You Must Read Building Thinking Classrooms in Mathematics By Peter Liljedahl

I have been a math educator for about twenty years and Building Thinking Classrooms in Mathematics by Peter Liljedahl has more potential to improve the way we teach mathematics than any other book I have ever read. These are not words I say lightly. This book is an absolute game changer for all math educators and everyone needs to read it.

I am writing this blog post for two purposes:

• to convince you why you should also read and implement what you learn from the book
• to have the many profound insights I noted in one place for me to come back and read again

Peter broke down what math teachers do into fourteen separate categories (called practices). He wanted to test how effective our traditional practices are so he ran experiments to test how much thinking happened when teachers did them. Then he had teachers do the opposite and compared how much thinking happened (for example, if we usually have kids work while sitting down then what would happen if they worked while standing up?). They continued cycles of modifying and testing alternatives to the status quo until they found what produced the most thinking. This book tells you all of those best practices so you can implement them with your own students. Best of all, his writing style is easy to read and not pretentious or know-it-all in any way.

I think of each practice like an infinity stone from a Marvel movie. You could just use one of them and it’s powerful on its own. However the more you combine, the more powerful it gets.
 

The fourteen practices were:

1. What types of tasks we use
2. How we form collaborative groups
3. Where students work
4. How we arrange the furniture
5. How we answer student questions
6. When, where, and how tasks are given
7. What homework looks like
8. How we foster student autonomy
9. How we use hints and extensions
10. How we consolidate (summarize / wrap up) a lesson
11. How students take notes
12. What we choose to evaluate
13. How we use formative assessment
14. How we grade

Think about how comprehensive this list is. It probably covers at least 90% of what we do as math educators. If you’re already doing what the research showed, you’ll feel so validated. If you’re not, wouldn’t you want to know what works best so you could consider changing?

I now want to go through some of the parts that most resonated with me. The book is FILLED with amazingness and my notes are in no way an adequate substitute for reading the book. What is below is me quoting, paraphrasing, or summarizing the book. Virtually none of it is my insight and is just me processing what I read.

Peter begins by defining “thinking” as the goal for what we want students to be doing in math classroom and contrasts that to what we often see instead in math class by using Darien Allan’s categorizations of “studenting behaviors” which include:

• slacking – not attempting to work at all
• stalling – doing legitimate off-task behavior (like getting a drink or going to the bathroom)
• faking – pretending to do the task but in reality doing nothing
• mimicking – mindlessly repeating what they have in their notes
• trying it on their own – attempting to work through a problem, regardless of whether they got it right or not

I would guess that pretty much every teacher has seen these behaviors, but I had never seen an attempt to classify them and found the categories useful. If you had asked me early on in my career which students were thinking, I would have for sure included the “trying it on their own” students. However, I probably thought that the “mimicking” students were also thinking.

If I’m being honest, I got through all of high school and graduated from UCLA with a B.S. in mathematics because I was a solid mimicker. More than half the time I knew how to get the right answer but had little idea what I was doing. So, acknowledging that mimickers were not actually thinkers would have forced me to acknowledge that I was also not a thinker, and I probably wasn’t ready to say that out loud twenty years ago.

This excerpt hit me right in the gut: “When we interviewed the teachers in whose classrooms we were doing the student research, all of them stated, with emphasis, that they did not want their students to mimic. Ironically, 100% of the students who mimicked stated that they thought that mimicking was what their teacher wanted them to do.” Well damn. That had to be what I would have said and what my students would have thought.

I really liked how Peter defined “problem solving” as “what we do when we don’t know what to do.” He goes on to say “problem solving is not the precise application of a known procedure. It is not the implementation of a taught algorithm. And it’s not the smooth execution of a formula. Problem solving is a messy, non-linear, and idiosyncratic process. Students will get stuck. They will think. And they will get unstuck. And when they do, they will learn.”

