Thoughts on Proof… and showing your work.

Suppose I give a group of sophomores this image and asked them to find the value of the angle marked “x”.

G.CO.10 - #2

Consider for a moment what method that you would use to solve this problem. (x = 121, in case that helps.)

Now, suppose I asked you to write out your solution and to “show your work.” What do you suppose it would look like?

I was a little surprised to see what I saw from my tenth graders, which was a whole lot of long hand arithmetic. Like this…

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and this…
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One-in-three had a mistake, which, in the midst of grading about 90 started to become an entire class worth of young people who were making mistakes doing a process that seemed fairly easy to circumvent (and by tenth grade, seems fairly cheap and easy to circumvent without much consequence.)

So, I asked why they were so intent on doing longhand arithmetic. The responses were fairly consistent.

1. Our math teachers have asked us to show your work and that’s how you do it.

2. It’s easier than using a calculator.

I will admit I was not prepared for either answer. (In retrospect, I’m not sure what answer I was expecting.) When I was asking students why they resisted the calculators knowing that they lacked confidence with the longhand, they said multiple times that they could show me that they “really did the math” without demonstrating the longhand. Also, one girl wondered why I would be advocating for a method that, as she put it, “makes us think less.”

They knew that I expected them to provide proof of their answers. Most of them were perfectly willing to provide the proof.

This student is starting to suspect that proof means words. So, he used words to describe the process.

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The conversation was pretty engaging to the students. A variety of students chimed in, most of them willing to defend longhand arithmetic and the only “true” work to show. I had shown them a variety of different looks at the longhand (the ones picture here, among others… including some mistakes to illustrate the risk, as I see it.) Then I asked this question which quieted things down quite quickly:

“Okay… okay… you proved to me that you did the subtraction right. I’ll give you that. Which of them proved that subtracting that 146 from 180 is correct thing to do?”

At first, they weren’t sure what to do with that. Although, quickly enough they were willing to agree that none of the work got into explaining why 180-146 was chosen over, say 155+146 = x or something.

I tried to convey that by tenth grade, I’m really not looking for proof that students can do three-digit subtraction. I would very much prefer discussing why that is the correct operation. They didn’t seem prepared to hear this answer. Apparently we’re even…

To be fair, there was one example where a bit of the bigger picture made it into the work. Check it out:

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I learned a lot today. I feel like I got a window into the students who are coming to see me. I ask them to explain, to prove, to show their work. Many of them willingly oblige, they just see an effective mathematics explanation differently than I do. It might be time to help the students get a vision of what explaining the solution to a math problem really looks like.

I would very much like your thoughts on this.

Policies of Fear

photo credit: Flickr user "Orange42" - used under Creative Commons - no changes were made besides resizing - http://creativecommons.org/licenses/by/2.0/

photo credit: Flickr user “Orange42” – used under Creative Commons – no changes were made besides resizing – http://creativecommons.org/licenses/by/2.0/

What are we afraid of?

In education? A lot of things, it seems. At least that could be the assumption if someone looked at the ways that we use policy to make decisions on behalf of our young people.

The causal onlooker might assume…

We are afraid that our young people (and their families) won’t value the education we provide. (That’s why we have laws that require students to come to school until they reach the age of 18.)

We are afraid that our young people won’t value mathematics. (That’s why we require students to take four years of mathematics in high school successfully completing at least Algebra II.)

We are afraid that our young people won’t value reading and writing. (That’s why we require students to take four years of English Language Arts in high school.)

We are afraid that our young people won’t value college readiness. (That’s why we require each student to be given the ACT test during their junior year of high school.)

We are afraid that our young people won’t value other cultures. (That’s why we require each student to successfully complete two years of a foreign language.)

We are afraid that our young people won’t value their own health. (That’s why we require them to take gym class. That’s why we require them to be served vegetables. That’s why we outlaw soda.)

So, our policies have helped us fend off that which we fear, right? Well, not so fast. We can legislate our students to be on site, and largely we have. The compulsory attendance rules are enough to convince the overwhelming majority of students to attend their classes almost daily. But it can’t make them value it.

Consider the likely state of a basketball team if each student was require to play a season. Coaches would probably spend an awful lot of their practice time deciding what to do about student who are literally only there because the state requires them to be. This would take its toll on the quality of the coaching that the program would be able to provide.

And because of the inherently competitive nature of basketball, we have no problem making decisions in the best interest of the quality of the programming. While I am not advocating for tryouts and cuts into the high school mathematics program, I think that it is fair to assume that the compulsory nature of the program has the same effect… and that the effect is undesirable.

As a math teacher, my job has never been so secure. The state has ensured my classes are full every hour, every year. Those kids don’t have a choice. They have to get through Algebra II and they have to take a math class their senior year if they should happen to go on ahead (that’s the law in Michigan, where I teach). They don’t have to value mathematics. They just have to earn credit.

So, we’ve won, right? Armed with 4 years of math, they are ready for college… Well, not so fast. Still much remediation is needed at the college level (at least in New York… and Washington… and California… and Illinois… ).

