The Hechinger Report asks, “Is it better to teach pure math instead of applied math?”:

In the report, “Equations and Inequalities: Making Mathematics Accessible to All,” published on June 20, 2016, researchers looked at math instruction in 64 countries and regions around the world, and found that the difference between the math scores of 15-year-old students who were the most exposed to pure math tasks and those who were least exposed was the equivalent of almost two years of education.

The people you’d imagine would crow about these findings are, indeed, crowing about them. If I were the sort of person inclined to ignore differences between correlation and causation, I might take from this study that “applied math is bad for children.” A less partisan reading would notice that OECD didn’t attempt to control the pure math group for exposure to applied math. We’d expect students who have had exposure to both to have a better shot at transferring their skills to new problems on PISA. Students who have only learned skills in one concrete context often don’t recognize when new concrete contexts ask for those exact same skills.

If you wanted to conclude that “applied math is worse for children than pure math” you’d need a study where participants were assigned to groups where they only received those kinds of instruction. That isn’t the study we have.

The OECD’s own interpretations are much more modest and will surprise very few onlookers:

• “This suggests that simply including some references to the real-world in mathematics instruction does not automatically transform a routine task into a good problem” (p. 14).
• “Grounding mathematics using concrete contexts can thus potentially limit its applicability to similar situations in which just the surface details are changed, particularly for low-performers” (p. 58).

BTW. I was asked about the report on Twitter, probably because I’m seen as someone who is super enthusiastic about applied math. I am that, but I’m also super enthusiastic about pure math, and I responded that I don’t tend to find categories like “pure” and “applied” math all that helpful. I try to wonder instead, what kind of cognitive and social work are students doing in those contexts?

BTW. Awhile back I wrote that, “At a time when everybody seems to have an opinion or a comment [about mathematics education], it’s really hard for me to locate NCTM’s opinion or comment.” So credit where it’s due: it was nice to see NCTM Past President Diane Briars pop up in the article for an extended response.

Featured Comment:

Chris Shore:

What is often overlooked in these kind of studies is the students who are enrolled in the various courses. The correlation between pure math courses and higher level math exists because higher achieving students are placed in the pure math classes, while lower performing students are placed in applied math.

Same thing is true for studies that claim that students who take calculus are the most likely to succeed in college. No duh! That is because those who are most likely to succeed in college take calculus.

The course work does not cause the discrepancy, the discrepancy determines the course work.

The internet has failed me.

In spite of following 150 creative math teachers on Twitter and subscribing to 750 creative math teacher blogs (including one blog that’s dedicated exclusively to creative math), I’m only now learning about Gordon Hamilton’s Unsolved K-12 Project. It’s creative. It’s math. It’s almost three years old!

Better late than never.

See, Hamilton convened a bunch of creative math types in Banff in November 2013 to a) select unsolved math problems and b) adapt them for use at every grade in K-12. Not a simple task, and I’m enormously impressed by their results. You can watch videos introducing the problems at this page or read about them in these slides.

Here are two of my favorites. (Click for larger.)

Grade 3: Graceful Tree Conjecture

Grade 10: Imbedded Square

These two problems have the capacity to develop fluency just as well as any worksheet or worksite. In working out their solutions, students will perform the same operation dozens of times — subtracting whole numbers in the third grade task and calculating slope and distance in the Cartesian plane in the tenth grade task. But these problems ask students to think strategically and systematically in addition to practicing efficiently and accurately. That’s no easy feat, but Hamilton and his team pulled it off thirteen times in a row.

Related:

• Lior Pachter’s Unsolved Problems with the Common Core
• Purposeful Practice & Dandy Candies
• Open Middle

Ann Shannon asks teachers to avoid “GPS-ing” their students:

When I talk about GPSing students in a mathematics class I am describing our tendency to tell students–step-by-step–how to arrive at the answer to a mathematics problem, just as a GPS device in a car tells us — step-by-step — how to arrive at some destination.

Shannon writes that when she used her GPS, “I usually arrived at my destination having learned little about my journey and with no overview of my entire route.”

True to the contested nature of education, we will now turn to someone who advocates exactly the opposite. Greg Ashman recommends novices learn new ideas and skills through explicit instruction, one facet of which is step-by-step worked examples. Ashman took up the GPS metaphor recently. He used his satellite navigation system in new environs and found himself able to re-create his route later without difficulty.

What can we do here? Shannon argues from intuition. Ashman’s study lacks a certain rigor. Luckily, researchers have actually studied what people learn and don’t learn when they use their GPS!

In a 2006 study, researchers compared two kinds of navigation. One set of participants used traditional, step-by-step GPS navigation to travel between two points in a zoo. Another group had to construct their route between those points using a map and then travel segments of that route from memory.

Afterwards, the researchers assessed the route knowledge and survey knowledge of their participants. Route knowledge helps people navigate between landmarks directly. Survey knowledge helps people understand spatial relationships between those landmarks and plan new routes. At the end of the study, the researchers found that map users had better survey knowledge than GPS users, which you might have expected, but map users outperformed the GPS users on measures of route knowledge as well.

