This is the second follow-up post to a previous one aimed at parents and other non-math teachers, The Real Power of Studying Mathematics, about the significance of the Standards for Mathematical practice.
For the first follow-up post, see MP 1, Part 1: Making Sense of Problems.
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In our last post, we talked about the first chunk of MP 1--what it means to make sense of a mathematics problem. Today, we'll continue with the second chunk of Practice 1: Persevering in solving problems. What does it mean to persevere in the context of a mathematics problem? What does perseverance look like? What does it not look like? And what can parents do to support students?
At the most literal level, perseverance just means to keep going and not give up. Sounds simple, right? Alas, for many students, especially as they begin to experience rich, challenging mathematics for the first time, it's not always so simple.
We almost don't even really need to talk about perseverance in the context of problems that students find easy. Not quitting is easy when you know what to do (or, at worst, a matter of self-discipline). No, the significance of perseverance in the math classroom lies almost entirely in problems where students do not immediately know what to do, or problems that are unfamiliar, or confusing, or where students get stuck. This is where perseverance comes into play.
Students who are well-versed in Math Practice 1 are able to struggle through a challenging problem. They might find it confusing. They might get discouraged. They might have to read it over and over again. It may take them a while to make sense of the problem and find an entry point. The problem may look different from others they've seen before. They might get stuck. They might get stuck over and over again. But gosh darn it, they don't quit. (Or, at least, their bar for quitting is a whole lot higher.)
But why not? What is perseverance, really? Let us dig a little deeper into what it means and looks like in the context of doing hard math problems.
Don't Give Up: Who, What, Why?
Students who persist in working on challenging problems aren't just innately smarter or better at math than those who give up easily. More likely, it's that they have a kind of tool kit they can draw on when they get stuck or confused, which might include things like:
- Strategies for making sense of a problem and finding an entry point;
- Strategies for organizing and keeping track of their thinking, even if they don't fully understand the problem yet;
- Strategies for what to do when they think they've made a mistake;
- Strategies for walking back through their process when they're stuck or unsure;
- A belief that math is supposed to make sense and that problems are solvable;
- A belief that the ability to solve hard math problems is something you can get better at through practice and effort.
(You might think of others; these are just a few off the top of my head.)
Some might be consciously aware of these ideas and habits; others may bring them to bear on a problem without even realizing they're doing it. How did they get this tool kit, though? Are some people just born with it?
Well; sort of.
As with many other personality traits, there is certainly evidence that our genes play a role in determining how persistent we are in the face of struggle1. There's also evidence that we're way better at persevering when we're intrinsically interested2 (a kid may persevere spectacularly at fixing her penalty kicks but give up on a boring school assignment in five minutes) or when the stakes are high3 (I am a master procrastinator of hard things but become a wizard of perseverance when there's a hard deadline).
But we also know that humans have a huge capacity for learning and adapting to our environments, and that includes things like learning how to keep at something even when it's super tough and we're not facing immediate dire consequences. We might start in different places, but we can all improve.
As I suspect is true in a wide range of pursuits, this "perseverance toolkit" consists mainly of
- 1) Skills that can be explicitly taught and practiced (How do I make sense of a confusing problem? How do I keep track of what I'm doing and thinking? What do I do when I get stuck?), and
2) Beliefs about mathematics and our own abilities that, again, can be explicitly taught and reinforced.
Let's stick our heads a bit deeper into each of these buckets, shall we?
If most people want to get good at playing the piano, they're going to need some explicit instruction and many, many hours of practice. It works the same way with challenging math problems. No one is just born knowing how to do it; it involves skills you have to learn.
I am not here to give a definitive, exhaustive list of these skills, but plenty of folks who should know have shared their personal favorites. The legendary George Polya, for example, laid out his opinions on the matter in his enduring classic, How To Solve It. I've also recently become a big fan of Grace Kelemanik and Amy Lucenta's Routines for Reasoning. My point here is not necessarily that there is One True Set of perfect, infallible problem solving skills that will work for every person and every problem all the time; just that there are in fact skills that can be explicitly taught, learned, and practiced. And having a solid toolkit of reliable problem solving skills to deploy makes it a lot easier to keep moving forward when things get tough.
"Ah, but some people are really good self-taught pianists!" I hear you saying. "Not everyone needs to be explicitly taught. Why can't it work like that with math?"
