Friday, November 15, 2013

Using Mastermind to Model The Life Cycle of a Problem

The following is part 3 of a recap for a workshop I led at the CMC-South (California Math Council) conference in Palm Springs, CA making the claim that educators and mathematicians should expand the definitions of proof in order to make proof accessible to elementary and middle school students.
Mathematics the verb is cyclic, an idea I outline in part 2 of my recap on the life cycle of a math problem  and explain to my sixth graders using the game of Mastermind.

If you're unfamiliar with the game, here are the basic rules .
If you want to play a few games against the computer, we have that too.

I use digits instead of colors, which means this can be played using an iPad drawing tool instead of having to go and fork out twenty bucks for a physical game. I suppose you could also use pencil and paper. I also keep track of "correct digit, correct place" and "correct digit, wrong place" instead of using white and black pegs (to be honest, I can never remember which one is which). Even using the digits 0-9 (equivalent to 10 colors), a game goes pretty quickly.

We start with some Avery vs. Class action, with some specific language used throughout.

Before the first guess: " _____, would you like to make a wild guess?"

After each guess: "Does anyone have a conjecture about this game?" We've used this language before, but you could introduce this language here. If so, the student states their conjecture.
"Do you have a proof for your conjecture?" After the student's proof, a concept that should be broadened, I ask the class: "Skeptical peers, does anyone have any questions or concerns about this proof?" If so, we try and resolve it by altering the proof or abandoning the conjecture. If not, we write it up as a theorem.

One important note: If students come up with incorrect conjectures that do not get detected, I still write it up as a theorem. I actually hope for this to happen at least once to reiterate to the class that they are the skeptical peers, not me. And I'm not a prophet (by any means) that will let them know whether they're right or wrong. They need to work at this. That said, this is a great medium for allowing incorrect theorems to live for a bit. After all, it's just a game of Mastermind and mistakes will quickly become apparent.

I then ask for more conjectures. If there aren't any, I call on another student for an educated guess. Rinse, lather, repeat. I always get great conjectures. "There cannot be a 2 in this puzzle." "There must be a 1 in the first or second place." These conjectures get more specific and more involved over time. In the end, this almost always naturally leads to students making conjectures about the solution.

We also talk about "the problem space" or the set of truths/axioms and how different Mastermind games will have similarities in their problem space (the rules of Mastermind), but also differences (different solutions). More on this in part 4 of my never-ending recap of my CMC-South session. This can lead to meta-conjectures and theorems about Mastermind such as:

The following information implies that none of the guessed numbers are in the solution.
? – ? – ? – ?

 The following information is impossible.
? – ? – ? – ?

We play as a class. We play in small groups. It's pretty amazing hearing middle schoolers say things like "I have a conjecture that..." and "Respectfully, I think I have a counter-example to ________'s conjecture." It's a great way to build a community of mathematics for the rest of the year.

Oh, and I also have some Mastermind puzzles. Below are a few examples.

Find the secret code with the following information if you can use the numbers 1, 2, 3, and 4.
4 – 1 – 4 – 1
4 – 3 – 3 – 1

Find the secret code with the following information if you can use the numbers 1, 2, 3, and 4.

3 – 3 – 2 – 1
4 – 4 – 1 –2
4 – 1 – 1 –3

Using this first clue, what is the maximum number of guesses you will have to make before finding the secret code if you can use the numbers 1, 2, 3, and 4.  Why?
1 – 3 – 2 – 4

This last puzzle is also a great problem to talk about lower and upper bounds, an important mathematical habit of mind. If you share solutions to the puzzle, please post a spoiler alert. If you have insights, suggestions, or opinions please post an insight, suggestion, or opinion alert.

Tuesday, November 5, 2013

Proof Doesn't Begin with Geometry: The Life Cycle of a Math Problem

The following is part 2 of a recap of a workshop I led this weekend at the CMC-South (California Math Council) conference in Palm Springs, CA making the claim that educators and mathematicians should expand the definitions of proof in order to make proof accessible to elementary and middle school students. Part 1 of my recap on redefining proof can be found here.

There is a life cycle to math problems (not to be confused with exercises) in my classroom. We start with a wild guess.

This is something that should be done quickly, individually, and even if students don't have all the necessary information to answer the question.

Immediately: Seriously. After five seconds everyone should be writing something down--or by the time you finish reading this sentence.

There's some awesome research out there about priming and anchoring. The general idea is that if you ask a group to make a guess, the value of the first guess will significantly affect later guesses. Imagine asking a group of students to guess the population of Istanbul. If the first guess is 100, even though most students will recognize this as way too low of a guess, this anchor will cause future guesses to be smaller. Students have been primed to give smaller guesses. On the other hand, if the first guess is 100,000,000, you would expect guesses to be too high. It would be great fun to try this out with a couple classes and collect some data on mean guesses and their relationship to the first guess. Let me know if you try this. Daniel Khaneman's Thinking, Fast and Slow has a slew of examples, some of which are much scarier than the example I just gave.
German judges with an average of more than fifteen years of experience on the bench first read a description of a woman who had been caught shoplifting, then rolled a pair of dice that were loaded so every roll resulted in either a 3 or a 9. As soon as the dice came to a stop, the judges were asked whether they would sentence the woman to a term in prison greater or lesser, in months, than the number showing on the dice. Finally, the judges were instructed to specify the exact prison sentence they would give to the shoplifter. On average, those who had rolled a 9 said they would sentence her to 8 months; those who rolled a 3 said they would sentence her to 5 months; the anchoring effect was 50%. (Englich, Mussweiler, and Strack in Khaneman p. 125)
There are other reasons to have students write down their guesses before they share, a practical one being you then know everyone has a writing utensil in their hand.

