It’s always fun to talk with Jessica Lahey, and I was honored she included me in the company of Steven Strogatz and Christopher Danielson for her last piece, “The Problem with Math Problems: We’re Solving Them Wrong.” I have been reading some of the comments and twitter chatter, and I’m so glad people are talking. Here was the original question from a parent:

My husband and I talked to our daughter’s pre-calculus teacher about her poor grades. He said that many students hit a wall at this point in math, moving from memorization — apply this theorem to this problem — to more abstract how-can-I-solve-this-problem thinking. I accepted that because that’s what is happening for her. What I thought later was that why can’t we find a way to help these many students get over that wall, instead of using it as a tool to weed out less developed brains? I really feel I have no way to have an impact on this teacher’s blind spot since it is shared by all math teachers and so many other teachers: If you don’t understand, it’s your fault.

Jess has to write under a strict word limit. Happily, I have no such limits here! I thought it might help the conversation if I posted my full comments to her, edited just a tad for this different format:

My first quick reactions to her email, which I typed from the Maine woods on my phone on a freezing cold day:

The entire idea behind reform mathematics is to eliminate this problem. Kids should not start with memorization they don’t understand. Kids should understand from the beginning. I just got to the woods where I’m going to walk the dogs in the snow. If you want to chat about it in more detail, I would be happy to talk with you later on.

A few minutes later:

P.S. Too cold to have fingers out of mittens for long, but I wanted to send one quick clarification. I am NOT saying kids should understand elementary math because they might need it for calculus. I’m saying if a student is going to learn something, she should understand it, whatever grade or subject. That is the first goal. And then, an important result is we would end up with students that have foundations upon which other powerful ideas can be built.

It was never a sensible idea to try to have students memorize first and understand later. Its just structurally flawed! So, I totally feel for these parents and this kid. This is what happens. Inevitable, if you try to build on a house of cards.

When I got back to the car, I wrote again:

I know there are many directions you could go with this question. It’s a fantastic question. For me, at the end of the day, this question exposes what is wrong. So many parents are saying, “Just give them the basics when they are young. They will learn how to do things with them later.” The math education world has to teach families why this approach doesn’t make sense. For example, most families do not understand that the ideas underneath algebra start very early in elementary school.

The next morning, I was able to write a little more in depth:

I woke up this morning thinking more about this question, and I’m finally off my phone and at a computer. I wanted to share a connection, not because it’s cute or quotable, but because I find it helpful for framing the situation.

When Sam started medical school, the dean gave a speech in which she told a version of this story (parable?):

A man and a woman were walking and talking along a river bank. Suddenly, they noticed a body in the river. The man took off his coat, jumped in the water, fished the person out, and administered CPR. Just as he was finishing up, he noticed another body floating down the river. Again, he jumped in the water to rescue the drowning victim. As he was dragging this second person to shore, he saw a third body in the river. At this point, the woman turned and ran away. He yelled, “What are you doing? I need your help! Where are you going?”

She yelled back, “I’m going upriver to see why people keep falling in!”

The dean’s point was that there’s a place in medicine for everyone. We need doctors to care for people who arrive in emergency departments, or who have contracted Ebola, or who have heart attacks. And we also need doctors who practice preventive medicine, research causes of disease, and engage in public health campaigns.

This letter from these parents, for me, is like the moment by the river. One response is to think about supporting students who have “hit a wall” so we don’t lose them–and they don’t lose STEM either. This response is especially important for underrepresented populations. I think this is what the parents were asking you to do.

The second response, though, is to think about why kids are hitting a wall, and whether that wall can be removed altogether?

We need good people working on both responses.

As someone who is inclined to run up the river and see why people are falling in, and as an elementary person, I tend toward the second response, which is what you got yesterday. When students leave my classroom, I want no weed-out walls in their futures. That’s why I’m on the team looking for systemic solutions to this problem.

Not sure if that’s helpful, but that’s how I think about their question. And I’m sorry. I hate that the odds are good this girl will end up thinking she’s “not a math person.” If she doesn’t, surely some of her classmates will. That’s why we have so much work to do.

As for what to do for this particular family, now, these were my quick thoughts:

1. For the parent. I am truly sorry that she and her daughter are in this situation. There are lots of us working to improve the system so it doesn’t happen. All is not lost yet, though! With support and advocacy, her daughter can learn the foundations and continue on in STEM! First step is working with her classroom teacher. What advice does he/she have about what would help her?

2. For you. This isn’t my age group, so I’m not as familiar with what teachers do for students who arrive without the foundations. BUT, I could point you toward some great HS math teachers who think about this kind of thing a lot, and they may have more specific advice. Would that help?

Middle and high school teachers are the ones fishing kids out of the river. I haven’t experienced the problem from that side as much. But, from either side, it’s not this student’s fault. It’s our fault.

