My Math Education Blog

"There is no one way"

Friday, January 5, 2018

Handling Wrong Answers

In my previous post, I listed questions to use in class discussions, or in conversation with a student or a group of students. Today, I'll discuss how to handle wrong answers. This is complicated and there is no single correct answer for all situations. I'll start by clarifying my goals:
  • broad participation by students in the conversation
  • progress towards better understanding for most
  • correctness determined by discourse, not by authority
Teachers often complain that it is always the same few students who raise their hands in classroom discussion. There are many possible reasons for this, such as not giving students enough time to think, not letting them practice their answer by talking with their neighbors, not asking the right questions, and so on. But one huge reason is students' fear of being wrong. If a wrong answer is met by ridicule from classmates or teacher, that is sure to cut down on participation. But this sort of intimidation can happen in more subtle ways: if we think of classroom discussion as a way to quickly get to the right answer, heap praise on students who supply correct answers, and move on, the message to students is that they'd better not speak up if they are not totally confident about their answer.

So it is important to not rush to the punch line. If something is important and difficult, there are probably misconceptions in the class, and rushing to the right answer will allow those to remain unchallenged. We should strive for an atmosphere where wrong answers are expected, and in fact appreciated. The classroom culture has to make it comfortable to be wrong, as it is really the only way to learn to be right. One way to bring this into being is to not praise correct answers, which are their own reward. Instead praise and encourage participation and risk-taking. "Lucy was brave, and raised her hand even though she wasn't sure her answer was right. Thank you Lucy!"

Still, there is a sting when one gets it wrong. One way to diffuse this is to ask the student who made a mistake to choose a classmate to help sort things out. This shifts their focus to this newly acquired power. Another approach is to routinely ask for many answers, whether the first answer given is right or wrong, write them all down, and discuss how one would sort out which one is right, perhaps after voting on them. Teacher mistakes (made on purpose, or not!) should be a frequent feature of class discussion, and being relaxed about them helps create the right atmosphere.

Many of the strategies I've mentioned rely on keeping a poker face, and using classroom discourse to address the errors. These strategies do support the three goals I mentioned at the outset. However, it is possible to overdo this, and to never ever make clear that an answer is wrong, instead falling into an awkward silence for fear of hurting student feelings. This doesn't work: students can read our body language in those situations, and in any case will sooner or later realize they were wrong. If you don't have a strategy to handle the situation, frankly, it's better to out-and-out acknowledge the answer was wrong, and thank the student for offering it. "Thank you Charlie! Other students almost certainly thought that was the answer, and not discussing it would not help, would it!"


More about teaching, on my Web site.

Saturday, December 16, 2017

Any Questions?

On the first weekend of December, the California Math Council held its annual meeting in Asilomar for the 60th time. (I attended for the 33rd time, and presented roughly the same talk I had presented in 1984.) Over the decades, the "must-attend" presenters have changed. Two of my favorites back in the day were Harold Jacobs and G.D. Chakerian. Both of them had written books* that influenced me mightily as a young teacher, and their talks were always interesting, albeit in different ways. Jacobs always did an amazing slide show of mathematics in everyday life, sharing photos and clippings he had gathered during that year. Chakerian gave lectures on specific mathematical topics I usually knew nothing about, often in geometry. I always walked out of their sessions inspired. Both of them are acknowledged in the front matter, and again obliquely in the text of my Algebra: Themes, Tools, Concepts. (Jacobs in Lesson 5.1, Constant Sums, and Chakerian in Lesson 8.8, Percent Decrease.)

Nowadays, at Asilomar, I usually try to catch Scott Farrand's presentations. He is a math prof at Sacramento State, and his talks are almost always interesting either for their math content, or for his ideas about pedagogy, or both. This time, he gave a talk on using the question "What solution to this problem would be the coolest?" or put differently, "If you were God, designing the universe, what would you want the solution to be?", or "What do you hope is true?" He gave the example of the volume of a right prism with a square base, with the top sliced off by a random plane. People pretty much agreed that it would be cool if the volume was the area of the base, times the average of the distance to the base from each vertex of the slice quadrilateral. (A 3-D version of what happens when you cut off part of a rectangle with a random line.)

