Thinking, Fast and Slow (Part 3)

Image result for LeeThatcher - "Thinking"

The first two parts of this series were created using Adobe Spark Page. I found while creating this third part that most of what was in it was text, and that a more stable place to host it was probably my blog. Hence, here it is. I’d encourage you to check out Part 1 and Part 2, if only briefly, before reading this part. 

We see in the last few chapters of Part I of the book that System 1 is a story teller. This system helps you make a coherent narrative of the world. This, like most things, has positive and negative aspects. I think the shortcomings of this tendency are important to understand.

The first problem with System one is that it’s relatively easy to manipulate it, especially since so much of it happens automatically. Take the following example from the book. Read the two words below.



In the second or two it took you to read those words and immediately following reading them you had a reaction. Some of it was physical, like the hair on your arms probably stood, your sweat glands were activated, your heart rate went up a bit. But you also likely sketched out a story that involved bananas causing the vomit (or in some other way being connected to the vomit). You did this automatically as system one is attempting to fit the input into a coherent story.

This, like most things, has positive and negative repercussions. It means that we are likely to seek out and find information that fits with the story system one is telling us. “Sally is lazy.” “James is smart.” “Maria is a hard worker.” Once we’ve put these narratives in our mind, system one tries to find information that fits the narrative. And while system two should be the hero here, always evaluating the assumptions of system one, it turns out that system two is a bit lazy. It’s much easier for system two to just go with the narrative. It takes cognitive work to constantly be evaluating everything system one is telling you, so often times that work is avoided by system two.

The key here is that we are aware of the narratives and stories we have in our minds. We need to be on the look out for information that both confirms our narrative (to be sure it does in fact confirm the narrative and that we aren’t overlooking something) and negates the narrative (so that we can change the narrative in our minds to better represent reality).

One major theme of the associative machine is this: when there is some sort of external input to your brain you’re not consciously aware of what’s going on in your brain. When you see an object or hear a sound or experience a feeling, you’re flooded with ideas which in turn activates more ideas. Only a few will pop up in consciousness and this flood of ideas is largely out of your control.

What does this mean for teaching?

When we deal with students we have to remember that fact. Much of the time they (and we, whether we care to admit it or not) are at the mercy of system one. Actions that you take or that other students take in the classroom can set off chain reactions in a student’s brain. The result could be positive or negative. You can imagine starting a lesson with some sort of introduction that primes students for learning, giving them some ideas that ignite other ideas (you might call this engaging prior knowledge) or put them in a positive mood. The latter idea is from Eric Jensen and is also described in the book and that is that being in a good mood helps your brain be prepared to understanding new concepts. You could also imagine taking some action that results in an unwanted behavior. The key to dealing with these behaviors is figuring out what the underlying cause of the behavior is. What is triggering their associative machine to result in the behavior?

The last note I want to make here is that we need to help students understand what’s going on in their brain. Just as I encourage anyone reading this to try to keep a check on the automaticity of system one, we should find ways to help students do this as well. I think doing activities that encourage metacognition is a critical step in that direction, but that’s a topic for another day.

Image is “Thinking” by Lee Thatcher. The original work can be found here.


The Absurdity of One-to-One Initiatives


As comes up every year, someone in our department suggested we go one-to-one. Of course, this sparked lively debate. So much so that do to the frequency of these debates and the cycle of outrage I invariably go through after each one, I’m motivated to write out the multitude of reasons that going one-to-one with textbooks is an absurd idea.

First, let’s talk about costs. A good textbook costs close to $100. Sometimes less, sometimes more, depending on how many you buy. If a kid has five classes, that means it’s going to cost roughly $500 dollars per student to go one-to-one textbooks. And it’s not just $500 one time. Of course not, because in a few years much of what’s inside the books will be dated. They will need to be updated and some of them will be so obsolete they’ll need to be replaced entirely. Do we want to go through the up front costs and then the future costs to update and replace them?

