Complicated Problems and Complex Problems

Matt Enlow shared this problem on Twitter this morning:

I had a ton of fun playing with it! It’s one of those problems that takes ideas that I think I understand — in this case, properties of equations and exponentiation — and turns them on their head, forcing me to think in new ways and helping me to better understand math I learned a long time ago. Play with it! There are more solutions than I thought at first. If you’d like a hint, check out the replies on Twitter.

My first instinct when I see something like this is to ask, “How can I engage my students with this problem?” I love math, and I love problems, and I want my students to experience the joy of solving problems. For a long time I would seek out problems like this one, problems I loved, to share with students. But many of those experiences were counterproductive, and I’d like to try to explain why. First, here’s another problem that I recently saw on Twitter and enjoyed playing with:

You meet a 3 headed 3 tailed dragon. You can cut either 1 or 2 heads, or 1 or 2 tails in one swing of a sword
If you
cut 1 head, 1 head will grow back
cut 2 heads, nothing happens
cut 1 tail, 2 tails will grow
cut 2 tails, 1 head will grow.
Can you slay the dragon?#mtbos #math

— Dave Martin (@d_martin05) December 14, 2018

Give it a shot!

Interlude: Complicated vs Complex

Atul Gawande writes in The Checklist Manifesto about the difference between complicated and complex. Sending a rocket to the moon is complicated. There are lots of little things that have to be figured out and designed and built and work right and lots of people who have to collaborate to put the pieces together. But once we get one rocket to the moon successfully, we can pretty well follow those steps and get another to the moon, and another.

On the other hand, raising a child is complex. There are lots of moving pieces, and lots of nuance and judgment, and raising one child does not mean that raising the next suddenly becomes a task of copying what was done before.

Working with something complicated involves coordinating lots of little things that have to be done right and add up to one big thing. Working with complexity involves much more judgment, subtlety, and responsiveness.

Back to Problems 

One reason to give students problems is to teach content. That’s important! But it’s not what I’m interested in here. The problems I give students also send messages about what it means to do mathematics. I worry that the first problem, with the factoring and exponentiation and all of the subtleties embedded in it, sends a message that practicing mathematics is complicated. It sends a message that math involves learning lots of little things and then piecing them together in unusual and contrived ways to figure out new things, but to be successful you have to remember all those little pieces and put them together in just the right way. I think problems like these play out in inequitable ways; students who already have strong skills and a disposition toward making sense of and persevering on a math problem are likely to get some positive reinforcement, and students already disaffected feel confused and left out of the conversation.

I think the dragon problem sends a different message. It invites experimentation and sense-making, and it can be represented lots of different ways, all from a very simple prompt. I think it sends a message that practicing mathematics is complex. Math isn’t easy; it takes originality, depth of thought, and a willingness to try new ideas and take risks. And it has value precisely because it’s not easy, and working through something hard can feel gratifying and fun. But that’s a very different message about the nature of mathematics, and why someone might want to pursue it in the future.

I love both of these problems, and the first problem was still fun for me. I still find it elegant and thought-provoking. I want to design some sequences of problems that get at similar ideas, where students can engage with the idea of exponentiation and the properties of equations. Those might serve a really useful purpose in helping to illuminate deep mathematical concepts that I often hurry past in the high school curriculum. But I only have so much time to engage students with problem solving for the sake of problem solving. For the purpose of helping students see themselves as potential mathematicians and illuminating the depth of what it means to practice mathematics, I think complex, inviting problems are where I want to focus my effort.

National Board Certification

Well, I did it. I’m a National Board Certified Teacher. Here’s my favorite of many tweets celebrating after the score release last weekend:

Attention beloveds~freshly minted National Board Certified Teacher here, hardest thing I've ever done professionally. Almost quit, twice. It doesn't mean I'm a "master teacher," it means that I'm committed to the craft + the scope and ambition it takes to be great for my students pic.twitter.com/xBU2TElo2A

— Tameka H. Marshall, NBCT (@equityandjust) December 1, 2018

I don’t think that being National Board Certified makes me a great teacher. I can give you dozens of ways I’m not. I do think it reflects that I care about the teaching profession, and that I’m working to get better. Board Certification is premised on five core propositions, and I think that these came through in my portfolio — but notice that these speak more to teachers’ growth than their expertise.