Lacking Information: This is another way to lower the stakes. If you ask students to guess something where they don't have all the necessary information (which I usually make explicit), then the stakes are that much lower for them to be right. Inspired by a talk immediately before mine by Brad Fulton, I quickly put this slide together and asked teachers to guess what number lay behind the black box.
There's no way you can know the answer to this (unless you went to Brad or my talk, and even then you can't be sure I didn't change the value). That said, your wild guess will most likely be a speed and it will probably be somewhere between 0 mph and 1000mph. I could have also asked participants to guess the answer to this problem and this is probably what I would do with students (creating a sense of what a reasonable answer would be before working on the problem versus reflecting on whether or not their answer is reasonable after working on the problem).

After making a wild guess, I ask students to use appropriate mathematical habits of mind such as estimating, bounding, and contextualizing to make educated guesses. These are still guesses (and students still may not have enough information to solve the problem), but they are guesses based on some initial reasoning and strategies.

Estimating: This broad strategy for guessing includes rounding (73*88 is relatively close to 70*90), chunking (the height of that tree looks to be about 8 of me, or around 45 feet), and disaggregation (dividing an estimation task in a number of smaller, easier estimations) If you're unfamiliar with Fermi problems, these problems will keep you busy for as long as you want. And anyone who writes off estimation as not being that important, tell that to the US government and BP who are currently wrangling over how much oil spilled from the Deep Horizon disaster in 2010, with ramifications to the tune of billions of dollars.

Bounding: I introduce the terms "lower bound" and "upper bound" to my students early in the year. Similar to making wild guesses, asking students to make a guess you know is too low and a second guess you know is too high keeps the stakes low and is accessible to every student. It's also a great way to build in intermediate success points while working on a problem. Students feel success when they narrow their lower and upper bounds. It's worth saying again: this is a really powerful tool. If you don't believe me, give the following to your students who have no idea what calculus is and watch what they come up with in terms of a lower and upper bound.

Contextualizing: Getting students to start thinking about reasonable answers before they even really start solving the problem will pay dividends. Consider the train problem above. If students spend the time to realize that the answer is going to be between 0 and 280 miles, they will think twice when they solve the problem incorrectly and get 400 miles as an answer. 

The next step is moving from an educated guess to a conjecture, or a proposition that you think to be true. This is the meat of mathematical problem solving, and applications of prior knowledge and mathematical habits of mind should be abundant. 

At this point in time, students who are convinced that something is true work on proving their conjecture. I talked (at length) in part one of my recap about expanding our definition of proof, and will delve into examples of types of proofs that are accessible to younger kids in a later post.

Finally (well not really), the student shares his or her proof with the rest of the class and the class plays the role of skeptical peers, respectfully looking for counter-examples and holes in logic. It's pretty amazing to watch this when the cogs are well-oiled. I'll talk about how I start to build this entire structure using the game Mastermind in a future post. Assuming all goes well (the class gives a stamp of approval), we now have a theorem. Woo hoo!

One important note: if the class let's something slide that isn't true, I do not jump in and correct them. I really want them to believe that they are the ones in control of determining truth. That said, when this happens I know I have some work to do in order to push the class in a direction that will allow them to see the mistake(s) they made.

So there you have it. The life cycle of mathematics, completed below with what makes it a cycle: tinkering and inventing new problems. So much fun.

Monday, November 4, 2013

Proof Doesn't Begin with Geometry: Redefining Proof

The following is part 1 of a recap of a talk I gave this past weekend at the CMC-South (California Math Council) conference in Palm Springs, CA. I attempted to make the case that educators and mathematicians should expand the definitions of proof in order to make this important aspect of mathematics accessible to elementary and middle school students, and more accessible to high school students.

A Humble Proposal: Redefining Proof

Currently, the class Geometry has a monopoly on proof in K-12 education. For the college football fans out there, Geometry is the Alabama of mathematical proof in school. Like Alabama, I hope that the stranglehold Geometry has on proof is on it's way out (take that, Crimson Tide). Geometry is too often the first time students are introduced to mathematical proof, and too often the last time students grapple with proof until well into college. I don't know how related this is, but the following graph represents a common refrain. "Oh, I hated math. I liked Geometry, but...ugh, everything else." 
To be clear, this is intended to be somewhat glib. Don't read too into how one would
quantify "what math class looked like" or be angry with my lack of axis labeling.
I do not see expanding the definition of proof as a hugely controversial proposal. After all, Merriam-Webster, a respected dictionary (at least it used to be), defines mathematical proof as "a test which shows that a calculation is correct." The Silver-Burdett Mathematical Dictionary I found on my classroom shelf doesn't even include a definition of proof. We can all agree that we can do better than that. 