Second, let’s talk about letting teenagers carry around several hundred dollars in textbooks. Have you seen the average teenager’s bedroom? Of course not! There’s too much stuff on the horizontal surfaces (and maybe even the vertical surfaces) to actually see any substantial part of the room. Are we going to let kids, who can hardly get a dirty tissue to the trash can across the room, be responsible for hundreds of dollars worth of school property? I think this is a nightmare that no administrator or teacher wants to deal with.

Third, I’d ask you if the cost is worth the benefit. Sure, textbooks have lot’s of knowledge in them. Give students books corresponding to the subjects they’re learning allows them to easily and quickly look up information, helpful diagrams, maps, and other media, but don’t we have teachers for that? Teachers have much of this knowledge, and if they don’t, then they can look it up in their book and deliver it to the students. What’s the point of a teacher if all the information is easily available to the students? Sure, the best teachers might use the books in coordination with their teaching ability to create near sublime learning experiences, but this would surely only be the most motivated of teachers, not the norm.

Fourth, it would be a logistical nightmare. Suppose your high school has 1000 students. Then we are talking about THOUSANDS of textbooks to keep track of. Not just keep track of, but record those that go missing and those that are damaged. Then schools have to make determinations about how much the damages cost. Then who pays for it? The students? What if it’s an accident? What if the student can’t afford it? What if they get lost in a house fire? Who’s on the hook for the bill then? And who does this burden of tracking fall upon? The library? The administrators? The teachers? There’s no good options. And then, who fixes them? Do we offload this responsibility on the already short staffed library personnel? We’d probably have to hire somebody to spend part of their day repairing textbooks, so tack that on to the bottom line.

It seems clear that the costs of trying to put the these resources into the hands of each student almost certainly outweigh the benefits. But this fight will never die. Year after year we’ll continue to hear how we should “give students access to the tools they’ll be using when they leave us”. Given what I’ve outlined above, I can’t see the logic that results in this being a good idea.

Update: I should make something clear. This is purely satire. I am simply trying to make the argument that when it comes to discussions of 1 to 1 technology I think the problems that are brought up are often ones that we have solved in other contexts. This situation never came up in my department. And even if it had, I would never throw them under the bus like this publicly. Once again, this is purely satire.

We should always be learning something new

Last week I started auditing a class on Hapkido, which is a Korean martial art. A friend asked if I’d be interested in attending the once-a-week class with him and I said I would. Having never done any martial art at any time in my life (I don’t even think I’ve watched any of the Karate Kid movies in their entirety), I was nervous. However, partway through the class I realized that there is clear value in learning something completely new.

I want to list a few feelings I had, as I think they reminded me of what it’s like to be a learner, as an adult or a teenager.

  • I didn’t want to make any mistakes. When the instructor demonstrated something, I wanted to do it perfectly. This notion is ridiculous because, as the instructor also pointed out, it takes thousands of repetitions before something becomes muscle memory. For as much as I preach the importance of mistakes in learning, I was shocked at how somewhere in my guts I still didn’t want to make them.

  • I didn’t want the instructor to come by me. Or at least if he did I wanted to be working on my right side (which I thought I was better at). I was afraid he’d find something I was doing wrong. Which I consciously knew would not be bad as it would get corrected and then I’d improve.

  • I compared myself to the people around me, unconsciously ranking myself. Better than that person, worse than those two, etc.

  • Frustration. I’ve never been particularly coordinated and I was consistently frustrated at knowing in my mind what I wanted my body to do, but struggling to make my body do it.

I walked off the mat at the end of class and my mind was reeling.

“Now I know why students are apprehensive to ask questions.”

“Now I understand better why a student might get uncomfortable while I hover over them watching them work out a problem.”

“I have to constantly remind myself to embrace the difficulty. That’s where growth comes from, but it’s difficult to do in practice.”

“Having an instructor that recognized we were all learning was incredibly helpful. He created an atmosphere where mistakes were not viewed as setbacks, but part of the process.”

My main takeaway was that these are feelings I need to constantly grapple with. I need to try to put myself in situations in which I’m the learner, with relative frequency. It helps me better understand where my students are coming from and I think will ultimately help me become a better teacher.

