My Brain on Lesson Planning

Okay. I’ve a got a few minutes. Where is what I did last year? Ah, that’s right. We did that activity, with some direct instruction following. Seems like I didn’t quite the point across when I closed the lesson. Like the kids still struggled with parts of this on the quiz. Maybe I should change it. Maybe I should just start from scratch. Did I leave myself a note or anything?

Check Google doc for comments

Nothing. Good job me. I’ve got to do a better job of that. But sometimes it’s tough to find time. Yeah but it pays off and saves time eventually. Like it would be saving time right now. Okay. I get it. Anyway. I don’t have time to totally revamp it. How can I tweak this to make it work better? Maybe I’ll start with a more open ended question. I read that’s a more effective way to start a lesson then just with direct instruction. Okay. So what’s the question or task?

Goes on the Internet. Checks the MTBoS search engine. Writes down 4 ideas.

Well the first two are probably too much work/time. I might be able to tweak the third though. That would give students a chance to discuss some solving methods before we do the lesson. But I know Sam won’t participate. Man, what is his deal? What is my deal with him? Did I make him mad at some point? I need to talk to him and try improve that relationship. Maybe he’d be more willing to work with his peers. But, he’s doing fine in class so maybe I should just leave him alone. Ugh. I’ll sort that out tomorrow. Anyway. I think this will work. But it’s probably going to take longer than last year. Yeah it’s definitely going to take longer. I really only wanted to spend a day on this. But if they learn this better because I spent more time on it, will it pay off in the future? I don’t think so. It’s not really a topic that builds on itself. But shouldn’t we try to teach every topic really well? Even if it doesn’t get built on later? Maybe. Otherwise why am I teaching it? Well some students will get it and remember it, just fewer than if we spent more time. Okay. So let’s do it, we’ll reduce the assignment a couple of problems and carve out 5 or 10 minutes tomorrow to wrap up anything we don’t get to.

Phone rings

“Yes, I’ll send her down when she gets to class. Thanks. Bye”

She really needs to be in class today. I need to remember to make a copy of the notes for her. She won’t be able to make up the discussion we’ll be having but I guess there’s no way around that.

Glances at papers to grade next to the phone

Ugh. I guess those aren’t going to get done today. Maybe I can do those on my prep tomorrow. Dang. I need to finish that lesson. I think I’m ready to update the weekly plans. I need to make sure I can accommodate this for my autistic student. Did I write down those notes on him yesterday? Nope.

Writes down notes in observation document

Okay. I can make this work for him as well. I need to remember to go through this the morning before we do it.

Adds it to to-do list

Well that should be all set. Just need to look at my other two classes and do the same thing… I hope those don’t need revision. They probably do. I mean how can you assume that they’re in their best form? You’ve been teaching for 5 years. They may not be but they should be in good enough form. They’ll have to be because I don’t have time to rework either of them. Man I hope we have school tomorrow. If we don’t then I can just……..

What I wish I could tell my students

Here’s a list of few things I want to say to students, but am not quite sure how to do it. I’ve said of some of these in whole class contexts and variations of some to individuals. But I’ve noticed in my career that sometimes I notice things about students that are difficult to tell them directly. Maybe it’s the natural human aversion to confrontation, I’m not sure, but here’s the list:

  • You don’t have to go to a four year college if you have no idea what you want to do with your life.
  • If you don’t get into that school, your life is not over. You will get out of college what you put into it.
  • I understand that you’re a bright student. There’s no need to demonstrate that to me and your peers at every opportunity. In fact, you risk alienating some of your peers if you keep doing this.
  • Your ACT or SAT score does not define you, as important as it seems right now.
  • You’re in a controlling relationship. You deserve to be in a relationship in which you don’t feel like the thumb of power is constantly pressing on you.
  • I understand that you’re introverted. The ability to communicate well is an essential life skill. When someone says “hi” to you, you have be able to respond with, at minimum, “hi”.
  • Learning is not a competition, so when you get your quiz back, resist the urge to see how you “stack up” against your peers. (Okay, I’ve actually said this one.)
  • You can break the cycle of poverty in your family, but not unless you make some significant changes to your approach to life and the people in your life.
  • You’re addicted to your phone. Not in like a “haha, I’m trying to talk you so stop snap chatting” kind of way. More like a, “I’m really concerned about how this is going to negatively affect the rest of your life if you can’t get it under control” kind of way.
  • You’re in “regular” math class (as opposed to honors) but that doesn’t mean you can’t be an engineer, computer scientist, etc. In fact, I think you’d be a damn good one.
  • The pressure your parents are putting on you to perform is unnecessary and probably doing more harm than good. Work hard, but don’t cry over test scores, college applications, or an A-.

