Have you ever seen studies like “Teaching General ProblemSolving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics” or “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching”?

There seem to be several conversations among math teacher bloggers and Tweeters about if and how they use “non-routine problems,” the role of asking vs. telling, whether it’s okay to give students hints or not, that often come down to a belief that sounds sort of like this:

*The best teachers say the least*

Math teacher hot-button terms like “productive struggle,” “high cognitive demand,” and even “problem solving” are lumped in with the two ideas or beliefs that the articles above debunk:

- It’s important to teach kids to solve novel, non-routine problems
- The best teaching is minimally guided — no hints, no lectures, no worksheets

Are you surprised that I think the two articles are basically right — not in the grand claims they make about how math should be taught, but in their specific claims of what the research shows about teaching and learning? I think that the math education research community has gotten pretty sloppy about both of these claims, which opens up a lot of good thinking and good research about how to teach to criticism it doesn’t deserve. Here’s why:

- I think, for the most part, kids don’t need to be taught how to solve novel, non-routine problems. Any generalized, transferrable, “higher-order thinking” skills that exist are so general that kids already have them, and from a young age. Research on Cognitively Guided Instruction shows that young kids are capable of reasoning about mathematical situations in order to get correct answers.
- If the best teaching were minimally guided, why would humans have organized kids into schools for centuries? The more technology, the more culture, the more stuff we humans produce, the more we need to ensure that our kids master skills, gain knowledge, and develop specific ways of communicating that let them be productive members of society — and it really is content knowledge that differentiates experts from novices.

Okay — so if kids don’t need to learn problem solving skills, and the point of school is to master content, then shouldn’t we just tell kids what we want them to know, and get on with it as quickly as possible?

NO! That’s where I differ a LOT with the authors of the articles cited above. I draw very different conclusions from the research about the importance of domain-specific content knowledge in problem solving, for several reasons:

- Experts in the field have organized their knowledge in highly useful, highly interconnected ways (schemas is one name for that, I think). There’s lots of evidence that, for generations now, most American students are not graduating with highly useful, highly interconnected deep content knowledge in mathematics. And I’m not talking about the knowledge needed to be a mathematician, just the knowledge needed to, say, reason confidently about percents to make smart decisions about spending money. It seems pretty clear to me that the worked examples and exercises based on textbooks methods that still dominate most math classrooms across the country aren’t creating experts, despite exposing students to lots of content knowledge and lots of worked examples. Maybe we’re using the wrong examples, but I think it’s deeper than that….
- There’s an interesting (and growing) body of work that indicates that learning from listening & seeing worked examples is not easy. Experts learn well from lecture, because they already have organized schemas for attending to the important information, and they already have questions in mind that they use to make sense of what they are hearing. They interrogate the material and their own understanding of it as they listen (and after they listen). Novices, on the other hand, may attend to non-salient details, they may think they understand something but actually be completely wrong, and they might misremember what they heard in the lecture to match their own misconceptions, rather than adjusting their thinking to take in new information (Derek Muller of Veritasium fame breaks down some of that research in a video, but if you love methods sections and analyses and stuff, you might check out his dissertation or a related journal article about how students learn more in settings that create confusion, rather than clarity).
- Students are perfectly capable of solving problems, and learning from their own problem solving. Books like Children’s Mathematics and Young Children Reinvent Arithmetic demonstrate how children can solve problems that they haven’t been instructed in solving or shown examples of, using everything from direct modeling to sophisticated numerical reasoning. I combine this idea that children come to us able to reason and think mathematically, with Manu Kapur’s work on productive failure and the idea that students seem to learn more from lectures/direct instruction after being put in a situation where they grapple with (and ultimately fail to solve) ill-defined problems for which the content they are about to be taught is useful.

The ideas that mathematics must be taught in a way that enables novice learners to organize it into a highly interconnected, coherent schema or body of knowledge, that novices don’t necessarily have the skills to learn the way experts do (let alone the motivation or disposition/mindset, both of which have been shown to have huge effects of ability to learn), and that novices do have problem solving skills that they haven’t been taught but are rather part of their birthright as thinkers, suggests to me that there are different conclusion to be drawn from studies showing that minimal guidance is not effective and generalized problem solving skills aren’t what separates novice from experts.

