## Fake-World Math, Real-World Engagement

December 4, 2013

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:

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.

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.

## Means Extremes – Balancing the Means and the Ends of Math Class

November 26, 2013

I have been seeing this play out in my geometry classes each of these past six school years. It’s been a tricky problem for me to figure out and once I started to see what was going on, it became even harder for me to communicate it. I think I am ready to try.

Each year, a fresh cohort of young people come in straight from Algebra I. In theory, I can assume that they are fully loaded with algebra skills and thought-processes that will support them through their study of Geometry. There is one glaring hole in their understanding which I attributed to the overall youthfulness of many of my geometry students. They are 13-15 years old and, for the most part, their math experiences lack a significant diversity. So, I am able to give them a pass on some of the ways they are still developing as math learners.

But then I saw the same deficiency in my calculus students that I am only seeing now because this is my first year teaching calculus. Suppose that I give a geometry student this problem and ask them to find the angle measures of each angle.

Taken from Holt Geometry – Pg 181

Or suppose I gave my calculus students this problem.

Let’s add in as evidence that the Geometry students are used to application problems that look like this:

Taken from Holt Algebra I – pg 478

And that the calculus students have spent a lot of time looking at pages like this:

Taken from Holt Algebra II – Pg 580

The primary difference between the work I’m asking my Geometry and Calculus students to complete and the work that they are used to in the Algebra I and Algebra II is in the latter the equations are provided and in the former, the students are required to write the equation.

This is no small point. No side conversation. I am not splitting hairs. I am convinced this is a big difference.

Let’s go back to our Geometry problem.
First and foremost, recognize that there is very little natural or intuitive about this set-up as a whole. There is very little reason why angle measures are represented with algebraic expressions. The variable “x” doesn’t represent any actual value and so, the students are left to their abstract understanding of how equations are built in order to solve this problem. Their previous experience hasn’t really prepared them for this. Overwhelmingly, their mathematical experience leading up to this point has trained them to know how equations are solved.

Let’s expand this to our calculus team (of 23) of whom I noticed only about 8 or 9 who seemed comfortable modeling volume and surface area with equations and then engaging the formulas. So, even among our most talented high school students, there is a problem with the use of equations as modeling tools. Once they have them, they can operate with them wonderfully, but they struggle when it comes to writing them to specifically to match a specific situation. And beyond that, checking the accuracy of the model and then making sense of the product once they are done.

Herein lies the major issue: the paragraph above highlights a variety of skills that students (at all levels, from what I can tell) struggle with. They seem to struggle with them because they aren’t practicing them. But those are the skills that actually make mathematics worth doing to EVERYONE. The ability to do complex arithmetic on a rational or logarithmic expressions is something that is going to come in handy to people for whom formal mathematics is going to extend into their post-secondary lives. This isn’t a high percentage or our students, but these skills constitute a high percentage of the problems in our textbooks.

On the other hand, being able to recognize a situation as linear, quadratic, logarithmic or rational and have a sense of how to model that in order to make some predictions? That is something that could be valuable to a higher percentage of people outside of school.

I think that we need to recognize that the specific skills that we are teaching our math students are a means, not an end. They are the tools, not the final product.

The real goal is for the students to explore a situation, recognize the mathematically significant parts and use their math tools to model the situation strategically to help them achieve their goal. In addition to our student being better, more confident, flexible and patient problem-solvers, it seems like we’d also hear “when am I ever going to use this?” a whole lot less.

## The Anxiety of Open-Ended Lesson Planning

November 22, 2013

It’s been about three years since I started weaning myself (and my students) off textbook-dependent geometry lesson planning and toward something better. I’ll admit the lesson planning is more time-consuming (especially at the beginning), but most of the time expenditures are one-time expenses. Once you find your favorite resources, you bookmark them and there they are.

As we pushed away from the textbook, I noticed two things: First, the course became more enjoyable for the students. This had a lot to do with the fact that the classwork took on a noticeably different feel. Like getting a new pair of shoes, the old calluses and weak spots aren’t being irritated (at least not as quickly). Out went the book definitions and “guided practice” problems and in came an exploration though an inductively-reasoned course with more open-ended problems (fewer of them) that seemed to reward students effort more authentically than the constant stream of “1-23 (odds).”

The second thing that I noticed, though, was that I had less of a script already provided. The textbook takes a lot of the guesswork out of sequencing questions and content. When the textbook goes, all that opens up and it fundamentally changes lesson planning. The lesson becomes more of a performance. There’s an order. There’s info that you keep hidden and reveal only when the class is ready. Indeed, to evoke the imagery of Dan Meyer (@ddmeyer) it should follow a similar model to that of a play or movie.

