With math scores lagging tremendously due to pandemic-related school, many of us may suddenly find ourselves responsible for supervising (if not tutoring) math.
If that’s the case, it’s important to keep in mind the big picture of math learning.
Of course the issue of conceptual and procedural learning apply to all subjects, not just math, but it especially becomes relevant when problem solving can become complex; symbols and abstractions must be used, and multiple steps for problem solving are necessary.
There’s an especially nice description of the differences between conceptual and procedural knowledge in math from Ruthie Sloan’s Teach Math Literacy blog.
Many of us learned math only through procedural learning. We didn’t learn “why” we did certain steps like “flip upside-down and multiply,” and as a result often learned only one way of solving problems with a narrow approach to understanding math is related.
Dyslexic and dyspraxic students are generally weak with procedural learning, so to maximize learning of math subjects, they must understand “why” and not just “how” certain math problem solving steps occur.
Some additional examples:
Actually, when students are working through different types of problems, the lines are not so sharp between conceptual and procedural knowledge. Students who only know math mainly rote and procedural memory, may actually do fairly well through traditional school, but they are less likely to generalize their knowledge to other situations.
So if we are supporting a student in math, how can we help if she/he is losing touch with the underlying concepts?
I recently checked out a book, Principles to Actions, which had a number of good examples for how math concepts could be taught with manipulatives, visuals and real life examples.
Here are some examples from the Principles to Actions book:
I don’t know how comprehensively this book goes through math topics, but I could see it being a useful companion for a tutor or homeschooling parent if a student’s curriculum was more procedural than conceptually-oriented.
In the example below, the authors below make the point about how math questions are structured can affect whether students are really experiencing high level math learning.
For example, if students are told to set up equations for both phone plans in the pattern of y = mx + b, then they will only learn a “cookbook” type pattern of following instructions. The tricky thing is many students may not know where to start. They need hints, but ideally these hints wouldn’t take the place of having students figure out how equations can be related to the information discussed in the problem.
Rather than telling students to set up an equation, a teacher or tutor can tell them to think about how much it would cost to have 1 text message in each plan.
Then the students can reason out how they can represent the different parts of a problem in an equation.
I found it interesting that the book used an example from Illustrative Mathematics. This program is no longer free, but it is relatively affordable and dovetails with district Learn Zillion subscriptions and ALEKS.
Although there can be language challenges with students trying to work with word problems; however, many dyslexic students benefit from them because they translate abstract math processes into real world scenarios. Some word problems can be written poorly; if you are working with problems that are poorly written, then rewrite them, avoid extraneous information and avoid passive tense.
Some updated curricula will have sufficient discussion and application of concepts; for the ones that don’t, though, supplementing with an appropriate example, manipulative activity, or video can provide a more solid foundation for understanding.
It may seem counter-intuitive if a student is struggling in math and perhaps having trouble keeping up with assigned work, to go to an outside source for math instruction. Textbooks and curricula vary widely in terms of their emphases on concepts and practical applications, and providing more conceptual context for the procedures that are being taught can tie cookbook parts of math problem solving together.
In some cases, such as if a homeschooling student is far behind in math from where they need to be eventually for college entrance exams, focusing on test prep may be a reasonable choice for a math curriculum. The key is being able to explain answers that students get wrong and choosing relevant YouTube videos or other supplements if an answer key is insufficient. If challenges with math run in the family, then seeking an outside tutor may be necessary. Fortunately, there are many options virtually.
Especially as higher math in secondary school gets more complex, providing context with outside materials, if necessary, can help ground the math in a real life way. I remember “getting through calculus” in order to fulfill my premed requirements, now many years ago, but today I can’t remember anything I did or any context for the math that I memorized simply by rote.
If only I had been able to watch a video like this one by Eddie Woo. 30,000 Likes for a Calculus video?
If only I had learned the context like in the video below! Some students only begin to grasp the “big picture of a topic after they are coming to the end of a class and have failing grades. They fare better the second time round, but might have avoided the initial failure if more information had been presented in context.
In college, some students have told me that beside asking for a reduced courseload and taking difficult courses over the summer, another way to help them through difficult electives is to audit a class that they plan to take the following semester or year. Sometimes this strategy also offers them a big picture overview without the stress of falling a course and having to retake it.
