Algebra 2 & relationships

We've been working on our algebra 2 curriculum over the past year, both to align it with the Common Core and make it seems less like a random collection of function things. Last year we had the idea to make it very data-based; the idea was to have students gather a ton of data, and then base our study of functions on those data sets.

There are a few problems with this approach. For one, our district decided that prep periods are a luxury that cannot be afforded, so we don't have the time to prepare such a radical departure from our current curriculum.

Second, and more problematically, I think this would put way too much emphasis on the applied part of mathematics, and would make it difficult to meaningfully teach any aspects that aren't as practical. If we're motivating the study of functions based on answering questions about real data, I'm not sure how I'm going to help students understand removable discontinuities. Or motivate them to solve difficult equations algebraically when a graphical method yields a solution quickly. And so on.

Mathematics is not only about answering practical questions about the real world and data. It is certainly useful for that, but it's also much more. Thus a solely data-based approach to teaching algebra 2 would be a mistake, in my opinion.

So we need a more flexible approach if we're going to be true to all of mathematics. Thinking about the essence of our algebra 2 course, and really algebra in general, what we're dealing with is relationships. Linear, polynomial, rational, radical, exponential, logarithmic, and trigonometric, to be specific. We learn about these relationships by looking them in different lights - algebraically, graphically, and numerically.

Currently, a major problem with algebra 2 is that, to students, it feels like a ridiculously long laundry list of things to learn. I get the feeling that they don't understand that the equations, graphs, and tables that we work with represent the same relationships. I try to teach the connections, but they're mostly lost on students. And it's not surprising - the focus of the course isn't on understanding the relationships, it's on the set of techniques and facts about them. Without understanding the relationships themselves.

So, my idea: refocus the course on the relationships. One way to do that might be to start numerically. For example, we might start with the following:

{ (1, -3), (10, 1019), (-1, -4.5), (?, 59), (2, -1), (3, 3), (0, -4), (0.5, ?) }

I think this would really put the focus on the relationship itself. We're dealing with two quantities, and there is some way to predict one from the other. However, it is difficult to figure that out simply with numbers, so translating the problem into different forms - equations and graphs - takes on more meaning. 

Our new semester starts in about a month; I'll let you know how it goes. Any wisdom you have to offer would be appreciated.

Rekindling why I teach mathematics

I haven't blogged for a long time, and I'll explain that in relatively short time. But in the mean time, I have a brief thought to share.

I've really, really been struggling lately to find my passion for teaching pure mathematics. I'm very passionate about teaching computational thinking skills and statistics - ideas and methods that clearly are relevant to developing an informed citizenry - but I've been questioning the point of teaching apathetic teenagers rather abstract mathematics.

Well, I've found the beginning of a cure in, of all places, a graphic novel. This weekend I've been reading Logicomix, a story about the life and ideas of Bertrand Russell.

For those of who don't know, or have forgotten, Russell's main role in the story of mathematics was his attempt to create a solid foundation of all of mathematics in his "Principia Mathematica", co-authored with Alfred Whitehead. This graphic novel does a beautiful job of connecting the ideas in mathematics to the questions that make our existence so wonderfully interesting.

I need to remember, I'm teaching ideas. And these ideas have a context; they are part of the story of real people attempting to more fully understand our place in the universe.

I haven't been teaching that at all.

[NBI] My teaching story

This post is part of the new blogger initiative.

Every teacher has a handful of characteristics that make him or her unique. While your teaching style may share characteristics with other teachers, it's the unique combination that makes you different. It's interesting to figure out where those come from.

Right before I started my first year of teaching, we did this activity:

1. Put a post-it note in the middle of a blank piece of paper.
2. Think of your four favorite teachers you've had.
3. Write their names on the paper around the edges of the post-it note.
4. Now think about each teacher individually. Choose one word that describes why you wrote that teacher's name.
5. Remove the post-it note.

Ta da! How similar does that look to your teaching style?

Here's mine:

Fun, faith, inquiry, & reach

Fun (high school Spanish teacher)
I wasn't convinced I wanted to be fluent in Spanish, but we had so much fun in this class that we didn't realize how much we were learning. I recognize that now with how much Spanish I still remember, even though its chief use is when we go out to eat.

Faith (high school computer science teacher)
He challenged us, a lot. But it was always obvious that he truly believed and knew that we could do what he threw at us. And you know what? We always did. After a while, I just started to figure that if he was assigning it, I could do it.

