# Increasing Engagement Through Art

A half-hour of drawing might make the difference to whether or not the data table gets filled out…

I have a teaching colleague to thank for this idea. He currently teaches all of our Algebra I students, mostly freshmen. We have noticed a struggle-disengagement cycles that is self-repeating and driven by inertia. The student begins to struggle, which leads to disengagement, which leads to more struggling, which leads to… you get the idea. However, my colleague created an art activity that might be a game-changer for some of these students.

It goes like this. First, create an image on a piece of graph paper. Whatever you want. Could be your name or a picture or whatever. But there are conditions. It must include at least 10 line segments. Endpoints must have integer coordinates. It must have one pair of parallel lines. It must have at least one pair of perpendicular lines. After the pictures were drawn and colored, then the students picked 10 line segments and found coordinates of the endpoints, slope, function rules in standard and slope-intercept form.

Doing all of those things ten times over is as good as a practice worksheet, except better! Better because, you activated the right side of the brain for those who need it (which is a healthy portion of our students). There is a greatly reduced temptation to cheat or shortcut because the products are unique (and the more competitive students will see to it that their picture isn’t copied). Plus, like a good billiards player, you are always thinking two shots ahead. By tossing in the parallel and perpendicular pieces, you are running the risk that the student might begin to make some conjectures about the look of lines and their slopes.

How much more Algebra could this student be learning now?

# Beyond Geometry (Help me please!)

A literary-minded student’s contribution to my classroom decor

In my fifth year teaching high school geometry, an opportunity has come to me that seems interesting, challenging, and worthwhile. So, in addition to integrating the Common Core’s version of high school geometry into our school’s math program, the lot has fallen upon me to do the necessary research for (what could become) our school’s new AP Calculus class. Well, actually, I kind of requested the job because I need the hours for my masters program and I would be interested in teaching the class once it is planned.

Okay, so confession: I don’t have any idea what I’m looking at or looking for. I took calculus as a high school senior several (or more) years ago. I took Calc 1 at Lansing Community College and Calc 2 at Western Michigan University. I’d like to think that I could give the students a better experience than I had. In that mindset, I’ve spent the last two hours or so looking at different unit plans, textbooks, labs, and syllabi. I even put “the best calculus textbook in the English-speaking world” into a Bing search bar to see what came up. (Nothing useful, by the way.)

So, here’s where you can help.

I need advice. What are the best textbooks? What should I look for? What are my red flags?

I’m very inclined to project-based, or at least student-centered curricula. What resources are you familiar with that will support the students in an exploration of calculus instead of a year-long teacher presentation?

Can you provide links to awesome problems or labs?

All suggestions and recommendations is welcome. Even a word or two of advice would be welcome. Load up the comments. If it’s longer, andrew.shauver@gmail.com is the way to get ahold of me.

Thanks for everything!

# Mathematical Creativity: Multiple Solutions to the Pencil Sharpener Problem

I enjoy watching students exploring a problem that forces them to come up with their own structure for solving it. Today, a group got a chance to mess around with The Pencil Sharpener Problem, which is a problem I posted a month or so ago. (I’ll leave you to read it if you are curious what the problem is.)

From my perspective, what makes this problem interesting for the students is the ease with which it is communicated and the complexity with which is it solved. It seems quite easy. The answer is fairly predictable, but the students quickly found out that if they were going to solve this problem accurately, they were going to need two things:

1. A way to organize their thoughts and,

2. a way to verify their answer.

As long as the solution process included those two things, the students ended up fairly successful in the process of this problem.

I submit to you four examples of student solution structures. They all look different, but they all have one thing in common: The students could tell you with absolute certainty what was on the page and that they were right. (I withheld judgment from the correctness, but I will say that their final answers were all pretty much the same.)

This one includes a bit of a floor plan

In this first one, shown above, this pair of students decided to draw a floor plan with each of the pencil sharpeners (and the bucket they were tossing their leftover pencil nubs). The solution process progressed from there.

Guessing and Checking

These two students are classic minimalists. They worked to guess-and-check, which requires a bit more skill than it might seem. They needed to decided what number they were going to guess (sounds like an independent variable) and how to check their answer (sounds like a function producing a dependent variable). They weren’t using the vocabulary. It didn’t seem to be a problem for them.

Guessing and Checking

These two were a little more formal with their guess-and-check process, and they were using the terms independent and dependent variable. For the record, the time was independent and the number of pencils sharpened was the dependent variable.

A graph of multiple equations (where the solution does not include finding a single intersection point)

These two struggled a bit to convince themselves that they were on the right track as they completed this graph. They were discussing the need for the graph to be accurate (essential), how to interpret the three sets of points (one set to represent each pencil sharpener). Eventually, the were able to find the spot on the graph when the numbers added up properly. The number along the x-axis represented the final time. (Once again, they didn’t call time the independent variable, but it looks so nice along the x-axis, doesn’t it?)

Each of these students would have likely looked at the others with confusion, yet their answers largely agreed. What it required for them to solve it wasn’t me, as teacher, telling them how. On the contrary, it required me, as facilitator, designing a problem they would engage with and then giving them the resources to make sense of the problem with each other.

# The Art of Geometry: My students’ first tries at Kolams

I recently turned to Indian Folk Art to help my students make sense of rotational symmetry. For some, it was a time for them to get to show-off their artistic abilities. For other this was a fantastic struggle for some as they built their understanding. It was awesome to see the different levels of understanding. First the students were able to recognize the definition and make sense of the written words. Then they were asked to recognize and interpret online images for symmetric properties. Finally, they were asked to create an original rotationally symmetric piece. And while not all of the art work captured the essence of the art of India, much of it demonstrated that the students were truly building authentic understanding of a spatial reasoning concept.Bottom line: I just built my own understanding. I’ve been told for years how the deepest understanding comes from having the students CREATE. It was awesome to see the process and it seems to be working.

# The new look of math class…

My students just did this… and I loved it. It was math class. It was math thinking. It was math questioning. It was math discussing. But it was holistic. There was social studies and art and math all getting talked about together. And best of all… there was no computation… and the kids loved it (mostly)…

and it looked like this…

and this…

… and I want to see them do it again.