Certainly, it can! Indeed, some people do become so fascinated with messing about on the piano from a young age that they figure some things out and get pretty good at it without any formal instruction; others become entranced with exploring number puzzles when they're small and likewise develop some skills for sticking with a hard problem on their own.
But not everyone. Not even most people. And I hope we can all agree that it isn't equitable to say to the kids who do become fascinated with math or number puzzles early-on, "Good for you!" while telling the others, "Sorry, kid, you're on your own."
(And quite honestly, speaking as a self-taught pianist, even a pretty good self-taught [whatever] can almost always still learn quite a lot from a good [whatever] teacher.)
If you've spent any time whatsoever in the education world in the last decade, you are almost certainly familiar with the work of Stanford psychologist Carol Dweck on fixed and growth mindsets. Her book site is a great place to get started if you want to learn more, but here's the briefest of excerpts from that page, just to give a sense of the idea:
- In a fixed mindset, people believe their basic qualities, like their intelligence or talent, are simply fixed traits. They spend their time documenting their intelligence or talent instead of developing them. They also believe that talent alone creates success——without effort. They’re wrong.
In a growth mindset, people believe that their most basic abilities can be developed through dedication and hard work——brains and talent are just the starting point. This view creates a love of learning and a resilience that is essential for great accomplishment. Virtually all great people have had these qualities.
When students understand they can get smarter, they exert more effort in their studies.
For a long time, the conventional wisdom that giving positive feedback to kids by saying things like, "You're so smart!" or "Wow, you're really good at math (or whatever)!" would encourage them to continue to work hard and be successful. We now know this isn't the case. Instead, this kind of praise actually causes kids to develop a "fixed mindset"--i.e., They come to believe that if you're "smart" or successful in math, it's just because you were born that way, like having green eyes or a pointy chin (and those who aren't, weren't).
Overwhelmingly, the current research from Dweck and others tells us that rather than encouraging kids to push themselves to try new and harder things, this kind of "fixed trait" praise instead puts kids in a place where they feel they have to defend and justify their innate intelligence or math abilities. They tend to be more worried and nervous about failing or appearing to struggle, and as a result are often less willing to tackle challenges or other kinds of tasks where they risk failing and being "exposed" as not actually smart or good at math. Which makes a lot of sense; if you've been told your success is due to the fact that you were just born smart or innately talented at something, it's easy to conclude that struggle or failure means maybe you weren't all that smart or talented to begin with. (See also: The dangers of labeling kids 'gifted'.)
Instead, we can praise students for their efforts, strong work ethic, and perseverance--"Wow, you must have worked really hard at that!" or "You did a great job sticking with it when you got stuck!" etc. This communicates to students that being successful or good at something is a flexible trait, something we can grow and improve at through hard work and practice. Students who develop this "growth mindset" tend to be the ones who enjoy learning for its own sake rather than for outside validation like good grades. They choose to take on more challenges, risk failure, work longer on hard problems before giving up, and worry less about whether people see them struggling. And the more kids are willing to struggle and push themselves, the more they learn.
Students who were praised for effort outperformed students who were told they were smart.
I do want to quickly debunk a couple of common misconceptions about growth mindset because they tend to pop up a lot:
- 1) "Growth mindset tells us that if we all believe in ourselves and try hard enough, we can all reach equal levels of performance/achievement."
FALSE. There will always be variation among students in academic achievement in any subject, due to a whole host of factors that are beyond the scope of this blog post. What growth mindset does tell us is that we are all capable of improving our skill and understanding and achievement if we are willing to work hard, practice, and learn from our mistakes.
2) "Growth mindset tells us that if we simply praise students for effort no matter what or how they do, they will improve."
Also FALSE. What it tells us is that effort is an enormous piece of the puzzle in terms of getting better at math and other cognitive activities, but that effort has to be directed toward meaningful practice. It's directing effort into meaningful practice that results in improvement, not the act of praising the effort. We praise the effort to communicate to kids that it's valuable and to encourage them to do it, but we still need to do things like provide feedback and instruction and other opportunities for learning and growth.