• Teachers are committed to students and their learning.
• Teachers know the subjects they teacher and how to teach those subjects to students.
• Teachers are responsible for managing and monitoring student learning.
• Teachers think systematically about their practice and learn from experience.
• Teachers are members of learning communities.

Some things were frustrating about the certification process. The feedback on my portfolio was hard to understand and not very helpful. The guidelines and rubrics were complicated and took forever to sort through. For Component 1, I had to drive two hours to the nearest testing center in Denver and sit in a cubicle staring at math on a computer for three hours; not fun. For Component 2, I had to figure out how to assess student learning at the beginning of a unit, use that assessment to differentiate and give feedback effectively within some uncomfortably prescriptive guidelines, and assess progress again at the end of the unit. For Component 3, I had to struggle to get intelligible audio and video of my teaching, throw out lots of bad clips, and then write something articulate about my teaching. For Component 4, I had to gather information from colleagues and students’ families about their learning, show evidence of how I design assessment systems based on student needs, and demonstrate that I’m learning outside of my school to meet those needs. This last one was a mess; it was hard to sort through exactly what I needed to do for each step and how the different pieces fit together. For the three portfolio components, I had to do a ton of pre-work planning when and where I was going to gather evidence and be prepared when things didn’t work out the first time. Then, I had to piece together what I wanted to communicate in my portfolio, and then actually write the thing. Luckily I like writing about teaching, but it was exhausting. 

I think part of the value of Board Certification is that it is a ton of work. It takes time, it costs money, it’s complicated. Teachers can’t start the certification process until they’ve been in the classroom for at least three years, and the credential isn’t worth much outside of schools. It’s not something people are likely to do if they’re on their way out of the profession. And all of the work is embedded in teaching; it’s not like writing papers for a master’s degree because so much of what I did was analyzing my actual teaching practice and talking about where I was working to improve.

Some folks say that the certification process is one of the best professional learning opportunities for practicing teachers. I don’t think this is true, but I feel incredibly lucky to have the  MTBoS as a space to share ideas on teaching, hear from others, and push my thinking forward. The NBCT community won’t replace that. But the MTBoS community is different. For one, it’s not all teachers. Lots of people I connect with work in curriculum, technology, instructional leadership, PD, and more. And that’s great! I went to NCTM in Seattle two weeks ago, and those were lots of the people I was hanging out with, and lots of the folks who read this blog. Hi! I appreciate you. The MTBoS is the best place I’ve found for engaging intellectually with teaching math, and I wouldn’t be the teacher I am today without it. National Board Certification dug into the practicalities of classroom teaching in a different way. It was messy and imperfect, but so is the reality of schools and teaching. I have no illusion that being Board Certified will influence my career the way the MTBoS has. But it serves as a symbol of my commitment to the classroom, and my commitment to improving my teaching in the classroom. 

My advice to other teachers: if you’re committed to teaching and your school or district is willing to support you financially, take a look at Board Certification. Be careful taking on too many components at a time. Learn to love writing. Know that the first lesson you want to videotape or assessment you want to use work samples from might not work out. Plan the logistics early. Know that it will be frustrating, the rubrics and criteria will be obtuse, and the portfolio will feel like a mountain of paperwork at times. Find someone you trust to look over your work. It’s less about being a brilliant teacher than it is showing off what you already do well, and being willing to reflect on where you want to improve. And you might fail — I didn’t pass by much, and better teachers than me have failed but pushed through.

