When I meet with parents at school functions, they want to talk about what their child is doing in class that’s unique, that’s cutting-edge. So often the question of the purpose of learning mathematics is brought into the public domain. I try to steer the discussion into the creative side of mathematics, how “thwarting” student assumptions and processes can lead to deeper mathematical connections. That message is hard to convey adequately. Most apologists of the historic math curricula give some derivation of the importance of learning logic as a great justification of learning mathematics. It’s a seductive argument, although when analyzed falls apart.
The discussion lacks nuance. Real world applications are tangentially connected to actual problems at best, at worst they’re deceptive to the point of turning knowledge into a set of oppressive devices.
And to flame the argument, when those with bigger platforms than most – the big voices in education – describe what they think of the purpose of mathematics, I tend to listen. So when the great Ed Burger recently posted his “Truly learning math makes wise thinkers.” my ears shot up. This got some press in the local Austin and Houston papers. Below is my FireJoeMorgan-ification of his piece. Original is in bold and italics.
The question that educators and legislators in Texas should be discussing right now is not whether high school students should be required to take two years of algebra. This is an excellent example of investing time, money and effort to thoughtfully and carefully answer the wrong question.
Man I love Texas politics. Where else can we get a young state senator to filibuster for 8 hours and get a shoe deal out of it?
The right questions for all of us are: What positive and profound lifelong habits of effective thinking are we offering within all of our math classes?
Oh… at first I thought we were getting rid of SeaWorld. But, brother, I hear you. Also, can someone get on that whole “I can use your English state test scores to predict every other test score you take?”
And if the content of the algebra curriculum will be quickly forgotten after the last required exam (or even before), then why bother to offer any algebra?
Well… I’m not sure that doesn’t disqualify Pre-Algebra, Geometry, Algebra 1, Algebra 2, Statistics, Pre-Calculus, Calculus and pretty much every class I’ve taken at the high school level. In fact, If the metric we’re using to evaluate a class’s worth is “Do you remember the content after the last exam”, I don’t know what high school class this wouldn’t disqualify. We’ve changed the discussion from “should we keep Algebra 2 as a class” to “What are we doing with this whole standards and objectives based curriculum?”
Currently, too many of our math classes — as well as other classes — focus on mindless memorization and repetition that is designed to game a system focused on scores on standardized tests that measure the ability to perform a certain act — an act that requires neither deep understanding of the content nor the necessity to make meaning of the material.
Got it, repetition and memorization = bad. Deep understanding and meaning = good. Let’s store these equations for a bit.
Like magic, the moment the final exam is over, poof, the material is forgotten and magically disappears. Think it’s a joke? Math educators know otherwise. The overlap in middle school algebra, Algebra I and Algebra II is conservatively around 60 percent, and more realistically around 75 percent.
This small truth hurts. I understand redundancy when we’re talking about building ships to fly people to outer space. But the redundancy between all the Algebra curriculum reminds me of the painful waste of time when put into an overlap argument.
Our curriculum acknowledges its ineffectiveness at inviting students to make meaning of algebra: Those who study algebra in school are doomed to repeat it.
DOOOOMED I SAY! Also, what?
We need to replace our current math classes with meaningful mathematical experiences that teach students how to think through math rather than simply memorize formulas about math.
Ok Dr. Burger! I’m in. What does that look like? Does it involve teaching kids calculus at 5th grade? Does it involve something playful like sending students to the Desmos carnival?
By thinking through math, I mean understanding the material in a very deep way so that the student can appreciate and (ideally) discover connections between seemingly disparate ideas. Discovering relationships and patterns is not only at the heart of mathematical discovery but also the requisite trait to innovate and create in any space — from big business to the fine arts, from sports to technology, from politics to education.
This paragraph is meaningless drivel. Not once does Burger give any kind of non-jargon examples of what real learning, creating or playing with mathematics looks like.
In mathematics, we need to delve deep into the simplest of ideas until we see how complex they truly are. Only then can we pull back and see the bigger picture more clearly. One of the greatest triumphs of the human mind is that, by intent, we can take our current understanding and challenge ourselves to understand that much deeper—and, of course, that’s at the very core of education. These habits of the mind are what we need to be instilling in our students to enable them to become wise and creative leaders in an ever-changing, multifaceted world.