The conclusion I draw is that problem-solving is a tool to be used in the service of learning content, and that content in turn becomes a tool to solve more and more interesting problems. I think that what students are learning in math is specific ways of solving problems (and learning what kinds of problems are interesting to try to solve, and what kinds of problems math can help with, and how to see the world as mathematically rich). I think students come to us with varying degrees of ability to solve problems in novice ways, and we help them develop their existing schemas and get more expert and efficient and solving problems by helping them see coherent “understanding stories” from basic understandings to highly mathematical, useful skills. For example, we help students move from modeling arithmetic problems with objects and counting to solve them, to making use of place value to fluently solve arithmetic problems in their head. We do this, not through minimal guidance, and not through worked example after worked example, but by carefully structured sequences of tasks that build on what we expect all students to be able to do, which we use to find out novice versions of increasingly mathematical ideas, and use tasks and worked examples and student-shared examples to scaffold students to develop more expert skills and content knowledge — knowledge that is connected and useful and feels relevant and important to the student.

Here’s why I think this carefully guided instruction works (and is hard, and is basically what everyone whose kids are learning is doing, albeit with a different style, whether their desks are in rows facing a chalkboard or their kids run around in fields in schools with no walls):

Mathematics is a coherent, highly interconnected body of knowledge, that includes both mathematical objects and (specific) mathematical ways of thinking. For example: algebra. Algebra is a set of thinking tools and heuristics (rewrite expressions to make them simpler, make strategic use of equivalence & its properties) and a set of master-able skills (use variables accurately, manipulate expressions & equations accurately, solve equations). There is a grammar and a vocabulary and a way of thinking. Students come to us with more and less novice versions of the thinking tools and heuristics (just ask them to go play with EDC’s mobiles), and not only do we need to help them master the skills and vocabulary, we need to link those skills and vocabulary to their existing schema around equivalence, while strengthening that schema.

I believe in the *learning theory *of constructivism — the idea that learners are actively engaged in trying to make sense of instruction* — AND I believe the research suggesting that they are not very good at making sense of what we want them to make sense of, whether it’s because of our instruction or whether it’s because they have unproductive beliefs about math and themselves, or a lack of motivation to learn, or they don’t know what to attend to, or they are assimilating what they are hearing into their own (wrong) ideas rather than shifting their thinking to take up new beliefs. So everything we want students to learn through listening and observation has to be presented to students who are primed to learn, in ways that support them to attend accurately, to recognize how the new material contradicts their current thinking, and to have chances to test their new (and old) beliefs against reality until they are solidified.

I also believe that students’ beliefs about things like who does math, what it means to do math, how math is learned, etc. are so important that pedagogy has to take them into account — even if that means sacrificing clarity or efficiency to support the belief that math can make sense to students, and good math ideas can come from anyone.

By now, it should be clear that while it would be cool if “minimally guided instruction that teaches students to solve any kind of problem” worked, it’s unlikely to. Carefully guided instruction, which is what I think actually works, involves thinking like:

- balancing the desire to help students master content exceedingly fast with the desire to provide experiences from which students can feel like “I can do this! I have the power to think mathematically!”
- being strategic about WHO provides worked examples — do they come from a carefully selected peer or from the teacher? Which will best support sense-making, engagement, and the learning culture the teacher believes these learners need?
- ensuring that students are ready to learn from instruction — that they have questions/curiosity to guide their listening, and a sense of what will be important to attend to and what to ignore
- sequencing the strategies that are shared from more novice to more sophisticated, to create a coherent storyline of how the formal math connects to student thinking.
- harnessing students’ existing abilities as problem solvers to create situations in which students are ready to learn math, rather than assume students aren’t problem solvers and try to teach them to be.

So… I would like math ed researchers to stop trying to prove that we can teach kids how to solve all math problems by teaching them a dozen strategies and turning them loose, and assuming that problem solving can replace learning content. And I would like them to stop contrasting minimally guided instruction with worked examples and direct instruction, and instead focus on helping teachers understand the kind of guidance and the kind of examples and the timing of examples and who should present examples so that the most learning happens.

I would like to see even more research that helps teachers answer questions like:

- what is the range of problem-solving skills that students present with at different age levels, and how can we support all students to have a basic ability to, say, understand a problem context and be willing to try stuff?
- what are best practices for supporting productive disposition and metacognition (which, it seems, can be situational or become more robust and actually transfer) — how do we help students be better math students?
- after arithmetic, where we actually have a pretty good sense of the strategies that young children invent, and how to support learners who need to see more examples and need more guidance to learn that from their peers in a culture that supports motivation and productive disposition (yes, I’m talking about CGI!), what are the conceptions that children are likely to come to use with, and what are the “understanding stories” that help move students from novice conceptions to a highly connected, more expert body of knowledge?
- what are the main factors teachers use to make teaching and learning decisions, and what is it useful to know about mathematics and learners to make better decisions? How do we help?
- how can we re-conceptualize the body of math knowledge so that it is clear to teachers and learners what the interconnections are, what kinds of problems each domain is helping to answer… essentially how do we make it clear the main, important thinking tools of math to ensure we are teaching students the most important ideas, as ideas, and making them super full of velcro-hooks for attaching more ideas to.