But, see… here’s the thing about the “performance” of the lesson plan. The students have a role to play as well… and they haven’t read your script… and they outnumber you… and there is a ton of diversity among them. So, when you unleash your lesson plan upon them, you have a somewhat limited ability to control where they are going to take it.

Therein lies the anxiety.

If a student puts together a fantastic technique filled with wonderful logical reasoning that arrives at an incorrect answer, you have to handle that on the fly. It helps to be prepared for it and to anticipate it, but the first time you run a problem at a class, anticipating everything a class of students might do with a problem can be a tall order.

Case-in-point:

The candy pieces made a grid. The rectangles were congruent. Let’s start there.

The Hershey Bar Problem was unleashed for the first time to a group of students. Our department has agreed to have that problem be a common problem among all three geometry teachers and to have the other two teachers observe the delivery and student responses (I love this model, by the way).

The students did a lot of the things we were expecting. But, we also watched as the students took this in a directions that we never saw coming. The student began using the smaller Hershey rectangles as a unit of measure. One of the perplexing qualities of this problem is that the triangles are not similar or congruent. Well, the rectangles are both. So, as we watched, we weren’t sure what conclusions could be drawn, what questions the students might ask, or how strongly the class might gravitate toward this visually satisfying method.

We didn’t want to stop her. We weren’t sure if we could encourage her to continue. We just had to wait and watch. That causes anxiety. It feels like you aren’t in real control of the lesson.

Adding an auxiliary line changes the look of the problem, “How were we supposed to know to do that!”

In the end, most of what we anticipated ended up happening. The team trying to estimate rectangle grid areas ended up seeking a different method for lack of precision and everything got to where it was supposed to. The experience is valuable. But the anxiety is real.

This student was trying to make sense of the perimeters and areas

And I suspect that the anxiety has a lot to do with why the textbooks continue to stay close at hand. When the structure leaves, the curriculum opens up. When the curriculum opens up, the task of planning and instructing becomes more stressful and (for a short time) more time-consuming.

If you are reading this and on the cusp of trying to move away from your textbook, please know that this is the right move. Your book is holding you and your students back. You can do this. I won’t say that there is less stress, but with more authentic lessons, there’s more authentic learning.

## The Hershey Bar Problem (#3Act Revised and Updated)

November 1, 2013

About a month ago, I posted The Hershey Bar Problem in which I discussed, among other things, the ways in which I rip off other teachers work. This is an example of that. If this is your first experience with The Hershey Bar Problem, I encourage you to go and read the original post.

This post is an update to that. I have reworked Act I and finished Act III.

As usual, all constructive feedback is welcome.

Here’s the rest:

Act III

Sequel #1

Sequel #2

## Proof and Consequences

October 28, 2013

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?

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:

… 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.

## What Should Professional Development Do For You?

October 22, 2013

What expectations should teachers have when they attend a PD session?

I’ll acknowledge that the job of a professional development facilitator is not easy. I’ve done it twice and each time, I felt like I didn’t do a very good job. So, I’ll be the first to admit that it is a lot easier to sit and criticize than it is to do the good job.

But…

I went to a PD session today that sold itself as a session on developing ”deeper understanding of mathematics pedagogy” and “the capacity to utilize an inquiry approach to instruction.” Add to that the Michigan Association of School Boards awarded the project Michigan’s Best Education Excellence Award just this year. That’s pretty good, right?

The morning session began with a 75-minute talk about the history of the program, the grant, and the logistics of continuing education credits. At this point, the energy was completely out of the room.

… and it never came back.

The afternoon was a bit better largely because they broke us up into groups of teachers and gave us a fairly rich task that included about an hour’s worth of discussion choosing an engaging problem to try with our classrooms and discussing how to pitch the problem, anticipating what methods the students might use to engage the problem, and the possible struggle points.

This isn’t to say that there wasn’t potential for some fantastic discussions. The facilitators brought some wonderful starters to the table. “What are some common traits among mathematically powerful students?” and “What makes a rich mathematical task?” are great hooks as long as the fish are biting. Unfortunately, by the time we got there, no one was in the mood to discuss anything.

But, it would be wrong to sat that I came away with nothing. I came away with this: Teachers are an overstressed and undercompensated group of people. Their time is incredibly valuable. So, when they step out of the classroom, the time needs to be effective, efficient, and potent.

