Tag Archives: students

On Curriculum: What Doesn’t Work

For some reason I feel compelled to write about teaching even though I just quit teaching. Quit? Yes, I resigned from my job this year and am taking a break from teaching high school math. Why did I do it? Honestly, I don’t think I can adequately articulate it, and I don’t owe anyone an explanation, but quite simply, I needed a break. I have a sneaking suspicion that I shall return to teaching again some day (probably sooner than I realize), but in the meantime, I’ve been pursuing some of my other passions and working on acquiring some new skills.

I’m still very interested in being part of the conversation on teaching high school math, and I still use Twitter every day to keep up with it. In fact, I have so much to say on the topic that I figured I might as well blog about it. I hope that blogging will be cathartic for me, helpful to other teachers out there, and helpful for me if/when I return to teaching.

So that was quite an introduction to a post in which I wanted to talk about curriculum.

I’m inspired to write about curriculum because the school where I taught had such a horrible, out-dated curriculum, and it was a huge burden for me. I’m pretty sure the curriculum pre-dated my own high school years, so I was shocked that I was required to teach it to my students. It was the most rote, procedural, and repetitive mathematics that I have ever come across. It made me think of the Cold War era, which I actually don’t really know anything about as I was born after that time, but if I could imagine it, I imagine different countries putting their young people in little school factories to see who could solve equations by hand the fastest.

Such was the imagery in my head because the entire curriculum at my school was built around solving equations algebraically. Here is the procedure for solving quadratic equations. Here is the procedure for solving exponential equations. Here is the procedure for solving trigonometric equations. And so on. Naturally, this led to an incredibly teacher-centered classroom. For each lesson, there were pages of notes that the teacher talked about. Then the teacher did some examples. Then the students were supposed to mimic the teacher exactly on a worksheet of 25 identical problems. It was brutal. I felt so sorry for… everyone involved.

Now, I don’t mean to say that we shouldn’t teach solving equations. The concept of what it means to solve an equation is a fundamental part of mathematics. During my first year of teaching I quickly realized the lack of conceptual understanding my students had as a result of our pathetic curriculum. Our assessments would be filled with equations to solve, but not a single student could answer the questions: What does it mean to solve an equation? What does it mean if a number is a solution to an equation?

When I discovered this discrepancy, I just felt terrible. Why were we making students do something that they didn’t understand? Hey kids, memorize exactly what the teacher did, regurgitate it on an exam, and then do it again. There’s no need to understand it. Heck, you can get an A+ grade without actually understanding anything. I quickly realized that no genuine learning was happening. It was sad.

My last two years of teaching I incorporated the two italicized questions from above into the first non-review unit almost every day. (Yikes, don’t get me started on how our curriculum wasted the first unit of every year on “review”.) Last year, I finally had more students answer the first one with something along the lines of “find the values that make the equation true” than students who said “IDK” or “get the answer”.

Besides a lack of understanding, our curriculum lacked efficiency and modern technology. Before becoming a teacher, I was first and foremost a mathematician, and I assure you that no mathematician was solving by hand some of the equations we made our students solve by hand. Mathematicians use technology. If I were to come across an equation that I knew I could solve by hand, but that would take me more than 60 seconds, I would turn to my computer or pick up my Iphone and use Wolfram or Desmos to find the solutions and then carry on from there. I don’t waste my precious time doing a rote procedure when a computer can do it so much faster. I spend my time on bigger and better, more important and more relevant mathematical ideas. Our students should be doing likewise.

Again, I’m not saying that students shouldn’t know how to solve equations or that mathematics is purely conceptual. However, telling students to memorize a specific procedure isn’t that important or that useful. Rather, let students explore equations, find methods that work for them, and develop fluency. As they do this, they will acquire and practice important mathematical skills such as: manipulate equations, model with equations, create different representations of equations, and solve equations efficiently.

Finally, this type of curriculum is also very boring. Be silent. Watch teacher. Work in isolation. No creativity. No thinking. No discussion. There is a complete lack of genuine student engagement. Those poor kids. It’s no surprise that so many dislike math/school. And poor teachers! Our curricula can set us up for failure or, at least, prevent us from seeing the successes our classrooms are capable of.

Wow, am I still complaining about my old curriculum? It appears so. I guess I had to get something off my chest. I should probably stop complaining and maybe write a post called On Curriculum, Part Two: Making it Better. Although I could probably write a whole book on that topic. Now there’s an idea…

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A quick anecdote on feedback

I passed back some Geometry tests the other day, and there was a problem on similar triangles in which students had to agree or disagree with a statement and explain why. While grading, I wrote “well said” or “nicely stated” next to any convincing explanations.