Of course, that sort of question is not always appropriate. Still, it can be added to one's repertoire, to be deployed in the right situation, whether working with an individual student, or in class discussion.

(Check out my previous blog posts on Scott Farrand's Asilomar talks.)

The question of what questions to ask must be in the air. In an online forum, mathematician James Propp recently offered these questions for everyday use in the classroom:
- What is the answer?
- Does this answer make sense?
- Is there another way we could arrive at this answer?
- Does this remind you of something else we've done?
- What do these things have in common?
- What question might this lead us to ask?
- Is there a pattern here?
- What mistake did I just make?
- How am I fooling you?
- Is this wrong answer the right answer to a different question?
- Are we using the right definition?
- Have I given you enough information to answer the question? What other information might you need?
- Can we think about this a different way?
- Is that a rigorous argument, or is there a subtle point that we're glossing over?
- How convinced are you?
- Can someone give a concrete example?
- Can we generalize?
- In plain English, what is this equation telling us?
- What kinds of mistakes do you think people are most prone to make when using this procedure?
 (Check out James Propp's monthly Mathematical Enchantments blog post.)

Jim Tanton adds these:
- Now that we see what the answer is, could we have seen that more swiftly?
- Was that approach to answering the question enjoyable? Could we have avoided doing XX?
- Do you think this solution would be a pleasure for someone else to read? Have we made it easy to read and understand? Is what we have presented inviting to read?
- What don't you like about the question? Shall we just change that and answer an easier question first?
- What do you wish was also in the picture/equation/...? Can we just make it appear?
- Oh bother. I don't see what we need in order to proceed. I think we should weep. (To spur the class on to do something with the issue/question at hand)
- Who says you have to do the questions in the order presented to you? (If part e is easier for you, do that part first!)
- Why would anyone want to answer this question?
- What would make this question interesting?
- Is there a three-dimensional version of this?
- What is it exactly that made this question hard?
- What did we actually do here? What is the general thing we've established?
(Check out Jim Tanton's G'Day Math Web site.)

Great lists. I'll just add a few more:
- We have several answers to the question. They can't all be right. What should we do?
- Would you like to choose a student to help you answer? (if someone is stuck, or blatantly wrong)
- Can you restate what X said, in your own words?
- (to the whole class) Can you give me the answer on your fingers? by "air graphing"? by pointing? (up / down or left / right, depending on the question) agree? disagree? don't know? (thumbs up, down, horizontal)

(Check out my Nothing Works article, which has a section on classroom discussion.)

Do you have any questions? Please share in the comments!

Sequel to this post: Handling Wrong Answers


* Harold Jacobs' Mathematics, A Human Endeavor has many wonderful ideas. His Elementary Algebra and Geometry: Seeing, Doing, Understanding are pedagogically not what I look for in a textbook, but they are filled with great cartoons, photos, and other connections with everyday life.

Chakerian, Stein, and Crabill's Geometry: A Guided Inquiry had an enormous impact on my teaching, as it pioneered the idea of group work, did not rank problems in order of difficulty, gave many answers right there in the margin, etc. It's probably not possible to use it as is any more, but it is a fantastic source of great problems. Their Algebra books though not as successful, had many of the same powerful pedagogical features. I have used their approach to trigonometry and to complex numbers ever since I came across it.

Thursday, November 30, 2017

Department as PLC

A PLC is a Professional Learning Community. In an ideal world, every math department is a PLC,but in reality there are some obstacles to that idea:
  • not all schools give teachers time to dedicate to professional learning
  • not all teachers are interested in professional growth
  • it is not clear what to do in a PLC, even if the first two obstacles are not a concern
In this post, I'll try to address that third obstacle.

In-house professional learning is not in contradiction to learning one can do online or by going to conferences. Learning from experts outside the school is a useful complement to what can be done onsite. However an in-school PLC has enormous advantages:
  • It is geared to a specific teachers, students, and school community. Outside Professional Development resources may or may not be a good match. 
  • It can be coherent, while the catch-as-catch-can off-site PD opportunities are usually disparate and unrelated.
  • It is a year-round opportunity for growth.
Collaboration in lesson planning is a very powerful approach to professional growth. As department chair, I tried to set things up so that different sections of a given course would be taught by more than one person. No one got to just teach one thing all day, so a bit more prep, but on the other hand, the preparation can be shared, everyone gradually gets to know and own the whole program, and teachers learn a lot from each other. Among other things, it allows the pairing of seasoned teachers with beginners, a much more effective approach to mentoring than occasional meetings about nothing in particular.

