# Tag Archives: math

## 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…

Filed under curriculum

## End of 3rd Quarter

Hi everyone.

It’s the end of 3rd quarter, and we’ve got a grading day. Actually half-day. So I thought I should blog since I got the time! No kids! It’s some sort of miracle. I’m fortunate at my school to have a prep period and a department planning period… but during my prep, I can expect to supervise 8 to 10 up-to-no-good-but-so-lovable seniors. Up-to-no-good is definitely putting a positive spin on it. During my plan period, I can expect to supervise two or three accelerated freshmen for whom school comes easy and are mostly bored with it, and two or three sophomores who care, but need me to give them 1-1 tutoring in Geometry.

In short, having some time to myself in my classroom is some sort of miracle. It never happens. Now if only I had something interesting and substantial to blog about.

The kids are the best part of the job though. Forget grading, planning, and prepping. I do what I do because I believe in those little punks. They’re beautiful, lovely, funny, and smart. They deserve the best.

OK, here’s something worth blogging about. I just taught right triangle trigonometry to my Geometry kiddos. I love introducing trig. This year it conveniently followed a similarity unit, so I introduced it with a quick lab measuring sides of triangles and computing SOHCAHTOA ratios. Huh, weird, for any 30 degree angle in a right triangle, the ratio of the opposite side and the hypotenuse is the same. Huh, weird. (Similar triangles, anyone?)

Then we do some boring, but straightforward practice. Then the next class we go on a field trip. I love to advertise this next bit as a field trip, even though we only go down two floors to the Commons.

I start by having them estimate the height of the ceiling in the Commons (we regularly do Estimation180 in Geometry). Then I have them take out their telly-phones and download a free clinometer app. The only issue is the kids who say, “but I don’t have any room on my phone”. Maybe if you deleted some of those dang selfies, kid.

I crappily, but enthusiastically, model what they’re supposed to do. (My teaching career is a work in progress, OK? Year two is better than year one, at least.) I pass out the awesome, giant tape measures that the math department owns. We disperse down to the Commons. Chaos ensues, naturally, but we’re on a field trip in math class, so it’s a good thing. Tape measures are being stretched out, kids are pointing their phones at the ceiling, and most kids are sketching a triangle and writing down some sort of trigonometric equation. It’s my favorite day of the year.

Eventually we return to the classroom. My least-focused kid (one of those with an ADHD star next to his name in Infinite Campus) happily sits down and gets to work solving trig equations. How could he not? I just let him run around the Commons for 10 minutes.

They’re beautiful creatures, ya feel?

8.4 trig lab

8.5 trig invest how high is ceiling (I think this was adapted from something from Tina Cardone @ drawingonmath ??? Not sure. But I definitely stand on the shoulders of giants. Thank you all.)

Filed under fun, Geometry, grading, trig

## Take Aways from Green Lake

Yesterday I had the good fortune of attending the Wisconsin Math Council Conference for the first time, and it was a lovely experience.

I went with my colleague who is also a first year teacher, and we ended up in some really great sessions.

My favorite session was definitely Get Up and Move, which was exactly how it sounds. I learned some great new strategies for getting kids out of their seats and moving, including Bucket Sort, Musical Math, Relay Race, and Clue. I think my kids sit too much, and I want to get better at doing fewer, chunked activities rather than long work times, so doing a practice activity where they’re out of their seats and moving sounds like a win-win.

I also went to Jo Boaler’s keynote. I’ve been following Boaler’s work for awhile now, so I didn’t really learn anything new, but her presentation was so lovely and her message so true. I can’t agree with her more, and I hope to see a major shift in mathematics education soon that encompasses her ideas on mindset, mistakes, and success in the math classroom.

One session that I wish would have existed is lesson planning or unit planning. I feel like maybe I should have learned that in teacher school, but whatever. I want to get better at planning.

Filed under collaboration, conversations, Uncategorized

## Fun with Expected Value

I just taught expected value in FST and really enjoyed it. The two main tasks I used were: The Carnival Candy Game and Dan Meyer’s Money Duck.