We can legislate them sitting in class. We can’t legislate that they value it. And they appear not to be valuing it.  An article from the APA discuss this very issue:

Teachers have observed that after second or third grade, many students begin to show signs of losing their motivation to learn. What happened to that natural eagerness to go to school and the curiosity to learn that is so apparent in preschool, first, and second grade students? Why do students progressively seem to take less responsibility for their own learning? This challenge only grows as students move from upper elementary to secondary school levels. (from Barbara McCombs, emphasis mine)

This seems like a natural consequence of us increasing the amount of non-negotiables that we a putting on our students. Think like a teenager: What’s the point of taking ownership if you are simply going to tell us everything that we have to do anyway?

The article continues to assert that motivation improves as teachers “provide meaningful choices” to help “students develop a sense of ownership over the learning process.” Also that, “motivation to learn is a thoughtful process of aligning student choices so that students see the value of these choices as tools for meeting their learning needs and goals.”

Finally, that we teachers should “involve students in setting objectives” and “appeal to student interest and curiosity.”

So, now we are faced with a very real conundrum. If ownership improves motivation, what is the natural consequence of legally forcing our young people to come to a place where most of their decisions are made for them?

Policymakers saw a problem with math achievement, so they require kids to take more math, removing a certain amount of ownership from the students. Well, when ownership drops, motivation follows. When motivation drops, achievement follows. When achievement drops, new policies are put in place that remove a little more ownership from the students. Then ownership suffers, and when that happens… you see where this might be headed?

And just like people simply never “got used” to prohibition, it seems unreasonable that students will eventually begin to self-motivate in a place where there is very little academic autonomy.

All this having been said, I don’t think compulsory education is going away. I don’t think math requirements are going away. So, what are we to do?

That makes the job of the math teacher to do everything they can to convince students that our math classes are useful. (I’ve talked about this before: In my “useless math class” series from earlier in the year.) Each course they take is an opportunity to learn how to model and solve problems of ever-increasing complexity and realism. Each course provides practice at modeling life situation mathematically because the math provides the means to focus on the essential variables of the problem. Each course provides an opportunity to develop a way to cope with the struggles that come with learning challenging material and being supported in the process of developing deeper understanding.

The education system is facing a problem in the area of mathematics. But low achievement is a symptom. It isn’t the problem. The problem is that our classes aren’t valued by our young people. That is a problem we can’t policy away. That is a problem we have to fix in the classroom by putting our students through valuable experience after valuable experience after valuable experience. The #MTBoS is working hard to develop those experiences for teachers at all levels.

Given the ways that the policies are undercutting our classes effectiveness, the conversation has never been more important.

Fake-World Math, Real-World Engagement

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Dan Meyer is currently leading a very engaging #MTBoS discussion regarding “Real World Math
” and it’s effects on student engagement with respect to completing (with quality) mathematical tasks. In general, “real world” is a term describing a task that attempts to emulate a task that might actually happen to someone in a non-school setting. The prevailing thought in many circles is that as a mathematical task becomes more “real world” it will become more engaging to students.

Many of us have plenty of anecdotal evidence to challenge that generalization.

Enter the “fake world” math tasks.

“Fake World” is a term used by Meyer to describe mathematical tasks that are engaging to students and encourage/require authentic mathematical problem-solving, but doesn’t attempt to emulate any actual action or task that one might use in a non-school setting. The Magic Octagon is an excellent example of a “fake-world” task. This is not a task that would EVER be asked of you in your family, work, or spiritual lives outside of school, but it is worth 20 good minutes of almost 100% engagement in a geometry classroom. These types of experiences cast doubt on the presupposed direct relationship between “real world” and student engagement.

As part of this, Mr. Meyer is attempting to do a bit of data collection to get a sense of what fake-world math activities we find engaging in our free time.

For me, it’s Flow Free on my iPad. I’ve seen this wonderful little logic game draw in 31-year-olds (my wife and I), high school kids (in my classes), and my 4-year-old daughter can get lost on it for an hour straight if I’d let her.

The task of a game is fairly simple. There are exactly two dots of each color on a grid. The goal is to connect each dot to its corresponding dot with a path that doesn’t intersect any other path. Also, each square on the grid needs to have a path going through it. No empty boxes.

So one solution to the above board would look like this:

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At this point, you can either try again to complete the same puzzle in fewer moves or move on to the next puzzle. As you might expect, the puzzles get progressively more difficult with additional colors added and the grids increasing from 5 X 5 to 6 x 6 and 7 x 7.

But what is it about this game that is so engaging?

The simplicity of the goal is a start. It takes very little explanation to begin playing. Also, the first couple of puzzles are quite easy to allow you to get the hang of the game.

The progressively more difficult puzzles is helpful as well. As you play, you start to develop some strategies and thought processes that you want to take for a spin on some harder puzzles. This game makes sure you get that chance.

Also, I like that the game has unlimited do-overs. If I had to do some guessing-and-checking to complete a puzzle and I want to start it over again, I can do that an unlimited number of times until I am happy enough to move on. Or I can move on right away.

It seems like those qualities could be integrated into math class. Consider an activity with a low entry point, a simple goal, some do-overs offered, and additional pieces that make the problem more difficult once the easier “levels” are solved. That would require us who design activities to take a more inductive approach to building engagement models. Look at what is engaging and see what elements they have in common.

I would absolutely encourage everyone to get involved in Mr. Meyer’s conversation. Do you have a “fake-world” math activity that you find engaging? Head over and tell the MTBoS about it in the comments.