In this monthly column, we speak with a notable member of the mathematics education community about their work and their perspectives on the teaching and learning of mathematics. This month, we had the pleasure of speaking with Max Ray-Riek, who will be presenting at this year’s Saskatchewan Understands Mathematics (SUM) Conference in Saskatoon.
Max Ray-Riek works at The Math Forum at NCTM and is the author of the book Powerful Problem Solving. He is a former secondary mathematics teacher who has presented at regional and national conferences on fostering problem solving and communication and valuing student thinking.
I would like to start off by asking you a bit about your background and your interest in mathematics. Was it a subject that you always enjoyed, or did something – or someone – hook you along the way? What drew you to teaching secondary mathematics rather than, say, research in mathematics?
“There’s so much being figured out right now about how math can be taught as a dynamic, engaging subject where everyone has unique ideas that matter… One of the most exciting problems facing the world today is how to teach math in a way that builds on students sharing their ideas.”
I had started off as a discouraged math student, fearing long packets of arithmetic problems that I was neither fast nor accurate with. I was lucky enough to have a 5th grade teacher, Ms. Allen, recognize that I enjoyed puzzles, problem solving, and thinking outside the box, and she invited me to try out some Math Olympiad problems. Even though I couldn’t solve a single one on my first try, she invited me to share the approach I’d used to start thinking about one of the problems, and that was when I realized that I could have math ideas that mattered to other people. From then on I was interested in math, and enjoyed doing math and talking about math thinking with other people. When I got to college, I realized that this was actually one of the most exciting areas to teach in, because there’s so much being figured out right now about how math can be taught as a dynamic, engaging subject where everyone has unique math ideas that matter. Reading math education research by people like Alan Schoenfeld, Jo Boaler, Ana Sfard, Paul Cobb, Jean Lave, and others helped me see that one of the most exciting problems facing the world today is how to teach math in a way that builds on students sharing their ideas.
These days, although you still visit schools and teachers to observe, co-teach and model lessons, you have transitioned from the high school classroom to work and blog at the Math Forum at NCTM. What sparked this transition, and what does this position involve?
I’ve been a part of the Math Forum since high school, when I asked a question of Dr. Math. In college I served as a volunteer “math mentor,” writing back to elementary students who solved our Problems of the Week, and then eventually as a research assistant on some of the grants that the Forum had received to study how to help college students like me improve as math mentors and learn about what kinds of feedback helped students reflect and revise. So I knew and loved the Math Forum’s community, classroom resources, and staff before I got to work there. As much as I loved teaching high school math classes, I knew I wanted to work for and with the Math Forum as soon as I could – and I only had to wait 2 years before they had a job opening I could apply for! The reason I wanted to join them was because I am passionate about the subjects of problem solving and building math learning experiences out of students’ individual problem solving methods. I wanted to spend time every day getting better at recognizing multiple methods to solve problems, recognizing students’ novice attempts to use different methods, and honing my skills at responding to students in ways that supported them in persevering, reflecting, and learning. And then I wanted to spend time practising these skills with other teachers: by doing problems together, by coaching, by creating online environments where we can practise in a less fast-paced setting than the classroom, in PD workshops – whatever it takes!
One of your roles at the Math Forum is writing support materials for the Problems of the Week. (The Problems of the Week, published on a two-week cycle to give time for reflection and revision, are challenging mathematics problems for Kindergarten to calculus students, ranging from applications of mathematics in daily life to improbable scenarios that nevertheless spark interesting questions.) In your view, what makes a mathematics problem a “good” problem? Do the criteria change in relation to students’ age and skill levels?
I think a good problem is one that, first and foremost, the students solving the problem understand – meaning that they understand the context the problem is set in (whether “real-world,” imagined, or mathematical), they understand what is being asked, and they can appreciate why it’s a reasonable question to have about this situation. My colleague Annie often says, “students can’t answer a question they haven’t asked.”