A growth mindset about learning mathematics goes a long, long way towards helping kids learn to persevere. Alas, although there are lots of fantastic teachers, teacher educators, and researchers out there doing the monumental work of trying to change things, the reality is that our current system is not super set up to foster a growth mindset when it comes to mathematics. Here are just a few of the damaging myths that prop up that "fixed mindset" system:
Myth #1: Bottom line, the most important thing in mathematics is getting the answers right.
- Reality: Bottom line, the most important thing in mathematics is to understand and make sense of things. Do we want kids to find correct answers? Of course we do. But finding answers without cultivating deep conceptual understanding of the mathematics that lies behind those answers is worse than useless. (See: Making sense of problems.) We should never fetishize correct answers at the expense of sense-making and real understanding.
And naturally, the flip side of that is...
Myth #2: Mistakes are undesirable and should be avoided at all costs.
- Reality: You literally cannot learn to do something new--if that something is in any way complex--without making mistakes! And yet, we've long had a math culture that demonizes mistakes and wrong turns. The best, most persistent problem solvers know that mistakes are not only inevitable but valuable gifts; not only do they help us to eliminate unproductive strategies or ways of thinking, but they often provide key insights into the very thing we're trying to understand. The fact is, we only learn deeply from making mistakes, and we've now got the brain science to prove it.
Myth #3: Being good at math means knowing what to do first/next.
- Reality: Interesting problems often require "tinkering." I first heard this term applied to mathematical problem solving by Cathy Humphreys in this video. A heuristic a lot of us grew up with in math is 1) memorize a bunch of processes for a bunch of different types of problem, 2) determine what kind of problem you've got, 3) do the right process. And that's just not always possible. Sometimes you don't really know what kind of problem you've got, so you've got to tinker for a while. This is uncomfortable for a lot of kids because (say it with me now) our system fetishizes finding the answer as quickly as possible with as little effort as possible.
Source: Inside Mathematics
Myth #4: Finding answers quickly means you're good at math and vice versa.
- Reality: Deep understanding takes time. Do we want kids to learn to work efficiently? Ultimately, yes. But worrying about the most efficient way to solve a problem before you've done the cognitive work of making sense is worrying about the upholstery before you've built the engine. Being able to regurgitate memorized facts or plow through an algorithm quickly is a neat party trick but doesn't say much at all about how well someone understands mathematics or how effectively they'll be able to solve a complex problem. You might be shocked to learn how many Fields Medalists (think the Nobel Prize of math) have described themselves as "slow at math"! (See Maryam Mirzakhani and Laurent Schwartz for starters.) In fact, there is a lot of evidence that emphasizing speed is actively bad for learning math.
Myth #5: Struggling to solve a problem indicates that you are not good at that topic/type of problem/math in general.
- Reality: If you're not struggling, you're not learning. I love the phrase "productive struggle" that has become popular in math circles in recent years. Understanding and getting good at something complex always involves struggle. In fact, struggling with something is an excellent sign that you're about to learn something! (Quick note: Navigating the fine line between productive struggle and unproductive struggle is part of the art and skill of teaching. Another post for another time!)
Stomping out these misconceptions and others like them would move us leaps and bounds forward in terms of helping kids learn to persevere in the face of challenging mathematics (and potentially in other areas as well).
"Part of the Process..."
By total coincidence, a math teacher blog that I follow (Math With Bad Drawings) published a post on getting stuck just a few days after I started writing this post. In it the author relates a conversation with legendary mathematician Andrew Wiles about what he tries to emphasize when he talks to the general public with math. While I don't agree 100% with everything in it, I found it hilarious and extremely relevant to a discussion of MP 1, in particular the following excerpt:
- “What you have to handle when you start doing mathematics as an older child or as an adult is accepting the state of being stuck,” Wiles said. “People don’t get used to that. They find it very stressful.”
He used another word, too: “afraid.” “Even people who are very good at mathematics sometimes find this hard to get used to. They feel they’re failing.”
But being stuck, Wiles said, isn’t failure. “It’s part of the process. It’s not something to be frightened of.”
Source: Math With Bad Drawings
When it comes to math, Wiles said, people tend to believe “that there is something you’re born with, and either you have it or you don’t. But that’s not really the experience of mathematicians. We all find it difficult. It’s not that we’re any different from someone who struggles with maths problems in third grade…. We’re just prepared to handle that struggle on a much larger scale. We’ve built up resistance to those setbacks.”
