I’ve been bothered for a long time by a common practice in math textbooks and classes. Because math is a logical system, it makes sense to write it out in logical order: start with the basics (definitions) and build up logically to the more complex results and applications. This makes sense for a rigorous math reference book. Even most high school textbooks reflect this order, and unfortunately, so do most teachers.
But is this really the order that makes sense for a first time learner? Why should they pay attention to some technical definitions when they have no idea why they should be able to work with those definitions? How exciting is it for most people to learn math this way?
I think it’s the wrong way to teach math. I am always looking for ways to create a “need to know” in my students by starting with the “interesting” problems (even if they are currently completely intractable to the students) and using that to motivate learning new definitions and skills.
For example: most of my precalculus students have seen logarithms, but so briefly that they barely remember the word logarithm (and that they don’t like it). So when we approach logarithms in precalculus, I like to start by working with exponential modeling problems. I ask students something like, “Okay, so you can tell me the population of bacteria at any given time. But can you tell me at what time the population will reach 8000?”
Solving this analytically requires logarithms, but they don’t remember that. So I let them try any method they can… guessing, using the graph or table of a calculator, or even trying some illegal analytic methods (taking the “nth” root of both sides). But then I try to instill that it’s bothersome that we can’t solve this by hand. I find problems that circumvent calculator methods or show them that guess and check is too inaccurate to be realistic. That motivates our need for logarithms, because logarithms are the inverse operation of exponentials and allow us to solve these equations.
Contrast this with the four or so precalculus textbooks that I have access to: they all introduce the definitions of exponential and logarithmic functions together and immediately start getting students to rewrite exponents as logarithms and vice versa (simple definitions). They don’t move on to applications or using logs/exponents together as inverse operations until a few sections later! It usually takes until the end of the unit to really “see” how logarithms and exponentials fit in together, and by then, many students are likely already lost and cemented in their lifelong hatred of logarithms.
The contexts don’t always need to be “real-world”, though that can certainly work. Sometimes the context is purely mathematical and helps students see why number theory is so much fun. For instance, my geometry students this year were incredibly intrigued by this question: Given an integer n, you have to create triangles with integer sides whose perimeter is n. For n = 1-10, how many different triangles can be created for each n?
I was honestly surprised by how involved the students got, and multiple students remarked how fun it was to complete a table and look for patterns. It was a great motivation to establish an often boring Triangle Inequality Theorem and I think the intuitive meaning stuck with my students much more than in the past.
I know as a first year teacher I often felt I had no choice but to follow the book, which was incredibly hard with the precalculus book we use because it is written this way. But the longer I teach, the more I am convinced that motivation and interest are more important than almost anything else in math. We need to find ways to introduce topics in context so that students have a reason to learn (other than grades, of course).