rawsonmath

During the past few weeks, I’ve had a teaching intern in my class. This has given me the opportunity to observe and listen to how my students talk about math. It has been a real gift to be able to listen. What have I heard?

First, let me say that students in my classroom typically work in groups. They are accustomed to working as teams and talking about math. I just usually don’t get a chance to listen to them all.

Two of my classes have begun a short unit on circles and properties of circles. They were investigating the relationship between central angles and their corresponding chords. The problem in the book asked them to prove that if AB = CD, then the measures of arcs AB and CD are also equal. Keep in mind that we use an integrated curriculum, so the last time kids had to prove anything geometric was in January. They were using whiteboards and drawing diagrams and pointing and talking math. While they worked the problem, I walked around and listened to what they had to say.

Here’s a sample from one group:

This was a typical interaction for this class. A couple of groups needed some teacher questioning to point them in the right direction, but most of them were talking about math and reasoning their way through the problem. Isn’t that the ultimate goal?

I’ve used whiteboards for math instruction for many years now, but the other day I learned something new on Kelly O’Shea’s blog post about whiteboard speed dating. Really cool idea. So, I’ve been thinking about how I can implement that idea in my classroom which, at times, has as many as 25 students in it. I don’t have the space to have 12 or 13 pairs of students (even my smaller classes would need 10 pairs). I also noticed Kelly’s comment about one group erasing what another had put on the board and just starting over because they couldn’t understand what the previous group had done. So this really got me thinking. My modification of whiteboard speed dating became the round robin review.

I had my students partner up creating 7 groups of two and one group of three. I used four review problems about creating and solving systems of equations that I modified from our text. Each of the four problems was given initially to two groups. They were given some time to begin solving the problem on chart paper. Then, after about ten minutes, we swapped the chart papers. This gave each group a new problem to make sense of and another group’s work to review, correct, or continue. Then we swapped once more. Finally, each group got their original paper back. The two groups with the same original problem teamed up to compare their final results, discuss any issues, corrections, or comments that were made.

When we had finished the entire process I asked my students how it worked for them. The responses were generally positive. They appreciated the opportunity to review in a different way. One student said that “it really helped trying to look for mistakes made by others because that would help her know what to look for in her own work.” Another student said that “it made him focus on communication.”

My observations and thoughts

I had one group doing all the work in the first round on notebook paper – and they were working individually – so they ended up with nothing on the chart paper. So, the empty chart paper with the other notes was passed onto the next group, who started solving from the beginning. I need to keep working on getting them past that idea that chart paper is for final drafts only – although I tried to make clear at the beginning that the chart paper was for doing the work. It was not something that was going to be framed and hung on the wall. They do seem to understand this with whiteboards.

I had a couple of groups “review” worked that clearly had mistakes in it, but made no comments or corrections. This indicates that these students are not invested in others’ work the way they are in their own. How do I change that thinking? Maybe that’s where Kelly’s mixing of the teams comes in. I’ll have to give those logistics more thought. How do I arrange the tables in my classroom to accommodate the complicated dance involving eighteen students?

I’m definitely going to try this again.

Once the blogger challenge ended and school really got going, I had little time for writing things down. Too bad, really, because some cool things happen in my classes and I’d like to record them. Take this class, probably last Thursday. My 10th graders are learning about matrices – what they are, how you can use them to organize information, how to do mathematical operations with them, and their properties and applications. The material all comes from our awesome textbook, Core-Plus Mathematics; the instruction comes from me, the person with the expertise.

Anyway, there we were, struggling with matrix multiplication. And, yes, I know that calculators do the multiplication. And, yes, my students have since learned how to use the technology. But, as I explained to my students, understanding how to multiply matrices allows them to be able to make sense of the results that the calculator gives them. So, they were struggling with the process. I gave them an example to work on in their groups. (Yes, my students work in groups.) They were to put their solutions on white boards. (Yes, we use group white boards.) Let’s say these were the matrices in question (the actual matrices aren’t important):

When the groups were finished multiplying these two matrices, they put their white boards up front. Two boards had

and the other three boards had

This was going to be interesting. Whenever we use white boards in class, the first question is, “What do you notice?” I don’t even have to ask it anymore. On this particular day, some students were saying that two groups had the wrong answer while other students were claiming that three groups had the wrong answer. I asked, “How do you know that any group has the right answer?”

