this has been established, a good exercise is to show expressions to your students, and ask: is this a sum or a product? For example:
x + 5 → sum
5x → product
4(x + 8) → product
4x + 8 → sum
... and so on, with increasingly complicated expressions. This is an essential prerequisite to talking about distributing and factoring, which really have no meaning without this foundation.

On "sum or product?", distributing, and factoring, see my Algebra Lab Gear: Algebra 1, Lesson 3.


Reading algebra is not just an issue for beginners. For example, older students sometimes get confused when reading algebraic expressions that include radicals or absolute values. Again, it is essential to discuss the meaning along with the "grammar". (See for example this activity: GeoGebra | TI-eighty-something.)

Part of what I'm saying in this post is that it's a mistake to put all your eggs in the "discovery" basket. Discovery, more or less guided, is excellent in many cases, but when it comes to arbitrary conventions, you will need to just tell students what they are. The pedagogical questions are: when is this telling appropriate? what sort of practice is interesting to help learn this particular convention? I have often found that it is helpful to make connections with manipulative or electronic environments (as you see in some of the above links), but you will have to figure out what will work in your classroom and curriculum.

One last point: do not overgeneralize! For example, a mathematician who is clueless about math education once wrote about my Algebra: Themes, Tools, Concepts: "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." That is ridiculous. There is much interesting work to be done to help students understand these laws, and treating them as arbitrary conventions would be doing a huge disservice to them. (Read my reply to this man here.)


Wednesday, September 28, 2016

Technical Writing

It is difficult to learn something new and challenging without ever putting it into words. This is just as true when learning math as it is when learning anything else. Thus, it is a good idea to make time for students to discuss their ideas with their classmates in pairs, groups, and as a whole class. It is also helpful to give students many opportunities to write about math.

In this post, I'll focus on expository writing, where students summarize and present key ideas. Typically, this will happen at the end of a unit, and will take the form of a report or poster. In this sort of assignment, the personal style and the self-centered focus of a personal learning journal is usually inappropriate. Moreover, while spelling and well-crafted sentences do matter, not everything the students learn from their English teachers applies to the writing of a math report. And the formal, ultra-concise style of mathematical proofs is also inappropriate. Some other guidelines are needed.

I first came across this idea many years ago when Carlos Cabana sent me the result of a discussion he conducted with his students and colleagues at "Railside High". They were shown examples of technical writing (such as users’ manuals), and asked to identify some of the key features of those documents. Here is the list they came up with:
*  Diagrams with labels
*  Numbers to separate text and show order of steps
*  Text with mathematical vocabulary
*  Telling and showing what you mean
*  Titles and headings
*  Arrows
*  Showing common errors
*  Use of boldface, capitals, italics, etc.
*  Color-coding
*  Bullets
*  Tables and charts
*  No naked numbers (i.e., all numbers include units or labels - e.g., 7 squares in each line x 3 lines = 21 square units)

In preparation for writing this post, I asked professional technical writers for any ideas they would add to the list. Here are some of their suggestions:
* Start by deciding what you're going to say
* Open with a summary, which may include ideas you are tempted to use as the conclusion.
* Organize the rest hierarchically, with parallel subtitles
* If appropriate, link to basic references such as a glossary
* Remember that the piece has a topic. It is not about its author.
* Be concise: avoid repetition, superfluous details, off-topic digressions, and self-evident information
* Keep sentences short
* Use plain language: short, common words
* Use terms consistently
* Use examples

I also asked Carlos to comment on that original list. He explained that in addition to the importance of clarity in communication, the purpose of these guidelines is to strengthen the writer's understanding, particularly in the case of multiple representations, a key ingredient in his (and my) pedagogy. Color-coding, arrows, "thought balloon" annotations, diagrams, etc. are largely about highlighting the connections between the representations. See for example this assignment (based on lessons in Algebra: Themes, Tools, Concepts), and check out the student work Carlos sent me (Glenda | Mariela.) These are examples from middle school students and English language learners, so you should adjust your expectations if you're thinking of high school kids. Younger students will typically need more help in understanding how to use the tools of technical writing.

Carlos adds that learning about technical writing in math class pays off in the humanities and science. Some of those skills (use of colors, annotations, etc.) are useful in helping students read, make sense of what they are reading, and mine a text for evidence. All the skills are also helpful in writing detailed lab reports in science.


Thanks to Carlos Cabana, the Railside teachers, Louise McFarlane, and Cate deHeer.

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!)