I am so glad Jess wrote a piece looking at the larger system. It’s only by stepping back and looking at it as a whole that we can think in terms of what’s good for kids in the long run.

It’s great to wake up in the morning and find such a thoughtfully written piece waiting for me to find through my twitter feed. I saw the NY Times piece and at the time wished that the ideas presented could be more deeply explored. I just got my wish.

What I am wondering about these days is when are children developmentally ready to understand the concept of ratios and relationships? Personally, I love having been able to memorize all my “math facts,” but I thought, really really believed that doing calculations was doing math. If I had understood that math was more about the discovery and description of relationships it would have made a difference. I think that this is where the Wall might be: when students need to shift from “getting an answer” to understanding relationships. The calculations mind-set is so deeply hard-wired into the brain that this shift never happens.

So I am wondering, from your work with young children, when do you think you can start talking about math as if it’s tool for discovery?

Thank you, Paula! You might also enjoy Ben Orlin’s post: http://mathwithbaddrawings.com/2015/04/08/the-math-ceiling-wheres-your-cognitive-breaking-point/.

As for your question, in our house, math is a tool for discovery from the beginning. It’s a way to notice, explore, and understand the world. It’s a way to ask and answer our questions. I recorded my girls talking about math they found in their world a few years ago: https://tjzager.wordpress.com/2014/11/10/maya-and-daphne-find-math-in-their-world/. I think it shows we talk about math a lot because the kids make mathematical observations and look for relationships all the time. As an example, I’m thinking about this conversation Daphne started: https://tjzager.wordpress.com/2014/10/25/the-same-amount-of-tall/ I think she was thinking about relationships right from the get-go. What’s a whole? What’s a half? What’s a whole and a half together? She wanted their names, yes, but I think she wanted their names because she was thinking about them as different things and wanted to be able to talk about them and the relationship between them.

So, that’s in our house. In classrooms, there are a few more constraints, but I take the same basic approach. Kids are young mathematicians who seek to understand and discover. We need to stop messing that up, you know? And the bonus is we can have much more fun teaching, too!

Yes – ratios and relationships…. how to get a feeling for these? And I’d include scaling too.

I can feel a mixing a particular green from yellow and blue paint sequence of lessons coming on. (Where did I see this mentioned recently?)

Simon, did you see the yellow/blue paint lesson on the Illustrative Mathematics site? This task was just recently presented https://www.illustrativemathematics.org/content-standards/tasks/100

Thanks Paula – that wasn’t the one I’d seen, but yes, that kind of thing. (Maybe it should be orange though – it’s hard to get a really strong green mixing blue and yellow.) I’m going to ponder the idea of using a line graph for representing it. We have used these graphs, to compare height and femur length, but it may be that this isn’t familiar enough to be something that really adds clarity. Not sure.

I think I might try it after our spring holidays. Would be good to think of other powerful ways of representing ratio.

(Sorry Tracy – I know your post is about a much more pervasive problem and general solution! And I’ve gone chasing after a little detail!)

I love the illustration of the river! I’m definitely going to use that whenever the conversation comes up– which is usually when parents find out I’m a math teacher and say “What’s up with this NEW MATH they’re making the kids learn?? I don’t understand it at all!!”

I find myself at both ends of the river. I’m certified to teach high school math, and when I’ve worked in high school classrooms I’ve felt like the panicked paramedic fishing students out of the river. If only I could go back in time and teach this to them in a way that they’d actually understand! But you can’t go back in time, so you just have to spend time reviewing old curriculum that they should have learned before. Hopefully they haven’t shut their brains down completely by that time.

Now I work as an online math tutor for a program that teaches grades 3-9, so I can get any grade at any time. I feel so much more powerful being in a position to teach children when they’re young. Here’s one reason why I think our program is successful– Math teachers for elementary students. Not just Elementary teachers who have had minimal math training, who have never gotten to the “conceptual understanding” point in their own careers as math *students*, who hate teaching math just as much as the students hate learning it, and who end up falling back on their comfort zone of teaching the way they learned it. Students shouldn’t have to wait until 6th or 7th grade to learn math from a teacher who actually went to school to teach *math*. The common core is trying to get teachers to focus on concepts instead of memorization, but change of standards MUST be accompanied with a large increase in teacher training, or else it will never work, and high school teachers will forever be fishing students out of the river.

“The common core is trying to get teachers to focus on concepts instead of memorization, but change of standards MUST be accompanied with a large increase in teacher training, or else it will never work, and high school teachers will forever be fishing students out of the river.”

I could not agree more!

Thank you so much for your comment.

The parable of the river is definitely a keeper. I also like that the point was to care for the whole course, not to divert attention from the people at the end.

I’ve thought about that river so many times over the years, in multiple contexts. I agree, it’s a keeper.