The day after I’d handed back a three-question quiz, a student marched into class and told me his parent had shown him I had marked his quiz wrong.
“That’s always a possibility,” I told him, pulling up a chair to my desk for him. “Let’s take a look.”
Here is the problem this was about.
The instructions read, “Find the length of the segment indicated. Round your answer to the nearest tenth, if necessary.”
He did a good job of walking me through his parent’s solution path. Since one of the circle’s radii is drawn perpendicular to the chord that it also bisects, we can deduce that each part of the bisected chord has a length of 10.9 units long.
Then, using the Intersecting Chords Theorem, his parent carried out the following calculation:
10.9 * 10.9 = 13.1 * x
After multiplying through and dividing by 13.1, the parent concluded that the correct answer should have been x = 8.3.
There was one small problem with this.
The Intersecting Chords Theorem doesn’t apply to this situation. It’s actually a totally different problem type.
This is a common problem of perception around which an entire new body of learning theory is emerging. This research has unfortunately not yet reached math education, but to my mind, it connects the essential goal of high school mathematics — cultivating an adaptive mathematical learner — with some of the most urgent problems that all current math instruction fails to equip students for. As one researcher puts it, “Adaptation can only begin when people discern new information worthy of a response.”
It turns out that, in addition to internalizing conceptual understanding and procedural fluency, math students who want to succeed in STEM fields also need to learn to braid these two skill bases with adaptive skills to be prepared to deal with nonroutine situations. This adaptive capacity enables learners to notice and name variations and anomalies in situations they encounter — a capacity that can open the door to innovative approaches to problem-solving.
So let’s go back to our quiz problem.
The first thing to notice is that of the two given radii in the diagram, the radius intersecting the chord is divided into two parts, only one of which is labeled as x. The other part is unlabeled, so let’s fix that problem and label it as y.
Since all radii of a circle are congruent, that means the length of our radius can be represented by the equation:
13.1 = y + x
My work as a public school math teacher is to get everybody over the finish line. I want all my students to have options, including advanced STEM studies and careers. That’s why in my geometry teaching, I focus on this skill of noticing and adapting to different nonroutine mathematical “situations.” A situation is basically a bigger system built out of smaller or more basic systems. Learning how to diagnose a particular constellation of systems is what unlocks possible solution paths.
In addition to internalizing conceptual understanding and procedural fluency, math students who want to succeed in STEM also need to learn to braid these together with adaptive skills to address novel, nonroutine situations.
To know what tools and adaptations are available to them in this case, a student has to be able to recognize and use the fact that x is only a part of the whole radius. For most students most of the time, this perceptual skill — the habit of spotting and naming parts and wholes — is an actual skill (known as “information pickup”) that students need to train themselves to use if they want to reach higher mathematics. Most high school students don’t even know that they don’t think about this. So we name it and teach it explicitly. It helps a lot. This is one of dozens and dozens of micro-components of becoming an adaptive math learner that even my strongest math students need to learn about and master.
Let’s go back to our problem.
Now that we see that x is only part of our situation, we can also see that the formerly unlabeled part of that radius – which we’ve named y and assigned its role as a missing part of a known value – is the far more useful part of that segment in terms of finding x‘s measurement. This lets us shift our focus toward finding y as a route for finding x.
This also allows us to imagine another radius in our diagram — one drawn from the center of the circle to the endpoint of our 10.9-unit segment, where it crashes into the circle. I’ve drawn this radius in orange:
This gives us a right triangle — a familiar situation that unlocks our whole toolkit of right-triangle tools. Now we can simply use the Pythagorean Theorem to solve for y.
So now we have:
13.1 2 = y 2 + x 2
And this enables us to determine that y = 7.3.
Once we know y, solving for x becomes a simple matter: x = 13.1− 7.3, which equals 5.8.
Which is not the answer my student’s parent had arrived at.
I loved the fact that he and his parent had done math together, regardless of the answers they’d arrived at.
This is the kind of thinking that advanced 15-year-olds need to learn how to do – the psychological subtle noticing you miss out on if you’re in a big hurry. Conceptual understanding and procedural fluency aren’t enough.
Sometimes the fastest way to hurry up is to learn how to slow down.
