Philosophy of Math and Math Education

Conception of mathematics.

How would you describe or define what math is?
Philosophers, mathematicians, and others have described the field of mathematics in a number of compelling and sometimes contradictory ways; my personal conception of the field draws from several of these:

• Mathematics is about measurement. It provides the tools necessary to provide useful descriptions (of a certain type) of the physical world
• Mathematics is about problem solving. It gives us a language that allows us to not just superficially observe the world around us, but to deeply analyze it.
• Mathematics is about logic and deductive reasoning. It provides tools of reasoning that often allow us to move beyond inductive thought, and come close to deductive certainty.
• Mathematics is about abstraction and modeling. It allows us to see similarities between disparate concepts through its underlying descriptions. Here's a simple example: once a mathematical model is developed to measure the velocity of a tennis ball, this same model can be adapted to measure the velocity of a basketball, or of any physical object moving through space.
• Mathematics develops intuition. At yet another level of abstraction, modeling velocity can provide an intuitive understanding of broader concepts: rates of change, relationships between variable quantities, etc.

Where do you think math comes from?

• I see the field of mathematics as a set of human-developed tool and philosophies for understanding and manipulating the world.
• I see the discovery and learning of mathematics as coming from multiple sources. Students are instructed in the language of mathematics by teachers and textbooks, but mathematical concepts can be self-emergent, as these students consider how to describe the world around them. Student observations can lead to questions, which can lead to mathematical discovery. Consider the "double your money every square" chessboard example: an attentive observer may, in wondering about the pattern of coin growth, discover for themselves a model for exponential growth.

How strongly do you feel about your conception of math (what it is and where it comes
from)?
I think that the way I've conceptualized the field is useful, but in no way would I suggest that it's the only, or best, way to describe mathematics in general. Certainly many pure mathematicians would have a different emphasis in their thoughts about the subject!

Math learning experience

Paint a picture of what a typical math class experience looked like in high school for you.
My high school math education followed a conservative model: students sat in rows of desks, and mostly listened to the teacher, opened their textbooks, tried to solve problems. Some teachers would occasionally ask students to share their work on the board, and AP calculus introduced the graphic calculator to the mix, but otherwise interactive moments were few and far between.

The sole speaker was the teacher in nearly all circumstances except "call and response" homework review. For this, the teacher would ask questions and expect student responses. Otherwise, the sound of math class was that of the teacher's voice, and of the furious scribbling in notebooks.
There was no formal groupwork or collaborative learning, though some of my teachers were relaxed enough during "be quiet and work through these problems" sessions that the stronger students could provide some unofficial assistance to their struggling classmates. The problems we worked on were either textbook problems (though never the applications problems at the end of each chapter!) or simulacra of the same, provided by teacher handout.

Do you think you learned math well this way? What worked/didn’t work for you?
Even after that somewhat-bleak picture of my high school math experience, my answer has to be "Yes". But my personal experience doesn't generalize well: I'm a very self-directed learner, and my high school knowledge and skills acquisition had very little to do with what was being taught in the curriculum. If I was disinterested in a subject, as I was (at the time) in American history, I'd skip the readings and the homework--and sometimes class! If I was interested in a subject, as I was with calculus, I'd jump chapters ahead and work through problems that weren't assigned.

But I still didn't pay a lot of attention in class, and I didn't complete much of the assigned work. Though, I can't speak to his motivations, my calculus teacher actually provided some support for this style of learning. When he found me designing calculus software for my graphing calculator rather than completing worksheets, instead of reprimanding me he listened as I explained what I'd created, how, and why. Ultimately, he actually recommended me for a math award at graduation for that work--despite my having almost never completed a single homework assignment for him!

I'm acutely aware that my experience, while not unique, is not particularly common. I was fortunate to not have to struggle with academic work, and was defiant enough to not be invested in the consequences of ignoring/not taking seriously "busywork" and mediocre instruction. So even in my worst classes, the teaching style didn't directly affect me in a profound way. In classes like American history, a good teacher might have made me love the subject and want to learn more, but at that stage in my development I still don't know if I'd have done the homework!

Your vision

As best as possible, paint a picture of how you think you'd want your math class to look.
I'm hesitant to sketch any firm lines around my math classes. I think the answer to this question must vary based on my students, their ages, abilities, interests, personal backgrounds, etc. I currently teach review math to classes of talented adult learners, and I'm happy with how my classroom is structured--but I can't imagine that my high school classroom would have much in common with this! A totally different student population, with totally different interests and needs.

Here are some general principles that I'd like to adhere to:

• High expectations. I don't like classes that get distilled into "lifestyle math"--how to calculate a tip or pay your taxes are important skills, but that's barely the beginning of math, certainly not the end goal! Good in-class examples and applications need to move beyond this superficial level.
• Extensive use of applications that emphasize not just modeling, but careful critical thinking
• Strong resistance to strict assessment-based teaching, coupled with an awareness of the very practical need of test preparation that most students will have.
• Some student-student learning opportunities are an appropriate extension of the traditional classroom, especially in classrooms where individual ability varies widely.
• An openness to innovation and non-standard techniques, but a cautious openness. Not every new idea is an improvement on what came before; I'd prefer to (try my best to) cultivate a careful mix of the conservative (what has worked) and the radical (what might work) in my classroom.

Describe what you believe should be taught in mathematics class.
This is a big question. Do I agree that the Algebra-Geometry-Precalculus-Calculus (for the privileged kids) progression is right or best for everyone? Certainly not!

At the same time, though, I'm highly resistant to the idea that "all math classes should be immediately practical", e.g. business math and the aforementioned "lifestyle math" classes. I don't believe that the key role of primary/secondary schooling is to prepare students to become workers, but there's no denying that aspect of education. In this context, I envision a model of schooling that provides access to a strong enough foundation of knowledge, skills, and critical thinking ability that most high school graduates could successfully explore nearly any career possibility and life path intelligently and capably.

This includes instruction in the language and use of mathematics, as it is used and practiced in the wider world. High school algebra and geometry provide useful bases for modeling the world and manipulating numbers and spacial relationships--most careers outside of the humanities, and many careers within, make use of this kind of knowledge. The more general critical thinking skills that a strong math education can provide are useful everywhere.

This certainly doesn't mean that I think the way these courses are taught is necessarily best, or even good! The European "Mathematics I-Mathematics II..." model might make better sense: I'm not well-versed enough in theory to have formed an opinion on that. Otherwise, there are at least two things I'd like to see done differently.

• There are many seldom-explored opportunities to integrate math with a broader curriculum; today it's too-often kept in its own silo, only occasionally interacting with chemistry and physics classes.
• One aspect of "practical math" that I do agree is sorely lacking in most secondary education is basic statistics. Interpreting data, and the numerical conclusions drawn from them, is a critical skill that many high school graduates lack.

Describe what you believe the role of contexts should be in teaching math. Explain why you
think this is a good way to teach math.
As I've stated, I think mathematics is, especially at the secondary-ed level, primarily about contexts/applications. I never liked how routinely teachers would skip the "Applications" questions at the end of each textbook chapter, out of a need to check boxes on a unit plan or imposed curriculum. I'm sure I'll face some of the same challenges and pressures, but I hope to be able to make relevant applications of mathematics a principal part of my instruction. Context allows us to connect with students in multiple ways--

• to engender interest in math through a recognition of the omnipresence of mathematical ideas in their own lives
• to suppress the "boredom factor" of stale textbooks and completely abstracted work
• to build students' engagement with their own communities and experiences
• • to extend the boundaries of students' knowledge of (and interest in!) the broader world