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.

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Proof and Consequences

A conversation was taking place over at Dan Meyer’s Blog (http://blog.mrmeyer.com/?p=17964) about proofs, which is a topic that I find myself faced with about this time every year.

This isn’t a new conundrum for me. I’ve been working for while now trying to make this idea of proof, which, when compared to the typical form of textbook Algebra I should be an easier sell. But it just isn’t.

Here are some discussions of my previous attempts to sell it. Posts from Nov 2, 2012, Nov 16, 2012, Dec 7. 2012 are a few examples of my thoughts from around a year ago when geometry hit this place last year.

The problem I have is that the academic norms seem to prefer deductive reasoning to inductive and use of the theorem names. Those two things seem important to decide on before starting the journey of proof. If you are going to prefer deductive measurements, it rules out using measurements in proofs and it requires a lot more formal geometric language.

The problem that I see is that to rule out measurements (at least from the very beginning) and to strongly increase the formal geometric language in a way that makes deductive proofs possible from the very introduction of proofs creates… well… what Christopher Danielson is quoted as saying in Meyer’s post… “one of the most lifeless topics in all of mathematics.”

In order to breathe life into the topic, from the experience I’ve had, you need to let students engage in ways that make sense to them at first. The target to start the process is simply to get them comfortable with the idea of designing a functional persuasive argument about a mathematical situation. This requires recognizing that they need to start with a clearly stated claim (preferably something that is provable) and then start supporting it.

I find it helpful to let them pull measurements from pictures first and use those in the proof. The idea of comparing two things by length and NOT measuring them to get the length seems to a lot of kids like we are making the math difficult simply because we want it to be difficult. If they sense there is an easier way to solve a problem, then the explanation for why that method is against the rules had better be very strong, or else buy-in is going to suffer some pretty heavy causalities.

Once they get the hang of making an argument, then we can start by having discussions about what kinds of evidence are more compelling than others. This is usually where the students can figure out for themselves that each piece of information needs its own bit of mathematical support.

Next we can start deliberately exposing the students to different ways of proving similar situations. Triangle congruence seems to be a popular choice. We can have conversations about proving a rigid motion or proving pairs of sides and angles. Eventually certain kinds of explanations become more and more cumbersome. For example, using definition of congruent triangles to prove that two triangles are congruent as shown here:

Do we really need to keep going to find the three pairs of congruent angles?

Do we really need to keep going to find the three pairs of congruent angles?

Then, we can start pushing into shortcut methods. Mostly because those angles are going to be somewhat tricky to find (and why do more work than you need to… the students DEFINITELY identify with that.)

By using this method, I am trying to create what I’ve heard Meyer call “an intellectual need” for additional methods to prove this claim. (Keyword: trying… not sure how successful it is, but I’m trying.)

Then, that transitions fairly smoothly into stuff like this:

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… where we standardize the side lengths of two different triangles and see how many different triangles we can make and in what ways they are different.

Now, the tougher question is whether or not you allow the class consensus following the “Straw Triangle Activity” (which was a gem that came out of Holt Geometry, Chapter 3) to count as proof of the SSS theorem. In an academic sense, now we should “formally prove” SSS theorem. To most of the students, it’s settled. Three sides paired up means the triangles are congruent. What are we risking by avoiding the formal SSS proof? Do we risk giving the impression that straws and string are formal mathematical tools? But wait… aren’t they? What do we risk by doing the formal SSS proof? Do we risk our precious classroom energy by running them through an exercise there isn’t a lot of authentic need for right now?

Am I able to say that this is the definite recipe for breathing life into geometry proofs? Not even close. I am sure there are students who are completely uninspired by this. I can say using anecdotal evidence that engagement seems significantly and satisfyingly higher then when we used to run deductive two-column proofs at students from the very beginning.

But, we’ll have to see what the consequences are as we keep going.

Circles from Cedar Street

A sloppy bit of construction work makes for an interesting geometry question.

A sloppy bit of construction work makes for an interesting geometry question.

