As a student, I was always great at memorizing formulas and processes. A couple of good cram sessions never failed me before a big test and always kept my grades in the acceptable range at home. That was a solid plan for me until Algebra II and Pre-Calculus hit and I found myself unable to understand any of what was going on without the basic foundation of number sense and algebraic reasoning required for such advanced mathematics. As a result, I have always claimed to be a proponent of conceptual teaching. I always gave my kids the obligatory 20 minutes of playing around with a concept before I said “ok let’s down to business” and pushed that conceptual piece straight to fluency and procedural skill in order to check off another standard on our curriculum map.

I teach a grade level Math 7 course, a collaborative math 7 course and Advanced Algebra I for 7th graders daily and although the content may be very different amongst them I have seen the same pieces of conceptual understanding missing over the course of the last week. In Math 7 we are working on understanding scale factor and as a result have had many conversations on why dividing by 2 is the same as multiplying by 1/2 or dividing by 2/3 is the same as multiplying by 3/2. It is a conversation we have had almost daily it seems as we work on thinking of scale as a multiplicative relationship and understanding how the value of the scale factor changes the size of the scale copy.

In my Algebra I class we are solving equations and inequalities but yet this understanding of the equivalence between dividing by a fraction and multiplying by its reciprocal is a barrier for many of my advanced students as well as we begin moving into solving more complicated and challenging equations. The whole experience has really pushed me to ask myself where this conceptual understanding has been lost along the way. The students I teach come from 5 or more different elementary schools and had a variety of teachers all of whom I have no doubt worked incredibly hard to teach this concept and so many others but yet like so many other concepts the students remain in the procedural rather than the conceptual world.

As teachers we all want 100% what is best for the students in our classroom. We want them to feel successful, to learn the content, and to feel like we have taught them everything listed in our District Curriculum maps and state standards. However, I think often times it is the conceptual understanding that gets lost amongst the ever growing list of things we need to cover and complete over the course of 177 days. As I said, I have always worked to give my kids “time” to work on conceptual understanding, but was it ever really enough? Conceptual understanding is hard…it requires time, perseverance, some creativity, and a lot of patience all of which can be hard to find time for in a 60 minute class. When my learners struggle I often find myself showing short cuts or breaking things down to simpler steps so that they can find success and comfort in the procedure rather than pushing them until they reach conceptual understanding.

My work with Core Advocates and MTBoS mentors have certainly pushed me and inspired me to dwell more in the conceptual and I have continuously improved as a teacher, however this year I seem to see the light more than ever. This year I’ve been lucky enough to get to pilot the new Illustrative Math curriculum from Open Up Resources and it has truly already challenged me in every way imaginable. I have really seen over the course of the last few weeks how much conceptual understanding opportunities I have taken from my students at times so we could get to the big idea. There have been lessons in the first unit that I thought to myself “are they ever going to get to the point” however I have held my feet to the plans provided. We are wrapping up that first unit now and here is what I’ve seen…the kids do eventually get there. I have given them the time to reason, struggle, make mistakes, draw pictures, and talk out their ideas and now here at the end of the unit I truly believe they know the procedure pieces better than they ever have. They now OWN why they are doing what they are doing instead of simply doing what I said they needed to do. When I posed a scale copy problem today, some kids grabbed a ruler and started measuring, others grabbed a piece of tracing paper and checked angles, others went for a protractor or index card, and some used simple reasoning but they all on their own found a way to prove two images weren’t scale copies. The time spent allowing them to really grasp the math was far more productive than any time I could have spent trying to procedurally teach how to identify scale copies or scale factor.

I get while writing this that it sounds much simpler than it is. If I knew conceptual understanding was key for years, why did it take this long to really embrace it? I think until this point I believed I was giving kids enough opportunities to struggle with the conceptual piece and until I really experienced an entire unit built around conceptual understanding first I couldn’t grasp how much time I really needed to give these ideas to develop and form in their head. I am excited now to see how their ideas continue to develop over the course of the year. The next IM unit is proportional relationships which will allow them to extend their understanding of scale directly to understanding proportional relationships as well. I am more excited to see their understanding and reasoning continue to develop than I have ever been. I am so excited I got to see the light.

Thanks for your message. It has helped me to reflect on my stages of development. What I realize is that I have continually, over the years, had to adjust what I thought was enough time. Yesterday I was astounded how long it took for my Thai pre service teachers to find the perimeter of all the rectangles represented by a graph of area vs one side length where the perimeter remains fixed. After 40 minutes we finished it with a student overjoyed at finding a way to find perimeter in an infinite number of ways with her own formula. (I wonder if she had ever had such an experience at school, or in her life for that matter).

We then had a discussion started by me re: the similarity and differences in pedagogy from their school age experiences. I pointed out how much time it took, that we hadn’t ‘covered’ even a third of the lesson I’d prepared, that there is pressure to complete curriculum in elementary and secondary education. I forgot to ask them: Was it worth it? Maybe they can’t answer that yet. For me I have to say yes. It’s almost like I have no choice. It was difficult to be that patient. But the alternative is to maintain the status quo into the future.

Are you familiar with debates surrounding the nature of multiplication that Keith Devlin helped stir up about a decade ago? https://www.maa.org/external_archive/devlin/devlin_06_08.html

Check this out, too: https://www.quora.com/Why-is-multiplication-so-important-in-math/answer/Alon-Amit?share=64962086&srid=oFL8

I love the evidence that students are learning to “use appropriate tools strategically”.

Very nice reflection. It resonates with my experiences too. I’m looking forward to exploring the illustrated math curriculum with my 8th grade students. We also is the Mathematical Vision Project materials for high school. It seems to have many of the same principles.

I think it’s easy to KNOW that conceptual understanding is important…but it can seem overwhelming as a teacher to come up with the right tasks and sequences that can help students grapple with the math. Very excited to give the IM curriculum a try when we get back from break.

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