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


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.




Speed Dating

I’ve heard this strategy being used by several teachers in the MTBoS, but I most notably remember Kate Nowak being the one I heard it from first. It was a total success.

If you don’t know already, here’s the deal. Set up the classroom so that students are facing each other. Create a worksheet with problems you need the students to study/review. After handing it out, I gave the students a few minutes to become “masters” at one problem (I assigned them each a problem). I had 22 students, so I had 22 problems. After this, each student will “teach” their problem to the person across from them – for my problem set this was about 3 minutes (for both students to teach). After the 3 minutes, one side of the students got up and moved one seat to the left. Now they were across from a different classmate and the 3 minutes would start again – and each new pair would teach their problem. Each new pair now had a fresh start on explaining their problem and understanding a new one. (Sort of like real-world speed dating.) This process repeated until the end of the period. Oh, and I put out whiteboards on the tables to help with all this.

I floated around as they worked and assisted as necessary, but I wasn’t really part of the picture all that much. I loved this! I felt a bit weird in that I wasn’t doing much throughout the period. Then I came to my senses: it was the power of student-centered learning taking over me.

Because each student was only required to “master” one problem, they weren’t overwhelmed. And because they had to explain that question several times over the course of the class period, they really became well-versed on the concept that their problem related to. Conversely, because there was a student walking them through a problem they hadn’t seen before, I was able to incorporate peer tutoring and bring the learning to them in a more native way. They were talking about math all period – teaching and learning from one another – and hardly realized it.

It was totally my fault, but I didn’t get around to getting in an exit slip to gauge their thoughts on the activity. In fact, the bell rang as we were closing up. But if I had to guess from the looks of it, they really liked speed dating.

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