I was driving to the grocery store. This particular trip took me down Cedar Street. I drove past this manhole cover. It caught my eye in such a way that I decided to pull into a nearby parking lot and, when the traffic cleared, tiptoe out to the middle of the five lane road and snap a photo of it.

So, my mind instantly went straight to rotations. (Which is on my mind because transformations are Unit 1 of the Geometry Course that I start teaching in, like, two weeks.)

What if I started out rotations by showing this picture and simply asking the students how much of a rotation would fix the yellow lines.

My goals would be for the students to explore how to investigate an apparent rotation, learn how to visually represent a rotation, and struggle through the task of explaining it out loud to another person.

It would be okay with me if we made it to the convenience of using degrees as a descriptor of how much something is rotated. That would be up to them. I do suspect that after a short time of “about this far” and “about that much” they’ll like to find something to ease the trouble of explaining the transformation.

If you can think of a way to frame this learning opportunity better, please make me a suggestion. I feel like this is a good opportunity that I don’t want to waste.

Penny Circles

This discussion represents the final Makeover Monday Problem of the Year. I must admit, I found this problem to be easily the most engaging for me simply as a curious math learner. (Here’s  the original problem posted by Mr. Meyer.)

Now, I will admit that this is going to be more of a discussion of my experiences exploring this problem and, if you get lucky, I might come up with a recommendation somewhere toward the end.

The problem presents a rather interesting set up: filling up circles with pennies, making predictions, modeling with data, best fit functions. Lots of different entry points and possible rabbit holes to get lost in.

First change was that my circles were not created by radii of increasing inches. For the record, I can’t quite explain why I chose to make that change. Instead, I used penny-widths. Circle one had a radius of one penny-width. Circle two’s radius of two penny-widths. Circle three’s radius… you get the idea.

It looked like this:

I started with 4 circles with radii based on penny width.

I started with 4 circles with radii based on penny width.

Note: I understand the circles look sloppy. I had wonderful, compass-drawn circles ready to go and my camera wasn’t playing nice with the pencil lines. So, I chose to trace them with a Sharpie freehand, which… yeah.

Next, I kept the part of the problem that included filling each circle up with pennies.

It looked like this:

Penny Area 1

Then I filled them with as many pennies as I could

Then I filled them with as many pennies as I could

Here’s where the fun began.

I saw a handful of different interesting patterns that were taking place. If we stay with the theme of the original problem, then we’d be comparing the radius of each circle to the penny capacity, which would be fairly satisfying for me as an instructor watching the students decide how to model, predict, and then deciding whether or not it was worth it to to build circles 5, 10, or 20, or if they can develop a way to be sure of their predictions without constructing it.

But, I also noticed that by switching to the radius measured in penny-widths, the areas began to be covered by concentric penny-circles. So, we could discuss how predicting the number of pennies it would take to create the outside layer of pennies. Or we could use that relationship as a method for developing our explicit formula (should the students decide to do that).

So, if I am designing this activity, I feel like the original learning targets are too focused and specific. I want the students to be able to use inductive reasoning to predict. Simple as that. If they find that using a quadratic model is the most accurate, then cheers to them, but this problem has so much more to offer than a procedural I-do-you-do button pushing a TI-84.

First, I’d start with them building their circles using a compass and some pennies. Build the first four and fill them in. I’m quite certain that the whole experience of this problem changes if I leave the students to explore some photos of the my circles. They need to build their own. Plus, that presents a low entry point. Hard to get intimidated lining up pennies on a paper (although, I have plenty of student intimidated by compasses).

Next, I think that I would be if I simply asked them to tell me how many pennies they would need to build circle five and to “prove” their answer, I believe we would see a fair amount of inductive reasoning arguments that don’t all agree. The students battle it out to either consensus or stale-mate and then we build it and test. It should be said, though, that as they are building the first four and filling them in, I’d be all ears wandering about waiting for the students to make observations that can turn into hooks and discussion entry points.

Then I would drop circle ten on them and repeat the process. This is where the students might try to continue to use iterations if they can figure out a pattern. Some will model with the graphing calculators (in the style the original problem prefers). I’m not sure that I have a preference, although I would certainly be strategic in trying to get as many different methods discussed as possible.