Board Certification doesn’t decide what good teaching looks like. But it does serve as a marker of commitment to the hard intellectual work and richness of teaching, and a step toward a profession that defines and regulates excellence and receives the respect it deserves.

Anxiety, Mindset and Motivation: Bridging from Research to Action

Lisa Bejarano and I presented last Friday at the NCTM Regional Conference in Seattle. Some resources: slides, collection of teaching routines, and a warm-up sheet. Below is a brief synopsis of the talk.

Mindset & Competence

Growth mindset is a hot topic in education right now, and teachers are often told to praise students for their effort rather than their ability. The catch is, in more recent research, changing the way we praise students doesn’t seem to actually influence many students’ mindsets. Carol Dweck has written about how growth mindset has been oversimplified and misused; lots of studies haven’t replicated the optimism of early research on growth mindset, and it seems like praising students a certain way or telling them to have a growth mindset is insufficient for actually changing their attitudes.

But why do students come to math class with fixed mindsets in the first place? They develop these attitudes toward math over years (for my students, a decade) in math classes that send narrow messages about what it means to be good at math. Lani Horn writes that “Schooling favors one type of mathematical competence: quick and accurate calculation” (Motivated, p. 61). Horn argues that we can value broader mathematical competencies — making astute connections, seeing and describing patterns, developing clear representations, being systematic, extending ideas, and more. Instead of trying to convince students to have a growth mindset, we can give students experiences in which they can recognize the different ways they are mathematically competent. As students see the value they bring to math class, they can start to develop a more positive identity as a math learner.

Routines  

About one-third of our students are likely to experience some type of anxiety disorder during adolescence.

Anxiety is a pretty rational response to the stresses of adolescence, both within and outside of school. While we may not be able to address many of the root causes, we can create classrooms where students experiencing anxiety, as well as the rest of our students, recognize their competencies. Routines are an opportunity for students to feel safe, to worry less about what’s happening next, and think more about the math. Within a routine, students can become more comfortable taking risks and sharing ideas. There are also a ton of routines out there. In our session we used Stronger and Clearer Each Time, Number Talks, Visual Patterns, Five Practices, and Stand & Talks, but these are just a few examples we are partial to, and other routines would work better in different contexts. Lisa’s blog and the Stanford GSE have plenty more examples.

Routines have value in creating spaces where students can take risks and feel comfortable thinking mathematically, but they also add value for teachers. As I use more routines, I become more comfortable with the structure of the routines, thinking less about what comes next in my lesson, and thinking more about how students understand mathematical ideas and finding more valuable conceptions that I can build off of.

Routines & Competence  

What’s the connection? Here’s our premise: routines are a valuable teaching tool, and every teacher already has routines in the ways we set up our classrooms and lessons, even if we don’t make them explicit. How do we start class? How do we launch problems? How do we have students practice? These routines send a message about our values. If our routines value a narrow vision of mathematics that causes students to focus on their deficits rather than their strengths, then negative feelings, negative mindsets, and anxious behaviors become entrenched. If our routines create rich and varied opportunities for students to recognize the ways they can be successful in math class, and to recognize those successes in lots of different ways, students who have felt alienated in the past can start, slowly, to change their perspectives. Stronger and Clearer Each Time values revision and improving ideas. Visual Patterns value different perspectives and unusual interpretations of a problem. Five Practices helps us to highlight every student’s thinking, rather than just loud students. 

This is slow, humble work. Students may come to class having been convinced for years that they aren’t math people and that math class is not a place that values their thinking. We can’t change that overnight with a flashy lesson or a quick pep talk. And routines aren’t the only way to create that change. But routines are one practical place that classrooms can find ways to value every student and help all students to see themselves as mathematicians.

What messages do your routines send about your values as a teacher? What is one opportunity to incorporate a new routine that broadens student conceptions of what it means to be smart in math class?

Minimally Different Problems

Meanings are acquired from experiencing differences across a background of sameness, rather than from experiencing sameness against a background of difference.