Over a few years, as teachers gradually cycle through the department offerings in different collaborative teams, ideas are shared, and everyone grows, especially if teachers take notes on what worked and didn't work. Some summer collaboration on the following year's program can help put that information into shared documents, thereby institutionalizing what was learned.

Collaboration meetings are not resented, quite the opposite. They feel useful and important. For more on this, see these slides, and this article.

But what about turning the whole department into a PLC, in a way that does not require years to percolate? Here are some ideas.  
Do math together
For example, if one of you attends a Math Teachers Circle, bring the problems back to the department. Or find teacher-suitable problems online. On my site, visit Teachers' Mathematics for some ideas. Here are some problems I shared on my blog: K-12 Unsolved, Taxicab Geometry, Scissors Congruence, Geobard Problems for Teachers. Also check out the problems page at the Julia Robinson Math Festival. Of course, there must be many other places to find good problems, for example math competition problems.

Explore learning tools
There are many learning tools that can enhance your department's program. Take turns learning and teaching each other. Some tools are electronic (e.g. Desmos, GeoGebra, online applets). Some are manipulative (e.g. Lab Gear, Pattern Blocks). Some are neither (e.g. function diagrams, the ten-centimeter circle.) See my article on a tool-rich pedagogy for some philosophizing on this topic, plus lots of links. Note that some tools will require multiple sessions of your PLC.

Read and discuss articles and blog posts
On my site, go to the page about Teaching. I especially recommend the articles on acceleration and assessment, either of which should trigger intense conversations. Or my very practical blog post on lagging homework, and the posts it links to. There are actually hundreds of math teacher blogs, some of which are totally worth talking about. I've occasionally read Michael Pershan's, and Dylan Kane's blogs, and enjoyed them. Find many others on the site. Or, if one of you is an NCTM member, you should be able to find articles worth discussing in the NCTM journals.

And more...
There are obviously other possibilities, such as Lesson Study, or rehearsing Instructional Routines. However I don't know enough about those things to say much about them. If you have links to information about those PLC practices, or other PLC suggestions, please share them in the comments!


Sunday, November 26, 2017

Geometric Puzzles at Asilomar

I'll be presenting a session on Geometric Puzzles at the Asilomar meeting of the California Math Council. (Saturday, Dec 2, Sanderling, 1:30pm The printed program says I'm in the middle school, but that is not correct. The app has the location right.). I will include material that I believe is relevant to teachers from kindergarten to tenth grade. Hoping to see some of you there!

Given that this is CMC-Asilomar's 60th anniversary, I thought it would be a good time to reminisce: this is nearly the same topic I presented in 1984. After rejections by several publishers, my first book (Pentomino Activities) had recently come out. It later got combined with two other pentomino books, and that combination in one big binder remained in print for about 30 years. It may still be available from, item # 0884883744. I highly recommend it. A new version, with fewer puzzles, can be purchased from Didax. It comes with an e-book version, so you can project any page from the book, and you can manipulate virtual pentominoes on the screen.

My pentomino obsession was followed by a series of puzzle books on polyominoes and supertangrams, all of which are now free on my Web site. After that, I moved on to other curriculum development projects, but I maintained an interest in a tool-rich pedagogy and a puzzler's ethic, both of which originated in this early involvement with pentominoes. The more observant among you may have noticed that my Web site logo is based on a pentomino P:

--Henri P

PS: I link to many of my geometric puzzle creations here

Thursday, November 9, 2017

Puzzles for the Classroom

In my last post, I shared some generalities about puzzle creation. Today, I will zero in on the specifics of creating puzzles for the mathematics classroom. I will do this by way of analyzing some examples.

