September 2023
Yvonne Lai (University of Nebraska-Lincoln)
Ji Y. Son (California State University, Los Angeles; CourseKata.org)
Dramatis Personae
A…………April the Applied
T…………Thea the Theoretical
Act I: Isn’t this what you believe?
April the Applied is taking a nice walk in the neighborhood.
Her neighbor Thea the Theoretical shoves open her front door. She glares at April the Applied.
Thea the Theoretical: How dare you! You accuse me of dragging kids through the useless doldrums of mathematics just to weed out kids who can’t passively endure rote manipulation of algebraic symbols. So you’re off peddling the magic key, to feed kids feel-good applications like videos of Pixar artists using trigonometry and pre-made apps for making fancy graphs. Do you think a few glitzy videos and apps will convince kids to see the usefulness of math and suddenly they will just understand polynomial functions? But you won’t teach them any fundamental mathematical –
April the Applied: How dare you! You think your wishful thinking is going to change the status quo in math education?
T: My wishful thinking? My thinking is grounded in –
A: Oh, let me guess. “Logic,” right? For you, “logic” is when kids mimic theorems already proven by people that have been dead for centuries. You think students forced to solve for some meaningless x over and over again will magically become more logical? I have bad news for you. Students learn all of this so-called “logic” as useless ritual.
A, T: It’s your fault that students don’t learn math!
Doors slam. They storm into their own houses.
Evening falls and a breeze rustles.
A and T both feel pensive and a little regretful. They come out on their respective balconies and gaze at the stars.
A: I’m sorry I yelled at you. Can we talk about what we actually believe?
T: I’ll try to listen.
A: Me too.
Act II: This is what I really believe
A: I want students to be curious about math. There are too many people who felt like their curiosities about the world were not welcome in their math class. Sometimes, a decade after they first memorized y = mx + b, they see a connection between linear functions and something they are actually curious about. Years after struggling with rote factoring and decimal multiplication as separate topics, they realize that factoring, division, and multiplication are systematically intertwined in proportional reasoning situations. Wouldn’t students and adults both be better off if they saw right away how math connects to the world around them, early and often?
T: It sounds like you want students to learn math…?
A: Yeah, I actually do! I just think that applications have a role in leading students towards more math. When middle school and high school students see patterns of data that don’t quite fall in a straight line, they might wonder, “Are there models that can predict this curve in my data?” Applications could motivate students to want to learn piecewise, quadratic, or trigonometric functions! Students might initially think they are just exploring the world by delving into a social issue or an engineering idea, but these applications could revive students’ curiosity about math.
T: But does it have to be applications? Mathematicians are curious about the interesting patterns in math itself! I’ve seen that students can get curious even in tasks that have no applications. Students are weirded out that whenever they construct three right triangles with the same hypotenuse, they all end up in the same circle!
A: Woah, that’s creepy! Those right triangle vertices really always form a circle? What?!
T: See how you feel curious? Even though it’s not an application, we can get students to be curious about Thales’ Theorem! Students can be and are curious. Real-world applied contexts can support students’ mathematical learning but even a highly abstract situation can serve as a context. The goal of using a context, problem, or constraints is to get students to be curious and ask questions. Highly abstract situations can motivate students to want to know why.
A: So you aren’t just trying to ram abstractions down students’ throats. You want them to ask why!
T: Yes, through deductive problem solving students can have an unshakeable understanding of why. That’s the elegance and power of math.
A: That sounds like an amazing math experience. Hey, crazy idea – what if we worked together?
Act III: What if … ?
T: What if designing students’ mathematical experiences didn’t have to be a zero sum game?
A: It seems as though we both want students to be curious about math!
T: Yes, whether students are curious about learning more math because of an applied problem or asking why some mathematical principle works, or both, we want all students to feel like their curiosity is welcome in mathematics, and even nurtured.
A: I’m in total agreement with you there. But I think we also agree that this isn’t what enough students currently experience. Too many students are still asking, “When am I ever going to use this?” and no one was asking for more theorems last time I checked. But I’m glad we can move together towards the same goal instead of fighting each other.
T: Agreed. And we need to keep in mind that too many students are still asking, “How am I ever going to use this?” even when applications are brought up. We have to find a way to make applications in a mathematics class not just relatable, but also mathematically substantive.
A: Yes, we can’t foolishly believe that using TikTok stats or placing a word problem in Wakanda makes math more fun. This illusion is nothing but, as Skew the Script says, “a veneer of relevance”. Just as solving a mystery to unveil the actual culprit captivates us, ultimately, the appeal of math must come from unveiling the mathematical concepts themselves!
T: When it comes to applications-based and more abstract motivations for mathematics, perhaps we are like a scatterplot and a sine wave looking for a fit between messy data and idealized definitions. Both of us are advocating for teaching and learning opportunities that increase students’ curiosity, motivate them to learn mathematics, and believe that mathematical ideas make sense. I think we both believe students can make sense of ideas through reasoning in abstract contexts or more realistic ones.
A: When we began, my scatterplot and your function were in different worlds. And as we listened to each other, our waves started synching up.
T: We’re in the same boat, my friend. Let’s row in the same direction.
Fin.
Epilogue beyond the fourth wall
April the Applied and Thea the Theoretical are avatars for Ji and Yvonne. Ji is a cognitive scientist who co-authored (with Jim Stigler) a high school data science textbook that uses simple functions students learn in algebra to model patterns in data. As she wrote in an op-ed with Jim Stigler, there need not be a false dichotomy between data science and algebra. Yvonne is an education researcher whose background includes hyperbolic geometry research. She is the research lead and part of the algebra materials writing team for MODULE(S2), a design and development project to strengthen mathematics courses for future high school teachers. Yvonne wants students and teachers to have many chances to experience proof as a part of a process of mathematical discovery.
Ji and Yvonne are troubled by how relevance and mathematical rigor have been construed as combatants in a zero sum game of educational policy. But most mathematicians and educators can agree that applications and deductive reasoning both play important roles in mathematics education and can be mutually supportive practices. Thus we wrote this together as the first of a series of posts that explore how the “relevance” of applications and the “logic” of mathematical thinking could work together. In this first post, our avatars say out loud some of the extreme accusations lobbed about each other but come to realize that we might both have the same learning goal in mind: students’ curiosities propelling them towards more about math. In our future posts, we’ll address specific strengths and weaknesses of application and deductive approaches and ways that they can work together.