Inquiry (high school physics teacher)
This was my introduction to a truly inquiry-based modeling class. He drove a lot of my peers mad because he wouldn't just give us the answers, but he was one of the first teachers to introduce the idea that we're fully capable of creating our own knowledge.

Reach (high school orchestra teacher)

I remember sitting in orchestra after a recent concert and getting the new music. We would look at it, try to play some of it, and sometimes barely make it through a measure. Follow this up with one of my most vivid memories from high school - the director tearing up on stage after finishing the last note. That's what I mean by reach - she showed us what we could do.

The really powerful part

I had a student come in during lunch last year, and she saw the post-it note hanging above my desk. "What's that?", she asked. I briefly explained it to her.

Her response? "Yeah, that really does fit you."

Just think about that. The qualities of my favorite teachers not only influence how I like to teach, but my students recognize them in my teaching. These teachers were so amazing that they're impacting my students.

That's the kind of teacher I want to be.

HELP! (or, be careful what you wish for)

You know how there's that saying, "be careful what you ask for - you might just get it?"

It just happened to me.

School starts on Tuesday, and my block algebra 2 class just got moved to the Mac lab. Every day. Instead of teaching in a normal, boring math classroom with 28 deskchairs, four off-white walls, and some corny math posters, I am going to be in a room with 30 gigantic iMacs, along with with a side room that has a half dozen circular tables with chair. I couldn't ask for a much better setup.

I have five days to figure out some kind of rough plan for how I'm going to approach and teach this class differently. Feel free to skip the next section if you don't care about what led to this.

The backstory

I was frustrated last year with the limited access to technology that my classes I had, and it didn't look like it was going to get any better. So I decided to take matters into my own hands, and I started collecting old computers from friends and family. I cleaned them up, installed Ubuntu or lubuntu, along with GeoGebra, Scratch, and matplotlib for Python.

That project is coming along. It would be great to have one or two more, but I should enough functional computers in my classroom for groups of four to be able to share a computer with GeoGebra, Scratch, and Python.

Then our tech person came into my classroom today. She asked a polite question about how my mini-lab was coming, and then said, "I don't want you to have to do this. I mean it's great that you are, but you shouldn't have to. The Mac lab is open block D - what class do you have then? Algebra 2? Ok, you're teaching that class in the Mac lab."

Umm, what? Really? This is going to take a day or two to sink in.

I want to underscore, I am ecstatic about this. I'm not complaining. It's just that school starts in five days, and I don't want to squander this. There is so much that I could do - I just don't know what that is yet, or how I'm going to do it.

Ideas

I don't want to just list the specific tools that I'm going to use. I want to develop a picture of what I want my students doing that is different than the standard paper-and-pencil based algebra 2. Then I'll figure out what programs, services, whatever that will help us accomplish that.

So, my list:

• Work with large data sets
• Programming
• Researching real problems
• Presenting with digital resources
• Writing
• Learning to become self-directed learners
• Live backchanneling
• Doing all of the above collaboratively

I could probably add more, but I think that's a decent set of ideas. I'll certainly try to give computational thinking a central role in the class.

Where I need help is making this happen in algebra 2. How can I adapt our unit on polynomials to a classroom with this kind of access to technology? What can I do with solving radical equations?

If you have any ideas, please comment, tweet, Google+, or email me. I promise to share what I do and give credit where it's due. The six units we teach are polynomials, rational functions, radical functions, exponential & logarithmic functions, trigonometric functions, and statistics.

Seriously. Whether it's just an idea you have or a lesson you've taught ten times, if you're reading this, please share. I've been given the blessing/curse of what I wanted, and I want to make sure I actually take advantage of it.

Please.

Transformations with Scratch

Students need to do more

I don't mean problems. I mean building. Creating. Writing. Designing. I think it can be amazing [read: tragic] how little students actually produce in a typical math class - no wonder math is often perceived as abstract and disconnected from the real world.

Having students do more is good for about a million reasons. They provide a firm foundation to build on their prior knowledge and experience, formulate questions, climb the ladder of abstraction, and experience how math is connected to the world they live in. And the best part? The result of their growth is concrete. It's not a completed worksheet, or a good grade on a test. Maybe it's a video, or a presentation, or a programmed simulation, or whatever. The point is that it is something much more real.

Challenges

That all being said, there are reasons why projects aren't the norm in math. I know some schools have taken bold steps and are letting going of pre-determined curricula in place of project-based learning and student-driven inquiry, but they're the minority. And not my school.