Multiple Paths

A characteristic of all classrooms is that they are constituted of students whose backgrounds and talents vary widely.  Offering multiple puzzles simultaneously can help, as it allows students to find their own way through the set, by selecting puzzles at the appropriate level of difficulty, and/or by pursuing partial discoveries. This addresses classroom heterogeneity, while having all students work on closely related problems. Here are some examples along these lines:
  • Staircases: find sets of consecutive whole numbers whose sum is 3, 4, 5, etc.
  • Egyptian Fractions: find three fractions with numerator 1, whose sum is 4/3, 4/4, 4/5, etc. For example, 4/5 = 1/2 + 1/5 + 1/10
  • Make These Designs: find linear functions whose graphs create these designs.
All three activities allow the students to find their own path through them. They avoid a common pitfall of curriculum development, which is the hubristic belief that one is capable of writing a single sequence of puzzles that will work just as well for all students. This is a common failing of both traditional and contemporary curricula. For example, the consistently brilliant Desmos environment offers teachers and curriculum developers the ability to craft one-path-fits-all sequential lessons in the Activity Builder. The best Activity Builder lessons, such as Marble Slides, incorporate many excellent puzzles. This is vastly better than most supposedly "intelligent" educational software, which tries to eliminate the need for teachers and is based on reductionist and insulting memorize-the-algorithm-and-practice sequences. Still, one can hope that a future version of the Activity Builder will allow the creation of choose-your-own-path activities.

Features of Effective Classroom Puzzles

In addition to the availability of multiple paths, the above three examples also share other properties.
  1. They are reversals of standard classroom activities. Instead of the mind-numbing request to "add these numbers", "add these fractions", "graph these equations", the questions are reversed: "find numbers whose sum...", "find fractions whose sums", "find equations whose graphs...". Reversal, in fact, provides a powerful mechanism for the construction of classroom puzzles: start with what you're trying to teach or apply, and reverse the question. Voilà! You've created a puzzle.
  2. They are non-random practice of important skills. Drill is not necessarily a bad thing, but random drill is boring and thus can be counter-productive. In these examples, drill is in the context of an interesting overall quest, and thus much more motivating. Also, unlike random drills, it lends itself to reflection, discussion, and generalizing.
  3. They are each a set of related puzzles, rather than one-of-a-kind puzzles that rely exclusively on "aha" insights. Therefore, solving some of the puzzles helps the student develop skills and intuitions that can then be applied to other puzzles in the set, and more importantly, contributes to their mathematical maturity. This also means that they provide an excellent environment for teachers to provide hints, and scaffold student learning. For example: "solving this easier puzzle will help you make progress on the one you that is currently frustrating you."
  4. They are interesting to both kids and adults. I have used these in the classroom with students at various levels, and in professional development sessions for teachers, and found that they are just as engaging for all. This is in part due to their "low threshold, high ceiling" quality: all include simpler and more difficult puzzles. Moreover, they suggest additional questions, such as the creation of similar puzzles, or the generalization of results, or the need for a proof.
  5. They involve significant mathematics and carry a substantial "curricular" load. They are about the math teachers and students already know they should teach and learn. Using non-math puzzles as a "change of pace" is a waste of precious class time, and gives students the wrong impression that "normal" math is no fun.
One cannot expect all these criteria to apply to every classroom-bound puzzle or puzzle set, but hopefully they are helpful guidelines for teachers and curriculum developers.

More Examples

Geometric Puzzles
As a young elementary school teacher, in the 1980's, I encountered geometric puzzles in Martin Gardner's books and columns. At the time, there were nice tangram-based materials for elementary school, such as a fantastic set of puzzles by EDC, but there was not much using pentominoes. I decided to create my own sets of pentomino puzzles, suitable for students. The key insight was that puzzles that did not require the use of the full set were much more accessible than the 12-piece puzzles discovered by Solomon Golomb and popularized by Martin Gardner. More accessible, but still interesting, and in many cases extremely curricular! I started with well-known puzzles from recreational mathematics, explored them on my own, and translated the fruits of that interest into classroom materials. This was an ongoing creative obsession over many years. You can read more about this work on my Geometric Puzzles page, though in fact this has infiltrated many other parts of my work as a curriculum developer. [Note to Northern Californians: I'll be talking about Geometric Puzzles in the Classroom at the Asilomar meeting, on Dec 2. See you there!]