Contrast this with how mathematics is usually taught: I’ll show you what to do and now you practice that skill. This is so disconnected from what really happens in life.

Peter describes three attributes of high quality problem solving tasks:

• low-floor task – anyone can get started with the problem
• high-ceiling task – they have enough complexity to keep people engaged
• open-middle – while there is a single correct answer, there are multiple ways to solve the problem

If you’re familiar with my work (especially with Open Middle and my real world problems), you won’t be surprised by how much I agree with these attributes.

One part that I did find surprising was that Peter stated that the problems he chooses are “for the most part, all non-curricular tasks. That is, very few of these tasks require mathematics that maps nicely onto a list of outcomes or standards in a specific school curriculum.”

It made me wonder how necessary it was to use the kinds of problems he mentioned and whether instead we could find suitable replacements that better matched the standards teachers were using. Realistically, it will be a hard sell to get teachers to do these practices if they are not tied to what they’re teaching. I forget where in the book he says this, but I recall Peter mentioning that when students are thinking well, everything else goes faster… so doing non-curricular tasks are investments that make everything else go smoothly. He goes on to talk about where to get problems like these as well as how to turn existing problems we use into rich tasks, so I don’t want to misrepresent what he’s saying.

That being said, I’m guessing we could get similar results with carefully chosen curricular tasks like Open Middle problems and from what I can see on Twitter, other teachers agree.

@openmiddle + @pgliljedahl ‘s “Thinking Classroom” = great student engagement during wonderful questioning. @JohnDrydenPS @deenickerson16 @Errs5 pic.twitter.com/MhInFc8c3u

— Al Savage (@TeachMath1618) December 3, 2019

Putting students into groups happens all the time in classrooms, but I never thought about it deeply. For example, what’s the ideal number of students in a group? How do you decide which students to group together? Honestly, I didn’t think it mattered much, but the research proved that I was wrong.

This quote really resonated with me about what it’s like for students in groups: “the vast majority of students do not enter their groups thinking they are going to make a significant, if any, contribution to their group. The are entering the groups in the role of follower, expecting not to think. That means that with the strategic groupings, other than those 10% to 20% who are accustomed to taking the lead, the rest of the students, by and large, know that they are being placed with certain other students, and they live down to these expectations.”

That’s exactly what happens. Most kids go in a group and sit there, waiting for someone else to take the lead and have time pass. This simultaneously surprises exactly no teachers AND is not at all what we want to happen when students are in groups. So how do we get around this?

There were many nuances to his suggestions but here are two summaries:

• The groupings had to be visibly random. Every student is going to think that you are purposefully placing them in a group regardless of how random you claim for it to be. The only way to get around this is to make it obviously and undeniably random. For example, there are websites like this one and countless others where you can enter names and it will generate groups for you. It can be done with offline methods like a deck of cards too. I would not have guessed how important visibily randomizing groups is in breaking down students’ perception that they were put into a group because of a specific reason which makes them more open to really participating.
• Three students was the ideal group size. He says “Groups of two struggled more than groups of three, and groups of four almost always devolved into a group of three plus one, or two groups of two.” Well that’s easy to implement and I had no idea. I almost always did groups of four.

This practice was perhaps the one I thought about least. Basically my whole life as a math student was spent writing at a desk. I don’t think it ever occurred to me that there was somewhere else we could do math, so that’s what makes the findings of Peter’s research on where students work that much more profound.

Simply put, having our groups of three students writing on a vertical surface like a whiteboard or poster paper generates a lot more thinking than having them work while sitting down at a desk. His findings are a lot more nuanced than I’m describing including who uses the marker to write, who uses what color, what can be erased, etc. For example, I probably would have given each student their own marker, but the research showed that “when every member of the group has their own marker, the group quickly devolves into three individuals working in parallel rather than collaborating.”

Sure, this will require some changes in the way we arrange our classrooms, but if it greatly increases thinking, I’m in.