In a typical public school, I think these are the main challenges for using projects in math: 

1. Finding projects that fit the unit.
2. Student expectations of what math class is.
3. Presenting the projects to students.
4. Assessing the projects.
5. Instructional time.
6. Planning time.

My plan is not to solve all of these challenges this year. My hope is to create some space and a framework that allows me to insert projects more frequently.

First project: Transformations with Scratch

I participated in a CS4HS workshop this summer and was introduced to Scratch. The best way to learn what it is is to play with it. Go ahead, it's a free download, cross-platform, and fun.

The first project of the year for my geometry students is going to be creating a 30 second animation with Scratch. I like this it introduces programming as well as gives students a deeper experience with transformations. 

Here's the handout for students that I will use. The front side presents the task and details to the student; the back contains the rubric by which the projects will be assessed.

Scratch project

I could greatly use some critical feedback before I give this a go. Design, language, rubric, whatever. The "product" part of the rubric definitely needs improvement, but I'm having trouble describing what it is that I want students to produce.

After this project, I plan on tweaking the setup and then using this as my template for the rest of the projects. Hopefully it will making creating new ones quicker and easier!

[NBI] Modeling Temperatures with GeoGebra

This post is part of the new blogger initiative.

The more that I grow as a teacher, the more I believe in teaching modeling with mathematics.

That might sound somewhat silly. I've always known, in some sense, that modeling is a good idea. They certainly encouraged it in my prep program and it's a significant part of the Common Core. However, when you're actually making instructional decisions for the day or week, modeling can be tough to choose. It takes more class time, and in some ways, can seem "softer".

But every time I have my students complete a good modeling activity, those thoughts just seem ridiculous. I see real struggle. I see connections get made and broken. I see students conjecture. I see them contextualize and decontextualize. Beyond the truly deep learning, I haven't encountered a better assessment tool.

Monthly temperatures in Wisconsin

One of the better lessons that I taught during my first year of teaching was a modeling lesson. We were finishing up a unit on trig functions and I *thought* that my students had learned the transformations. I wanted them to see that what we learned really does show up in the world, so I grabbed the last five years of monthly temperatures in Wisconsin from the Online Climate Data Directory. I then entered that data in the spreadsheet of GeoGebra, created a list of points, and distributed the GeoGebra file to my student. (I would love for my students to be able to handle the data, but I made an executive decision based on time-constraints).

I know that it's supposed to look like this, but it still kind of amazes me. 

For the activity itself, I distributed the following halfsheets to guide students through the techy aspects of the activity:
Temperature Directions

Basically, I wanted students to find an equation that fits the data as accurately as possible, interpret the different parts of their equation in the context of monthly temperatures, and present their work in a professional looking document. As we had done very little work of this type, I created a template with Google Docs to use.
Temperature in Wisconsin

Results

It wasn't perfect, but it was good. The insights into students' thinking were amazing. For example, it turned out to be very difficult for my students to interpret the period as one year, or that the midline represents the average temperature. I had a number of very good conversations trying to help students make these connections.

Additionally, I really enjoyed watching students while they were fitting equations to their data. I had considered setting up the equation for them with sliders, but I'm glad that I didn't. There was value in having them manually type in and change constants in the appropriate places in their equation. Students were checking their notebooks to figure out how to change the amplitude, or just tinkering to see if it worked. Really, either was fine with me. 

In the end, I hope trig was a bit more concrete, and I had a much better idea of what my student understood.

Next time

Instead of ending the unit with this, I will start with it. Trig would then come from the real world and students would be playing with all of the transformations before I teach the vocabulary. We could then discuss the ideas of trig transformations with a solid context to refer to and draw comparisons to other places in the world - "How do you think the amplitude of the monthly temperatures in Wisconsin compares to Cuba's? Why? What about the period?"

Next time.

Geometry summer curriculum work

This summer our department was given some curriculum hours to develop/reshape our geometry courses in light of the Common Core State Standards (CCSS). I'll assume that you know something about this - I just want to share the process we went through and where we are currently.

At the beginning of the summer, we had a consultant give a short workshop for math teachers in our district. To be perfectly honest, I didn't find it incredibly helpful - possibly because I'm not that long removed from my certification program, where the CCSS were the only standards that we worked with.