Algebra Manipulatives
One of the features of the lessons I developed for algebra manipulatives in the 1990's involves a crucial re-envisioning their role in the classroom. The standard algebra tiles lesson is based on the idea that the tiles illustrate what is going on with the symbols. In my Lab Gear materials, I turn this around. Start with a geometric puzzle: arrange these blocks into a rectangle. Then interpret what you accomplished with the help of the rectangle model of area. This is more fun, more accessible, and in the end more effective. I also introduced a whole genre of perimeter puzzles (e.g. use an xy-block and a 5-block to create a figure with perimeter 2x+2y+2), and visual patterns based on these blocks (what is the 10th figure in the sequence? the nth?)

And yet more examples
I will not comb through my (freely downloadable) Geometry Labs and Algebra: Themes, Tools, Concepts to find all the puzzles they include sprinkled throughout, but I should mention my puzzle-based approach to geometric construction, which I presented in multiple blog posts and on my Web site.

Puzzles Throughout?

As you know if you've read this far, I'm a big fan of puzzles in math education. However, there is no one way: while puzzles are an essential ingredient in effective teaching, they are not everything. There are very interesting and fruitful explorations that cannot be described as puzzles. Still, even topics as dry as factoring a sum of cubes, or function behavior, or rate of change, can be turned into puzzles! Teachers, curriculum developers: stay alert to those possibilities!


Sunday, November 5, 2017

Puzzle Creation

John Golden asked whether I had written about my approach to puzzle creation. I've only written a brief post on the subject, five years ago. Yet I believe that my work as a curriculum developer is largely based on my involvement with puzzles: solving them, constructing them, editing them.

Of course, puzzling is not the only ingredient in my approach to curriculum development. As I pointed out in response to a request for a “pedagogical framework” some time ago, I believe that actual classroom teaching (and thus high-quality curriculum) cannot be imprisoned into a single framework. Teachers are eclectic, and curriculum developers need to learn that flexibility. That said, a math consultant I know, who often uses my materials when working with teachers, insists that contrary to my claims to the contrary, I do operate within a pedagogical framework. For example, I have zero interest in creating pages of random drill exercises. Fair enough, but I don’t think generalities like “guided discovery” and “student-centered” are particularly helpful.

I'll say that if I do have a pedagogical framework, there are different overlapping but distinct ingredients to it. One ingredient, for example, is what I call a tool-rich pedagogy. Another ingredient, largely in line with the emphasis NCTM has been championing for decades, is putting problem-solving at the core. In fact, problem solving is where the connection with puzzles is most obvious. As I see it, not all instruction should be problems, and not all problems are puzzles. Still, even non-puzzle activities and problems gain from being created with a puzzle constructor's approach. That is what I hope to address in this post and the next.

Puzzles as relationships

A puzzle is a relationship between the puzzle constructor and the puzzle solver. There is an unwritten contract between the two. Here are some of the contract’s clauses:
  • The puzzle must be solvable and fair.
  • The puzzle must be challenging.
  • The solution must be satisfying.
Of course, these requirements depend on context. The same puzzle may be too easy to be satisfying for one solver, while another solver might deem it unsolvable, and yet another may consider it "just right". Still, these guidelines may be helpful to puzzle constructors, as they provide some direction on how to think about this. For a puzzle to be solvable, it must be possible to imagine some path to the solution. Fairness is harder to determine, as it depends on matching the puzzle difficulty to the solver’s probable skill and experience. What complicates matters is that insufficiently challenging puzzles are not as satisfying to solve. The purpose of a puzzle is for the solver to “win”, but not to win easily.

Alas, these guidelines do not provide a blueprint. Here are some ideas that may (or may not!) help the actual process of creation.
  • The puzzle should be interesting to you, the constructor, even if you consider it easy to solve. If you’re bored, the solvers will be bored.
  • You should mentally inhabit the mind of the solver, and imagine how they might get to the solution. If there are multiple paths to the answer, all the better. If there are partial solutions along the way, those help to keep the solver engaged. (Alas, not all puzzle solvers appreciate partial solutions. Some completists would rather not ever have tried a puzzle they cannot fully solve...)
  • You should also try to imagine what a frustrated solver would feel if they break down and look up the answer. Would they think “Darn, I should’ve gotten that”, or “How did they expect me to figure this out?” (The first reaction is the one you want.)
Admittedly, those ideas are abstract and general. I will try to make them more concrete with an example from cryptic crossword construction, which is one of the things I do when I'm not doing math education. (I co-construct the puzzle in the back of The Nation magazine.)

