modeling – mathteacherbarbie.com

I love volunteering with math groups at our neighborhood elementary school. One of the groups I work with right now is a group of second graders who need math challenges that can recapture their imaginations and keep them engaged in the subject. Often now called an extension group, this group of students has grade-level mastery of the current math topics in the course and are ready to extend their math understanding -- either into a deeper insight into those same topics or into new math explorations (or both when I do my job best). This was originally published on mathteacherbarbie.com. If you are reading it elsewhere, you are reading a stolen version. Go deeper, not wider What I don't want to do is to let them learn so far ahead that they're even more "bored" with the classroom math next year. Instead, ideally, I help them learn to play with the math ideas and explore a depth of understanding that they can continue to pursue as they grow as learners. Much of the modern discussions of fixed and growth mindsets in education focuses on changing self-talk along the lines of "just not a math person" or "I'm bad at math" or the like. Unfortunately, we see strong fixed mindsets that limit high-achieving students just as much. These students (and I was one) often have a "I'm smart" or "I'm good at math" idea that is held central to who they are as a person. When this is true, any wrong answer, mistake, or even uncertainty begins to threaten their self-identities within that mindset. Learning to "play with math" encourages a willingness to explore, take chances, and grow in ways that expand their horizons and build truer confidence in the student's internal abilities to overcome challenges rather than allowing them to block their path. Go so deep that you run up against mistakes, uncover misunderstandings, and develop explorations and models that help do these for you. Allow these students a safe place to learn that correcting course, playing, and running up against things they don't know doesn't mean they're not smart. It simply means that they have now an opportunity to gain even more knowledge. Guide the Students' Natural Curiosities Following the interests these second graders expressed, we've been discussing multiplication. They had already picked up enough understanding to fairly quickly figure out multiplication facts using the numbers 1 through 5. They'll gain more experience with this and get them quickly into ready memory in the next two grades. So, instead of focusing on fact learning, which they'll get naturally over the next few grades, we turned to deepening our understanding of what multiplication is as well as meeting one student's request that we learn about squares and square roots. (Yup, someone has already introduced this second grader to this concept. I've also seen him doing some simple algebra.) We had already observed that most of the multiplication facts we talked about had come in pairs: $2\times 3$ and $3\times 2$ for example. We also observed that a few of the facts weren't part of a pair, and moreover, these facts all had matching factors: $3\times 3$ and $4\times 4$ for example. Expand upon natural and known models So how to deepen understanding, give them a tool they can access for future experimentation and deepening, and even explore square roots? Closed arrays to the rescue! Using those same 1s through 5s facts, a piece of grid paper each, and our pencils, we learned to draw rectangles that were, for example $3$ squares long by $2$ squares high, count the squares inside the rectangles, and associate them with the multiplication problems they represent. After a couple rounds of practice and gentle corrections, the four of them together drew rectangles representing almost all of the multiplication facts we'd already discussed. This crew was quick to observe that the rectangles of related (paired) facts were just rotations of each other and so basically the same rectangles. A little more prodding got them to notice that the unpaired facts -- those facts with matching factors -- were unique and different from the other facts. These were the only facts that generated perfect squares! Take opportunities to correct misunderstandings In fact, this square business led the students to doubt themselves when they first started drawing the rectangles for these facts. These students, like most, do not seem to internalize the fact that squares are just a special type of rectangle. Since I had asked them to draw rectangles, when the problem started turning out a square, they thought it was wrong, believing that a square was not a rectangle. Just a note to watch out for and help correct this frequent misunderstanding! Step back and let them explore These were, in fact, simple closed arrays. Two days later, their teacher informed me that they had been drawing these rectangles and counting the squares relentlessly since, and some of the students reported showing it to their families. They insisted we do more together. (One even tediously worked to figure out the factors of $321$. I don't remember his conclusion, and I believe there was an error somewhere in the counting, but I do know he was close!) Be their mentor and coach My role here was a guide and mentor, not an instructor. My work involved providing frameworks, models, and games that allowed the students themselves to explore the ideas and concepts, not to teach them the concepts. Even in students this young, I did occasionally have to be the voice of emotional support when they ran into something they didn't immediately know, encountering that barrier of self-identity as a "smart person." These moments were some of the most challenging for both me and the students. My role became a safe place to land and a coach to help them move through those emotions to a place of re-engagement and growth. It's never too early nor too late to learn that being "smart" is not the same as always being "right." This goes as much for us as it does for our "bored" students. You've Got This!