But, I did take one quite useful idea from the workshop - a process for breaking down the standards into something that we can get a better grasp on. I turned that process into a sensibly designed Google Docs template to facilitate our summer work.

So how did this work? Well, as a starting point, we decided to base all of our work of the suggested "pathway" from appendix A of the CCSS for mathematics. Within each unit, we looked at the different clusters. We decided that, for the most part, we're going to teach mini-units based on clusters.

For example, here is the first cluster:

Reading through this it makes sense, but this is where is becomes more real. How we turn this into what we're teaching? We considered just making these our direct standards, but there was some messiness there. Concepts overlap, others would be difficult to assess, and so on.

This is where breaking them down became quite helpful. For each cluster, we created a document that broke down each standard into the what and how. Looking at these all together, then, we would create our standards. We used some standards pretty much directly. Others we would combine, tweak, and interpret. Some standards we decided were things we would do/teach in class, like G-CO.4, but we wouldn't make them standards for assessment.

For the above cluster, this is how we broke the standards down, and these are the standards we developed:

1. Geometric definitions
   Identify & precisely define angle, circle, perpendicular, parallel, & line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
2. Transforming figures Reflect, rotate, and translate figures.
3. Identifying transformations Identify sequences of transformations that carry a given figure onto another.
4. Transformations as functions Describe a transformation as a function in the coordinate plane.

Now these are standards that I'm more confident that I can teach and assess.

What we're currently working on & where we're going

We're done with everything above, and we're finishing writing small formative assessments for every standard. We're then going to pull them together, tweak them, and create the summative assessments. Our school implements a 80% summative, 20% formative grading policy, so we're somewhat forced to break things down like this. I probably wouldn't otherwise.

This was all rather mundane work, and I have a hard time calling it "important", especially considering what we might have done with our time otherwise. However, whatever gripes you may have with national standards, and I have several, we are legally obligated to teach them, are students are going to be judged on how well they know them, and we're likely going to be judged on how well our students do. So, in some manner of speaking, this was important work to complete.

There's clearly something missing from this model. It treats math as the sum of the standards, when it should be so much more than that. At the beginning of the summer, we were hoping to include a "cluster question" within every cluster of standards that would be an interesting problem, application, project, or whatever that uses the main idea of the cluster. We didn't have time to develop them, but those could help.

Incorporating the practice standards will help. I'm going to try to use them daily to guide my teaching. But what I really want to do is use this work as the base for turning geometry into a project-based course. We'll have a solid grasp of what our students need to learn, along with some of the how. With this foundation laid, I think we could do something really special while respecting our commitment to teach the standards.

[NBI] Focus on culture

This post is part of the new blogger initiative.

My primary goal for the first week of school is the same as it was last year - begin to build a classroom culture that supports exploration, creativity, safety, and fun. I'm hoping to fail less this year.

Last year I failed for several reasons. First, I had no control over the physical environment. I was a traveling teacher without my own classroom, and the classrooms I was in were, well, boring. My wife is also a teacher, and we've sometimes discussed (read: grieved over) the fact that elementary school classrooms just look more inspiring that high school classrooms. Why is it that the typical second grade classroom looks like Google's offices and high school classrooms look like oversized cubicles? This year, thanks to a "bold" move by my school district, I will have a classroom. So that will give me more control.

However, that was far from the main reason that I failed. My chief failure last year was that I approached the building of a classroom culture as solely my responsibility. I didn't involve my students in its development. I planned activities, made seating charts, and listed important values, and then tried to project those onto my classes. In retrospect, I kick myself for thinking that I could build culture top down with any real success.

It's especially annoying since this is a lesson I thought I had learned from observing our wars in Iraq and Afghanistan for the last ten years. Knowledge transfer is hard.

The bottom line is that I cannot impose a culture on my classes. Culture is developed by groups of individuals who share common values and practices. I can help guide the development of our culture and be intentional about which values are emphasized, but I cannot create it on my own.

So, concretely, what am I actually going to do? I've got about two weeks to figure that out. Here are my current ideas:

1. Find a solid, collaborative, & creative task for my classes on the first day. I would like to begin our class with my students doing something
2. Figure out a way to spark a discussion around the ideas of collaboration, creativity, risk-taking, and fun. Round it out with an explicit mission statement of our class.
3. Have my student help design the physical setting of the classroom - decoration, seating arrangement, workspace design, etc.

I'd be very interested in hearing your ideas or words of wisdom!

Papert - What comes first, using it or 'getting' it?