Wanna play?

*or, if they aren’t trying to make sense of math, at least they are trying to be successful in or cope with school — students often come to realize that this means making sense of the teacher’s motivations, expectations, grading schemes, etc. and not the content itself; my favorite example of this is in Jean Lave’s “The Culture of Acquisition and the Practice of Understanding” on page 29, which, sadly, is not part of the Google Books preview, but is summarized here

Max, love this post. It deserves several reads, but on my first go-through it’s this paragraph that jumps out at me:

“I also believe that students’ beliefs about things like who does math, what it means to do math, how math is learned, etc. are so important that pedagogy has to take them into account — even if that means sacrificing clarity or efficiency to support the belief that math can make sense to students, and good math ideas can come from anyone.”

This is a core belief of mine, informed by my experiences growing up, learning math in very traditional, teacher-directed classroom settings, and coming away with nothing but a major case of math anxiety, and feelings of fear and shame. The sacrifice you call for here is well worth it. It will make a tremendous difference to the kids who spend math class wishing they were invisible.

Thanks Joe, for sharing what jumped out at you. I would love to hear more teachers talk about the values they have about how they teach, how they feel about math and how they want their students to feel, in addition to goals about mastering content. So thank you for sharing your own stories and values and how you put them into practice!

See also: http://www.ams.org/notices/201310/rnoti-p1340.pdf

Thanks for adding this to the conversation, Barry. It seems to me like the non-routine problems you’re referring to invite students to do “sense-making” — to understand a problem context and the relationships in the context, and work to represent the problem until they have a representation that, as you said, “makes the non-routine problem routine.” Over time, students will build up repertoires of representation, and comfort trying different representations, and a library of problem types that are helped by different representations.

A question I have is about the use of problems that are non-routine at the beginning of a lesson or unit, in that they are problems that the student can solve and represent, but they don’t have any tools for representing them efficiently or as a mathematician would. For example, most students who are familiar with fractions, multiplication, or subtraction could answer this question: you have 7 cups of cat food. You use 2/3 of a cup each day. How many days will the cat food last? They don’t need to be taught fraction division to solve it, even though it’s a routine fraction division problem. However, treating this problem as non-routine at the beginning of the unit, gathering student methods, and then using pedagogical tools that encourage students to compare strategies, generalize, and learn efficient approaches from peers, might be a way to teach students to be competent fraction dividers.

One of the factors that should make us automatically suspicious of “minimal guidance” as an organizing principle is that expertise (in the sense I think you mean) in this domain requires integrating a dazzling amount of convention into whatever general problem-solving schemata you might bring to the table. Convention being—almost by definition—ungeneralizable. Immune to sense-making.

“For example, we help students move from modeling arithmetic problems with objects and counting to solve them, to making use of place value to fluently solve arithmetic problems in their head.”

But that move requires leaving the realm of the literally reasonable (material relationships that are, as best we understand the observable universe, persistent) and entering the realm of the merely (?) conventional (an arbitrary mapping from material states to a system of representation). That requires guidance. Community input. Direction. “So we’ve already sort of decided to represent these relationships using a left-to-right positional number system with two hands’ worth of fingers’ worth of symbols that are shaped like this…etc., etc.” And that’s just arithmetic.

I think it’s tough to build fluency/expertise without, at a minimum, the guiding and telling and directing that necessarily attends the perpetuation of a culture (which is essentially what we’re talking about when we talk about math education). To me, the more interesting problem is how we can simultaneously foster in students the idea that they have the agency to do more than receive mathematical culture. That the conventional/arbitrary parts don’t limit the scope of the inventible/discoverable ones. That anyone who claims membership in the community has the power to shape it.

This reminds me of Piaget’s concept of social knowledge vs. logico-mathematical knowledge vs. physical knowledge. And also Ben Orlin’s most recent post about a trick vs. a technique. One way of thinking about designing math instruction that I’ve really enjoyed over the years is to try to think about what is *possible* for a group of learners to reason through and come to own as their own thinking, and what is *reasonable* for a group of learners to reason through & own, and what is *important* for a group of learners to reason through & own.