Now, having said that, I don’t mean that problems always have to come from familiar contexts – there are lots of ways to help students get into problems and make sense of them. For example, there’s a PoW I really love about tagging salmon at a salmon hatchery. Some students might have never heard of salmon, or they might know it only as a food and not realize it’s also a live fish. Even if they can picture it as a fish, they might not understand the wildlife management context or be able to picture the act of tagging or understand why someone would do it (or care how fast it can be done). To me, that doesn’t mean we shouldn’t use the problem, just that we shouldn’t assume that it automatically makes sense to students. Can we work with students to find YouTube videos of salmon being tagged at a fish hatchery? Is there a hatchery nearby we could take a field trip to, or one further away that we could Skype with a worker at? Could we read more about fish hatcheries and tagging and make a movie in our mind of what’s happening? Could we act out the problem using manipulatives until we understand it? Could we treat the problem as a scenario by leaving out the question and having students generate the questions they want to answer about the story themselves?
The other qualities a good problem has, besides being understood by the solvers after they have put some effort into understanding it, are that it is:
- Able to be tackled with a variety of methods;
- Worth talking about after you have an answer to compare methods, to look for generalizations, to analyze errors, etc.; and
- Connected to important mathematical ideas, so that it can be extended and/or made use of later.
In 2013, you published the book Powerful Problem Solving (in collaboration with Math Forum education staff Annie Fetter, Steve Weimar, Richard Tchen, Suzanne Alejandre, Tracey Perzan, Ellen Clay, and Valerie Klein), and have spoken about fostering problem solving at many regional and national conferences in the United States. In your view, what does it mean to be a “powerful problem solver,” and why is this an important skill for students to develop? Without spoiling the punchline of the book, what are some of the suggestions you offer readers in your book to foster students’ problem-solving skills?
There are many reasons we want students to be good problem solvers. However, I tend not to focus on the idea of problem solving as a life-long skill or important for 21st century careers. I tend to focus on problem solving as an inherent part of learning math. For example, think about how we teach students to solve equations in Algebra class. If we aren’t careful, it can become a series of steps that students feel they need to memorize. They can solve problems like 2x + 5 = 12 but struggle with 12 = 5 + 2y because they’ve focused on superficial features of the steps rather than on making sense of them. At its heart, though, solving equations is about finding simpler, equivalent forms of the same relationship until a solution becomes obvious. Whether the problem is written as 2x + 5 = 12 or 12 = 5 + 2y, good algebraists look at the situation and ask, “What’s making this hard to see the answer? Oh, it’s because there are two things happening to the unspecified quantity. How can I turn this into a simpler equation without changing the relationships?” Then they rewrite the problems as 2x = 7 or 7 = 2y by thinking about the relationships in the problem.
“What we’ve noticed is that getting good at guess and check is like learning to crawl before you learn to walk. While some people do skip the guess and check stage, it’s really much better, developmentally, to go through it.”
- Figuring out what is unknown in the problem
- Deciding whether you need to guess for both of the unknown quantities or whether guessing a number of llamas will let you figure out a number of ostriches based on your guess
- Figuring out what calculations you can do with the guessed quantities
- Figuring out all of the constraints in the problem so you can check your guess
- Repeating the guesses
If students were to write equations to solve this problem, they would need to:
- Figure out what is unknown in the problem
- Decide whether they need a variable for both of the unknown quantities or whether assigning a variable for the number of llamas will let them write an expression for the number of ostriches based on the number of llamas
- Figure out what calculations they can do with the variables
- Figure out all of the constraints in the problem so they can set their expressions equal to something
- Solve the equations
Becoming a good problem solver using guess-and-check means doing 80% of the work needed to solve a problem algebraically. Students tend to naturally gravitate towards using more and more symbolic notation in their guess-and-check work as they get more confident in their problem solving abilities, and often end up making a very smooth transition to algebraic problem solving the longer we’ve let them choose to use guess and check.