But I’ve seen some pretty effective professional development models. I am thinking of the examples put forth Dan Meyer (@ddmeyer) and the MathTwitterBlogoSphere (#MTBoS). I got to be a part of one of Dan’s “Perplexity Sessions” this summer and the #MTBoS and I have a longstanding relationship that includes some excellent professional growth. So, why are these PD’s more effective than the one that I attended today? Well to start with they…

1. Start with a bang! Potential buy-in is never higher than at 8:00 am when everyone is still an open book to what may or may not happen over the next few hours. Plus, everyone is nicely caffeinated and a little energized from the change of scenery. If you waste this energy, you won’t get it back. #MTBoS usually hooks people in because educators come looking for help solving a specific problem. The first experience is an effective one. Dan received a bit of an intro from the hosts, but once he got the floor, he wasted no time getting the entire group started DOING something so that he could…

2. Model effective instruction! Dan did this all the way through his session (which lasted 6 hours). The #MTBoS nails this one too. Many, many members of this group are willing to open their classrooms up to the masses through their blogs. I love the question: ”What effective teacher moves did you see me do just now?” In order to do that, you mustn’t have the group looking back at your introductory lecture that lasted longer than the class periods that any of those teachers teach.

3. Jam pack the session: Fill it end-to-end with rich tasks one-after-another. I was floored at the amount of ground covered in the Perplexity Session, #MTBoS handles their business in this respect as well. You can go from blog to blog, twitter feed to twitter feed for days on end. You should at least be able to put enough rich tasks in front of the group that keeps them actively engaged practically the entire time so that you can…

4. Draw as much as you can from the group. This one sort of goes back to modeling effective instruction, but that having been said, no one knows the classrooms of the group members like the group members, so build into your individual segments repeated think-pair-shares where the central ideas are being developed, expressed and generalized by the group members. After all, it’s their classes that this material has to go back to. Developing these ideas can help the facilitator, too, because you should…

5. Do a tight, concise debriefing session at the end of each segment that not only ties together the main objective of that segment, but relates it strongly to the rest of the overall session. This needs to address a couple of main points: Why will this help my students, why will this be worth the effort to implement. Dan actually gave the group members a chance to share their “secret skepticisms” to draw out the roadblocks and address them.

It is to these 5 ideas that I will commit. If I should ever facilitate a PD session, I will do my very best to meet each of these five expectations. If you should be there, please hold me to them, because it seems like you should be able to expect your PD to do those things for you.

October 11, 2013

A tool is just that. Tools are inherently valueless. They aren’t good. They aren’t bad. They are tools. They are perfectly meaningless and useless separated from a user. This is true of classical tools as well as much more recently-invented tools. Pencils, hammers, pillows, frying pans, coffee mugs are all just tools that could be used to do productive things or destructive things.

Twitter is an example of a tool that I have found incredibly productive. The #MTBoS has helped to define and demonstrate Twitter as a tool for professional development, peer review of the goofy things that rattle around my head, and a warehouse of effective resources for the effective instruction of mathematics. However, it goes without saying that those are not the only uses that Twitter has. In fact, the general student population seems fairly inexperienced with Twitter as a academic networking tool. Their use of Twitter has been largely social, which isn’t necessarily bad, but reflects a missed opportunity to take advantage of a resource that could be helpful and supportive.

Last night, I got to see this on full display. I ran my first (EVER!) Twitter review with my geometry students. I invited the students on for an hour and tried to lead a Twitter discussion. It’s the first time I’ve ever done that and it was the first experience many of the students had had with that type of reviewing structure.

In general, the discussions were teacher-to-student, which has it’s value. I’d ask a question, they’d respond. They’d ask a question, I’d respond. And despite my best efforts, I had difficulty fostering those meaningful student-to-student interactions to push this chat to the next level of group learning.

But then this happened (I apologize for the grainy photos. I’m still working on learning some photo editing software… that blurring out trick would be killer! Anyway… ):
It started with a question that I tossed out in response to a student’s statement of confusion:

Then that student-to-student started happening…

Notice what happened: The student got some advice as to where to find the definitions directly from another student. Then UH-OH! The geometry binder got left at school. But this is the 21st Century! That is TOTALLY not a problem, because…

She went on to send her both sides of handout G.CO.4 as well. This is a powerful moment. It is a moment that sets a precedent. Marissa just found a resource that she can trust. Twitter is beginning to be redefined in her mind as a potential academic tool.

This is a fantastic example of what is possible. This is what we in the #MTBoS have been doing for a while now. We are used to networking, leaning on trustworthy sources, and asking for help (in 140 characters or less). Most students aren’t used to using twitter in this way. Students need to be taught how to use their networks for more than simply social needs. If the academic world is becoming more digital, then let’s take the digital world and make it academic.

Last night’s review session gave me a glimmer of twitter hope that this social media machine that is being driven by the young people in our classes can be as effective a learning tool for them as it has been for me.