Teaching more than what we know we need
An Economist looks at Math Ed
Freakonomics co-author Steven Leavitt recently re-ran an episode of “People I Mostly Admire” which includes his criticism of US Math Education and what he would like to see changed.
I was surprised by my reaction as I mostly agree with his conclusions.
It was like my reaction to the Nathan Myrhvold book on bread which criticized bakers for not using his modern and more effective techniques. Myrhvold’s book didn’t contain anything that working bakers didn’t know - but they are constrained to produce a consistent product at a scale that Myrhvold’s techniques wouldn’t support – certainly not at the price that bakers would have to charge for such loaves.
I first taught math more than forty years ago. I’ve met some bad ones here or there but I’ve met a lot of really good, caring, thoughtful ones. I’ve lived through and witnessed reforms and many of the ideas and intents are good and many don’t think through what it would take for implementation.
The power of the textbook lobby is huge and some states adopt statewide and have undue influence.
Richard Feinman tells a story of serving on the board for adopting a new series of textbooks in California and the publisher sent notes to the board members apologizing that the book wasn’t ready in time. The group was sent fakes - they were covers bound around blank pages. His fellow board members spoke in support of the book without realizing they couldn’t have evaluated it because they weren’t sent the actual book.
Boards and control of curriculum by those who don’t now the subject and standardized testing that evaluates the same old thing are more to blame than teachers. Most teachers care deeply about their subject and their students and would love to make things better.
What to change
Leavitt argued against the “Geometry sandwich”. In the US we teach a year of Algebra, a year of Geometry, and a year of Algebra II. I’ve hated that separation since I was fourteen and took a year of integrated “Maths” in the UK. I taught in Newton, Mass where we combined the two subjects but the students had to carry a Geometry and an Algebra textbook because nothing was set up to support this combination.
He also argued that we should teach more Data manipulation. Students should be able to work with data sets and come to conclusions. They should be able to read conclusions from others and reason about them.
I love this idea as well.
I hated his justification.
He argued that in surveys a very low percentage of people report working with Calculus or Geometry in their everyday life. We should teach math that is more useful.
That is a standard that Math is held to in ways that no other subject is.
The math we teach must be useful.
I would hate for us to argue that literature, art, or music should not be taught because it isn’t useful.
Maybe in English class we should teach people to read and write tweets and FaceBook posts - they’ll be much more likely to use that than they will use the plays, poems, and novels we have them read now.
Three decades ago I was invited to a Math/Science retreat in Breadloaf and one person explained that the college science curriculum was designed to prepare people for graduate and professional schools. The high school science curriculum was designed to prepare students for college. The middle schoolers were being prepared for high school and the elementary students were given what they would need in middle school.
The presenter argued that our elementary curriculum was designed for the few that would end up in graduate school or med school.
So it’s not that things don’t need to be changed. I just worry about “useful” as being our guide.
What’s the Use
In the introduction to his book “What’s the Use”, Ian Stewart asks, “What is mathematics for? What is it doing for us, in our daily lives.”
Although I’m a huge fan of Stewart’s I discovered this book in a post that thought that Stewart was making their point.
Stewart responds to his question by saying, “Not so long ago, there were easy answers to these questions. The typical citizen used basic arithmetic all the time, if only to check the bill while shopping. Carpenters needed to know elementary geometry. Surveyors and navigators needed trigonometry as well. Engineering required expertise in calculus.”
He then argues that none of these uses exist anymore so we could conclude that “mathematics has become outdated and obsolete.”
At this point most readers of articles about the book stopped reading. This was their takeaway - math is outdated and obsolete.
But, says Stewart, “that view is mistaken. Without mathematics, today’s world would fall apart.”
There are obvious applications of math to our every day world. This is not really surprising or sexy. A member of our iOS group has designed an app to help plumbers who need to bend pipes to make them fit in certain situations. The fact that his app uses elementary trigonometry is not a surprise - that’s what trig was invented for.