This practice was probably the most postdictable to me in that I would have never thought of it on my own, but immediately after reading it, it was obvious that he was right. For example, imagine coming into a large auditorium for a lecture in college. The way that room was set up gave us the signal that there was not going to be a lot of groupwork and conversation going on. Our role was to listen to whoever was at the podium.

So, my question to you is how would would you place students in a classroom to show that they would be doing the thinking or NOT doing thinking? Maybe rows of desks all facing the front of the classroom would be closest to a lecture and signify that listening is more important than collaborating here. So how would you rearrange the class to show otherwise?

Ultimately, what Peter found was that teachers “only needed to defront a room in order to also destraighten and desymmetrize it, as long as we defined defronting as ensuring that every chair in the room was facing a different compass direction.” The reasoning is that when there is a front of a classroom, that is where the knowledge comes from. The teacher is generally at the front of the classroom, so the message we’re conveying is that the teacher is where the knowledge comes from. Defronting the classroom removes that unspoken expectation.

Here’s an example of what that might look like:

✅Defronting the classroom!
Diving into implementing #thinkingclassroom in Grade 8 Math. Thanks to @pgliljedahl for the inspiration!

Even though it’s the end of the day the room feels ready!
✅Whiteboards (VNPS)
✅Visible Randomized Groups
✅Open Middle Thinking Questions. pic.twitter.com/sR50XIbanh

— John Stephens (@CTEPEI) March 22, 2022

 
Will my OCD tendencies enjoy a defronted classroom? Not initially. Will it be worth it if it gets kids thinking? Hell yes.

This paragraph really shocked me because it was showing the unrealized flaw I used to do: “Thinking is messy. It requires a significant amount of risk taking, trial and error, and non-linear thinking. It turns out that in super organized classrooms, students don’t feel safe to get messy in these ways. The message they are receiving is that learning needs to be orderly, structured, and precise.” Instead of straight and symmetrical classrooms helping students, they were placing unspoken expectations upon the thinking that was encouraged in this classroom. Damn. If only I had known that my efforts were having that effect.

This chapter reminded me a lot of my bench presser and spotter metaphor: the spotter’s goal is to give the bench presser the least amount of help needed to lift the weight. Similarly, our goal as teachers is not to answer every student’s questions. Instead, we need to be strategic about the questions we respond to.

The first big insight for me was his categorization of the types of questions students ask. He writes: “As it turns out, students only ask three types of questions: proximity questions, stop-thinking questions, and keep-thinking questions.” He breaks down these categories very well, but a rough explanation is that:

• proximity questions are ones that students tend to ask only when you’re near them and are generally not that important
• stop-thinking questions are ones where kids don’t want to think and they’re asking something to either get you to do the thinking for them or give them permission to stop thinking entirely.
• keep-thinking questions are ones that are legitimately helpful in continuing their thinking

He goes on to say how “it turns out that of the 200-400 questions teachers answer in a day, 90% are some combination of stop-thinking and proximity questions.” Then he continues by saying “Answering these proximity or stop-thinking questions is antithetical to the building of a thinking classroom. The only questions that should be answered in a thinking classroom are the small percentage (10%) that are keep-thinking questions.”

While this makes perfect sense, I’m sure I’ve answered proximity and stop-thinking questions far more than I should have. He goes on to share great ideas for avoiding answering the wrong kinds of questions including how to avoid having students revolt because you’re not being helpful enough.

The research suggests that we give tasks to students as quickly as possible (rather than later in the class), and that we do it while huddled around a central area rather than sitting in our seats.

What blew my mind and continues to be hardest for me to accept is what the research showed was the best way to give students a task. So you can play along, rank these methods for giving students a task from most to least effective.

1. Projecting a task
2. Giving it pre-printed
3. Writing it out on the board
4. Giving it verbally

I don’t know what order you picked but I knew for sure that giving it verbally would be dead last. So it made it all the more shocking to me when I read: “Nothing came close to being as effective as giving the task verbally. This was a shocking result. Not only does it go against decades of norms, it also goes against teachers’ instincts.”