So to me, being a good learner of math requires us to use habits of mind that we develop through problem solving, such as identifying unknowns, likely calculations, and constraints; or looking for simpler ways to write the same relationships; or looking for patterns in numbers or repeated calculations. The more we can emphasize and support students in becoming aware of and developing these thinking skills, the easier our job of teaching content can be.
In Powerful Problem Solving, the way that we emphasize building these thinking skills is through specific activities that draw students’ attention to a problem solving strategy or habit of mind, and then having them learn from their own work and each other’s work ways that they could get even better at that strategy or skill. A lot of our activities emphasize listening to peers share their thinking and then reflecting and revising based on those conversations. We also offer a lot of support for teachers and students to learn questions that help get us started in our thinking, such as:
- What do you notice?
- What do you wonder?
- What is an answer that would definitely be wrong? Why?
- Is there another way to say/write the same thing?
- What has to be organized?
- What must be true? What might be true? What can’t be true?
- What makes this problem hard?
Do you feel that there is a conflict between problem solving and procedural fluency? How might one support the other?
“Procedures without meaning have to be memorized separately and kept separate, even as you learn more and more algorithms. Therefore, they are prone to all sorts of bizarre errors.”
These algorithms are competing with each other because students were focused on memorizing what to do when a problem has decimals or integers in it. They didn’t learn tools for reasoning quickly about integers and decimals, didn’t focus on estimation and reasonable answers first, didn’t build up from their prior knowledge of place value, addition, multiplication, and opposites.
Imagine instead if students had been given the opportunity to learn to multiply decimals by building on their understandings of place value, multiplication, and fractions; if they’d started with problems like 20 * 2.5 and reasoned in contexts such as adding up 20 payments of $2.50 each, or scaling up a 20” picture two and a half times. Imagine that students had offered methods as diverse as:
- reasoning that 2 groups of 2.5 is 5, so 20 * 2.5 is the same as 10 * 5
- reasoning that 20 groups of 2 is 40 and 20 groups of ½ is 10, so the total is 50
- reasoning that 10 groups of 2.5 is 25, and so 20 groups of 2.5 is 50
- reasoning that doubling 20 is 40, and ½ of 20 is 10, so 2.5 times 20 is 50
- repeatedly adding 2.5 twenty times
- drawing a picture using arrays or rectangles that shows 2.5 by 20
Teachers could then scaffold students to share, learn from, and get good at these methods, particularly ones that relate to general algorithms such as partial products. They could engage students in solving more, and more difficult, decimal multiplication problems, until students had made sense of decimal multiplication, connected it to their ideas of place value and multiplication, and ultimately developed fluent strategies.
The connection from doing multiplication in a meaningful context to having procedures that make sense and don’t interfere with addition procedures is to have built up to them through students’ informal methods. For teachers of younger students or students learning arithmetic, books like Children’s Mathematics (http://www.heinemann.com/products/E05287.aspx) and Intentional Talk (https://www.stenhouse.com/content/intentional-talk) have been helpful in laying out how that actually works.
More recently, you’ve been discussing the role of online communities in professional growth. The online math teacher community – referred to on Twitter as the MathTwitterBlogosphere, or MTBoS (see https://exploremtbos.wordpress.com/ for more information) – certainly has exploded in the past few years, with hundreds of teachers and teacher educators choosing to share ideas and to discuss issues in mathematics education online with colleagues around the world.
In your view, why do you feel that mathematics teachers are drawn to these (ever-growing) communities, and what are the benefits of becoming involved? What first steps would you suggest to teachers who would like to join the online conversation?
On the flip side, during your Ignite talk at the latest NCTM Annual Meeting and Exposition (https://youtu.be/j28BHpIzFzg), you argued that although blogs can be a great source of ideas and provide a great space for discussion and reflection, teachers shouldn’t rely solely on them in lesson planning. Why would this be a problem, in your view?
“With an online community, we get to focus on our passions and our strengths and be involved in a community who shares those passions and strengths.”
