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):
![[ \begin{array}{ccc} 2 & 3 & 1 \end{array} ] \left[ \begin{array}{cc} 1 & 2\\ 0 & 6\\ 4 & -3 \end{array} \right]](https://s0.wp.com/latex.php?latex=%5B++%5Cbegin%7Barray%7D%7Bccc%7D++2+%26+3+%26+1+%5Cend%7Barray%7D++%5D+++%5Cleft%5B++%5Cbegin%7Barray%7D%7Bcc%7D++1+%26+2%5C%5C++0+%26+6%5C%5C++4+%26+-3+%5Cend%7Barray%7D++%5Cright%5D++&bg=ffffff&fg=333333&s=0&c=20201002)
When the groups were finished multiplying these two matrices, they put their white boards up front. Two boards had
![\left[ \begin{array}{cc} 6 \\ 18\\ -1\end{array} \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B++%5Cbegin%7Barray%7D%7Bcc%7D++6+%5C%5C++18%5C%5C++-1%5Cend%7Barray%7D++%5Cright%5D++&bg=ffffff&fg=333333&s=0&c=20201002)
and the other three boards had
![\left[ \begin{array}{cc} -2 & 25 \end{array} \right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B++%5Cbegin%7Barray%7D%7Bcc%7D++-2+%26+25+%5Cend%7Barray%7D++%5Cright%5D++&bg=ffffff&fg=333333&s=0&c=20201002)
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.
rawsonmath 