Then, when they have come to an agreement on circle ten, I’d move to circle 20 and offer some kind of a reward if the class can agree on the right answer within an agreed margin of error. Then, we’d build it and test (not sure of the logistics of that, but I think it’s important).

All that having been said, I see an opportunity to engage this in a different manner by lining the circumference of the increasingly-bigger circles with pennies so that the circle was passing though the middle of the pennies and seeing how, as the circles got bigger, how the relationship of the number of pennies in the radius compared to the number of pennies in the circumference. I suspect it could become an interesting study on asymptotic lines.

It just seems like this particular situation has, as Dan put it, “lots to love” and “lots to chew on.”

Reflecting on the Common Core, Part II

Creativity, flexibility, are were on the rise this year.

Creativity, flexibility, are were on the rise this year.

Around Halloween of 2011, we began to prepare to rewrite our geometry class to align with the Common Core. This enabled us to do a couple of things that I had been wanting to do for a while. First, we ditched the textbook. Then we began to move toward Standards-Based Grading (Shawn Cornally (@thinkthankthunk) has some great stuff on this). We also decided to reconsider Algebra as the backbone as had been the previous practice in favor of a more visual, experiential approach.

We made the decision to embrace the CCSS’s Standards of Mathematical Practice because they made so much sense. We imagined building a class around patient problem-solving with strategic use of a variety of materials, student-to-student geometric arguments making use of geometric modeling and repeated reasoning. We imagined a course that drew the student into an experience. We didn’t want to see student memorize facts. We wanted them to experience the relationships, explore the different figures and circumstances, and draw conclusions about the significance of their observations. The Common Core enabled us to do that.

It was a lofty goal. I thought we tried our best this year. We didn’t do all that we were hoping. Our course didn’t live up to my standards. The students still memorized. The students didn’t explore enough. I told my students too many things. They told me too few. (Dan Meyer (@ddmeyer) would say that I was “too helpful.”) Needless to say, we have some work to do and I look forward to all of you coming along side of me for year two. Your support has been unbelievable so far.

Common Core puts a premium on student-to-student discussion

Common Core puts a premium on student-to-student discussion

So that’s that. The 2012-2013 school year is over and with it, the first try at creating a geometry class filled with patient problem-solving with strategic use of a variety of materials, student-to-student geometric arguments making use of geometric modeling and repeated reasoning. Those are our targets. I get 12 weeks to catch my breath and version 2.0 gets released to another collection of eager young minds.

And to me, that’s the essence of Common Core.

I understand the political conflict that exists when a single set of educational expectations are being enforced coast-to-coast. There are a lot of different ideologies, a lot of different beliefs, a lot of different communities. There isn’t much hope of finding something that everyone is excited about.

But, Common Core or not, a class depending heavily on patient problem-solving with strategic use of a variety of materials, student-to-student geometric arguments making use of geometric modeling and repeated reasoning, is something that I suspect most people can get excited about.

Reflecting on the Common Core…

photo credit: flickr user "Irargerich" - Used under Creative Commons

photo credit: flickr user “Irargerich” – Used under Creative Commons

A lot has been said about the Common Core State Standards in the last year. Some of it has been by me. Some has been by guys like Glenn Beck who is not a big fanMost (if not all) states have some sort of a “Stop Common Core” group. There is even a #stopcommoncore hashtag on Twitter that turns up quite a few results (although some use that hashtag as a means of highlighting objections in the arguments of CCSS opponents.)

The pub isn’t all negative. Some groups, The NEA among them, have come out in favor. Phil Valentine has some good things to say in support.

It is possible that both sides are probably overstating the impact that the CCSS will have. That being said, I will admit that I have some opinions on the CCSS. This year is our first year introducing a new geometry curriculum that we designed around the CCSS. I’ve written a few pieces before this one that have chronicled my journey through a CCSS-aligned geometry class. For example, I’ve documented that the CCSS places a greater emphasis on the use of specific vocabulary that I was used to in the past. I have also discussed (both here AND here) that the CCSS has present the idea of mathematical proof in a different light that I have found to be much more engaging to the students.

