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.



Why I wouldn’t tell my kid to become a teacher

teacher-309403_1280Right now, in the Michigan legislature, they are working to pass a bill that would gut the defined benefit retirement system for newly hired teachers, replacing it with a 401k system.

And it would cost tax payers $28–$33 billion over the next 30 years ($1.6–$3.8 billion over the next 5 years).They’ve been working on this for more than a decade, but this is a bold step in a direction that would basically finish it off.

Shortly after I hired in to my school district, in 2011, I had to choose how I wanted my retirement to work. I didn’t know anything about anything in regards to my retirement so I did about ten minutes of research and made a selection. I’m currently in the hybrid program, which is a combination of a defined benefit and 401k plan. I also

I’m fine with this. I realize that if I’d hired in ten years earlier that my retirement benefits would be significantly better, but it could be worse. For instance, I have up to 90% of my health insurance premiums covered when I retire, assuming I continue to pay 3% of my pay throughout my career.

However, if I was hired in the fall of 2012, I’d have no health insurance after I retired. Essentially all I could do would be to put money into a 401k that is designated for health insurance premiums after I retire.

Welcome to the real world!

Okay. Fine. I understand that defined benefit retirements aren’t that common. I understand that most people aren’t going to have their health insurance premiums subsidized after they retire.

However, that used to be part of the deal. A career in teaching wasn’t an awful finiancial decision because you knew that, despite dealing with a low salary for a large chunk of your career, you would be covered on the back end.

The health insurance guarantee is gone. Now the legislature is working hard to end what’s left of any defined benefit retirement for new hires.

What’s left?

I understand that you don’t go into public service to get rich. I’m fine with that. But low wages, no retirement, and no retirement health insurance makes this gig a tough sell.

Oh, I almost forgot. My out-of-pocket for health insurance tripled this year.


And there’s no sign of these trends reversing any time soon.

So yes, if an 18 year old kid asked me what I thought about becoming a teacher, I would say to take a long, hard look. It’s not what it once was. The legislature would be wise to pay attention to what this will do to the profession in the years and decades to come.

Finally, the burden is also on them to explain to taxpayers how this is a fiscally responsible decision for our state.

Real Questions

Every year in precalc, two things happen. First, the exponential and logarithm unit gets squished (education jargon, I know). Second, we do the “student loan” blog post, which Steve came up with when we were first working on precalculus together. The former drives me crazy because I think that unit is more useful than some of the other things we spend time on. The latter I look forward to because it’s such a great learning experience for students.

The problem goes something like this:

Suppose you take out a $5000 student loan every year for four years. How long would it take you to pay back the money? Please make sure to include research on government subsidized and unsubsidized loans and private bank loans.

Yeah, it’s vague. There’s no rubric, besides the standard blog rubric. The openness of the problem drives students a little crazy, which I’ve decided isn’t a bad thing. Let me explain.

The students in this class are, most likely, college bound. I don’t teach in a particularly affluent part of the state so most of my students are going to have to take out loans to pay for college. This means that the question, how long will it take to pay back loans, is painfully relevant.

Here are my main objectives for this assignment:

  • I want students to understand the different types of loans and some of the verbiage they’ll encounter, if only at a surface level.
  • I want students to understand the connection between exponential functions loans.
  • I want students to understand how important interest rates are in total cost of a loan over time.
  • I want students to realize that in this fairly common scenario, it’s not unlikely that they’ll be paying back loans for 20ish years.

I should also note that I don’t make students run these calculations by hand. I encourage them to use loan repayment calculators. I want them to understand that there is mathematics at play here, but I don’t want them to get too hung up in computations.

For many students this is an eye opening project. I’ve had students say they plan on paying it back in five years and they want to be a teacher. I had to gently explain to them that it was unlikely they’d be able to afford that. Steve has had students in tears doing this project. It has many great opportunities for students to learn about life, and also mathematics.


I wrote the beginning of this post when students were working on the project. I’m writing this part after grading them. Here’s a couple things I’m going to connect year to make it go better.