An example

A cryptic crossword, of course, is a puzzle, but each clue therein is its own mini-puzzle. That structure already allows for multiple paths, as solvers can decide the order in which they solve the clues. This means that there are different entry points for solvers with different skills and backgrounds. Moreover, each individual clue contains three paths to its solution. For example, consider this clue:
Tech pioneer: "I know A-Z, but in a different order" (7)
(The 7 indicates that you're looking for a seven-letter word.) Let's say that you already have
W _  _  _  _  _  K 
in the diagram. The unusual letters at the start and end of the entry may suggest the answer. Or, you may get it from the definition of the answer ("Tech pioneer".) Or, you may get it from the wordplay part of the clue "I know A-Z, but in a different order", which to a solver of cryptic crosswords suggests anagramming (rearranging) the letters IKNOWAZ. One of those three paths to the answer, or more likely a combination of two of them, or all three will lead you to the solution: WOZNIAK.

As a constructor of cryptic crosswords, I have some choices. For example, I could make the solution easily researchable: replace "tech pioneer" with "Apple founder". But that would not be satisfying to the solver: whether they already know it or look it up, the answer is obvious, and they would not need either of the other two paths to the answer. Or, instead of "but in a different order", I could write "anagrammed", but that too is just too blatant, and moreover it would take away from the humor of the clue, which is part of what makes the solution satisfying. So I would say that this clue hits the sweet spot, and satisfies all the guidelines I suggested above.

But we should get back to math education. Constructing puzzles for the classroom brings with it additional complications and challenges. In my next post, I will discuss specific examples of classroom math puzzles, and explore those to help flesh all this out for the readers of this blog, who are probably not particularly interested in cryptic crosswords.(If you want to find out more about cryptic crosswords, go to my Puzzle Page, and scroll down to Cryptics: How to.)

To be continued...


Tuesday, October 24, 2017

Scissors Congruence!

I attended the San Francisco Math Teachers' Circle last weekend. It was facilitated by Paul Zeitz. The topic: showing two polygons have equal area by cutting one into pieces, and rearranging them to cover the other one with no gaps or overlaps.

The assumption was that the area of a rectangle is length times width. The warmup was to use that assumption and scissors to derive formulas for the area of the triangle and the parallelogram. This reminded me of the activity Rachel Chou and I prepared for our Hands On Geometry workshop last summer. We asked participants to find a formula for the area of a trapezoid using a strategy based on adding and subtracting pieces. We had come up with nine different ways to do it, and if I remember correctly, one of the workshop participants found a tenth.

Anyway, last Saturday, I ended up spending most of the time trying to show that if two rectangles have equal area, one can be cut into pieces that can be rearranged to cover the other. I was certain that the solution depended on superposing the two rectangles so that they shared a vertex, like so:
As it turns out, there is a solution based on that arrangement, but I didn't find it. Still, I enjoyed my various attempts. As Paul put it, it was "funstrating"! See if you get further than I did.

Later that day, I remembered that I had solved a related problem long ago, in GeoGebra: start with a rectangle, cut it into polygons, and rearrange those to make a square. Sure enough, that solution was still in my computer. Spoiler: here it is! Two cuts, three pieces.
 Nice, no? Why does it work, you ask? Understanding that requires familiarity with the theorem of Thales, and what I call "the three triangles". (The relationships created when dropping an altitude onto the hypotenuse of a right triangle.)

See if this figure makes it clear:

Next SF Math Teachers' Circle is December 9. Katherine Sanden will lead us in an exploration of fair cake sharing.


PS: I will lead the East Bay Math Circle in Hayward on November 1, on the topic of Geometric Puzzles (tangrams, pentominoes, supertangrams, rep-tiles.) It should be fun, and it will serve as a rehearsal for my Asilomar talk on that topic (December 2.)