Tips Moving From Math Procedures to Understanding

Good At Math or Good at Memorization?

Tips Moving From Math Procedures to Understanding

I opened my inbox to the following email this morning:

Hi Kyle,

Scott Miller here. We met at the DVC Math Conference in February.

In reading/watching your presentation “Two Groups of Math Students I Created in My Classroom” have you found students that are Not Good at Memorization, but do not struggle with unfamiliar problems?

Have you gotten flack from parents that their child does not learn “that way” because you are “not teaching” them? I have used the terms Good at Procedures and Not Good at Procedures. I have found that by creating more challenges for a student to think through has benefited more of the Not Good at Procedures because they try different things. The Good at Procedures student shuts down and is unwilling to try because he or she does not recognize a memorized pattern.

What Scott is describing is very common when you start making a shift to teaching for understanding in service of building procedural fluency over time.

I would agree that challenging my students to think is initially more more beneficial for my not good at memorization or unwilling to memorize students. I see this group of students giving up when they are presented with an algorithm that they must follow. However, when I challenge them to solve a problem without attempting to give them a procedure, they are more willing to get creative in their thinking. The opposite appears to be true for our good at memorization friends likely because they are aware of their skill to replicate a given set of steps. When we don’t offer up those steps, they are concerned that they might not get the right answer on the first try and thus, they wouldn’t be considered “good” at math.

I believe this issue has a lot to do with mindset. I often find that the typical good at procedures group has a fixed mindset (I’m good or not good at something and I don’t have much control over it) and they have consciously or unconsciously figured out the “game” of getting good grades in math class. It is much easier for someone who can memorize easily to follow a procedure without having to understand what they are doing. Even in a classroom where a teacher is introducing topics through investigation, if the assessment routinely requires only the use of an algorithm, students naturally pick up on this. Why bother with understanding the “why” if I can just remember this handy set of steps?

If my assessments focus primarily on steps and procedures, then we are subconsciously telling our students to treat math concepts like a job on an assembly line:

  1. Grab two bolts from the bin.
  2. Fasten one (1) bolt to the threaded hole at the top left of the vehicle.
  3. Fasten one (1) bolt to the threaded hole at the bottom left of the vehicle.
  4. Apply grease with the grease applicator to the hinges on the front passenger door.
  5. Repeat for each vehicle on the line.

The worker may or may not understand why she is doing those steps in that particular order to complete her job successfully. Observing the worker might make one believe that she has a deep understanding due to her competence in completing the steps. However, only when you move her to another random job on the assembly line without any training will you truly observe how much she knows about building a vehicle.

This assembly line approach is what I see in my math classroom on a regular basis when we allow students to rely too heavily on an algorithm. Unless a problem looks familiar to what they have tackled previously, students lacking a deep conceptual understanding will experience difficulty. While it is difficult to pinpoint the specific cause, my hypothesis is that students in the good at procedures group have not been asked to think critically often enough in the math classroom.

I think there is a connection between what we are describing here and all of the posts on social media that describe “new math” as a simple question with a solution that appears more complex than necessary vs. “old math” where the same problem is tackled with a seemingly simple algorithm. Parents often get more frustrated than the student because they believe they understand math, but can only tackle addition/subtraction and multiplication/division using the standard algorithms.

When a good at procedures student is frustrated because I do not give them an algorithm to memorize, I make sure that I discuss why I am teaching from a task based, inquiry approach over simply giving them the steps. The frustration will not go away immediately and we shouldn’t expect it to as we have literally ripped the mathematical carpet from under them. You can imagine how a student must feel when they come into my grade 9 math class and quickly realize that they’ve been misled into believing they are “good at math”. The first instinct of the student and his/her parents is to blame the teacher because they have made it through eight years of elementary school math without a hitch. However, I find that you can get both the student and the parents on your side by being clear about why the student has been successful in the past. I try to help the student and the parents understand that knowing procedures alone is like being a “mathemagician” with a limited number of math tricks that are only useful in very specific situations. Helping students develop a deep understanding by thinking critically on a daily basis is the only way students can truly understand math and apply their knowledge to unique situations.

We can also help ease any anxiety that this change to learning math might cause by using assessment practices that promote student learning and growth. Since learning math is a process and every student is at a different place on their learning journey, offering multiple opportunities to demonstrate learning will help the student understand that they will not suffer if they struggle with a concept initially.

Have you had a similar experience in your math classroom? Have you found a way to help students understand why you don’t want them to simply memorize? Help us out by leaving a comment below!

Here is a timely YouTube video that I came across just hours after I published this post. Fits in nicely.

Can’t see the video? Click here.

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  • Sandy

    Teaching 10th grade geometry I hear all the time from students that they are just “better at algebra than geometry”. While there are differences in the 2 subjects, I find more often than not that they were only good at memorizing the procedures in algebra and lacked any deep understanding of why these procedures worked (I call this “recipe math”). Even when their teachers didn’t specifically teach them procedures or recipes, they inevitably created their own. In geometry, it’s much harder for them to come up with a procedure for every problem since it really relies on being able to see the bigger picture. I think it’s a little easier to get them to buy into the fact that memorizing procedures isn’t the way to go since they quickly see it doesn’t work for them.

    • I like the term “recipe math” – that’s a new one I haven’t heard yet! I agree that we unintentionally teach students to memorize when we are assessing using questions that can be reproduced via a straight forward set of procedures. It’s never our intention, but students pick up on it and learn “the game”.

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  • Kathy

    My students know that I won’t allow them to use “tricks” unless they can explain the math behind the trick and when it will break down. I know that many are frustrated because I won’t “help them” by telling them what to do. However, I struggle to assess these sorts of things well. When I test skills, I feel like I’m testing memorization. When I ask more open ended problems, I can’t figure out how to allow enough time for students to problem solve (without writing a test that only has 3 questions.)

    • I also struggle trying to figure out what I’m assessing. Some days, I feel like I have it locked down, then the next I start to question myself. Any of my traditional tests were pretty much assessing memorization in most cases. If a question wasn’t familiar (i.e.: haven’t seen this one before, Mr. Pearce) then students would give up. Now, I’ve moved to a weekly assessment every Tuesday. Usually only 3 or 4 problems and students have the entire period. If they finish early, they can work on mastery tasks. It seems to work pretty well. I give them the same talk every week that explains the purpose of the weekly task is to see where they are on their learning journey and there should be no stress associated with the assessment. Students seem to enjoy the style and I’m enjoying the useful information I gain from providing them feedback each week.

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  • Ross Cooper

    Great post. I love the assembly line analogy that you use to discuss the ideas of conceptual understanding and transfer. Only when students apply content to a unique context can we truly assess their level of understanding.

    Also, as you state, it’s not so much “old math” vs. “new math,” but rather more of an emphasis on inquiry. You bring up such an important point when you mention how we need to be transparent with students AND parents in regards to what we are trying to accomplish. It’s all about first setting the stage and promoting a culture of problem solving, rather than “blindsiding” stakeholders with a new approach.

    Well done!

    • Thanks Ross!

      In Ontario (and much of Canada) the suggestion is that “new math” is inquiry based. Many believe that inquiry means students just sort of work on “stuff” at their own pace and eventually, everything will work itself out. I’m only recently realizing that inquiry is very structured (or at least structurally planned) in order to try to predict what might happen when you throw certain situations at students. Like you say, we want to create that culture of problem solving and there is no problem solving without inquiry.

      Thanks for stopping by!

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