  • Clear up the guidelines. There’s explanations in the video and a synopsis on the website. They’re both slightly different. I also plan on making them clearer.
  • Listing things for students that are wrong and that students have done in the past. Things like only running one scenario, only computing a loan for $5000, and not providing explanations for their numbers.
  • I will include a question asking them to explain what they learned or gained from doing the projects.
  • I will look up some useful repayment calculators as suggestions. I noticed that this can be a bit help or hinder ace depending on where google leads them.

Other than that, this activity will be used next year.

I’ve underestimated the importance of vocabulary

"Words" by Shelly on Flickr

I thought for a long time that I could get by teaching math while deemphasizing vocabulary. Obviously we would discuss the meaning of words, especially the ones that come up frequently. But I thought that if I was able to help students get a feel for the math, and show kids how to do math, without getting too caught up in what the new vocabulary meant, that would be success.

Part of this was time. Or rather to save time. Spending time helping students really understand vocabulary takes more time, especially if it’s something that is more easily shown/practiced. For example, I feel like one of my struggles with helping students understand domain and range is that I don’t do a good job at really helping them understand the words. In algebra II, if I present a new type of function to them and ask them to find the domain and range, they often struggle until they see a few examples. It’s as if they’re simply replicating the process for each type of function.

At risk of this turning into a domain and range post, let me explain a bit further. When we study quadratic functions I tell students the domain is always “all real numbers”. The student thinks, “Sweet. Whenever I see a question over domain on the quiz, I’ll just write ‘all real numbers’.” When we learn a new family of functions they have no understanding of how to find the domain, beyond “that’s something with the x values, right?”.

It’s not just that topic. In fact, the concept that propelled me to write on this topic was grading a quiz over factoring polynomials and finding zeros in polynomials. Way too many of my students don’t know the difference between factors and zeros and constantly get them confused. My most significant observation was that I find students are trying to get by with the least amount of vocabulary understanding, and I don’t think I’m helping things by demphasizing it.

Since I’m having this realization at this point in the school year, I think the fix going forward will be trying to find and develop small activities to help reinforce vocabulary. Simply emphasizing it more is a start. I’ve also done some activities, like concept maps and “functions back-to-back” which help with vocabulary understanding. Next school year I’d like to take a more systematic approach and deliberately build in vocabulary activities into each unit.

Drop your favorite vocabulary activities in the comments below or send them my way on Twitter. Thanks!

Image Credit: “Words” by Shelly on Flickr

How to Factor Quadratics

After years of teaching how to factor quadratics and then getting in my car and banging my head on the steering wheel, I decided that enough was enough. I was going to spend some time finding a better method. I took to my favorite community of math educators, the #MTBoS.

Several different ideas were thrown my way, but the one that was most attractive was Mary Bourasa’s method, sent to me by Helene Matte.

Last year I had tried the “diamond” method, which worked a bit better than simply guessing and checking, which I’d done in previous years. The first problem I ran into was that I had trouble remembering what went in the top of the x and what went in the bottom. In videos I watch online teachers did it different ways. I think this was because of the second problem I ran into, which is where the hell did this giant “x” thing come from anyway? It’s not a “trick” really, but it does seem to have no connection to other things we do in math.

I might as well have said, “Today we are going to factor quadratics. Draw a random shape, fill it with numbers in the recipe I give you, then get your answer.” And kids weren’t that good at it.

Enter Box Method (or area method, or whatever)

A few years ago I learned about the “box method” for multiply polynomials, binomials included. Put the first polynomial along the top, the second along the left side, multiple rows by columns. Very similar to multiplying actual numbers with this method.

Example with Numbers

Example with Binomials

The approach to factoring using this method is attractive because it feels like working the box method, in reverse. If students are familiar with the box method for multiplying binomials, it’s a natural extension to use this method to factor them (as I often talk about factoring as the reverse of distributing).

Step one

The first step in this process is writing down a*c (M), the coefficient on the middle term (A), and then finding two Numbers that multiply to give you M and add to give you A.