I’m sure I’ve forgotten some, but this is a pretty good list. Some of these are positive, but I struggle with how to explain them to students in a straightforward way. One that doesn’t sound preachy. I’m curious as to how other teachers approach situations like this with their students.

6 Reasons This is My Favorite Lesson

I want to share what might be the best lesson I’ve created and a few reasons why.

I actually wrote about this a couple years ago but since we’re doing it right now I thought it useful to reflect on and share it again.

This lesson came from the following problem I was struggling with:

I had spent a lot of time thinking about how to help students understand the connection between trig ratios on the unit circle and the graphs of trig functions on the Cartesian plane. Despite a couple activities and practice I was convinced, mainly through questioning, that they didn’t fully understand it.

My solution to this was to make a giant unit circle and cartesian plane and have students use them to work out problems. This would allow us to literally walk to specific angles and equivalent places on the cartesian plane. The hope was that this would help students solidify the connection between the two.

The details of how the activity works are in the original post and the materials are linked at the end of this post, so I want to emphasize the aspects of the lesson that really make it effective, in a convenient list.

Assessment

The activity is broken into two parts. The practice portion and the assessment portion. The assessment requires students, working in pairs, to come into the hallway and work through five problems (like these). This portion is vital for the following reasons.

Like any assessment, it helps me know what they know.

It makes students take the practice seriously. The assessment mirrors what the practice rounds were like. They take it seriously and practice until they’re confident.

Students work in pairs, sometimes disagree, and then must convince each other of their reasoning. Tremendous mathematical conversations come from this time.

It puts students in a position in which the teacher is there, but can’t help. This is true of assessments in general, but the format of this one means students must convince themselves and each other that their answer is “their final answer”.

No calculator. No notes.

On the assessment, and likewise on the practice, students cannot use a calculator, notes, their unit circle, or anything besides their brains and a whiteboard. This means students don’t have any crutches with which to rely on. These problems are not algorithmic. Each one is slightly different from the other ones. This means that the only way to be successful is to truly understand what is going on in the math.

Engagement

This is my third year doing this activity and every year there’s nearly full engagement. Now, this is precalc and while I wouldn’t say that all of these students want to be there, it is an elective. But it’s difficult for me to get this level of engagement from them.

This is, in part, because they know there’s a test coming after they’ve practiced. But I think it’s also because each problem sparks at least a little bit of curiosity. “How do we figure this out?” Initially many students don’t have a clue about how to approach something like sec(2pi/3) with only their brains and a whiteboard. But with a good understanding of trig they can figure it out.

And figuring it out is satisfying. Students are proud of themselves when they solve one of these problems correctly. I love seeing high fives in my classes, and this is one of those activities where they happen.

Embodied Cognition

I’ve written about embodied cognition before so I won’t go into too much detail, except to say that it’s incredibly valuable if you can incorporate it effectively. There is something fundamentally different from paper and pencil when you can stand there with a student inside of a unit circle and discuss these problems. It’s something that is hard to describe, but once you’ve tried it you clearly see the value.

Purposeful practice without a book assignment

A few weeks ago students initially learned how to do these problems via a lesson and practice problems. If that was effective, then I wouldn’t have needed to do this activity. What ends up happening in this activity is that students end up doing a bunch of practice problems, that I never assigned! I just tell them they can do as many practice rounds as they feel they need. Then they work until they have convinced themselves they’ve mastered it.

Partners

The test and practice require students to work in pairs. This is incredibly valuable as students are constantly conversing and helping each other understand. Once again, the knowledge that there’s an assessment plays into this, but who cares? From my observations students are rarely begrudgingly woking through these problems. They seem to enjoy them.

I probably see more learning and teaching happening between the students in this activity than any other lesson I do, for any class.


I understand that without seeing it happen it might be difficult for you to implement this. I’ve included some images below to give you an idea of the set up. Feel free to contact me with any questions you have. I’d encourage you to look for opportunities to use embodied cognition in your classes as I think it can be an incredibly useful teaching tool.

Here are the resources for doing the activity

Description Sheet

Possible Problem Bank

Practice Cards

Assessment Cards (Yeah, I’m not posting these on the web. I, shockingly, sometimes have students read my blog. But if you reach out to me I would be happy to email them to you and save you the time of making them.)

Assessment Rubric

X-axis “Tick Marks”

“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.

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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.

Tripled.

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.

Reflection

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!