Also, here’s one more thought that has popped in my head recently and probably doesn’t need an entire blog post, but fits with the theme in this one. I’m in my fifth year teaching precalculus, AP calculus, and algebra II and I can feel myself having less empathy with my students, with people learning the concepts for the first time. The first year I taught these courses I think I had a better understanding of their struggles as I was solidifying my understanding of the concepts prior to teaching them as well. I’m not entirely sure what this means for my teaching now, but I think awareness of it is important.

“Everything Springs from That”

I don’t listen to many political podcasts. In fact, only one. Dan Carlin’s show, Common Sense. In his latest podcast he interviews James Burke, a science historian, documentary creator, broadcaster and all around smart dude.

This episode flirted with politics, but was more focused on how technology affects society and how the rate of change often has unforeseen ripples. It’s a fascinating interview, but the best part for me comes at the end of the interview. Dan presents Mr. Burke with a hypothetical (which I’m paraphrasing).

Suppose the leaders of the country call you up and ask for your advice. What would you tell them in regards to the absolute most important thing to focus on in the future?

“I’d say put a massive amount of effort into the educational system. Everything springs from giving people the kind of education that allows them to think more clearly and express themselves more clearly. Everything springs from that.”

I’ve been thinking about education a lot lately. I recognize that might be like pointing out that a historian has been thinking about history a lot lately. But I’m talking about the big picture of how we educate our society. With the appointment of charter school evangelist Betsy Devos to the head of the Department of Education and recent moves by the Michigan congress to weaken the teaching profession and cut funding, I worry greatly about where we are headed.

The election of Donald Trump, the proliferation of fake news, the gravitation towards soundbites, the lack of empathy, and constant decrease in social capital mean that having a society that can’t think critically could be (already is?) disastrous. If there was any time in our history that we should be focused on education, it should be now.

We can’t have a society of mindless drones that will believe the headline and first two lines of any article that comes across their news feed. We can’t have a society that can’t take another person’s perspective. We can’t have a society that fears change. We can’t have a society that doesn’t understand the value of civil discourse.

An education system that’s working on all cylinders can help prevent this.

We should be focused on how to graduate great teachers. We should be focused on how to help teachers become great. We should be looking to other education models and schools that we want to emulate. We should be focused on making teaching a profession that our best and brightest want to pursue. We should be working to get away from standardized test scores as the sole measurement of a quality education.

As Mr. Burke mentions in the podcast, if we put as much energy and money into education as we did into the Apollo project it could have countless dividends for our society.

Order – How Mathematics is Life

Humans are in a constant pursuit of order. We try to develop schemas to help us deal with frequently occurring situations. We constantly look for patterns. We try to make our lives somewhat predictable.

The brain doesn’t like to think. Thinking is hard. So the brain naturally gravitates towards pattern finding.

This is mathematics.

Mathematicians look around the world for patterns. Looking for truth. They take things they know to be true, and build on them. Constantly growing the body of patterns we know to be true.

The difference between me noticing that whenever it’s cloudy out I’m a bit gloomy and that the derivative of a parabolic function is linear, is that the latter is true always. It’s a fact that exists regardless belief, mood, perspective, or measurement.

I wrote the idea for this post down months ago, but it seemed relevant as this week I embarked on teaching my algebra II class how to factor polynomials. Something that nobody does, with the exception of math teachers and their students. (And I mean that quite literally. I went to the twittersphere and came up empty.) My advice to students was similar to other seemingly obscure content we learn in mathematics.

Treat these problems like puzzles and look for the patterns.

Because pattern finding, curiosity, and creativity in problem solving are all skills that are valuable and can be improved with practice.

Nobody does a puzzle and while they’re doing it says, “This is never going to help me in my life.” I don’t claim to be an expert on the motivation of puzzlers, but I did puzzles just to figure them out. I enjoyed the mental exercise.

This is how I want my students to approach math problems. I want them to enjoy and appreciate the pursuit of solving the problem. I know that’s abstract and might be difficult for teenagers to grab onto, but I’m not sure of any other justification for some of the concepts we teach.