Step Two

Once you have your numbers, fill the box. The upper left corner and lower right corner have to contain the squared term and constant term respectively. Fill the upper right and lower left with the two numbers you found in step one.

Step Three

The last step is to factor out the GCF of each row and column. Then you’re done. You have the factors that multiply to give you the quadratic.

A couple notes

This method fails miserably if you don’t factor out any common factors at the beginning. For instance, if you have a 2 in each term that can be factored out, you have to do that first before using this method.

I still have to grade the quizzes that cover this section but the kids seemed to respond a lot better than they have in previous years. I’ll update this post once I know more.

The Classroom I want to Visit (and someday have)

You walk in and are immediately taken by the number of students either focused and working independently (often with ear buds in) or quietly collaborating. The teacher is difficult to find at first but then you find her, huddled around a whiteboard working out a few different ways to approach a problem involving polynomial equations. The furniture is easy to move and comfortable. Small tables for small groups, single desks scattered around the room, with oversized chairs scattered around as well. The walls are neutral colors, not the standard white that bounces fluorescent light almost as well as a mirror. As you look around a brief, friendly, argument erupts in the corner over why long division of polynomials is more pure than synthetic division. The teacher then stands up, walks around the room checking on students, snapping pictures of student work with her iPad. She then projects some mistakes she found students had done and the class discusses the thinking that led to them. There are rugs, art on the walls, a laptop cart in the corner, and a projector screen towards the front (or what you assume is the front) of the room.

I read an article recently called “Why the 21st Century Classroom May Remind You of Starbucks”. This got me thinking, again, about learning environments. This topic sparks a few questions in my mind:

  • What are environments that I prefer to learn in?
  • What makes an environment conducive to learning?
  • How do you develop an environment that can be easily transitioned from independent work to collaborative work to whole class work and everything in between?
  • How much is my classroom layout getting in the way of learning?

To at least partially answer these questions I don’t think Starbucks is a bad model, in some respects, for what a great learning environment looks like. Obviously Starbucks is more conducive to independent learning, but I like some of the big ideas.

Learning environments should be comfortable

I can see that if you wanted kids to avoid falling asleep you would make the seating uncomfortable. I’d rather make the classwork engaging enough that students don’t fall asleep. I’m not saying we should all work in bean bag chairs. I’d hate doing real work in a bean bag chair. But I’m not everybody and I don’t hate the idea of having options like that for students that do prefer to work in the type of seating.

And comfortable learning environments go beyond just the furniture. Rugs, art, music, lighting, and the teacher’s attitude all contribute significantly to the environment.

Learning environments should be flexible

As technology changes the way content is delivered and the way that students interact with content, the classroom should change. The amount of time a teacher spends lecturing to the entire class should probably be decreasing. This means that the work done by students in class will be more fractured. Some students may need to watch instructional videos. Some may be writing blog posts. Some may be working on a group project. Some may be using computer graphing technology. The teacher may need to work with some students that have been absent. The teacher may need to give a lecture to the entire class.

This is the future of learning. The class setup needs to support this.

Learning environments should be safe

I don’t mean that students shouldn’t feel like someone is going to physically hurt them, although that is obviously true. I’m saying that students shouldn’t walk in and feel like they’re in a place where mistakes are not valued, their opinion is not wanted or their thoughts are better kept to themselves. This doesn’t have much to do with a “Starbucks classroom”, but I thought it worth noting.

In all seriousness, I want to visit a classroom that has these characteristics, so if you teach in Michigan and have a classroom environment similar to the one I described, then I’d love to observe a lesson!

Any other thoughts on classroom environment? Anything I missed? Drop a comment below!

4 ways to leave School at School



If you had told me that it would take me 5 years of teaching to figure out how to mentally leave work at work then I might not have continued in this career. I’ve gotten incrementally better at it each year but this year I’ve committed to prioritizing it. Here’s a few things I’ve learned that help me do that. I hope you can, especially if you’re just starting out, find a piece of advice that will help you live a more balanced life.