–Ference Marton & Ming Fai Pang

One common thing I do in class is have students practice something. Some students get bored quickly, some work happily along, others struggle. This post is an attempt to design practice in a way that supports the learning of all of these students.

Students have been introduced to arithmetic series and need to practice. Here are two sets of problems:

Which sequence of problems better helps all students?

I’m going to argue the first. Three reasons:

1. First, each problem only varies in small ways from the previous problem. Students’ attention is then focused on these small changes, and they are more likely to make sense of the components of an arithmetic series problem, rather than having to start from scratch for each problem. When the problem changes in only one way, students can better understand the impact of that change on the mathematics.
2. There is more potential for extension. The structure of the problems means that students can find shortcuts, using one answer to more easily solve another. Then, we can return to those ideas to review as a class, providing more opportunities for discussion than typical practice.
3. There is more opportunity to scaffold success. A student who is struggling might have trouble at first, but varying only one element of the next problem makes it more likely that they can use what they figured out right away and better consolidate their understanding.

This idea comes from Variation Theory, which Craig Barton talks about in How I Wish I’d Taught Maths. He writes:

By working through carefully chosen sequences of questions, students have to carry out procedural operations, thus engaging in vital practice. But through connected calculations, they also have the opportunity to consider the deeper structure. Such variation allows students to anticipate, notice and then generalise, instead of permanently playing catch-up (249).

I think there is more potential in these sequences of problems both for students who already have strong skills and have the opportunity to notice new connections, and to students who are struggling with the concept and can benefit from only focusing on the essential differences between problems. But the problems above are only one very narrow type of question. What about when students need to distinguish between similar problems?

If you do not know what English is and you hear 100 people speaking English, you will have no better idea of the meaning of “a language”. If you do not know what “a lively style of writing” is, and you read 100 articles, all of them written in the same lively style, you will still not know what “a lively style of writing” means.

–Mun Ling Lo

Let’s say I want to help students distinguish between arithmetic and geometric series, and as a secondary goal practice identifying the common ratio of a geometric series. The above sequence of problems strips away any unnecessary ideas, and gives me a great chance to see exactly where student thinking breaks down, and to address those breakdowns. I don’t think all practice should be structured as variations of one problem; after these six problems, I might start with a new sequence focused on different ideas and asked in a different way. But by only varying a single element of a problem at a time, I get more precise information about what students know and don’t know, and can facilitate a more fruitful discussion of the problems. 

I think there is one more possible use for this type of minimally different problems. Let’s say I want to introduce students to sigma notation. I often struggle to explain sigma notation concisely, and a few examples can go a long way. I might give students a few examples of sigma notation to notice and wonder about. But with too much variation, it just looks like Greek alphabet soup. By only minimally varying problems, I give students more to latch onto, and make it more likely they notice what I would like them to notice:

I really like these sequences of minimally different problems. They still serve goals I had before. But now, students’ attention is more focused on the essential ideas of a topic, sequences of problems scaffold success for more students, and I open up natural opportunities for differentiation as students can make new connections and generalizations. While I’m only starting to experiment with minimally different problems, I also think that over time these problems could help students to see that math can make sense and isn’t just a collection of disconnected ideas. As students see more sequences of problems like these, they might start to believe that they can find shortcuts and new strategies for problems, and develop a disposition to look for patterns where they might not have before.

Lessons from Learning Abstract Algebra

I’ve been trying to teach myself some abstract algebra the last few weeks from this great free online text. I’m really enjoying it! It’s fun to learn new math, and I like seeing new ideas as a learner. I never took any abstract algebra in college, and before starting this adventure I knew there were things called groups, rings, and fields but had no idea what they were. Now I know more! Along the way, I’ve been thinking about what learning math can teach me about teaching. Two lessons stick out.

