Albert Vilalta
Building Thinking Classrooms in Mathematics (K-12), by Peter Liljedahl, is a great book. No advertising is needed: every teacher I met in NJ this September had read the book along the summer. It is unusual to see such a success from an educational math book.
What’s its secret?
Probably, the shortest answer is the one Robert Kaplinsky pointed out, “because it has more potential to improve the way we teach mathematics than any other book I have ever read”.
Let’s see why!
It all begins with a flight to Peru. This summer, Laura, Marc and I, from Innovamat, were traveling to the Inter-American Conference of Mathematical Education (CIAEM), one of the most relevant congresses in our field.
Despite the thrilling destination of our trip, it was a 13-hour flight. Presumably boring. Luckily for us, Laura had other plans.
The plane is still leaving Barcelona when she opens her bag and asks, “Do you remember Peter Liljedahl?”
“Yes! We quoted his work about teaching math as storytelling in our NCTM workshop.”
“Right! You saw that he recently published this book, right? Want to leaf through it?”
Next thing, “This is your captain speaking. In fifteen minutes, we are landing at Lima Airport.”
We had spent hours and hours engaged in the reading, discussing its main ideas, and sharing our experiences and points of view. This was a beautiful example of the “optimal experience” described by Csíkszentmihályi (and quoted by Liljedahl himself within the book, p. 147): “Whenever someone had an optimal experience, they lost track of time”.
As we discovered first-hand, this book is a great tool to foster discussion. It presents research-based results in a very friendly style and structure, with real applicable examples and questions to reflect on, covering most of what we do as math educators. Moreover, it is strongly aligned with our view at Innovamat: we also want math students to think. As simple—and complex—as that.
This article is a summary of the most interesting big ideas from the Introduction and Chapter 1. You’ll find them explained through our interpretation and experience. Although we ultimately agree on Liljedahl’s view of what is a thinking classroom, we may add nuances to some of his examples, and above all, lay out questions to keep discussing and learning.
If you already read the book, I hope you find our reflections useful to delve into it. If you haven’t, I hope you feel invited and join the conversation.
Let’s go!
Big Idea 1. Redefining success
Think about a successful student in your class. What makes him or her successful? What is success? Many teachers, families and even students would say success is about passing tests, since the system itself promotes this twisted view.
Imagine you had a supernatural memory, and you were able to memorize all the German-English dictionary. Would that make you proficient in German? You’d know the words and their meanings, but not how to combine them, how to pronounce them, how to play with them, how to create literature with them. Would you be able to pass a test on German words and definitions? Possibly. Would that make you successful? And what is more important: would you love the language? Possibly not. Learning math through memorization, without thinking, focusing only on content, only through what Liljedahl calls “mimicking” and “now-you-try-one tasks”, is like reading the dictionary.
Tracy J. Zager, in the prologue of the book, writes this beautiful sentence (p. xvi): “Success means getting more students thinking in math class, for longer amounts of time.” Thinking is key. From our perspective, thinking somehow condenses in a single word all the mathematical processes (or practices) defined by the NCTM (2000)—problem solving, reasoning and proof, connections, communication and representation. Learning and doing math consists in exploring the content while developing such thinking skills, that is why we foster them in our lesson guides.
As an example, take this processes-based tip from our “Remainders Race” guide, a lesson in which we play a board game to consolidate the importance of the remainder in a division:
Liljedahl goes even further, claiming that “Thinking is a necessary precursor to learning, and if students are not thinking, they are not learning.” (p. 5) Considering math, we mostly agree to this emphatic argument. Nevertheless, we’d feel more comfortable saying that “thinking is a precursor to understanding”, because learning has a more vague definition—someone could even argue that memorization is also a way of learning, facts at least, in a superficial way.
In any case, reconsidering success beyond passing tests is a necessary first step that has the potential to transform our approach to teaching, our expectations, and our assessment strategies.
Big Idea 2. Delving into problem-solving
To interpret state standards, math practices, teaching guidelines and programs that put problem-solving front and center, it is fundamental to agree on what a problem is within math education. In a recent article, we explored them, and we compared them with exercises through some examples.
Like Kaplinsy, we really like how Liljedahl defines problem solving by summarizing what other authors—like Pólya (1945) or the NCTM (2000), whom we also regard as references—wrote in the past: “Problem solving is what we do when we don’t know what to do.” (p. 19) This means, as we always say, that tasks are a problem when someone is trying to solve it without knowing directly how. Eventually, once the problem is solved and mastered, it dies as a problem to become a routine task. And since we’re on the topic of tasks, let’s conclude by taking a look at the kind of tasks we’re interested in.
Big Idea 3. Types of tasks
What kind of tasks are the best ones to foster thinking classrooms?
Although later in the book he adds nuances to his view, in the first chapter Liljedahl is very conclusive about this: “What a task needs to do is to get students to think.” (p. 19)
Does this mean that we should get rid of every routine task? What about exercises that help consolidate procedures or algorithms? At Innovamat, we clearly stand for building any procedure or strategy from through thinking tasks, but alongside that, in our experience, a few weekly minutes of reproductive practice is needed.
In fact, Liljedahl aligns more closely with our view later in the book: “The goal of building thinking classrooms is not to find engaging tasks for students to think about. The goal of thinking classrooms is to build engaged students that are willing to think about any task.” (p. 159)
We couldn’t agree more.
What is worth discussing is the concept of what Liljedahl calls “non-curricular tasks”. He defines them as “highly engaging thinking tasks used without the concern of curriculum”. And then he puts some examples like card tricks or word problems such as:
I buy a video game for $10. I then sell it for $20. I buy it back for $30. Finally, I sell it again for $40. How much money did I make or lose? (p. 21)
Calling such a task “non-curricular” seems misleading. Not only because solving it requires bringing into play curricular standards like 4.NBT.B.4—Fluently add and subtract multi-digit whole numbers using the standard algorithm—, but also because it is directly targeting the math practices—making sense of problems, reason quantitatively, etc.— which are also curricular. As we saw in the first big idea, we firmly believe that math processes and practices are at least as important as content itself. Keeping them out of the equation when talking about the curriculum seems risky, because it may invite teachers to belittle such processes and practices. In any case, as you can see, Liljedahl’s idea of a thinking classroom is strongly aligned with ours: we feel that it truly takes into consideration the math processes and practices. It’s just a matter of being explicit and including them any time we talk—or think—about the curriculum and the standards. This is precisely why we always try to foster them through our guides, like in the following example from our logbook (see the full lesson and its teaching guide here):
In Conclusion
If you haven’t had the chance yet, read the book. If you know any other math teachers, recommend the book to them. As I hope I’ve conveyed to you, this book is a fantastic tool to start discussions, try new things, confirm intuitions, and consolidate great math teaching practices.
It is very satisfying and encouraging to see that we are all in the same boat—a boat sailing towards thinking classrooms.
Bonus track!
f you’re eager to read one last big idea about students’ engagement, here it is.
Laura Pavón
Innovamat