For example: I can’t think of a good reason to teach kids how to solve equations as though it were a procedure (social knowledge) and just logical reasoning. BUT that’s only in the context where kids understand all of the notation of algebra. I don’t think kids can invent for themselves algebraic notation (for one thing, the reasoning preceded the notation by centuries, and the notation was probably harder to spread) — but I also think they can be the ones to come up with what they want to say in this notation.

It’s like how writing instruction has shifted over the years from telling kids both what to write and how to write it [copy these vocabulary words about our trip to the beach from the board and use them to write a paragraph about what you liked about the beach], to working with kids on the writing process, letting them say what they want to say and helping them with coming up with ideas, researching ideas, justifying their arguments, and also how to actually nail down the grammar, spelling, and rhetorical techniques.

As we teach kids algebra, sometimes we fall into the trap of telling kids what logical reasoning to do and what math ideas to have, as well as how to express them — and we’re not even sure the kids are seeing what we’re seeing. That’s why I love all the ways people have come up with to get kids doing algebraic reasoning and coming up with the need to differentiate between constants and variables, the need to express multi-step calculations unambiguously, even the need for a symbol other than “x” for multiplication! Then it’s easier to teach kids the formal way to write “rules” or functions, and how to read the rules/functions of others.

And I think there’s also room in math, when conceptualized as passing down a culture, to teach content-specific techniques because they’re cool and useful. I like learning about generalizable mathematician tools, especially when they’re being offered to me in connection to my own ideas — right now I’m trying to learn more about the relationship between the strategy “add auxiliary lines” (which seems about as useful as “solve a simpler problem” used to feel — how does that help? doesn’t it just add confusion? how do I know I solved the *right* simpler problem?) and the geometric concept of dependencies. Is there a way that thinking about dependencies can help me add auxiliary lines strategically? I’m at a point where I’m ready to read, listen to lectures, see worked examples, etc. I want to be part of the culture of mathematicians that are good at solving Geometry problems. Apprentice me!

I agree that I need to read this a couple more times! Max, I feel like you’ve laid out one of the Grand Challenges that NCTM (et al.) are soliciting! Have you thought about submitting this to them?

I agree that you’ve highlighted some important elements not just in the cognitive domain but also in terms of the interaction with students’ conceptions of identity and their motivation to engage (and how the two sides of this coin in turn lead to some of the careful balancing that we must do as teachers).

Especially on this note, I wonder what you think of Phil Daro’s comment that we should stop treating math as “peculiar” in our schools — and start teaching more like we do other disciplines, e.g., writing. (To me, the parallels seem clear: have students produce, a lot; have students make sense, especially collaboratively; give plenty of feedback, have students revise and improve their work, and generally use a full range of formative assessment strategies.)

Sandhil, thanks. I’m not sure if NCTM’s research committee is still soliciting Grand Challenges, but if they are, I’d be happy to share this with them. Or, better yet, share this article that Michael Pershan shared with me: https://books.google.com/books?hl=en&lr=&id=W4GnocmF02IC&oi=fnd&pg=PA763&dq=problem+solving+and+modeling+lesh+zawojewski&ots=YKI0lvLkGS&sig=qoQZPycXuDlvKVbspTeqyglSAxc#v=onepage&q=problem%20solving%20and%20modeling%20lesh%20zawojewski&f=false (just a preview, sadly). It has a review of the literature on problem solving and a more research-oriented framing, but seems to echo some of what I was thinking about.

And YES to the idea that we should stop treating math as peculiar. I’ve been learning more about the transition in the 60s and 70s to a workshop model in teaching kids to write (the work of Donald Graves, Lucy Calkins, Susan Sower) and it seems to parallel what is happening, or could happen, in math: acknowledge students as mathematical thinkers, give them chances to do math, and give them lots of very informed, useful feedback & opportunities to revise. And authentic audiences of readers!

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Hi Max,

Is a Dalton school not the perfect way to teach kids math and solve problems? As we all know Helen Parkhurst abolished the teacher’s nightly preparation of classes and the child’s nightmare of homework. This could work also with Math? Originally I am from the Netherlands and my kids go to a Dalton school. As you can see on wikipedia for example, the Dalton Plan is very populair in the Netherlands and we do have already hundreds of Dalton schools.

It is great to see that children are getting freedom and opportunity to develop initiative and self-reliance.

Do I have a point?

[...] term goal and increasing understanding of the subject is the long term goal.” Max Ray’s post from 2015 really hammers home the wavering perspectives of problem solving in American classrooms. One [...]