On Thursday after school, we had a Staff Meeting. Our vice principal, Moojean Seo, facilitated a math professional development session, which later inspired me to send this tweet.
Thanks to @moojean_seo for leading such a wonderful Math PD session at our staff meeting today. My head is still spinning. I think I have at least 3 blog posts brewing from today. Thank you for helping me think differently (& more)!
— 𝘼𝙫𝙞𝙫𝙖 𝘿𝙪𝙣𝙨𝙞𝙜𝙚𝙧 (@avivaloca) October 17, 2019
This post is one of these promised blog posts, and depending on how it evolves, it may actually encapsulate all three posts.
To start here, I need to give a little background information. Our principal and vice principal always share the Staff Meeting PowerPoint presentations in our Staff Team Drive, so while I was waiting for the meeting to start, I checked out the Drive. I quickly perused the presentation, and realized that we would need to answer three math questions during this PD session. I’ll admit it: I read the questions. The first question didn’t bother me too much: six divided by three equals?? I could do this one. I could even show it. But what about the next two questions? These next two linked division with fractions: 1 divided by 2/3 and 3 divided by 2/3. Two thoughts immediately came to mind.
1) Could I “phone a friend?” 🙂 Jonathan So, and his love of fractions, immediately came to mind.
2) Why did I feel as though visual spatial skills would come into play here? As I’ve blogged about before, in Grade 2 I was identified with a non-verbal learning disability. I struggle with spatial reasoning, and while I’ve learned strategies to compensate over the years, drawing diagrams, manipulating shapes, and all things mapping, are my biggest struggles. While I’m the first one to joke about my parking skills — or lack there of — the spatial reasoning required for parking makes getting in between the lines one of the hardest things that I do each day. I will admit that just knowing that a diagram was likely going to be involved in at least a couple of these problems, had my palms feeling sweaty and me taking a few extra deep breaths.
Cue the start of the Staff Meeting. It turns out that Moojean was going to randomly pair us up with other staff members based on our birthday months. Fantastic! I could work these problems through with a partner. Except, wouldn’t you know it that my partner had to leave early for a dentist appointment, and so the last two questions would be for me alone.
I kept trying to think back to conversations that I had with Jonathan in the past about what division of fractions actually means. I was sure that I needed to draw one rectangle and divide it into thirds. Then I coloured 2 of the 3 pieces. But 1/3 couldn’t be the answer, could it? While I struggle with visual spatial skills, I have an excellent memory, and remembered the standard algorithm: invert and multiply. The answer then should be 1 1/2. Was I wrong about how to draw the diagram? What was I missing here? Since we were all working around the room on vertical whiteboards, I started looking at what others were doing. I even began chatting with the educators beside me.
One of the teachers on my left was the first one that I asked for help. I noticed that she drew a picture similar to mine. Were we both wrong? We started to go back and forth on how we would represent the answer in another way when Moojean called all of us back together. At this point, I was completely lost. I was sure that my diagram was wrong, and I couldn’t even explain it even it was right. So I did the only thing I could think of doing: I drew a big X over my work. Okay! I was frustrated. Maybe hearing someone else’s explanation would help though.
Moojean took us through the answer, and while I looked around the room, I saw some head nodding and a few ahas. But all I had were these wonders:
- How does 1/3 become 1/2?
- Why do others not seem confused?
Now to begin our next problem. Thank goodness for the kindness of colleagues. The teacher on my left patiently walked me through 3 divided by 2/3. She showed it to me in different ways. She helped me see the connection to repeated subtraction as well as the visual model. While I have no doubt that my questions slowed her down, this teacher still remained kind. Not once did she say, “We’ve already done this.” A teacher on my right, watched and listened to our discussion, and even tried drawing a diagram to help me understand this more: the extended rectangle. I know that I’ve seen this before, but I still don’t completely understand it.
- Why do we extend the rectangle?
- Isn’t the original rectangle the one that we’re working with?
And so, when Moojean brought us back together again, I had a correct answer and a diagram on my whiteboard, but no real understanding of exactly what I just did.
I’d love to say that this is my first time feeling this way, but the lack of understanding actually brought me back to just about all of my high school and university math classes, of which 93% was my lowest mark. I remember years of students asking, “Why?” Many of my classmates were frustrated by their lack of understanding the algorithm or how to approach a problem. What was going through my head? Almost always? Just use the formula! I was all about algorithms, and I still have most memorized. My mental math skills are strong. In theory, I have what could be a teachable in math (for up to Grade 10), but I’ve never taken the Basic Additional Qualifications Course because I know something else about myself: I may be able to work with the numbers, but I don’t have a strong enough understanding of the concepts to develop the thinking skills and answer the multitude of student questions that will be coming in these higher grades.