Before you leave work

I use a to-do list like it’s a religion, so before I leave school I go through my list (in Todoist) and do three things.

  • I check off everything I completed but forgot to check off during the day.
  • I look for anything on the list that can be taken care of in less than two minutes. If time permits, I do those things. I schedule times to do the other tasks.
  • I look for any task I can do now, that will save me exponentially more time in the future.

This leaves me cognitive space for the way home and when I get home. I don’t have to worry about when I’m going to tweak that lesson, write that letter of rec, or grade those papers. I may have 20 tasks to do in the next 24 hours but each one has a time pinned to it.

The drive home

First, if you have more than a few minutes to drive from work to home, leverage that time to mentally leave work. I’ll do a few things to do this, depending on how the day went.

  • If I it was an especially busy day and I have lot’s on my mind I will ride in silence and simply let my mind think about whatever it wants in regards to school. It’s like letting a cold run it’s course. I just get out of the way and let my mind go. Although I am thinking about work, this gives my mind a chance to empty. If I don’t, like if I turn on a podcast or audiobook, I find that although I distract myself during the ride, my concerns about work pop back when I get home. If I empty my brain in this way, I don’t think about work nearly as much at home.
  • After I’ve done the above, or if I don’t think I need to, I often listen to stand up comedy. Laughing puts me in a good mood and also helps me walk through the door with a positive attitude. I sometimes listen to novels or music that I’m really into. Anything that puts me in a good mood and provides for a transition from work to school.

Embrace the moment

Yeah, this is cliché, but if you take it to heart and try to frequently exercise this idea you’ll be in a better place at home (and at work). The success of this, at least for me, depends a lot on the previous two points. When I’m home I try to focus on my kid, my wife, my dog, cooking, or whatever I’m doing. This is important for me because I want to enjoy the limited time I have to spend with them. But it may be more important for them to have a husband/father that is present in the evening and not thinking about why 8 of my students blew off the activity from 3rd hour. (And by the way, I’m not gonna be able to fix it at that moment anyway, so it’s wasted cognitive energy.)

Understand you can’t fix everything

Somewhere deep in the dredges of my brain I used to think that if I just spent as much time as possible thinking about how to fix the problems that arose at school (or in education in general) then I’d be able to fix them.

This is obviously not the case.

I’ve found that teaching, not unlike many professions, is a push and pull between idealism and pragmatism. The fact is that there are a lot of problems that I can’t influence. And many problems I could influence, but only with a large amount of time and effort, that would end up with uncertain results.

The takeaway for me was that I need to pick and choose carefully the problems I’m going to tackle. And then only take on those problems that I have space in my cognitive bandwidth to deal with. If it means that I’ll be sitting at the dinner table trying to figure out how to solve the problem of poverty in my community or big money in high stakes testing, my resources are probably misdirected and infringing on my life in other ways.

They’re problems. I’d like to solve them. But I don’t have the time, skill set, or resources to do it. So I let them slide out of my mind.

I see teachers all the time that are just surviving. Trying to get from one hour to the next, getting beat up along the way, and then dragging all the stresses from school home with them. We are inclined to do this. We go into the profession to help kids and we love to see them succeed. When they aren’t, we take it personal. So it invades our personal lives. But I’ve been there, and I can tell you it’s not worth it.

You can’t solve all the problems. There will always be kids that don’t meet their potential, think your activities are boring, have awful home lives, and otherwise break your heart. I’m not saying I’m okay with it, I’m just saying that I’ve gone the route of devoting an inordinate amount of mental and physical energy to worrying about it, and guess what. I still had those problems. And by trying to solve them all, all the time, I harmed myself (mentally through stress) and my relationships (stress spillover, not being present, etc.).

Take care of yourself and the time you do devote to your students and profession will be more effective.

This is something I’m passionate about. I know people always say, “post your thoughts in the comments”, but I really am curious as to where people are on this. Do you have other ideas for going home with a clear mind? Do you struggle with other aspects of work-life balance that I left out?