WHAT?!

Now I should absolutely clarify that he goes into great detail and clarification about what it means to give a task verbally including saying “verbal instructions are not about reading out a task verbatim.” And gives a great many practical implementation tips. But as he wrote, it goes against my instincts and I’m still struggling to process this.

The reality for most classrooms is that homework is not helpful for most students. Peter categorizes most students’ experience with homework as either they didn’t do it, got help, cheated, or tried it on their own. Of the ones that did do it on their own “the vast majority completed the homework by mimicking from either their notes or the textbook.”

Accordingly, very little real thinking is coming from homework. He shared that the “data on homework showed that 75% of students complet[ed] their homework, only about 10% were doing so for the right reason. When completion is the goal, it encourages, and sometimes rewards, behaviors such as cheating, mimicking, and getting unhelpful help.”

So what do we do? Peter suggests that the solution is to switch homework from being done for teachers to being done for their own learning. To make that switch they “stopped calling it homework and started calling it check-your-understanding questions.” This paired with several other changes including: not grading homework, not punishing kids for not doing it, etc. resulted in significant increases in thinking. I’m not doing justice to the numerous research-based tips he suggests, but this chapter is great.

Remember that with our existing practices, they’re already not working. So while this new approach might sound very different than our own experiences, having some students doing real thinking is better than most students doing little to none of it.

You’ve probably been in a classroom where kids hide their work and answers from one another so no one cheats off them. Not surprisingly, there’s probably not a lot of collaboration happening in this classroom.

So, Peter suggests strategies that helps empower students to take control of their own learning rather than relying on you to be the source of all their knowledge. The strategies seemed to validate what I was already doing and most seemed rather intuitive.

This chapter was both really intuitive AND seemed like something I could spend the rest of my career on trying to get better at. Let me explain.

Have you ever been in the zone where you were so into something you were doing that everything else around you kind of faded away? Some people call it “flow”. Well imagine that happening in math class where students are so into what they’re working on that they get into the zone.

First, it’d be hard to get them there to begin with but it’d also be hard to keep them there. If it’s too easy or boring, they will fall out. If it’s too hard or confusing, they will fall out. Figuring out the just right amount take a lot of skill. He goes into great detail as to both the theory behind this as well as practical tips for keeping your own students in the zone.

I really like this quote he shared: “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.” So simple yet such a profound shift.

I’m still processing this chapter to blend with what I already knew. Specifically, I’m a huge fan of 5 Practices for Orchestrating Productive Mathematics Discussions by Peg Smith and Mary Kay Stein. In fact, Thinking Classrooms and 5 Practices are probably the two books that have most impacted me as a math educator. So, I’m trying to wrap my head around the contrast he suggests.

He says: “Whereas Smith and Stein do both the selecting and sequencing in the moment, within a thinking classroom, the sequencing has already been determined within the task creation phase – created to invoke and maintain flow. What is left to do is to select the student work that exemplifies the mathematics at the different stages of this sequence.”

I can see what he’s saying, but I would push back and say that most teachers who use the 5 Practices already have an idea of the student work they hope to find and the order they hope to share it in, ahead of the lesson. They are then going through the room hoping to find that and or nudge students in that direction. So in that respect, I think it’s fairly similar.

That being said, Peter also mentions “another difference is that, whereas Smith and Stein have students present their own work, in the thinking classroom the decoding of students’ work is left to the others in the room.” This is interesting because it gets at the heart of what happens when a student presents to the class. Is everyone checked out? Are they engaged? How do you manage this?

He unpacks it better than I can, but if you’re a fan of Smith and Stein, I think you’ll appreciate this chapter even more.

