I don’t have time to get into the full 13 (? iirc) steps of Liljedahl’s Thinking Classrooms approach, but it’s exactly designed to meet the needs of students like you. Some highlights:

• Students are randomly assigned to a new group of 3 daily
• All students work on vertical whiteboards, or equivalents
• The teacher presents a math task that starts easy-ish, but requires some work/thought to figure out
• If 30% of students in the room understand the task, then it will quickly trickle between groups
• The teacher circles exemplars of great thinking; students are not allowed to erase these until the next debrief
• The teacher regularly cycles back to get students to explain their work to the class, showcasing and explaining the bits the teacher circled
• Start over with a more advanced task/“next step”

It’s an incredibly effective teaching method for secondary math. And there’s clear motivation every step of the way for what you’re doing and why it matters.

And the teacher only explains about 5-10% of the material; everything else is explained by the students as the carefully curated progression of activities guides them through discovering the math themselves.

Yes, examples like that are good, of course. But, frankly, abstract examples like that won’t do much to motivate the students who need the most help to get motivated learning math.

I like to interject little anecdotes like that, too. One of my “go tos” to “why are quadratics useful” goes something like “Well, they come up a fair bit, so I could give you some examples—and I will, as we with through the unit, but the real reason we teach quadratics is because they’re the simplest non-linear function. This is the first steps into looking at functions that aren’t a straight line. And the tools you use to work with quadratics are super important for understanding all the really cool functions you get to learn on the next couple of years…”

That’s basically your example, but one step lower and more directly applicable to students, imho. The Taylor Series thing I usually only drop in grade 11/12 (pre)calculus classes, mostly as a hook for the math nerds that they have really cool things to look forward to learning in post secondary. It’s a terrible application to use to try to motivate learning about polynomials for a student who couldn’t care less, lol.

Really, we need to intermix all approaches, depending on the students in the class. At private prep schools, leaning into academic needs works well. In a non-academic math stream, both your example and my examples will go over like a lead balloon.

But, regardless, motivating students to be excited for math, and the excitement of finally figuring out a tricky concept/problem? That’s what we need more of.

If by “practical application” you mean “motivation for learning the skill”, which is I think the way you’re using it, then yes. But that’s not the usual definition in math education, and not what most people mean by it.

Like, for example, to introduce quadratics, a good progression might be to challenge students to build a table of values and graphs for x², then x² + 3, then graph x² – 5 without a table of values, then 2x² vs. 5x² vs. ½x², –x², etc.

And if you have a Thinking Classroom, every student in the class is working on figuring out that progression collaboratively in small groups. The teacher guides students to discover the math themselves through a series of examples, and mostly interacts with the students by asking questions, never giving them the answers.

That’s not “a practical application of quadratics”—at least not in the usual definition—that’s a learning activity sequence (paired with a set of interrelated pedagogical practices).

A good, practical application of quadratics is more like a Dan Meyer “3 Act Math” lesson on predicting the trajectory of a basketball shot. Also cool, good teaching. But not a great way to introduce quadratics.

(P.S. Yes, I use and like em dashes. I’m not a robot.)

Citation needed.

Seriously, though, that’s not what the research is showing. Peter Liljedahl’s research, for example, supports that a very effective way to teach mathematics is by having students actually think about math, instead of just passively receiving info dumps (as is common in most traditional math classes). See Building Thinking Classrooms for details but, in short, it’s a method of getting students playing with math concepts for almost the entire class time every day.

No “practical applications” needed. Counterintuitive, but it’s a highly effective practice.

What’s core to practical applications working is student motivation, and practical applications are one way to induce motivation. But it’s often not the best option, especially for inherently abstract skills.

That kinda breaks down in practice, though. Math is hard for a lot of students. Adding an extra layer of domain-specific application on top of an already confusing topic just makes it worse.

Like, we need polynomials for huge swathes of higher-level math. My favourite application of polynomials is that most continuous functions can be approximated by a Taylor series, which makes some functions that are otherwise impossible to calculate a derivative or integral trivially easy. It’s elegant, beautiful, and deeply practical.

And completely useless for a grade 8 student learning about polynomials for the first time.

Sure, there’s lower-hanging fruit for practical uses for polynomials, but they’re either similarly abstract (albeit simpler) or contrived. Ain’t nobody making a sandbox with length (3x + 5) and width (2x – 7), eh?

I could go on. At length.

Point being, yes, practical applications are better. BUT (and this is a big but) only when there are simple practical applications.

Instead, recent math education research supports teaching fluency through playing with math concepts and exploring things in many ways: symbolically, graphically, forwards and backwards, extending iteratively with increasing complexity, etc. This helps students develop intuition for math concepts and deeper understanding. Then, and only then, teach the standard algorithms and methods, as students will appreciate the efficiency of the tool and understand what they’re doing and why they’re doing it.

Thank you for listening to my TED Talk.