As we discussed last week, Practice 1 has often been presented together with Practice 6 ("Attend to Precision") as one of the two "overarching" practices that work in conjunction with all the rest, which makes a certain amount of sense.
I also like the way that Grace Kelmanik and Amy Lucenta have presented it, with Practice 1 as a sort of "umbrella" practice that becomes a gateway to the others.
(Kelemanik & Lucenta, 2016)
In any case, I think most people agree that the other practices don't much come into play if you can't make sense of a problem to begin with, and persevering through it is a key part of eventually solving it, or even just learning something new from it, even if you don't ever arrive at an answer. Last week we talked about making sense; this week, we'll discuss what it means to persevere in solving a math problem.
For many students, especially as they begin to experience rich, challenging mathematics for the first time, perseverance can be a serious challenge, particularly with problems that are unfamiliar, confusing, or otherwise nonroutine. 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 they don't quit. (Or, at least, their bar for quitting is a whole lot higher.)
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:
Don't Give Up: Who, What, Why?
- 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.)
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?
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 struggle. There's also evidence that we're way better at persevering when we're intrinsically interested (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 high (many of us become wizards of perseverance when a deadline is approaching). That said, humans have a huge capacity for learning and adapting to our environments and learning new skills (like perseverance). We might start in different places, but we can all improve. This "perseverance toolkit" consists mainly of
- 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
- Beliefs about mathematics and our own abilities that, again, can be explicitly taught and reinforced.
If most of us want to learn to play piano, we'll need some instruction and many, many hours of practice. Learning to persevere is the same. No one is just born good at it; it involves skills you have to learn. I doubt there is One True Set of perfect, infallible problem solving skills that will work for every person in every situation, but there are many fantastic resources out there to try, including George Polya's classic How to Solve It and Grace Kelemanik and Amy Lucenta's Routines for reasoning, to name a couple.
"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?" It can! 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.)
You may be 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 to briefly summarize 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!" 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.
"Part of the Process..."
As legendary mathematician Andrew Wiles said:
“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.”
Kids sometime think that when teachers give them a problem where a solution strategy isn't immediately obvious that we're tormenting them on purpose, or else just being 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.
Next time: We dive head-first into Practice #2, "Reason abstractly and quantitatively"!