I had the wonderful privilege of talking with Ira Flatow on Friday, August 26, 2016, on the Public Radio International show Science Friday, along with guests Andrew Hacker and Maria Droujkova. What a great ride! If you’d like to listen….
Overall I love how the interview went, but there were a few things I either didn’t have the time to clarify or couldn’t help listeners visualize because of the non-visual format.
In the interview I mentioned three groups of folks who, because we come from different vantage points, often talk past each other—not communicating, even though we think we are. These different perspectives impact the conversation in ways that keep us spinning and are not helpful. During the show, I only had time to describe two groups. I’ll re-describe them here with some extra illumination, and also include the important third group. All of these perspectives are important in the conversation about improving math education.
These are in no particular order. I am hoping you all will help me refine these descriptions and find short catchy names for the groups. Short names, because it will make them easier to use in conversation. I know we are all unique and that categorizing is always fraught with generalizing problems, but nonetheless, I think it can be helpful to consider how we might be miscommunicating. These categories are broad but distinctive, the useful kind.
In this group are the folks that despite traditional teaching actually constructed mathematical relationships in their heads and then used those connections to solve problems.
When the teacher told them, “To add, line up the numbers and start right to left (small numbers to big numbers)”, they intuitively did NOT do that when they added something like 20 + 50 (they just thought about 20 and 50, not 0 and 0 and then 2 and 5, etc.); and also with problems like 99 + 37, they thought about it like 99 + 37 = 100 + 36, since 99 is so close to 100, just get there and add what’s left over.
These folks saw the math behind the directive to rote memorize rules. When the teacher said, “Show your work,” they often asked, “Why? It’s just so obvious.” or “I don’t know how to write what I just did and what I just did makes more sense then all those steps you’re trying to get me to write down.” Importantly, they think that everyone is this way—that all of us are able to see the math behind or in spite of the rules.
Due to this thinking, they often don’t see the need for the reform conversation: “It worked for me. Why change it?” Ironically, it only “worked” for many of these folks up to a certain point. At some point, they started to memorize the slope formula or polynomial long division (even though those things are totally figure-out-able!) and then they typically lost interest as they were not able to build the relationships on their own anymore. If this group would have had active help all along the way, they would have built the same relationships faster and gone farther. What would be a short descriptor for these folks?
- Mathy folks (nah, cuz we can all be mathy)
- Thinkers even though you’re telling me to rote memorize folks (too long)
- Kim (she is the first of this group that let me see insider her head) (but no, not general enough of a descriptor)
- Picked it up on their own folks (too long, and also, sometimes these are the folks whose granddad was always playing games with them and they talked about their reasoning, so they weren’t really on their own….)
- Cameron (he’s my oldest son and he challenged me to think about numbers as soon as he started to talk)
- ??? Help! What would you call this group?
In another group are the folks who bought into the fallacy that mathematics is a bunch of rules, procedures, and facts that are to be rote memorized and regurgitated—what I call “fake math”. When the teacher said, “Show your work”, this group heard, “Show the teacher’s work. Mimic the teacher and you’ll get credit for showing all of the teacher’s steps.” These folks nodded and did their best to remember which rule to do when and the correct order of the steps and to circle the answer at the end. They dutifully flashed card after card and subtracted the same thing from both sides, cross multiplied, inverted and multiplied, and knew that “ours was not to reason why”.
These folks went as far with fake math as their memories could handle, before it all started leaking out of their ears or all of the rules became mixed up and intermingled. I was in this group! My memory lasted longer than many folks’. I made it all the way to being a high school math teacher, not really owning very much real math at all. After they do some real math, these folks are often flabbergasted to find that they could have understood all of that stuff they memorized. They feel like they got the shaft – they followed the rules, but had they been actively helped to create the mental relationships, they could have learned much, much more math. And enjoyed it more… because they would’ve been doing real math. A moniker for these folks?
- Rule followers (not a bad thing, but it doesn’t bring to mind the nature of real math vs fake math)
- Fakers (just kidding)
- Many (not all!) math teachers (that will raise the hair on too many necks)
- Pam (that’s me – I was in this group)
- ??? Help! What would you call them?
This is the group I didn’t have time to talk about on Science Friday. These are the folks who just cannot do something unless they understand it. They might not have the best memories or it might be because they just see so little purpose in doing something that doesn’t make sense. With no one actively helping them make sense of real math (not how to rote memorize, but actually build the relationships), they often flounder. They want to make sense, they want to understand the relationships, the why and how, but because they can’t or haven’t been helped to, they try to memorize and spit it back out.
The result is a nonsensical jumble. They mix up steps in crazy ways—not because they are crazy—but because none of it made sense in the first place. Or it makes such little sense and seems so irrelevant that they are not willing to play the game. They do their best to mimic and they fall further and further behind.
Ironically, sometimes they own a ton more real math than group Z (see – I wish I had a name here), because in the process of ignoring the rote memorized rule (because it doesn’t make sense) they often figure out ways to solve the problems using real math! However, unlike group X who are doing the same things and thinking that everyone thinks this way, or group Z who doesn’t even know that real math is possible, this group somehow sees real math as less than, inferior, not the right math because it’s not the steps they are told to perform. I meet so many of these folks who look at me sheepishly and say, “Well, I can do 36-19 but only with these funny things I do, not the right way.” Then they tell me how they use relationships like 36-19=37-20 “because it’s easier.”
THIS is real math, but they feel bad about it! When folks tell me, “I was never good at math. I’m just not good at math,” my response is, “I bet you are (or could be) good at real math. You were probably just not good at fake math.” The relief and tentative hope I see in these faces are sometimes heartbreaking. No system should work so hard to create this disenfranchised group. What could we call this group?
- non-memorizers (not helpful, doesn’t say enough and isn’t always true, sometimes they just don’t memorize what doesn’t seem helpful or relevant)
- incorrectly identified as failures (too long and hurts too much)
- my daughter, Abby (bright and beautiful at real math, doesn’t give a lick about fake math and refuses to play the fake math game. When she tries to do fake math, it’s never pretty.)
- those without the math gene (NO! Everyone has the math gene. Everyone. EVERYONE.)
- ??? Help! What would you call this group, as I’ve described it?
When I talk to the group X folks, they often don’t believe that there are people like me (Z) or group Y. When I talk to group Y, they don’t even want to engage in the conversation because anything about math is just painful or dumb. When I talk to group Z, they don’t know (yet) what real math is.
What do you think? Where have I missed something? Do you identify with a group? Tell me your story. Love to hear from you!