This is where the magic began. One student said, “Can I explain how we did ours?” Like I’m going to say no to that. Then he asked, “Can I go to the board?” Of course. As he was explaining his group’s thinking, other students were clamoring to respond. Immediately after he was finished, another student said, “I have a rebuttal.” Very quickly, my students were debating the correct way to multiply matrices. Debating! Respectfully! In fact, one student explained her group’s process

Once everyone was convinced that three groups had found the correct solution, we reflected on what had just happened. We learned

• interesting things can happen when we disagree
• we can disagree respectfully
• the teacher doesn’t have to tell us if we’re right, we can reason through it ourselves
• time flies when you’re having fun

That’s right. My students said that the class was fun!

That was the first day of matrix multiplication. Do they continue to have struggles? Of course, but we chip away at it a little bit every day.

I began this year by having my students tackle one of Dan Meyer‘s 3-act math tasks. In fact, all of the teachers in my department did. We wanted to pick a problem that wasn’t too difficult mathematically so we could focus on the problem solving process. We picked “The Incredible Shrinking Dollar” and we had all of our students solving the problem on day 1. I teach one 10th grade integrated math class, two 11th grade integrated math classes, and one 12th grade pre-calculus class. Here’s what happened:

• Watched the video a couple of times.
• Asked the question, “How big is the dollar after 2 (or 3) shrinks?” (The class determined how many times Dan shrank the dollar in the video.) Groups worked and presented their process on white boards.
• Some groups interpreted “big” as referring to side dimensions, while other groups determined that “big” meant area. Several groups correctly calculated the new dimensions (using four different strategies). One group determined that losing 25% two times meant that the dollar would now be 50% of it’s original size. What a fabulous opportunity to address this common misconception! Groups interpreting “big” as area were split depending on how they chose to calculate the reduction. Some shrunk the area (typically incorrectly), while others shrunk the dimensions and then calculated the area. The differences in these results made for some very rich discussion.
• Asked the questions, “How big is the dollar after Dan shrinks it 9 times? Draw your guess. Will you still be able to see it?” This time all groups made correct calculations. Some of the “area” groups multiplied the original area by 0.75^18, while others used (0.75^2)^9. Cool, huh? But, I wasn’t convinced that they saw that these dimensions weren’t even close to the estimated pictures they had drawn. In fact, several drawings suggested that the groups thought that the dollar would be 1/9th the size in area. So I gave them rulers and made them draw their results. This gave us another great opportunity to address misconceptions: units were in millimeters (“which side is millimeters”), is 1/9th of the dollar (in area) the same as shrinking it 9 times, and so on.
• Asked the question, “How many times does Dan have to copy the dollar for it to become invisible?” A couple of groups thought this was a trick question: “The dollar will never be invisible – it will always exist.” Other’s asked, “What do you mean by invisible?” I turned the question back around to them so they had to define the term and then answer the question accordingly. One group looked up “too small to see with the naked eye” while others made more arbitrary decisions. We had a great discussion about how different assumptions can lead to different, valid results.
• The pre-calculus class had to use a different problem because one student had already solved the dollar problem. I chose a different problem on the fly. I looked down Dan’s list of problems and Coke vs Sprite caught my eye. This was true problem solving for this class. The students looked to me for help, but I hadn’t done the problem, either. All I could do was point them to the problem solving guidance that our department created. The problem poses a very simple question that my students spent 75 minutes trying to answer. The best part of the whole process was that they didn’t believe their results – because the results didn’t agree with their initial prediction. How awesome is that?

I’m really happy that we decided to start the year this way. It really set the tone:

• Different approaches can lead to equivalent results
• Different assumptions can lead to equally valid results
• Equally valid doesn’t necessarily mean the same
• We can learn from everyone and from everyone’s mistakes
• Mistakes are opportunities for discussion and clarification
• Initial thinking can be wrong; you have to do the math to find out

What a terrific way to start the year!