Stories

I became interested in learning about groups after reading Patrick Honner’s October article, The (Imaginary) Numbers at the Edge of Reality in Quanta Magazine. It’s a great read, and it positions group theory as part of a larger story, framing different number systems in terms of their connections to physical problems and sharing the stories of the mathematicians who first worked with them. I became fascinated by quaternions, and I’m lucky that the text I’m learning from uses quaternions as an example in different contexts and keeps me connected to a narrative beyond the math itself. 

How often does this happen in math classes? Not very often in mine. Now I’m thinking about how I can find ways to position the math that we’re learning as part of a larger story. I don’t think this needs to be a radical change; it can be a quick addendum of historical context, a narrative about a relevant mathematician or mathematicians, an interesting application of a topic, or just taking a moment to share how different concepts are related, framing where students have been and where they’re going. But humans learn from stories, and are motivated to learn from stories, and I think this is something that is underused in my classes.

Examples

An example is worth a thousand definitions. You can define “ideal” as carefully as you like and I’m still going to be confused the first time I learn about it. Share a handful of well-chosen examples and non-examples and all of a sudden it makes sense. In math we love definitions. I find many definitions elegant and beautiful. I spend lots of time thinking about how to explain concepts in ways that will make sense to students. These things are important, but it’s possible to overestimate their importance. Examples work with explanations to create students’ mental models of concepts. Examples give something concrete to latch onto, and they can illuminate boundary cases and subtleties that might not make sense in an explanation or be clear from a definition. And as I see more examples, I start to create new generalizations and come up with explanations that make sense to me. When I first read about ideals, they were nonsense. Now I think about them like a magnet — they’re this set of objects that pull other objects in, and once you’re in, you can’t get out. For instance, if I’m working with integers, once something becomes a multiple of 3, no matter what you multiply it by, it stays a multiple of 3. That might not make sense to you, but it makes sense to me. And the more I see new examples and incorporate them into my mental models, the better I can apply that knowledge. Examples give me a chance to test my understanding and see whether my ideas make sense in a new context.

I don’t think I do this very well with students. Student understanding often happens within the paradigm of my explanations and my ways of looking at mathematical ideas. There’s a place for that, especially to minimize confusion and misconceptions. But there’s also an place to give students lots of examples to work with, to ask them to come up with explanations that make sense to them, and to embrace their ideas and perspectives. I can explain the end behavior of rational functions until I’m blue in the face talking about top-heavy and bottom-heavy functions, and students are often still confused. Offering a set of well-chosen examples and letting students come up with language and analogies that make sense with their experience could be a much less painful way to do it.

There’s nothing groundbreaking about either of these ideas, but as someone who knows a lot of math and isn’t often in the position of learning new math, they’re easy to forget. A constant challenge of teaching is the curse of my own knowledge, and learning something new, even when it’s hard, is a great way for me to see learning from a new perspective and push myself to teach in ways that are accessible and engaging for all students.

Mindset and Routines

“I just can’t do math.”
-Lots of students since forever

When I started teaching, my typical response was to blame the student. Of course they can do math. Anyone can, with the right support and patience. Why can’t they just have a growth mindset?

Here’s the question I’ve started asking: why does that student have a fixed mindset in the first place? We work in an education system that excels at communicating to some students that they have intellectual promise, to others that they don’t, and that there’s nothing anyone can do to change it. This message is implicit in the ways students are sorted into tracks from an early age, to the way we talk about professionals in our fields and promote or discourage representation, to the grades we stamp on students’ assessments. And all of those messages accumulate over time to exactly the opposite mindset that we might hope students develop.

Growth mindset suffered from the mistaken idea that someone’s mindset can change just by being praised for their effort rather than their ability. In reality it’s a little tougher than that, and researchers have struggled to design interventions that consistently change mindsets. But understanding where mindsets come from helps in having some humility about changing a student’s mindset. It’s definitely possible, but it’s also definitely harder than we might like to think.