"The principle is called the power principle or "what comes first, using it or 'getting it'?" The natural mode of acquiring most knowledge is through use leading to progressively deepening understanding. Only in school, and especially in SME is this order systematically inverted. The power principle re-inverts the inversion." 

-Seymour Papert, An Exploration in the Space of Mathematics Education

A phrase I often hear is "teaching for understanding", which is a personal goal of mine. I don't just want my students to be able to do, I would like them to understand. So I try to introduce new material with exploratory activities where they wrestle with the material, apply what they already know, and try to create something new. My success varies, and I'm sometimes at a loss for why.

It might be something as simple as this: I'm asking to students to understand something before they know what it is. I would imagine this is difficult indeed.

Transforming geometry

I never wanted to teach geometry. It bored me in high school, and I could never get myself particularly excited about the side splitter theorems or, well, anything about it. Ick.

So naturally, geometry was and is one of my main course assignments as part of my first teaching position.

Goal: make geometry interesting to teach and learn. Or, at least suck less.

There are two reasons that this might actually happen - The Common Core State Standards and computational thinking.

Common Core

 The first one is that the Common Core State Standards (CCSS) changed the focus of high school quite significantly by giving transformations a central role. Our department made the decision to embrace this direction and go with it. This summer we've been taking the suggested sequencing from Appendix A, and then interpreting and translating this into a curriculum that we feel we can teach. If you want to a gander/copy/steal, our work is here (and still changing):

• Unit 1: Congruence, proof, & constructions
• Unit 2: Similarity, proof, & trigonometry
• Unit 3: Extending to three dimensions
• Unit 4: Coordinate geometry
• Unit 5: Circles
• Unit 6: Applications of probability

Now these alone aren't going to make geometry awesome, but I think that they do provide an improved sequencing and emphasis than a traditional geometry course. I'm also just getting a start on writing some assessments which is highlighting how different the approach is. I'll post more about that later.

Computational thinking

This one has me much more excited. I already wrote a post about what it is and why I've decided to use it. At a high level, think of computational thinking as combining critical thinking with the power of computing, an incredibly relevant approach.

Now that may not sound like it has much to do with high school geometry. High school geometry is traditionally a very abstract course where students are first introduced to mathematical rigor and proof. Unfortunately, we can fall into teaching geometry as a means to end (teaching proof), and we lose some of the utility and concreteness of geometry. After all, geometry is shapes!

So with that motivation, I want to consistently incorporate computational thinking practices into geometry. I want to make geometry a more hands-on and practical course. I'll still teach proof & rigor, but I want students to associate geometry with geometry rather than proof.

Computational thinking

Now that I have a year under my belt, I would like to think that I'm a little wiser. I know how to be really, really inefficient while planning lessons. Goal number one of next year: be a little less inefficient.

A chief cause of the hours I would spend planning a poor lesson was an overload of ideas and resources. I would spend far too long reading through ideas, half starting, and thinking. Then I'd run out of time, underplan an inquiry-based activity, and run a bunch of copies of worksheets. During the lesson, I'd get frustrated by the students' lack of initiative and interest (caused by a lack of support and clarity).

And repeat, day after day. Lots of time spent planning lessons that could have taken ten minutes to plan.

I would like to be a little less inefficient.

My plan is to focus on one specific framework. Instead of trying to do many different things because their cool, I want use a more focused set of ideas to both give my teaching more consistency and my planning more efficiency.

Framework: Computational Thinking

What is computational thinking (CT)? Combine critical thinking with the power of modern computing. This is not trivial – it is a significant shift in how we develop students' ability to solve interesting problems.

There are several ways of breaking down CT. Google and ISTE both offer definitions and resources. Some computer science curricula, Exploring Computer Science and CS Principles also offer approaches and resources, but are more computer science centric. The point is that there are resources available and many smart people who think this is a valuable approach.

But, to make this really clear, I think it's best to present an activity, and contrast a more traditional approach with a CT approach.

Activity: Review solving linear equations

Traditional approach: Model solving a few. Perhaps have students justify the steps to underline the idea of equality. Then have them practice – either alone or in groups. You could mix it up and play the mistake game if you're feeling different, or play a row game.

CT approach: Write a linear equation on the board with one unknown. Have students tell you how to solve it, step-by-step, but be an absolute pain in the ass. Do only exactly what they say. Make them justify their steps. Ask if you're done after every step. Do a couple of problems like this.