It is the cases where math is applied to completely different fields that is so surprising. As an example, Stewart tells the story of Euler working out a puzzle of people walking on bridges from the early 1700s that asked if every bridge be traversed exactly once on a walk. Nearly four hundred years later the solution has application to kidney transplants.
We develop basic Math and Science with no clear application and then are prepared when the need for it arises.
The speed at which the COVID vaccines appeared were because of research into mRNA that had been going on for years. (This being a non-Math example, I need to note that I could be wrong here.)
We can’t let “use” exclusively determine what we spend our time and resources on.
A few years ago Maggie found a special program for me to get training as an expert in Gerrymandering cases. They were looking for people with a combination of advanced Math background and journalism and so I applied. I wasn’t selected but the topic really interested me.
There are maps like these two that clearly feel wrong.
Take a look at Texas’ 18th Congressional District.
There’s a hole in the middle and there are big bites taken out of the bottom left and bottom right.
There’s no way you can look at that district and not suspect that areas were deliberately carved off and added on to achieve some particular end.
My mom and sister live in Ohio’s 4th Congressional District.
This is the so called duck district because - well look at it.
Districts like that don’t draw themselves. This was clearly drawn to achieve a political aim.
These two districts illustrate the two main strategies in drawing lines for political advantage.
My mom and sister’s town have been split off from any population that shares their political views. This is cracking. The goal here is to separate like-minded groups that are in the minority so that their voice is diluted and surrounded by enough of the majority party. If four out of every ten people disagree with you then you want to group those four people with the six people on your side as much as you can and then you win every district.
Sometimes, however, there are pockets with two many of the minority in them. That brings us to the second strategy which is called packing. Instead of splitting those local majorities across multiple districts so they would be more accurately represented, we pack as many of them together as we can and we give them a single district. That’s what’s going on around Houston.
And that’s why the former president and his supporters were so vocal about the urban vote. The effect of those areas was minimized as much as possible in Congress by those who drew the congressional district maps, but Senate and presidential races are very much swayed by these voters - if they are allowed to vote.
As Marc Elias said, “Republicans believe that they cannot win elections fair and square. They can’t win them by attracting the votes of the majority… so they are left to change the rules of the game - to make it harder to register, to make it harder to vote.”
But we can’t just point to a map and say, “that doesn’t look fair.” We need a way to quantify it and come up with a measure that we can use to judge the fairness.
The next section uses math - you may find it difficult, but you can do it.
Playing with string
Suppose I give you a piece of string that has length 100.
“100 what, Daniel?” you ask.
I don’t care - depending on where you were raised you can think in inches, centimeters, miles, kilometers. You pick the unit.
Arrange the string in a shape that captures the most area.
It turns out, the best you can do is a circle.
Now the actual problem I have in mind is laying out Congressional districts in the US. You can’t really fit circles together without there being overlaps which could lead to people in more than one district or gaps which leaves some people in no district at all.
So let’s restrict the shape to be rectangular.
You may remember from your distant math past that the perimeter of a rectangle is twice the width plus twice the height (just work your way around the rectangle adding up the sides).
Stay with me here.
So since the perimeter is 100 we can say that 100 = 2 * w + 2 * h or 50 = w + h. This means h = 50 - w.
The area is the width times the height so A = w * h = w * (50 - w).
In other words we can think of the Area as being a function of width: A(w) = 50w - w^2.
The graph of this is a downwards opening parabola and if you took and remember calculus, it’s maximum point is when the derivative is zero. This happens when
0 = 50 - 2w or in other words when w = 25.
This is a square since h = 50 - w which is also 25.
So this first basic analysis says that one measure that a district is not fairly drawn is that it isn’t roughly circular or at least close to a square.
It turns out that there are many mathematical ways for analyzing district lines. Stewart discusses several of them in “What’s the Use.”
Most of the methods use math that was invented without this application in mind.
We don’t drop history for courses in doom-scrolling. We don’t drop literature for lessons in FaceBook FOMO. We don’t teach only the math that we can directly use in everyday life.
Essay from Dim Sum Thinking Newsletter 77. Read the rest of the Newsletter or subscribe