“Right and Wrong” vs. “Good, Better, Best”

So, I’m finding that I am becoming hooked on showing student work back to the students for them to explore as an opportunity to develop a deeper understanding of mathematical topics. But I’m learning there are a couple ways to handle this situation.

I could show this picture…

Geom #4-D

… and ask if the reflection is right or wrong. Which would activate a certain type of thinking. But it is a short and stifling line of thinking, especially when you consider that most hand-drawn student work isn’t perfect so “wrong” is the most likely answer. Then it risks becoming an annoying knit-picking session, which might have a negative effect on engagement.

A different approach would be to acknowledge that which we all know (that nobody’s perfect) and ask the question differently. Suppose I show the follow four photos together…

Best Reflection

… and ask “Which of the reflections is the best?”

That question opens up a lot of potential thought-trails to wander down. As I did this activity with students today, the class settled on three criteria for rating these reflection attempts. The first is that the image and pre-image should be congruent. Attempt A won that battle (with B at a close second). The next thought was that the image and pre-image should be the same distance away from the line of reflection. Attempt B was the closest (with D pretty good, too.) Finally, the students thought that the segment connecting the image/pre-image pairs would should be just about perpendicular to the line of reflection. Attempt B took that contest quite comfortably.

The students concluded that attempt B was the best reflection and I was able to confirm that by showing them this photo, which seems to agree quite strongly with that conclusion.

Best Reflection - Act III

If you want to try this activity, go ahead. I’d love to hear some feedback on how it went in your class. I’m still dealing with some quality control issues with some of my multi-media projects, so I apologize for that. I didn’t notice it to be too distracting when I was showing it to the students, but there is still room for improvement.

Best Reflection – Act I

Best Reflection – Act II (reflection distances)

Best Reflection – Act II (reflection angles)

Best Reflection – Act II (segment lengths of image and preimage)

Best Reflection – Act III

Bankshot 3-Act Revised

In my previous post, I presented Act I of a 3-Act geometry lesson that I called Bankshot. I also called out to the #MTBoS to support me by offering feedback of which I received two excellent comments that were rich in suggestions, particularly in the aspect of improving the student experience.

If you want to go back and look at the first draft and/or read the comments that got left, then feel free to check it out.

Now, all three of us agreed that the action begun and ended too quickly. Danny Whitaker (@nemoyatpeace) suggested that I slow the video down. In addition to agreeing with Danny’s suggestion, Dan Meyer (@ddmeyer) suggested that there was more I could do to set the situation up and make the clear what was going on before the I roll the film.

Danny also suggested that I cut the film before the ball bounces into the wall. Another interesting suggestion that Dan made was to include multiple attempts. I thought that those are both interesting suggestions.

You will see my attempts to respond to the feedback in this second draft of “Bankshot”. I start with a more direct set-up of the situation. I also added a second attempt from when I was filming to give the students something to think about. Now they have at least four guesses they could make. “Both hit target,” “both miss target,” “first hits, second misses,” or “first misses, second hits.” You will notice also that I slowed the playback down which cost us the sound. It’s possible that a bit of background music will be needed in a third draft, but that is something that we can talk about.

All right, #MTBoS, here is my second draft of “Bankshot.” I look forward to more feedback.

Bankshot (I give 3-Act Lessons another try…)

Feel free to read this post, but the video I present at the bottom has since been revised. After you read this post and watch the video, I’d encourage you to read the comments. Then head over to Bankshot 3-Act Revised to check out how I interpreted the feedback.

I recently attended a workshop led by Dan Meyer (@ddmeyer) which I found to be incredibly valuable. In the workshop, Mr. Meyer broke down the 3-Act lesson design model that he describes on his blog. He demonstrated it and showed a few of his own examples as well as some of Andrew Stadel’s (@mr_stadel)I have been trying to make sense of the 3-Act model for a while now. I had continually felt confused by a handful of the aspects. I tried to do some activities but found that I was often either giving the students too much information or not nearly enough.

Bottom line, I was confused about the basic point of the 3-Act Model. It is designed in a way that maximizes engagement and allows you to raise the bar while being more inclusive (which is tricky business). The first act, you set the students up to be curious about the situation. You prepare a scenario where a handful of outcomes seem likely and ask the students to choose which of the outcomes they suspect will happen. This is a short amount of time. You get the students curious, you have them choose a camp and then move on.

In act 2, you start leading the students through the mathematical processes that will allow the students to rule out focus on the outcome(s) that seem to be the most supported by the math. This is where the students begin to explore the variables of the situations, determine the appropriate modeling mechanisms and choose which tools they are going to use. This might also be where you lecture them a bit if they are trying to use models that they are unfamiliar or uncomfortable with.

In act 3, you reveal the answer and the allow the students to make sense of any differences that “real life” has with mathematical modeling.

In an earlier post (the value of face-to-face) I commented on how powerful I find in-person, face-to-face interactions and how the #MTBoS, as powerful as it is, is unable to accommodate for this particular shortcoming. That is only more solidified in my mind now, after seeing how much better I understand the overall approach and value of the 3-Act model having gotten to interact with Mr. Meyer face-to-face, ask him questions, and hear his responses.

So… I decided to try again.

For my first Act I, I decided to go with a geometry/physics topic. Also, the Act II is still in production. So, it probably isn’t quite ready for implementation yet, but I want to see if I am actually making progress in being able to understand, deliver, and (hopefully) create 3-Act lessons.