My job is to find ways for students to recognize the ways that they are mathematically competent. One way we’ve convinced students of their lack of mathematical ability is by valuing a narrow vision of mathematics that emphasizes computational speed and accuracy. If I’m deliberate I can send new messages that, over time, broaden that student’s idea of what it means to learn and practice mathematics, and help them to recognize their mathematical competence. Doing so helps to change mindsets, and to create classrooms that communicate the values of mathematics and help every student to see themselves as a potential knower and doer of math.

I try to do this through routines that value different competencies. Routines, repeated over time, help students to become comfortable practicing mathematics in new ways and give them opportunities to practice and recognize their own brilliance in new ways. I come in with the perspective that every student already has great mathematical ideas; my job is to create space for students to share and recognize those ideas. Routines are the healthiest space for those ideas to grow. And routines help to build enduring messages that communicate new ideas about students’ mathematical potential.

One important result of this shift in perspective is that it helps me to be patient. Instead of becoming frustrated that my students aren’t ideal math learners, I try to understand why they feel the way they do, and to feel like I have some concrete tools to help them shift their perspective. I know that nothing will change overnight, but I also know that I’m playing a longer game that can help students to see themselves in a new light, based on their successes in class each day.

I’ll be presenting on this topic with Lisa Bejarano at NCTM Regionals where we will practice a few routines that allow students to see their own brilliance and discuss many more while considering how they can be adapted to fit your own classroom.

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Come join the discussion in our session at NCTM Regionals in Seattle:

Anxiety, Mindset, and Motivation: Bridging from Research to Action
November 30, 2018 | 9:45-11:00 a.m. in Washington State Convention Center, 606

Developing a supportive class culture and growth mindset can reduce students’ anxiety, allowing learners to engage thoughtfully with each other around mathematics. Participants will discuss the challenges of shifting mindsets, experience routines as learners and leave with resources and ideas to implement these structures in their classroom.

Problems

What is a problem? And what types of problems are most useful for helping students learn to love math?

I struggle to define “problem” despite always having tons of them rattling around in my brain. It seems like a decent way to define it is to offer a bunch of examples. Here are some favorites:

Split 25

(Play With Your Math)

Self-Aware

(Play With Your Math)

Cows in Fields

(Singh)

To Cross the Bridge

(Singh)

The Census Taker
During a recent census, a man told the census taker that he had three children. The census taker said that he needed to know their ages, and the man replied that the product of their ages was 36. The census taker, slightly miffed, said he needed to know each of their ages. The man said, “Well the sum of their ages is the same as my house number.” The census taker looked at the house number and complained, “I still can’t tell their ages.” The man said, “Oh, that’s right, the oldest one taught the younger ones to play chess.” The census taker promptly wrote down the ages of the three children. How did he know, and what were the ages?
(Batchelder & Alexander)

Long Division
The following long division problem has a unique solution, despite providing just one digit. The Xs can represent any digit, and the problem is an 8-digit number divided by a 3-digit number producing a 5-digit number and dividing evenly.

 

(Gardner)

105
In how many ways can 105 be expressed as the sum of at least two consecutive integers?
(Liljedahl)

Circle in a Parabola

There are many circles that will “fit” inside a given parabola. What is the largest circle that will do so? Why?
(Lockhart)

7.11
A guy walks into a 7-11 store and selects four items to buy. The clerk at the counter informs the gentleman that the total cost of the four items is $7.11. He was completely surprised that the cost was the same as the name of the store. The clerk informed the man that he simply multiplied the cost of each item and arrived at the total. The customer calmly informed the clerk that the items should be added and not multiplied. The clerk then added the items together and informed the customer that the total was still exactly $7.11.

What are the exact costs of each item? (Assume that they multiply to 7.11 exactly, with no rounding.)
(Brumbaugh)

No Trigonometry Required!

(Brilliant)

November Nonagon
The figure below shows a square within a regular nonagon. What is the measure of the indicated angle?

(Five Triangles)

Why Problems? 