Thoughts on “GPSing our Students”

In June Dan Meyer posted Your GPS is Making You Dumber and What that Means for Teaching. In it he makes the argument that providing step by step instructions for math concepts results in students being able to get from point A to B, while not understanding much about the concepts they’re supposed to be learning. His argument can be summed up with this paragraph, and is somewhat inspired by what Ann Shannon wrote in what teachers should Look for in the CCSS Mathematics Classroom.

Similarly, our step-by-step instructions do an excellent job transporting students efficiently from a question to its answer, but a poor job helping them acquire the domain knowledge to understand the deep structure in a problem set and adapt old methods to new questions.

I would tend to agree. I do give students steps occasionally but it’s often in order to simplify concepts and, if I’m being honest, to some degree avoid students truly struggling and grappling with the concepts.

I’m curious as to what others think about his post and the notion that GPSing students leads to less learning.

Panera Bread and Learning

In the last month I’ve become a Panera regular. We’ve been doing a lot of traveling and Panera was as close to fast food as we were willing to go. Until the last month however I’d only been there a few times in several years, without being particularly impressed.


image credit:

Take a look at the menu above. This is what you encounter when you walk into Panera Bread for the first time. Seven sections of menu packed with different dining options. For the less food savvy among us, not only is the menu packed with stuff, but it’s packed with a lot of stuff that is unfamiliar. In my first couple of visits I stood there bewildered for a while, let my friend order, then picked something with turkey in it.

Because I was pretty sure I knew what turkey was.

Well, I was wrong. I mean, I think the thing had turkey in it. But everything else did not complement the turkey in a positive way.

The next time I went there with a few colleagues I ordered a caesar salad. Why? Because every time I’ve ever ordered a caesar salad in a restaurant I haven’t been surprised by what came out. The same was true for this visit.

Sweet. I found something on the menu without holding up the four people behind me.But after that I avoided Panera for a while. A long while. I could get a caesar salad from a lot of places. The notion of standing there in line trying to find something different felt as though it would only bring frustration.

Then my wife and I were looking for something quick and healthy to eat, so we decided, to my slight dismay, to give it another try.

This time however, on the way there my wife looked up the menu on her phone and read off several items, describing them from their website. When we walked in I knew several of the items on the menu, how it worked (that “pick 2” thing for example) and was basically ready to order when I walked up. We’ve been there a couple of times since and now it’s one of my favorite restaurants. Since I’m confident in my understanding of the menu, I try new items, ask questions about different types of food, etc.

How does this relate to learning?

Think about the first time you encountered an unfamiliar topic and tried to learn it. This was difficult at first but I can certainly think back to math lectures (here’s lookin’ at you, Linear Algebra) in which there was new notation, vocabulary, and concepts and it all felt unfamiliar.

However, once I got with my peers and we began working through the problems (encountering the “menu” multiple times) the concepts began to feel more familiar. I realized that I knew more about them than what I thought (romaine and kale! Hey, I know what those are…). The more I worked individually and with my peers the clearer the concepts became.

If I avoided doing the exercises or only did it individually, that feeling of everything being foreign never really went away. Usually parts of the notation would be confusing. Or the instructions around a problem wouldn’t make sense. Or I could start a problem, but get lost in trying to solve it, etc.

A couple points can be pulled from this. First, a student’s first exposure to a topic is incredibly important. If you drop seven new vocab words and a gaggle of new notation on students at 8:00am on Monday morning you’re bound have a large group of students not wanting to come back to the menu you just presented them.

Even if they should know most of the food on there.

So we have to think carefully about how students first engage with content. The second point is helping students understand that multiple engagements with a concept will (usually) alleviate this feeling. I think many students never go back to the restaurant because they don’t want to be embarrassed for not knowing what their peers may already know. We have to help students be comfortable with this phase of unfamiliarity (my study group in college was a place I felt comfortable being wrong), and develop tasks that help them engage with the concept in safe, productive ways (investigating the menu on the way).

For a great post on how students see unfamiliar mathematics check out Ben Orlin’s recent post entitled What Students See When They Look at Algebra.

Alternative Titles

“What’s on that sandwich? Oh… will that be on the test?”

“How Avocado Ruins Education”

“I don’t even like lettuce so why should I be learning about salads?”