“Habits of mind” is utter filler and nonsense. “Habits of mind” is a generic approach that give nothing to looking at mathematics as something interesting or creative. “Habits of mind” is a desirable by-product, not the end goal.
Algebra provides a perfect example of this.
Algebra is a set of tools. Learning Algebra has little to do with making wise and creative leaders in an ever-changing multifaceted world. Learning when it’s appropriate to use those tools should be the outcome of mathematics education.
Calculus is one of the most beautiful constructs of humankind (of course, as a mathematician, I’m slightly biased).
Sure Calculus is beautiful. Let’s just start kids with that and then talk about the hyper-specialized cases that arrive in Algebra 2, right?
However, the whole subject revolves around just two basic ideas. So why do masses of students every year struggle with and eventually give up on calculus?
Kids ditch calculus? Or kids never had the chance to take calculus on account of Algebra and pre-calculus being so mind-numbingly boring?
The answer is because they never made meaning of the basic ideas of algebra.
Oh… I guess we agree here.
Even after manipulating the same equations for years in algebra, those students never were exposed to a curriculum that invited them to think through those equations and make them sing in their minds.
Ok Dr. Burger. Here’s where you’re getting me down. You are a textbook writer, the winner of many awards, and you’ve published a ton of material that has been some of the most popular written by a math educator. When you say students were never exposed to an interesting curriculum, this is where you put your money where your mouth is. What’s the interesting curriculum? What gets kids thinking? Where’s your advocacy really going to be powerful?
In my nearly 4,000 online math videos, I have attempted to make those ideas meaningful and, ideally, intuitive.
Awesome! We’re getting somewhere. I don’t really mind plugging your own work if it’s amazing. But then I tracked down a few of the videos where Burger tries to make ideas meaningful and intuitive.
Here’s an example:
“Something x plus something y equals a number.” “Standard form.” “x-intercepts”
Burger simply shows us how to graph the equation. There’s nothing playful here, there’s no higher wisdom to be gained. It is merely a process. It’s a context-less problem. It’s a rote memorization like he railed against at the beginning.
Damn it, I know where someone’s already going with this. Dr. Burger’s videos and general sense of style made him famous. His talks are incredible to go to, hell I’ve been to three in Austin myself. But these videos are dull and lifeless. They’re showing you how to do a hyper-specific process, and I can’t find anywhere where effective thinking comes into play.
This point can be applied to other subjects as well. In music classes, for example, students can simply memorize the finger movements in a piece.
Or watch a video of someone else doing it and try to copy them.
Or they could learn to hear each note and understand the structure of the piece.
Yeah, or that. That’s what the videos are for! Understanding structure.
My real beef is that so many talented minds go into the “let me show you how to do that” field, whether it’s making videos, apps or tutoring. And who can blame them? That’s where the money is at. No one is paying people to scaffold playfulness in math classes. They’re paying teachers to transfer information and skills to students.
In history classes, students can memorize basic facts about the Civil War such as the names of the generals. Or they could try to understand the background, competing forces and evolving social values that ignited the conflict.
Straw man alert! Who is saying history classes should be simply built around rote facts? What happened to the Algebra 2 framework which drew me in?
When teachers give assignments, they should always be asking themselves “What permanent benefit — what habit of thinking — will students get out of this exercise?” Teachers should craft assignments that promote long-term goals such as understanding deeply, learning from mistakes, asking probing questions, and seeing the flow of ideas. In other words, instilling lifelong habits of effective thinking.
Dude! Did you think you came up with that? Why is there no citation to Dewey, Piaget, Vygotsky or anyone else who have been saying the same thing 100 years ago?
And he came dangerously close to shamelessly plugging his book on effective thinking (which is on shelves near you).
Sure, I would be happy to see more students become math majors in college.
Pander to me baby!
But it is even more important to me that they learn to become wise, original and creative thinkers.
This is an empty wish. How exactly do students do so within any context, be it mathematics or somewhere else? I guarantee you, it doesn’t start with curriculum design or choices. It starts with teachers, parents, administrators and students all focusing on empowering students’ thinking. Without that, you could choose the most interesting content in the world, and it’s a lost point.
I’m not here to bash Dr. Burger. Like I said, I’ve cleared my schedule to hear him speak locally. What I’m objecting to is his empty jargon and recommendations around math education without even the slightest mention of what best practices actually look like. As a big voice in education, I dig the memorization = bad equation. But what else are you offering us? What’s your alternative?