A student saw this comment, and asked me, “Is this supposed to be sarcastic or what?”

I was surprised. “No… I meant that. I thought it was a good explanation.”

The kid responded, “Oh, well it was in red so I thought it was bad.”

So that was interesting, and it has me thinking about different types of feedback. What does effective feedback look like? How do kids perceive feedback?

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Filed under culture, grading

A cold day, followed by a beautiful display of student initiative

Yesterday we had a cold day! It’s like a snow day, except it’s really cold out. With wind chill, temperatures around here were -35 F. The timing was good because somehow I ended up being really sick yesterday. So I didn’t particularly enjoy myself on my day off (in fact, I felt terrible), but thankfully I could nap by the fire, drink tea, and spend the day recuperating.

Anyway, I wanted to post about a proud moment from my FST class today. These kids are used to a lot of hand-holding and spoon-feeding, and many of them rarely do independent work (unless I really hound them). Most days, I’ll hear this from at least one FST student: “I’ll be honest, Ms. C, I’m not gonna do this.”

These kids are mostly seniors who’ve been placed in “lower track” math classes their whole life, so changing their mindset isn’t easy. But they did elect to take 4 years of math in high school, plus they’re all good kids, so I know it’s worth it to keep trying.

Today, I told them I would walk them through one example of each type of problem (unit circle stuff), but that was it. No more.

A few kids said, “Aw, can’t you keep going.”

“Nope. I said that was all I was going to do as a class.”

Here is where one kid said, “We can keep doing them as a class, I’ll just go up there.” And he did.

The awesome thing was this kid didn’t know how to solve the problems. But he was willing to go up there and try to figure it out. It probably helped that he’s in the drama club and is an anchor on the school announcements.

So he starts to play the role of the teacher. “Ok, so let’s do problem 2: 495 degrees. We need to find an equivalent rotation between 0 and 360 degrees. How do we do that?”

Miraculously, the rest of the kids played along.

“It’s 45 degrees.” “No, it’s 135 degrees.” “How’d you get that?”

The 135 degree kid explains his thinking, the kid at the board follows along, agrees, and writes down 135.

I quickly snap out of my state of shock and try to remember good techniques for facilitating student discussions.

So I ask, “S, could you please repeat how you got 135?”

So he does.

“Thank you. Can someone summarize or rephrase what S just said?”

Someone does.

And, oh man, it was beautiful. Students were participating without any prodding from me. I managed to remember to ask good questions (Who can rephrase that? Who did it differently?) and to occasionally ask for a collective pause to let something sink in for everyone before moving on. Most importantly, I remembered not to interrupt too much.

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Filed under classroom management, collaboration, culture, FST / Algebra 2

Motivating 1/x

We’re deep in a functions unit in FST (year two of a decelerated Algebra 2 course), and I love it. I love the concept of a relationship that takes inputs and produces outputs. I love visualizing functions with graphs. I love that functions feel natural and intuitive. I’m trying hard to share this enthusiasm with my students. They’re doing well with it so far, and it’s interesting to see how they think about functions.

Last class, I wanted to introduce f(x) = 1/x. I love this function. I love the discussions about division by zero and division by really large numbers and how the graph represents those ideas visually. My colleague shared a fun investigation with me, and I am so glad that I tried it out. At first I was hesitant because I know that I don’t explain directions well, but I focused on being very explicit and modelling each step. The kids investigated the breaking point of spaghetti. I wish I had some photos, but the students placed a dry spaghetti noodle over the edge of the table and hung a paper cup on the end of the noodle and added pennies one by one until the noodle snapped. The fun factor was definitely there- the kids enjoyed predicting when it would snap and liked watching the pennies crash to the floor.

Besides being fun, the activity modelled the function effectively. The kids recorded their data (length of spaghetti vs number of pennies), and I used Desmos to display some class data.

hyperbolaPenniesSpaghetti

Voilà, a hyperbola. The investigation gave the kids a good understanding of how the function behaves and why the graph looks the way it does. In retrospect, I should have done more of a “Noticing and Wondering” activity with the graph, but instead I just asked some questions like “What happened as the length of the spaghetti got longer?” and “What happened if the length of the spaghetti was really small?” which probably did too much of the thinking for them, but oh well.

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Filed under FST / Algebra 2, fun, graphing, group work