If I had to use one gif to summarize what it feels like to take notes in math class, it would be this one:

Personally, I rarely take notes because when I do, I struggle to also process what is being said in real time, and truthfully I almost never look back at my notes anyway, so why bother? I doubt any of this is shocking to you, so the question then is that if we all agree that the status quo for note taking is not great, what are our alternatives?

What Peter figured out is beautiful in its simplicity: they wrote “notes to their future forgetful selves.” They asked students “What are you going to write down now so that, in three weeks, you will remember what you learned today?”

He also experimented with all sorts of graphic organizers that made note taking feel more manageable and less overwhelming. There were countless things whose brilliance was obvious only after he described it, because I was never going to consider and study it on my own. This is definitely a section worth diving into.

Have you ever seen a teacher write detailed and thoughtful feedback to students on a graded assignment so that the student could learn from their work, only for them to look at the grade and read little to none of what was written? I’m guessing you have.

Choosing what work to evaluate and how to evaluate it such that students actually grow from the experience is tricky. Sometimes it fails because we’re trying to treat it as both a formative AND summative assessment at the same time… and it does neither particularly well. Sometimes it fails because the way we convey the feedback is not received as we intended.

To combat these realities, Peter shares a variety of revised rubrics we can use to help students reflect on their progress. I especially appreciated the nuanced breakdown of the strategies they tried but revised along the way. It helps to not only see what was the best option but also some of the steps along the journey to get there. For example, instead of having a rubric where every column had a descriptor, you could have descriptors at the beginning and end but with an arrow pointing in the direction of growth. This helped students shift from seeing where they are as a fixed to seeing where they are as a signpost on their journey.

This chapter was also full of really interesting ideas to help teachers collect information in real time that they can immediately use to adjust their teaching. While there were many strategies that you could use, I’ll share one I had never heard but immediately loved.

He wrote: “At the end of a unit of study, ask your student to make a review test on which they will get 100%. If they can do this, then they know what they know. Then ask them to make a review test on which they will get 50%. If they can do this, then they will know what they know and they know what they don’t know.” This is fascinating! I’ve never tried this with students but I’m so curious how they’d respond.

Have you ever thought about how subjective grading is? For example, when calculating GPA, an A is 4 points and an F is 0 points. So if a person gets an A and an F, they have an average of 2 points which is a C. With percentages, if someone gets an A (100%) and an F (0%), that averages to 50% which is an F!

When the same scores can give you different final grades, something isn’t right. Even more challenging is that the grades students have may not reflect what they know. Often things like participation and homework are factored in, which could lead the grade to misrepresent what their knowledge.

For example, consider these students who all get the same C grade at the end of the year:

• One starts the years with all As and ends the year with all Fs
• One gets a C on every single assignment
• One starts the years with all Fs and ends the year with all As

How do you feel about where each student is at? Does each of their C grades seem to match what they are currently demonstrating? While these are my examples, Peter is making a similar point in that the way we’ve traditionally graded students is lacking and it’s worth considering better options.

Peter advocates a shift away from collecting points to discrete data points that no longer anchor students to where they came from but more precisely showed where they currently are. There are a lot of benefits, but perhaps my favorite is that it gets teachers and students on the same page about where the child is at and incentivizes them to always keep learning rather than give up when it feels like improving their grade is hopeless.

Peter wraps up the book by sharing his findings for the question of “What’s the best way to implement these 14 practices or does order not matter?” He concluded that grouping the practices into four “toolkits” gives the teacher milestones where they should focus on one set of practices before moving on to another. He even goes as far to share what order to do them in after your first year when they’ve become more familiar to you.

I hope you’ve enjoyed my reflections on Peter’s book. I’d love to read about what resonated with you in the comments, and I’d especially appreciate you pushing my thinking on any points I may not be fully processing.

I should add that one part I haven’t mentioned is that each chapter ends with an FAQ with questions Peter often gets about the practices as well as questions you can talk about in a book study or on your own. Even if I didn’t have my own questions after reading about a practice, I valued reading what others asked because they were often quite good.