As I read the different articles that are being written, it seems like the beliefs about the inherent goodness or badness of CCSS has a lot to do with how you view the most beneficial actions of the teacher and the student in the process of learning. It’s about labels. Proponents call it “creativity” or “open-ended”. Opponents call it “wishy-washy” or “fuzzy”.

I suspect they are seeing and describing the same thing and disagreeing on whether or not those things are good or bad.

To illustrate this point further, a “Stop Common Core” website in Oregon posted a condemned CCSS math lesson because the students “must come to consensus on whether or not the answer is correct” and “convince others of their opinion on the matter.” The piece ends with “What do opinions and consensus have to do with math?”

The authors of this website are objecting to a teaching style. They are objecting to the value of a student’s opinion in the process of learning mathematics. Fair enough, but that was an argument long before the CCSS came around. I can remember heated discussions during my undergrad courses about the role of student opinion and discussion. (My personal favorite was the discussion as to when, if ever, 1/2 + 1/2 = 2/4 is actually a correct answer. One of my classmates rather vehemently ended his desire to be a math teacher that day.)

The CCSS have become a lightning rod for a ton of simmering arguments that haven’t been settled and aren’t new.

Consensus-building and opinions in mathematics vs. the authority of the instructor and the textbook. Classical literature vs. technical reading. The CCSS have woken up a lot of frustrations that are leading to some high-level decisions such as the Michigan State House of Representatives submitting a budget that blocks the Department of Education’s spending on the CCSS.

It is a little strange thinking that I am making a statement in a fairly-heated national debate every time I give my students some geometry to explore, but it seems like I do.

And I am prepared to make that statement more explicitly as I continue this reflection.

When measuring is okay…

I was raised in the math field to believe that measuring with ruler and protractor have no business in deductive reasoning. “Don’t trust the picture,” they’d say. I once sat at a PD session instructed by a man who told me that he would purposely give his students ridiculous pictures that had  no bearing in reality so they, “learned to only trust the given information.”

This bothers me.

It bothers me because the official stance involves deliberately providing a misleading visual to students.

It bothers me because it excludes students whose logical sense is still developing.

It bothers me because it sends the message that there is no place for the skill of observation and intuition in the world of deductive reasoning. It is as if observation is peasants work while the real men and women learn to reason deductively.

I teach geometry (as the web address of my blog might suggest) and so, I teach the students as we develop the art of “The Proof,” which is treated like some secret membership card. Those who can “do proofs” are those who are deemed worthy to enter the realms of upper level mathematics.

Up until this year, I’ve always taught proofs the same way I was taught. Start with the “two-column” method and flood them with theorems, properties and postulates. Then teach them to jigsaw them together in ways that make the learning look more like classical conditioning than deep understanding.

This year, I scrapped that. I started with what it means to “prove” something first. Let them write in their own language. Lots of words; paragraphs, arguments like that.

Then, when they get tired of so many words, I teach them the shorthand and the notation.

Then, when they get tired of making the same arguments over and over and over again, then I teach them about theorem and postulate.

You know what? Engagement is soaring! Students aren’t afraid. They are willing to try. They are willing to listen to the feedback. They are willing to learn. Pieces of this story are documented in several previous posts here and here.

So, if the model that I am using is indeed the improvement that it seems, then what do the students do before we get to theorems?

The students have to be able to use some sort of evidence to support their developing sense of deductive reasoning. We are going to start them where they are. And use need and convenience to justify making changes. We start with rulers and paragraphs, because deductive reasoning is hard enough to pick up new. Why throw in new structures, notation, and vocabulary in with it? Once they begin to understand deductive reasoning enough to begin worrying about style and quality, then we can move to notation… and on and on.

The ruler and protractor are already starting to become obsolete to some of my students. Why? Because protractors are a pain-in-the-you-know-what when all you’d have to write is “Vertical Angles Theorem.”

But I didn’t tell them that. They are making those decisions on their own. That sounds like understanding to me. The understanding I could never give them in previous years as to why rulers and protractors weren’t used much in deductive reasoning.

But, it required me letting them use them at first and still being okay calling it proof.