I think math is worth learning for lots of reasons. I want students to be quantitatively literate in a world that increasingly requires mathematical knowledge to be an informed citizen. I want students to understand math to open doors for them in the future, as mathematicians or in any number of other disciplines that rely on mathematics. I want students to cultivate skills of abstract reasoning, recognition and generalization of patterns, critical argument, precision, and structure. I want students to see math as a subject full of challenges that they are capable of overcoming, and for math to help them recognize their intellectual potential.

But from my perspective, the most important piece is for students to get a sense of the beauty and joy of mathematics, and to experience the “a-ha moments” that characterize our discipline. English has great literature. Science has the mysteries and wonders of the natural world. History has the gripping narratives of the past. Math has problems.

Two Things

I want students to experience the a-ha moments of problem solving as a catalyst to help them understand the discipline of mathematics and their potential as mathematicians. But not all problems are equally useful for creating these moments. I’d like to hypothesize two elements that allow a problem to facilitate students’ love of problem solving.

Insight vs Experimentation 

On one end of a spectrum are insight problems, like “November Nonagon,” “To Cross the Bridge,” and “No Trigonometry Required!” These problems lend themselves to certain representations and strategies, but the approaches one takes at first are unlikely to be successful. Solving the problem relies largely on an insight: a change of perspective that illuminates a path to a solution. A solver might end up staring at the problem, making no progress, for some time. With some luck, the insight will whisper itself at an opportune moment, and the problem will be solved. On the other end are experimentation problems, like “Circle in a Parabola,” “The Census Taker,” and “Split 25.” These problems lend themselves to trial and error and don’t require any large leaps of logic or intuition. A solver can try a number of different approaches, stepping back to look for patterns as necessary, on a much more well-defined path to a solution. That’s not to say that these problems are easy, just that they are more likely to suggest plausible pathways than dead ends.

Success

A second spectrum is how quickly a solver is likely to experience a feeling of success — whether or not they solve the problem, can they make some concrete progress early on? The problems “Cows in Fields,” “Circle in a Parabola,” and “105” allow a solver a quick taste of success, where one or several examples are readily available, although finding all of them still requires a great deal of persistence and ingenuity. These successes can act as springboards to the rest of the problem, rather than experiences of frustration from the beginning. Alternatively, problems “Self-Aware” and “7.11” resist easy wins. One could try a few ideas, but they don’t lend themselves to quick strategies, and a successful solver will likely have to muddle through a significant amount of failure, trying unsuccessful ideas, to get to a solution.

What Makes a Problem Useful?

I think that the best problems to teach students a love of problem solving allow for experimentation facilitate early success. Experimentation allows multiple access points, gives students half-formed and informal ideas to share and argue about, and gives a sense that, while the journey may not be easy, it is at least possible. Early success builds motivation; feelings of success help students understand that problems exist for the pleasure of solving, rather than to frustrate and bore them.

These aren’t necessarily static properties of problems. A teacher could facilitate experimentation in “November Nonagon” with the suggestion that a solver try adding auxiliary lines, or in “No Trigonometry Required” with the hint that the angles can be rearranged (without changing their size) to try to make a useful shape. Similarly, “Self-Aware” could be modified to make early success more likely by prompting students for 5- and 7-digit self-aware numbers in addition to 10-digit ones. These small changes, combined with choosing problems thoughtfully given students’ knowledge and motivation, can make a big difference.

This isn’t to say that problems without these characteristics are worthless — they can be fantastic fun for students who have already developed some interest and joy in doing math. But to create that a-ha moment that shifts a student onto the path of being a math lover, I think these two features are critical. Staring at a problem with no clear paths forward or ideas to try is likely to result in frustration for many students. And even when there are clear ideas to try, without some positive reinforcement of early success a student is likely to give up before they get to the good stuff. Not all problems fall neatly on one side or the other and no problem is perfect, but I do think these two features make a problem much more useful for all students, rather than just those who already like math.