The Issue of Formality

The Online Free Dictionary's definition of mathematical proof represents what I see as the status quo: "a formal series of statements showing that if one thing is true something else necessarily follows from it." Starting with an agreed upon set of axioms (truths), proof is the process of showing that other things are also true. I don't want to change this; I simply want to relax our idea of "formality" to be more age-appropriate. Showing that something works for three examples will never be a proof. Saying "because it seems to be true" won't cut the mustard, hack it, or be up to snuff either. But what about the following argument related to a certain number trick?

First, the trick.

  1. Pick a number. 
  2. Add 5
  3. Double the result
  4. Subtract 4
  5. Divide by 2
  6. Subtract your secret number
And your answer is....

*drumroll please*

3! Magic? Nah.

And my proof? Let's say your secret number is box.
Is this formal enough?


The Social Aspect of Proof

Our above definition of proof says nothing about who decides whether or not a proof is adequate. For a professional mathematician, this work is done by the rest of the mathematical community, a community of skeptical peers. This shouldn't be any different in the classroom, and no, the teacher shouldn't be playing this role. Supporting, yes, but the community is the other students in the classroom. So here's my definition of proof I give to my 5th and 6th graders:
Convincing your skeptical peers that a mathematical statement is true
And then I put my money where my mouth is. When a student claims to have a proof of an existing conjecture, they give their argument using whatever medium makes sense, ranging from a verbal argument to an interpretive dance. The rest of the class takes on the role of (respectful) skeptical peers. Once everyone is convinced, the class has a new theorem. Giving a cogent argument is hard work. Looking for flaws, counter-examples, and unclear logic is hard work. All of my students are working hard. The hardest part of this culture is not letting yourself, as the teacher, play the role of prophet. Students have to know that you'll let them create theorems that aren't true (at least for a bit). Otherwise, they have no incentive to play the role of skeptical peers. They'll just wait for your cue, whether it be "Sounds great! Anyone have any reasons to question that proof?" or " that always going to be true?" Remember, we want the students to be analyzing the proof, not the teacher's response.


Expanding Proof Techniques

I learned inductive proofs in Pre-calculus. Geometry was the class where I learned two-column and indirect proofs. I didn't know you could write proofs in paragraph form until I was in college and proofs without words didn't hit my radar until after I was teaching. There are a few problems with this, the first of which is that two-column proofs are a structure for proving, not a proof technique.

Two-column proofs

The blog Math With bad Drawings has a cheeky, but poignant post titled "Two-column proofs that Two-column Proofs are Bad" that addresses many of the issues with this being the sole vehicle for proof in school. While the structure is intended to scaffold the idea of deductive proofs, it too often obfuscates the important idea of using a set of assumptions to show that something else must be true.
 Two-column proofs are also not a proof technique, but a way of organizing proofs. Imagine students going their whole life thinking that filling in data in a table was statistics.

The other major issue with two-column proofs is that they are completely disconnected to how people make arguments in real life. I don't think the following has ever happened in the history of the world (but please let me know if I'm mistaken).

Alternative Proof Techniques

I believe there are many proof techniques that students can begin to grapple with in elementary and middle school.  Prove something using words. Draw pictures. Acquaint students with the ideas of indirect reasoning, induction, and parity. Introduce or create simple axiomatic systems and prove other truths using deductive reasoning. I'm telling you. These can all be done at a much earlier age than they are currently done. This is big, and a lot, and will be flushed out in part 3 of this tome post.


Unanswered Questions

I have a list of unanswered questions. You will most likely have more.
  1.  What is the appropriate level of formality? I assume we can all agree that "I tested two examples" won't cut it as a proof at any age and that formal logic is overkill for students, there's a great expanse between these two extremes. 
  2. How do you create a culture of skeptical peers in a classroom? How long, and to what extent, do you allow incorrect theorems to live? 
  3. With an expanded definition of proof, what is the difference between an explanation/solution and a proof? Is this a skill/concept divide where work for demonstrating skills != proof while arguments for demonstrating concepts = proof? Or does this difference have more to do with the problem space? Or is a "proof" that 34*21=483 just too small of a problem space to call this a proof? We are typically ok with students saying "The sum of the interior angles of a triangle is 180 degrees." Should we instead be asking them to say "The sum of the interior angles of a triangle is 180 degrees within the problem space of Euclidean Geometry"?  
  4. Even if we agree that it feels silly, is there harm in using the vocabulary of proof, theorem, etc. when talking about 78*53?4. Is there harm in using the vocabulary of proof, theorem, etc. in a very small problem space (i.e. a particular game of Mastermind vs. Euclidean Geometry)?

Some thoughts to ruminate and questions to ponder. In the end, I think this boils down to my belief that we will have a higher level of mathematical thinking and discourse if we allow ourselves a lower level of formality related to proof. Please feel free to disagree. You may submit your arguments (proofs?) in the comment section. 

Part Two: The Life Cycle of a Math Problem