Which I think beautifully illustrates an important point about perseverance in the face of being stuck: Kids sometime think that when we give them a problem where a solution strategy isn't immediately obvious that we're tormenting them on purpose, or else just being infernally lazy ("Aren't you the teacher? WHY AREN'T YOU TEACHING US??"), but in fact, what we're really teaching them to do is to think like mathematicians.
Mathematicians, or scientists, or doctors, or lawyers, or HR professionals, or mechanics, or anyone else who ever had to figure out how to solve a problem that wasn't 100% clear and/or exactly like some other problem they'd solved before.
And that's a huge part of the value of MP 1, not just because kids need the skills and mindsets that help them persevere in solving math problems, but because odds are good they're going to have to persevere through challenging, ill-formed problems in other, perhaps more utilitarian areas of their lives at some point, and for better or worse, those habits and ways of thinking transfer more than you might think.
What Can Parents Do?
(**Caveat**: It's always a little tricky to make suggestions to parents because a) not all parents have the luxury of being available regularly to support their kids with homework; b) even for those who do, or can shuffle things around to make the time, it often comes at great opportunity costs; c) parents have a limited amount of control over what and how much work kids are being assigned, how it's being presented at school, etc.; and d) when kids are struggling with school work, emotions between parents & kids can run high & parents trying to help can start to feel like torture for both parties.
So I don't want to come across as if I'm suggesting you're responsible for doing your kid's homework, or teaching them what they didn't learn in class for whatever reason, or even sitting patiently next to them night after night until they're done. But if you do have interactions with your kids around their math work, particularly if they're struggling with something, here are a few things to keep in mind that help support students' ability to persevere in solving math problems.)
- Emphasize growth mindset. Explicitly fight the attitude that some people are "just good at math" and other people aren't and that's that. Remind them that struggling through tough problems is a) the only way we really learn anything, and b) takes practice to get good at just like anything else. Exterminate the idea of being a "math person" or "not a math person" with extreme prejudice.
- Encourage them to revisit sense-making. Sometimes kids struggle to persevere because they don't have any strategies for what to do when they get stuck. A good one is to step back and say, "Let me remind myself of the big picture. What am I trying to figure out? What do I know? What have I done so far? Can I retrace my steps?" (A comprehensive discussion of strategies for getting un-stuck is probably its own separate blog post.)
- De-emphasize speed. Remind your student that most math problems--and some might even say the best math problems--can't be solved quickly using some recipe or trick. Instead emphasize effort and patience, and yes, even, "Why don't you go do something else for a while & come back to it later." (I think this is especially important if kids get super emotional and/or stressed and/or agitated about a problem.) If it helps you can even remind them that a lot of the best mathematicians describe themselves as "slow at math."
- Celebrate--rather than demonize--mistakes. I know, this is really, really hard! If you're still working toward being able to celebrate mistakes as invaluable gems of insight, you might try starting with trying to communicate to your student that mistakes are an inevitable and necessary part of learning something new and getting good at it. (Remember, your brain grows when you make a mistake, not when you get something right.) Being terrified of making a mistake, especially in front of others, is a fixed mindset worry.
- Don't be afraid to contact the teacher. If a kid has really, really tried and they're still stuck and you're not sure how to help them and it's becoming a nightmare for everyone involved, don't be afraid to get in touch with your kid's teacher and/or encourage them to contact the teacher themselves (depending on the age and the kid) & say in a very very kind and collaborative and non-blaming way, "Listen, s/he worked super hard on it, s/he can turn in what s/he was able to do and/or can s/he have more time/a bit of extra support" and if it's becoming an every day- or every week-kind of issue, "Maybe can we sit down and chat about this sometime and come up with some sort of different plan." Teachers want kids to succeed and they generally hate the thought that the work they're asking kids to do may be contributing to a deep and abiding hatred of their subject. None of them want you and your kid up at midnight madly googling the internet or calling any math nerd you can think of in fear of a bad grade/homework shaming. Trust.
Next time: We dive head-first into Practice #2, "Reason abstractly and quantitatively"!
(Thank you for reading this far! I consider these posts to be living works in progress, so please do feel free to share any thoughts, suggestions, or questions. I want to make them as accurate and useful as possible!
1, 2, 3 Related, I've been looking for civilian-friendly, non-super-technical references for these three little factoids, so I'd love to add them if anyone has a good suggestion.)