reflection – Page 19

For a long time

For a long time, I taught my students in a way that I thought was effective.

During the last couple of years, I’ve now discovered that I was all wrong. I actually made this revelation two years ago while “flipping” my classroom.

Student learning is best when it comes from complex, indefinite situations and then, after contemplation, taken to broader ideas and concrete generalizations. When learning begins, students should be confused and perplexed, or at least unsure about what is going to happen during a lesson. The problem comes first and the solution/generalization later. I think this really stems from how we, as humans, learn on an everyday basis.

Let’s say I’m confronted with a problem, like back pain. When my back starts to hurt, I immediately begin thinking about why. I’ll probably ask myself many questions and if I injured myself during that dunk I had over Lebron James. Or was it Carmelo? Either way, I’ll try various solutions like adjusting my sleep patterns, changing my exercise routine (no dunks), and using my knees more instead of my back – all to try and alleviate the pain. Let’s say that I struggle for a while and nothing seems to work.

Over time, I begin to realize a pattern. I notice that every day I wear my old, worn out sneakers, my back hurts at the end of the day. And on days when I don’t wear them, I feel fine. So I conclude that my sneakers are the problem (and not my dunking). They seemed to have caused my joints misalign causing a chain reaction to my back. I toss them and get a new pair and my back pain goes away. Also, I learned that moving forward I should replace my sneakers more than once every 7 years.

That was a weird example, but whatever. It still sort of frames how “normal” learning happens.

When confronted with a problem we use our inherit problem solving abilities to find solutions. It’s natural to be perplexed initially and to later understand. In no way is someone going to come along and immediately present a solution for my back pain. Similarly, neither should I, as a teacher, initially provide clear theorems or concepts to students as solutions to problems I will soon give them on an exam. In a math class, we should have them identifying problems and use problem solving abilities to find solutions and generalize ideas. This doesn’t necessary mean real-world problems, just critical thinking situations overall.

I didn’t approach teaching and learning in my classroom like this for a long time. Sometimes now, even though I value the approach, it’s still hard to for a bunch of reasons. But now I try to do everything I can to teach discovery-based, problem-based lessons.

Oh, and I DID dunk on Lebron. Once. Then I woke up.

The Quotient ~ 11.14.14

Whew.

What an incredible few weeks it has been. A school year of four preps, among other things, has finally caught up to me. Yes, four preps. Here are the things that having running around in my head over the last several weeks.

1. Student questions. When I get a lot of them, I’m pleased because I know my students mind’s are stimulated. But how to handle all those questions can be a challenge at times. What about when there are no questions?

2. Checking for student understanding during the lesson and adjusting on the fly. Basing the direction of my lesson on their level of mastery on any given day.

3. Relations with administrators, new and old. I’m learning a great deal about this dynamic and it’s impact on me and the overall school community.

4. Listening is a very underrated skill.

5. Including the proper scaffolding into each lesson. Specifically, having multiple entry points for students at the onset of learning.

6. After an assessment, the value of data analysis. Numbers never lie. Also, how structured analysis (time and protocol) can really help reveal the story behind the numbers.

7. Related to above: point biserial.

8. Always remember that kids are kids. Relate to them, laugh with them, have fun. They’ll enjoy your lessons a lot more – and work harder for you too.

9. I let one of my students borrow one of my ties to wear at his interview with Harvard University. Very cool. Awesome kid.

10. How much my use of Desmos has increased. I’m literally using it every day.

11. Reflect as much as possible. You’ll grow in ways that you otherwise cannot. Even just 10-15 minutes of concentrated reflection is substantial.

Simplified Probability Bingo

I teach an Algebra 1 course and the other day we were studying experimental and theoretical probability. I saw a Probability Bingo activity on Dave Ferris’s and Sarah Hagan’s blogs and wanted to try it. The problem was I saw it the day of the lesson and didn’t really have a bunch of prep time. In fact, I had about 15 minutes. (I changed my previous plans at the last moment.)

What do teachers do? We adapt at the last second. Here’s what I did and it took about 10 minutes of prep.

I colored several pieces of paper in different colors.

I also created 2×2 squares in Word and printed them out on 1/3 sheets of paper. During the lesson, I put all of the colored paper in a small cup. I told the students I was going to pull out two pieces or paper, one at a time. I asked the students to predict what I was going to pull out by filling in their tables by putting two colors in each corner of the table.

I proceeded to pull out the pieces of paper from the cup. If the combination they wrote down was pulled from the cup, they crossed it out. The first student to have their 2×2 grid entirely crossed out wins.

We played twice. Afterwards, we discussed the probabilities of choosing each combination of colors. They then dived into some practice problems on probability. It was great because the formal “learning” about probability took place after I had them engaged in the activity and not the other way around.

I found that the students thoroughly enjoyed the activity. It was a simple game and they didn’t even care what they won (which was nothing). They just wanted their colors pulled from the cup. Plus, it was an awesome hook into basic probability….especially since it only took 10 minutes of actual prep time. Next time, I may try and go with full-blown bingo.

Height Problem from Graphing Stories

This week in my Precal class, we were working on increasing, decreasing, and constant functions. I showed them a Graphing Story. These problems are perfect for analyzing graphs of functions. I chose the one that asks the viewer to graph height vs. time based on a lady swinging on a rope in Costa Rica.

If you haven’t already, first watch it for yourself here.

I’ve done some of the other graphing stories before and I love them. We did this one, discussed various aspects of the graph, and eventually hit on increasing, decreasing and constant intervals of the function. Its a great activity, but an interesting debate arose regarding the given answer.

You notice that the answer (i.e. graph) has a relatively large increase during 7-10 seconds.

When I initially displayed the solution to my students, they debated whether this interval was accurate. They found it intriguing that, according to the solution, she swung roughly 32 meters higher than her original height on the platform. Some of them said yes, it was possible for her to swing 44 meters in the air. I mean, looking at the video her upswing does look VERY high – much higher than the platform she started jumped from. Some students said no, that the camera angle was playing tricks on us. It was such an awesome discussion, and to be honest, I told them I wasn’t sure about who was right, but I’d do my best to find out.

After school I spoke to a physics teacher at my school, Shane Coleman (who these students also have), about the problem to get his scientific opinion. He informed me that it is physically impossible for the lady to have swung as high as 44 meters, unless she was pushed hard, had extra momentum when she left the platform, or pumped her legs on the swing, none of which happened. He told me for the footage in the clip, her kinetic energy would not have exceeded her potential energy. In other words, she wouldn’t have been able to swing higher than her original height; it would have been less and less with every swing. I probably should have known this, because its sort of common sense, but whatever. So then I brought all this back to my Precalc kids and it made for a pretty good discussion while we ironed out the actual solution. Later that day, the students probed even more while in Shane’s class. Awesome.

Now here’s my question: have my students, Shane, and I missed something here or is the solution inaccurate? If so, my follow up question is what assumption(s) could we make about the swing that would make the given solution valid?

On a slightly different note, isn’t it also interesting that her height is 0 at two different instances, yet she never reaches the ground?