Open Set

When I was a kid, I sang Frere Jacques in French, even though I didn’t speak any French. When I was just a little older, the Madonna song Like a Virgin was on the radio all the time. Looking back, it’s clear that I had no idea what it was talking about. If I had, no way I would have walked around school singing it. Clearly, it was possible for me to know all the words and sing these songs without an understanding of what they were about. 

Did I understand these songs? In my own way, yes, sure. They made me happy, the tunes and the tempo were nice. But there is a whole dimension of meaning that I was missing. Compare that to the way I get hit with a dose of nostalgia when I hear “Fifteen” by Taylor Swift. I think about being a teenager, confused and in love. Or when I hear “Respect” by Aretha Franklin, and I know about being respected and not being respected in my life. Those lyrics connect to experiences and ideas, and it builds a world in my mind. A far cry from Frere Jacques. 

There is a difference between being able to (re)produce words (or repeat procedures) and those words having meaning to you. You can memorize, but that is not the same as understanding. 

Conceptual understanding in math class seems to be a controversial topic. I’ve met many people who tell me that concepts in math don’t matter. Others tell me that conceptual understanding is the same thing as answering the question correctly, or that the concept they need to teach is simply, “follow this procedure.” I don’t believe it. But before I talk about math, I want to just build the concept of “concept.” I don’t know how to build a concept without giving you some examples. 

I remember somewhere in middle school, we were introduced to the concept of “theme.” Books have themes, separate from their plots. That was confusing to me. “The theme is what the book is about,” said Ms. Miszuk. Sounds like plot to me! “No, it’s a message the author is trying to send.” You mean the moral? “No, morals are related, but a theme is an idea the author is exploring.” Huh? I remember that I just didn’t get it. Later in the year, when we read Flowers for Algenon, we had discussions about whether it’s morally right to use technology to change your personality. And I started to understand how the author was raising a question that was bigger than just the plot itself. The year went on, and with more examples, I was able to slowly build this concept of theme. 

A.P. Economics in 11th grade was a very hard class, widely agreed to be the hardest in the school. Opportunity cost was our first big concept. If you make more guns, then you have less resources available to make butter. Guns and butter, that’s what we were constantly told to think about, the trade off between having people to work on a farm and having them to work in a factory. Using the land to let cattle graze and using the land to mine for ore. Guns and butter. Opportunity cost. Not just the money you spent, but what else you would have done with the money. And the trade offs are not always so even: the opportunity cost for the exact same item changes over time and based on the scenario. Mr. Irgang would constantly tell us to go back to the guns and butter examples, think about it that way. I did, and it was really helpful, that whole year. When I took econ in college, the professor did not talk about guns and butter, but my mind went right back to that example the first day of class. Except, after seeing a few other examples, I didn’t have to go back to guns and butter. When I saw this concept a second time, and heard other examples, the idea of “opportunity cost” had its own place in my mind. Just one example is not enough, you need to see what different examples have in common, but how they differ too. You need to see connections between different related ideas.

What does it tell us? Concepts are big ideas that you understand through multiple examples. Even if you can articulate a concept in a concise sentence or two, telling those sentences to a group of students rarely finishes the job. Students need to see how it plays out in multiple examples, and explore it from a variety of directions. Usually, they need to have their own ideas about the concept, and try to put it into words, and then refine those thoughts over time. 

The concept of “concept” is no different in math than in other subjects. Math has its range of concepts, and understanding needs to develop over time. But it can play out differently because the language of math is often symbolic or graphical. Sometimes you can work your way through a concept by talking about it, but other times, you need to do it in the context of solving a problem. 

For example, I know plenty of students who can multiply decimals blindly. They have the procedure down pat, they move the decimal point correctly. Maybe they even enjoy doing it, knowing the rule and getting it right. But it has no more meaning to them than Frere Jacques had to me when I was six. What is involved in the concept of a decimal? Maybe we need to talk about measurement: that 2.6 cm is smaller than 3 cm but larger than 2 cm. Maybe we need some place value:  0.03 is smaller than 0.2 and bigger than 0.025, and if you are counting by tenths, then you should not say “3.8, 3.9, 3.10, 3.11…” We can talk about the concept of multiplication, the connection to repeated addition and having multiple groups of items. With that background, you are ready to explore the idea of multiplying by a number less than 1, taking a portion of a group instead of multiple groups. Eventually you get to the concept of decimal multiplication.   

In this case, the concept is not particularly related to the algorithm. Even if you understand the concept fully, you are probably not going to come up with the algorithm on your own. And I don’t think in this case that understanding the reasoning behind the algorithm is the most important part of the journey. Something similar happens with using the quadratic formula in Algebra I/II. You can have plenty of conceptual understanding of quadratic functions and equations, two x-intercepts for a parabola, or maybe no x-intercepts. But that will not help you derive the quadratic formula. And I’m not convinced that all students need to see the derivation, even if they are expected to use it. 

But when we are talking about solving equations, it plays out very differently. If you have any concept of what an equation MEANS, what the equals sign is all about, and how x’s and 1’s are different types of objects, then the process involved in solving equations makes sense. In this case, teaching the concepts and teaching the procedures are tightly linked and can be hard to distinguish. Now, I know plenty of students who have memorized the procedures but have no concept of what is happening. Certainly you see it when a student can solve   2x + 4 = 16 but they can’t solve   4 + 2x = 16   or even   16 = 2t + 4. But then there is a student who can even solve 4 + 2x = 16 correctly, going step by step, using their inverse operations, and arriving at x = 6 with no hitch. But when you ask them what number x would have to be in order for the equation to be true, they have no clue what you are talking about. “These are the steps you follow, this is the answer you get. That’s what the teacher told us to do” It’s like me singing “Frere Jacques” all over again. 

This is enough for one post. Please let me know your opinions and reactions!