Another huge way Twitter and blogs are useful is by connecting teachers to resources we can trust. As we come to trust other online colleagues, we eagerly check out or save for later anything they share about the topics we teach. Teachers learn together about content, share their best lessons, share student thinking and collaborate to adjust based on student thinking – all kinds of different collaborations are possible, and lots of those collaborations lead to activities, lessons, or whole sequences of lessons that teachers can use in class.
If you’re just thinking about how to join online communities of teachers, here’s what I always tell people:
- It’s okay just to “lurk” – make a Twitter account and start following some math teachers so that you have some activity in your “timeline.” You can go to my page, http://twitter.com/maxmathforum, and look at who I’m following to find over 2,000 math teachers on Twitter. Alternatively, go to http://twitter.com and enter one of these “hashtags” in the search field to find people tweeting in your grade band or subject area:
- You don’t have to Tweet to be part of the community. And “liking” or “re-tweeting” is a great way to show that you’re here and you care about what’s being said, even when you don’t feel like writing your own tweets.
- If you do decide to Tweet, try joining a conversation. Hit the “reply” button on Twitter to respond to a tweet – the person you reply to, even if they don’t follow you, will see your message and will probably write back to you. It’s not too hard to find the time to write 140 characters back, so Twitter often works better than email for getting a reply. Note that this doesn’t necessarily apply to super famous people like Jo Boaler 😉
- It’s also okay to just read blogs and not start one of your own. Commenting on other people’s blogs is a great way to, again, show that you value their thinking and are learning from the community, as well as maybe even putting your own ideas out there.
- Communities are made up of lots of roles: appreciaters, users, question-askers, question-answerers, sharers, creators, etc. Embrace your current comfort zone and know that online, we can shift roles really easily as we find things that play to our strengths and spark our passions!
Now, on to the strengths and pitfalls of “community-curated” lessons.
The benefit of these “community-curated” lessons is that we know who came up with them, why they were invented, and how the lesson was designed to meet its goals. That makes lessons we get from our online friends highly likely to go pretty well – we’re more likely to adjust them to match our own goals in useful ways, and/or more likely to implement them faithfully because we understand not just the hows and whats, but also the whys of the lesson.
“The trouble with teaching students entirely from lessons we pull from a wide range of sources all over the Internet is that our lessons can lack cohesion.”
I don’t think the situation is hopeless, though. First of all, I think that teacher teams can do more work together to make sense of the curriculums they already have and make sure they understand the hows, whats, and whys before they replace one lesson with another. And I think that blogs and Twitter can help with that work – working with a team to dive in and understand curriculum is a very fun project that should totally be live blogged/Tweeted for everyone to learn from. Secondly, I think curriculum writers (and the various systems that fund them, from grants and public funding to private publishing companies) can do a lot more to make their process open and transparent, sharing more about the whys during the process of making decisions about the whats and hows of their curriculum.
Our readers will likely be aware that you will in Saskatoon this November to present as a keynote speaker at our very own Saskatchewan Understands Math (SUM) Conference. (We can’t wait!) We don’t want to spoil the surprise, but could you give our readers some insight into what you will be discussing during your sessions?
I always focus on two things:
- Doing math ourselves, as teachers, to help us see and appreciate more connections – connections to students’ informal methods, connections between different strategies and methods, and connections to other areas of mathematics.
- Looking at student work to help us hone our skills of anticipating students’ thinking, seeing connections in student work, posing purposeful questions, orchestrating discussions, etc.
So please come prepared to do some math (and hopefully try out a new strategy that you wouldn’t normally use, like Guess and Check or Draw a Picture or Act It Out or Make a Mathematical Model…), to look at examples of student thinking about that math problem, and then to talk about how doing math and looking at student thinking can be a part of your planning routine, whether you do your planning online or in-person with your colleagues.
Thank you, Max, for taking the time to have this interview! We are eagerly looking forward to continuing the conversation at SUM 2016 in November.