Why? You're going to focus on solving linear equations as an algorithm. Ask the students to write an algorithm to solve any given linear equation. Explain what you mean by an algorithm – a very precise step-by-step set of instructions that a trained monkey could use to solve an equation.

Teaching in Wisconsin

The Background

In the past year and a half, teachers in Wisconsin have been in and out of the national spotlight after our governor's attack on collective bargaining rights. The bill sparked massive protests for weeks around the state capitol, and ultimately passed through a budgetary process. The law that was passed, known as Act 10, severely diminished the powers of unions and increased contributions to benefits. This was a leading cause of the unsuccessful recall election that took place recently.

After Act 10 became law, there was a great deal of uncertainty about how districts would use their new powers. As it turns out, my district is going to be the first one to really put them to use and experiment. It was announced after spring break that in the future, nearly all high school teachers now are teaching the entire school with no prep period. In the core departments, this lead to approximately one out of every four teachers in the department being laid off. The remaining teachers will receive a bonus stipend of $14,000. In many respects, we now have a spotlight on us from across the state.

My Experience
I don't write all of this to complain. I simply want to relate my experience of becoming a new teacher and completing my first year during this time as I think we are at a crossroads in education in Wisconsin, in more ways than one - how we teach, how we're judged, and what society believes about teachers.

The climate has been decidedly negative. Very few people encouraged me to become a teacher, but more people than I count have questioned me. Or simply told me it's a poor choice. These voices have been inside and outside of the school, and the story they have been telling is the same: "You don't want to get on this train. It's breaking down and heading toward a cliff. There are better things to do with your life."

Perhaps. There are changes coming, definitely. I doubt the wisdom of many of them, but I also know that this where I am supposed to be.

Teaching is a strange profession. It seems that people either (a) think we're doing a truly noble service, or (b) think we're parasites who deserve less than what we earn. I have a lot of support - parents, students, friends, and community members who genuinely thank me for my daily work. And then I balance that with the looks of condescension and judgment, or people shouting "f@#$ you" out of their car window at me on my walk home because of a pin on my bag.

At the end of the day, and the end of the year, I haven't decided how I feel or what I think. There's not a single way that I feel about teaching in Wisconsin right now. I love teaching - I know that - but teaching in Wisconsin right now is too complex to relate without several hours and a few pints. I don't know how long I'll be able to deal with the negativity and pressure.

It's hard serving in a community that proudly voted nearly 3:1 for a governor that has extremely little respect for educators. I try to remind myself that I'm here for the students, but it is going to be my blood, sweat, and tears that make these new changes successful. What I fear is that my success is going to be used as proof of the validity of their ideas, even though they have handicapped me in an insulting manner.

But that's not going to stop me, because I love this state. We have an amazing tradition of progressive thinking and high quality education (see: The Wisconsin Idea). Further, the people of this state are genuine, friendly, warm, and hard working, even if that doesn't seem to always be reflected in our politics. Politically, we are severely divided, and I am clearly in the slight minority. There are many, many people with us, but just a few more who disagree with us.

I don't know what the future holds. I'm anxious about the teacher evaluation system that is now being piloted in a number of schools. I'm anxious that Wisconsin's commitment to a nation-leading public education system is slipping. I'm anxious about the quality of minds and characters who are going to volunteer to teach in this climate.

I'll be here next year. I am committed to this state and want to believe that Wisconsin will keep bending toward justice (great article by John Nichols). But I do have some serious doubts now, and that is new.

An equation is a problem...

I have been teaching algebra 2 for about three weeks now, and I've noticed some serious deficiencies in my students' understanding of algebra. We have been reviewing the basics - solving linear equations and inequalities, graphing, and working with absolute value. Students are consistently making the same mistakes - their typical response is a vacant "oh", followed by a quick fix, and I'm struggling to help them understand what is happening.

I've been scratching my head, rather anxious about our prospects of success. Our algebra 2 course is essentially a class about the major functions of mathematics - polynomial, rational, radical, logarithmic, exponential, and trigonometric. For each unit, we do the same set of things - graph, solve, and model.

After working with them for a couple of weeks, I developed a suspicion that they don't really understand what we're working with - specifically, the equals sign. They can somewhat reliably graph and solve basic equations, but I've seen little evidence that they understand what these actually represent.