So your GPS does an excellent job transporting you efficiently from one point to another, but a poor job helping you acquire the survey knowledge to understand the terrain and adapt to changes.

Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.

I’ll take that trade with my GPS, especially on a dull route that I travel infrequently, but that isn’t a good trade in the classroom.

The researchers explain their results from the perspective of active learning, arguing that travelers need to do something effortful and difficult while they learn in order to remember both route and survey knowledge. Designing learning for the right kind of effort and difficulty is one of the most interesting tasks in curriculum design. Too much effort and difficulty and you’ll see our travelers try to navigate a route without a GPS or a map. While blindfolded. But the GPS offers too little difficulty, with negative consequences for drivers and even worse ones for students.

2016 Jun 17. The two most common critiques of this post have been, one, that I have undervalued step-by-step instructions in math, and two, that this GPS study offers very few insights into math education. I respond to both critiques in this comment.

Henry Pollak, in his essay, “What Is Mathematical Modeling?”

Probably 40 years ago, I was an invited guest at a national summer conference whose purpose was to grade the AP Examinations in Calculus. When I arrived, I found myself in the middle of a debate occasioned by the need to evaluate a particular student’s solution of a problem. The problem was to find the volume of a particular solid which was inside a unit three-dimensional cube. The student had set up the relevant integrals correctly, but had made a computational error at the end and came up with an answer in the millions. (He multiplied instead of dividing by some power of 10.) The two sides of the debate had very different ideas about how to allocate the ten possible points. Side 1 argued, “He set everything up correctly, he knew what he was doing, he made a silly numerical error, let’s take off a point.” Side 2 argued, “He must have been sound asleep! How can a solid inside a unit cube have a volume in the millions?! It shows no judgment at all. Let’s give him a point.”

What a fantastic dilemma.

Pollak argues that the student’s error would merit a larger deduction in an applied context than in a pure context. In a real-world context, being wrong by a factor of one million means cities drown, atoms obliterate each other, and species go extinct. In a pure math context, that same error is a more trivial matter of miscomputation.

The trouble is that, to the math teachers in the room, a unit cube is a real-world object. They can hold a one-centimeter unit cube in their hands and, more importantly, they can hold it in their minds.

The AP graders aren’t arguing about grading. They’re trying to decide what is real.

What a fantastic dilemma.

Featured Comment

Shannon Alvarez:

Whenever I wanted to give students the most amount of partial credit, my coop teacher would ask me the poignant question, “What exactly are you assessing?” I found this was a great question to continue asking myself. So, in the example you gave, are you assessing students’ ability to perform mathematical functions correctly or are you assessing their ability to connect those math functions to the real world?

Benjamin Dickman alerts us that Pollak’s piece is online, free, along with a number of his modeling tasks.

Thanks for making the trip all the way out to the San Francisco Bay Area, team. It’d be a treat to run into any or all of you during conference week. Here are my whereabouts.

Association of State Supervisors of Mathematics

Math, Education, and Technology: Reasons for Pessimism and Optimism. Monday. 9:15AM. The Metropolitan Room.

My optimism and pessimism are currently balanced at about 50/50.

National Council of Supervisors of Mathematics

Beyond Relevance and Real World: Talking with Teachers About Engagement in Mathematics. Monday. 1:30PM. Grand Ballroom EFGH.

I’ll summarize some of what I’ve learned from five years of offering workshops to teachers on student engagement in mathematics.

National Council of Teachers of Mathematics

Task Makeover Techniques for Grade 6-12 Math. Friday. 11:00AM. Exhibit Hall.

This is one of those informal “Networking Lounge” sessions. I’ll bring some textbook tasks that are eager for our help. We’ll brainstorm improvements.

ShadowCon. Friday. 5:00PM. Yerba Buena 7.

Zak Champagne, Mike Flynn, and I recruited another set of six amazing speakers from across K-12: Burrill, Bushart, Fletcher, Gutierrez, Kaplinsky, and Turner. We’ll film and post all their talks. They’ll all be eager for your conversation afterwards.

Beyond Relevance & Real World: Stronger Strategies for Student Engagement. Saturday. 9:30AM. 134/135 (Moscone).

This is the last time I’ll give this talk. I’ve been polishing it over the last 15 months, sanding off rough bits, tightening its scope, and adding more practical strategies. I’m really happy with it. I will also record it and post it here shortly after NCTM, so I welcome you to enjoy any of the 57 awesome other sessions NCTM has wedged into that timeslot.

Looking for Some Extracurriculars?

• Desmos and Mathalicious are hosting their fourth annual happy hour and trivia contest on Thursday at 6:30PM. First round is on us.
• From 1:00PM to 5:00PM on Friday you’ll find the Desmos Teaching Faculty in the Desmos booth answering any questions you have about our activity builder or calculator. (My slot is 2:00PM.)

Whose sessions are you excited to see? (Here is the unofficial #MTBOS track.) Recommend some new names, fresh voices, or first-timers in the comments below.