Dear Mathematicians, Parents, Students, Educators, and All Interested Parties,

Is it a sign of weakness for a teacher to admit perpetual confusion on the best way(s) to administer instruction? Although I have been teaching for about 15 years, only a few of them have been spent teaching math at the middle school level. Since making the glorious move to middle school, the distinct advantage of pouring all of my extra time and energy into one subject has both reinvigorated my purpose and sent me down a path of wonder.

In my quest to prevent any student from truly thinking he or she does not have the math brain, the amount of articles consumed by me is, to say the least, staggering. Can I remember who wrote most of them? Not usually. I peruse for content. Only after multiple exposures from the same author do I start to take notice. This is why a few names have made their ways to the corners of my cerebrum where the long term storage of my memory lives (Thank you Sousa). I usually refer to the information annoyingly as, “I read an article that stated…” Yes, I have turned into one of those people.

My favorite pastime is to research lesson structure ideas as this is my professional focus. Some of the names that continually pop up in my consumption within that topic are Jo Boaler, Dan Meyer, Andrew Stadel, and Yeap ban Har. Each of these math gurus share a common thread, which is that mathematics is a subject that spans beyond mere procedure. Although I could not agree more that math is not strictly procedural, each time I read an article I find myself asking, is there still a place, and furthermore, a need to teach procedure(s) in a math class?

If it is true that the best teachers steal from the best, in some small way, that categorizes me as the best. I have “stolen” lessons from my teaching counterparts, Dan Meyer, and Andrew Stadel. The stolen lessons have been glorious experiences. However, I do not believe any of the stolen lessons would have been successful if students had not possessed the background knowledge on procedures as well. Now I wonder, did I enhance their conceptual learning or detract from it with that viewpoint?

Our district was blessed by the personal teachings of Yeap ban Har. I spent a good month after that momentous training opportunity trying to design my lessons just like him. This was not easy to do with only one real half-day of training, but I really gave it my all. Some lessons went astonishingly well, others, not so much.

What I do know is my goal is to do better every single day. This is where I feel as if I am on the giant hamster wheel of math instruction.

In my mind, if students do not learn the concepts behind the math, the procedures for any and all algorithms will be meaningless. They will learn a series of steps, study them for a quiz or test, regurgitate them, and then quickly dump the total experience from their memory. Obviously, this reality is not true for all students. Those students who are excellent at rote memorization might remember the steps, but will they have any idea why they are performing them? If they don’t, can that be considered effective math teaching or learning? On the other side of this paradigm, sits many students who demonstrate conceptual learning but struggle with the rote procedures. For example, several students in my class this year forgot how to subtract opposite signed numbers using an algorithm, but when I placed a number line or integer tiles in front of them, they knew how to solve the problem immediately and could explain their thinking. Is their learning inferior because they cannot demonstrate their understanding in an algorithm?

The articles I have been reading lately push my questioning even further. I believe Jo Boaler flat out posited whether or not it is necessary for students to memorize their times tables. Is this type of thinking correct for educators, and more importantly, for students beyond the classroom?

Here is where I flat out ask the community for feedback. Is there an appropriate balance needed in our classroom between concepts and procedures? Are procedures completely out of date or still necessary? Do we need to argue the opposite ends of the spectrum, or consider that the ideas are not opposing but supporting of one another? I ask you, in a growth mindset sort of way, to reflect carefully. Perhaps someone out there can inspire me to jump off of the hamster wheel, if only for a moment.

Sincerely,

A math teacher looking for answers.

Re:

“Is it a sign of weakness for a teacher to admit perpetual confusion on the best way(s) to administer instruction?”

No. If anything, I respect this question a lot. The simple fact that you’re reflecting on this question is powerful. Furthermore, it’s great that you’re considering this whereas I’m sure it’s easy for many teachers to stick with one way od doing things, even if they’re not sure it’s the best way to do things.

That said, from my experience, it’s all about moderation. As with anything in life, I’ve found that too much of one (good or bad) thing doesn’t necessarily give us the balance that we typically need. Specific to education, I think a balance of

You have baseline data now that you’ve tried different types of activities. I’m glad your students had “possessed the background knowledge on procedures as well.” to be successful. You have to start somewhere. Now that’ve taken the plunge, you’re moving toward the stage of refinement. In other words, next time you try a task, you can compare it to the first time you ran a task. Or you can launch the learning with the task and teach the procedural fluency part after once students have a better conceptual understanding. Again, it’s all about finding that balance. I think that’s the most important part, is working toward finding that balance between conceptual tasks and procedural-based learning.

You asked, “is there still a place, and furthermore, a need to teach procedure(s) in a math class?”

I say yes. There are times when we simply need to tell students what to do, within reason (and moderation, of course). For example, dividing fractions is a concept I continue to work with teachers on. Many teachers love exploring the conceptual side of dividing fractions, mainly because they never understood it before Common Core, so it’s fascinating and interesting to them as teachers. This doesn’t mean our students will have the same excitement. However, I’ve seen some cool activities/tasks that these teachers are facilitating with students to help build a better conceptual understanding. Does it mean that all students now “get it”? Not necessarily. Furthermore, are these activities the most efficient way to divide fractions? No. However, the activities lend themselves to a few things: patterns, conjectures, and procedures.

The activities are structured so students can look for patterns when dividing fractions. When students see patterns, they can attempt to make conjectures about the mathematical relationships. These conjectures can be tested. And if we’re successful, maybe one or two of those conjectures turns into the procedure where we multiply the first fraction by the reciprocal of the second fraction. And at the end of the day, we teachers have the authority to simply say, “look kiddos, if you struggled with looking for patterns or making conjectures, others have explored division of fraction before and have found a procedure for multiplying fractions.

Yes, provide students with conceptual learning opportunities. Yes, provide students with procedural learning opportunities. Find that balance. Find that moderation. Or at least work toward finding that balance and moderation. I tend to fall back on sports analogies I can relate with. Basketball athletes still need to practice lay-ups just as much as they need to learn offensive plays. They don’t practice lay-ups for one entire practice. They don’t practice learning plays for one entire practice. If a basketball coach knows where they want their players to be, then they find that balance between practicing skills and practicing the theory of the game. We, as teachers, are often charged with the same task.

Hope that helps.

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I love your sports analogy, although I admit, my analogies tend to be related to cooking. 🙂 I tell my students’ parents that the procedures are important and the algorithms are still an important goal. After all, they are often (not always) the most efficient way to solve math problems. The issues I have can be found in the hotly contested educational world where we practice our craft. There are common core haters and common core supporters. There are the misinformed and the over-informed. There are the fixed mindset people in both camps. I am doing my best to find that balance you speak of and I aim towards, but whatever I do, I know my teaching can’t cater to everyone’s expectations. My greatest concern is shortchanging the students in their instruction and in their experiences while they are in my room for the year. It is what drives me to constantly change, improve, start over, and question everything I ever thought I knew about teaching. I am most grateful for your comments and your online support. You are making me reach higher and, for that, I can’t thank you enough.

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