So, I took five minutes from class yesterday and had them answer a few questions. Here are the questions with some typical responses:

• What is an equation?
• A problem that can be solved mathematically.
• Something that has an end solution through multiple applications.
• A problem you solve to get an answer.
• It is a sequence of #'s and variables, and your job is to find the right answer to it.
• Numbers or letters used to represent or solve something.
• What is a graph?
• A grid where you plot points.
• A typically square thing you draw a line on.
• Where points are plotted and to show increase and decrease.
• A graph is a visual equation.
• A graph is like another way of showing your work when you solve an equation.

While there are some glimmers of hope in there, I think this is going to be a barrier for us. Students will not be able to reach the depth of understanding that I desire unless they understand what that damn equals sign actually means. And I think this boils down to looking at operators, relations, variables, constants, and expressions. Maybe not using that language, but building on those ideas.

So, here's my plan. I'm going to students do an open sort with the following:

• Relations: =, <, >, ≤, ≥
• Operators: +, -, ÷, ×
• Variables: x, y, z
• Constants: 0, 1, 2, 3
• Expressions: 3x + 4, 5 - 1, 4ab
• Equations/inequalities: 2x ÷ 5 = 4, 2 - y < x, 5a ≥ 8b, 1 + 1 = 2

After the items have been sorted, I'll encourage student to create a hierarchy of their groups. I really want to focus on the groupings - what do things have in common, how are they different, and what do they represent.

In the end, I'm hoping that my students have a better understanding of equality as a relationship, and that an equation is simply a statement that two expressions have the same value.

I would really appreciate any feedback before I give this a go!

The Mulligan

I don't love the block schedule. I'm not sure I even like it, and I'm not sure it's good for teaching math.

But, as a first year teacher, there are two significant benefits:

1. Reduced number of preps
2. I get a mulligan halfway through

I'm pretty excited about #2 right now. Now that I have half a school year under my belt, there are about a million things I want to change, and some of the changes just work better when you get a new class. Here's a short list of changes I'm thinking about:

• Change the purpose of the openers
• Start student portfolios
• More mid-block formative assessments
• Make whiteboarding and sharing an everyday part of class
• Better cooperative group activities with individual accountability
• More emphasis on communicating reasoning
• Preparing for student mistakes instead of helping them avoid them
• More consistency is the handling of assignments
• Focus more on connecting with student
• Focus less on the real word justification of the mathematics
• Connect mathematical reasoning to students' lives more often
• Follow through with consequences more consistently
• Make better use of our weekly trips to the computer lab

The new term starts next Tuesday. While I'll certainly try to work on all of these, I'm going to pick two or three over the weekend to really set my sights on and reflect on in more detail. More to come later.

Becoming A Better Mathematician

A year ago, there were two Project Euler problems that really gave me headaches (#26 and #34, if you're really interested). I remember spending hours on them to no avail. This was frustrating, especially with how I work. I ended up skipping them and moving on to other problems, but that kind of thing doesn't sit well with me.

So I took them up yesterday, and surprisingly, solved both without a terrible amount of trouble or time. I could chalk this up to simply taking a break and seeing the problem in a different light - which has worked for  before - but I think it's more than that. Teaching high school mathematics has made me a much, much better mathematician.

While this certainly gives me a personal sense of accomplishment, I'm more interested in the specific things that have made me a better mathematician and how I could use these in my teaching. Reflecting on my work yesterday, there were a few significant changes in the way I was working:

• Make big problems small
• My go-to approach is to plug some stuff in a small example, see how it works, and look for some kind of pattern.
• Slowing down
• I was solving for slowly - once I had an idea that I thought might work, I took the time to think/write out my reasoning before getting started on the solution.
• Starting over
• I was much more ok with completely scrapping something and starting in a new direction.

These are nothing revolutionary. Far from it. They are things I've heard and read before, and things I (sometimes) try to incorporate into my teaching. But there is something about experiencing this first-hand that really drives it home.

What's my take away? This needs to be a more conscious part of my teaching. It's not going to change a student in a day. It might now show up on a test. But at the end of the year, if I've done my job right, there will be a whole new level of problems that a student will now have the ability to take on. The process is slow, but without fruit.

I also need to keep doing mathematics. Whether it's inside the classroom or out, it's not an indulgence - it's a professional development necessity.

New Year's Resolutions

1. Run - three to four times a week. It makes me a happier, healthier person. I just need to commit to the time.
2. Read - at least one book a month.
3. Blog - one new post a week, even if I don't feel like I have much to say. Because when I start writing, I find that I always do.