Means Extremes – Balancing the Means and the Ends of Math Class

I have been seeing this play out in my geometry classes each of these past six school years. It’s been a tricky problem for me to figure out and once I started to see what was going on, it became even harder for me to communicate it. I think I am ready to try.

Each year, a fresh cohort of young people come in straight from Algebra I. In theory, I can assume that they are fully loaded with algebra skills and thought-processes that will support them through their study of Geometry. There is one glaring hole in their understanding which I attributed to the overall youthfulness of many of my geometry students. They are 13-15 years old and, for the most part, their math experiences lack a significant diversity. So, I am able to give them a pass on some of the ways they are still developing as math learners.

But then I saw the same deficiency in my calculus students that I am only seeing now because this is my first year teaching calculus. Suppose that I give a geometry student this problem and ask them to find the angle measures of each angle.

Taken from Holt Geometry - Pg 181

Taken from Holt Geometry – Pg 181

Or suppose I gave my calculus students this problem.

Soda Can Problem

Let’s add in as evidence that the Geometry students are used to application problems that look like this:

Taken from Holt Algebra I - pg 478

Taken from Holt Algebra I – pg 478

And that the calculus students have spent a lot of time looking at pages like this:

Taken from Holt Algebra II - Pg 580

Taken from Holt Algebra II – Pg 580

The primary difference between the work I’m asking my Geometry and Calculus students to complete and the work that they are used to in the Algebra I and Algebra II is in the latter the equations are provided and in the former, the students are required to write the equation.

This is no small point. No side conversation. I am not splitting hairs. I am convinced this is a big difference.

Let’s go back to our Geometry problem.
Taken from Holt Geometry - Pg 181 First and foremost, recognize that there is very little natural or intuitive about this set-up as a whole. There is very little reason why angle measures are represented with algebraic expressions. The variable “x” doesn’t represent any actual value and so, the students are left to their abstract understanding of how equations are built in order to solve this problem. Their previous experience hasn’t really prepared them for this. Overwhelmingly, their mathematical experience leading up to this point has trained them to know how equations are solved.

Let’s expand this to our calculus team (of 23) of whom I noticed only about 8 or 9 who seemed comfortable modeling volume and surface area with equations and then engaging the formulas. So, even among our most talented high school students, there is a problem with the use of equations as modeling tools. Once they have them, they can operate with them wonderfully, but they struggle when it comes to writing them to specifically to match a specific situation. And beyond that, checking the accuracy of the model and then making sense of the product once they are done.

Herein lies the major issue: the paragraph above highlights a variety of skills that students (at all levels, from what I can tell) struggle with. They seem to struggle with them because they aren’t practicing them. But those are the skills that actually make mathematics worth doing to EVERYONE. The ability to do complex arithmetic on a rational or logarithmic expressions is something that is going to come in handy to people for whom formal mathematics is going to extend into their post-secondary lives. This isn’t a high percentage or our students, but these skills constitute a high percentage of the problems in our textbooks.

On the other hand, being able to recognize a situation as linear, quadratic, logarithmic or rational and have a sense of how to model that in order to make some predictions? That is something that could be valuable to a higher percentage of people outside of school.

I think that we need to recognize that the specific skills that we are teaching our math students are a means, not an end. They are the tools, not the final product.

The real goal is for the students to explore a situation, recognize the mathematically significant parts and use their math tools to model the situation strategically to help them achieve their goal. In addition to our student being better, more confident, flexible and patient problem-solvers, it seems like we’d also hear “when am I ever going to use this?” a whole lot less.

2 thoughts on “Means Extremes – Balancing the Means and the Ends of Math Class

  1. I agree with your assessment. It is one of the reasons I like the 1-2-3 or PBL approach. The students are given a chance to identify a situation that can be solved, model the solution with some math tools, and find one or more solutions — on their own! ( okay, with guidance and good questions from us). This is a whole lot more satisfying than just being given blind algorithms. Students are not generally taught this skill until geometry or algebra II. Makes that ‘model with mathematics’ practice standard a lot more critical, doesn’t it!

  2. I totally hear what you are saying here. Yet, I give my Geometry students the Algebra “solving” problems because they need the practice with the series Alg 1, Geo, Alg 2. Helps to ask them to give you the rationale for setting the two expressions equal to each other, this seems to at least give them a foundation for doing the work in the first place. Just found you, will read more!

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