I still think about my junior level teaching experiences. The geometry units usually required me to spend hours sitting at a table with my step-dad patiently teaching me strategies to work through the content so that I could move beyond my learning disability and help students as needed. I know from my years in Grades 5 and 6, that many children are stronger in these geometry units than in some other strands. Different experts emerged here, and a new love for math also seemed to emerge. For me, I was back in Grade 12 Physical Geography and that map of Canada.
- Where do the countries belong again?
- What if I needed to do more than just memorize the content?
While I was desperately trying not to, I was feeling the same way as I tried to represent division of fractions in a diagram. A visual model worked so well for even the vast majority of adult learners, but here I was as one of the 2%. I kept wondering, if there was a reliance on understanding back when I went to school, would I have continued with math all the way through university? Would I have been able to meet with as much success? Maybe though, my bigger wonders should be, if I could meet with success and not understand what I’m doing, am I really that successful after all? Is a 90+% meaningless when thinking, application, and communication were really never at play? I think maybe it was, but I also wonder, if there’s a way to make someone like me — that student where maybe that visual doesn’t work — still get more than just the algorithm. I want to believe that we can truly support every child in learning more. I’m now one of the biggest proponents of authentic math, learning through problem solving, and moving beyond just computations, but on Thursday I struggled. On Thursday, I was that child that needed more intervention, but I still don’t know what would have worked for me. What would you do? Thanks Moojean for inadvertantly making me struggle enough to actively seek out a better solution!
I’d like to offer some ideas here for you to think about.
Conceptual understanding is awesome, and I’m so glad that we teach that now. Learning those is what made math go from a boring drudgery as it was for me in school to this beautiful thing that I constantly want to learn more about as an adult.
Don’t discount your ability to use an algorithm. That is an important piece of the overall picture. Sometimes you just need to use the math to get from point A to point B and if your algorithm gets you there, then you’re doing well.
At some point, it’s time to move from conceptual understanding to applying that proficiently and efficiently to get to the next level. I think that’s a skill that sometimes gets lost in math today. Ultimately, different people can appreciate the beauty in different areas of mathematics, and it’s important to present a balance and a variety of approaches so that everyone gets a chance to engage in the areas of mathematics that they enjoy (algorithms being a very valid choice in that regard).
Thanks for the comment, Melanie! Like you, I actually don’t think that there’s anything wrong with using an algorithm and using it well. In fact, as kids move up in the grades, adding, subtracting, multiplying, and dividing with a bunch of manipulatives or a group of pictures would likely not be very efficient. But I think that what this activity taught me is that drawing this diagram — or not being able to draw this diagram — made me realize how little I understood about the math problem. I may be able to get the correct answer, but I didn’t know why. Picture or algorithm, I think there’s something to be said for this understanding, and are my struggles stopping me from getting that deep understanding? Is there something that might help change this?
Not sure this comment is completely relevant to what you’re discussing BUT….
I just want to give you a big hug. This blog hit close to home. Not as a grade school teacher, as I work with toddlers who wont be doing any math that complicated for a little while but as an individual who struggles immensely with Mathematics.
Every year was the same for me. Spending 2,3 sometimes 4 or 5 hours at the dining room table after school EVERYDAY while my step dad tried to explain my math homework to me. “Once you know it, it’s all the same” he’d say to me. However no matter how many times I did a particular unit in math, I would always forget it by the next time it came around again. I’ve never been so embarrassed or frustrated by a single school subject. Some teachers dismissed me as a lost cause while others worked so hard to help me understand. I hope with everything I have inside of me that my children won’t ever struggle with a school subject or course the way I did with math.
A teachers flexability, attitude and patience makes all the difference. The one rule that even applies for me as a preschool teacher is
“If a child can’t learn the way we teach, maybe we should teach the way they learn.”
I repeat that beautiful quote every day. It’s a powerful one!
This comment is so very relevant, and I cannot thank you enough for sharing your story! The quote at the end really hit close to home, as it’s the one that I include as my email signature. These are words that I stand by, and that drive Paula and I as we think about how to plan for kids or maybe reach that child that is struggling in a specific area. My background is working with kids with language learning disabilities, and I’ve learned a lot from both of my parents — a retired Speech Language Pathologist and a teacher — that all children really can learn. I hope that your children don’t struggle either, but I will share something that I’ve learned over the years from struggling: my own challenges helped me connect with kids that might also struggle, and this empathy, also helped as we worked through these challenges together. On some days, I wish that I could read that map with ease, park in a few less attempts, and attack a geometry problem without the need for a deep breath first, but in the end, learning disability or not, I ended up doing exactly what I’ve always wanted (and dreamed) of doing. Anything really is possible! And maybe, as a teacher now, I’ll learn a way to better understand that math that I didn’t quite understand as a child.