The Carnival Candy Game

You’re at a carnival and you get to pick one piece of candy from a bag. The color candy you draw determines how much money you win. I used starbursts, and I set it up like so:

The students didn’t win money; rather they won that many starbursts. (I had a different bag of starbursts for prize winnings because I made sure that the candy drawn was replaced each time to keep the probabilities the same for everyone.)

This was enjoyable because naturally all the kids wanted to pick the purple one. Not surprisingly, most picked pink, yellow, or red, but I have 45 FST students (two classes), and the 44th student did pick the purple one.

Then I asked them to calculate the expected value for their prize winnings when playing this game.

Then I said, suppose it costs \$5 to play this game. What does that mean for the player? What does it mean for the carnival game host?

Money Duck

Love the Money Duck. The students were very engaged by the idea of the money duck. I basically followed Dan Anderson’s lesson plan for this one. Like Dan’s students, and as I commented on his post, my students also wanted to determine the possible/impossible distributions based on what they saw in the video instead of in theory. I slightly fixed that in my second class by stopping the video after the first \$1 money duck, explaining that the video was made up, and stressing that we were interested in what is possible, not necessarily what the company actually does.

Like Dan, I had my students come up with company names, probabilities, and price. They then had to compute expected value and their profit. I also compiled the data in a spreadsheet, but didn’t really do anything with it, unfortunately. If I did it again I would like to have the students do some more sharing and comparing between groups.

And then things got even better. Today was the grand opening of a new Cabela’s nearby my school, so several of my male seniors told me how they all skipped class this morning (well, some of them probably had open campus 1st period… I hope) to get in line at the new store because the first 500 customers received a gift card up to \$500. One of them said, “But Ms. Cummins, they didn’t tell us how many were for \$500″. It turned out that they all got \$10 gift cards except for one who got a \$25. It was perfect. I told them I was going to write a test question about that.

Filed under FST / Algebra 2, fun, probability

## 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?

1 Comment

## 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.

1 Comment

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

## Plans for the New Year

Happy new year! What a wonderful winter holiday this has been. I think I really lucked out as a first year teacher getting a two-week break from school this year. It was been a period of relaxation and rejuvenation, as well as a celebration of family, friends, and good times. And it’s not even over yet!

As a result, I’ve had plenty of time to reflect upon my teaching experience so far, and as a result, I’ve developed some ideas and plans for the new year and next semester’s classes. I still have two weeks left to wrap up before finals week, so while I may implement some fresh ideas now, I might not get around to all of them until the new semester starts.

Here are some thoughts I’ve had, in no particularly order.

1) Change up the seating arrangement. This one I’m going to save until second semester because I don’t want to throw off the kids right before finals week because I swear I’ve read somewhere that a person tests best in an environment that he or she is familiar and comfortable with. Anyway, my plan is to arrange my students in pairs. Right now the kids are seated in small groups of four to facilitate collaborative learning, but the tables are simply too big for the kids to work across. I encourage them to stand up and move to the other side, but sometimes they’re reluctant to do that. Additionally, partner work has been more effective than group work in my classroom so far. I’d love to do more group work, but it’s a dream in progress, and I think the days would just run more smoothly with students in pairs.

2) Figure out a good system for warm-ups. I have to decide what I want my expectations to be for warm-ups, and I think they’re going to be different for my Geometry classes and my FST classes. For Geometry, I think I might have the students do a weekly warm-up sheet (a la Fawn Nguyen, etc.), but for FST I think I’m going to have them do a daily half-sheet that is either prepared by me with review of some Algebra skills that will be needed for the day’s Functions, Stats, or Trig concept OR that is some sort of writing task. Which  leads me to idea number 3.

3) Incorporate more writing into math class. Still have to think about this one, but I love, love, love it. The ability to communicate is so important in mathematics (and in life, as my mother would say).

4) Continue to build relationships with my students, my school, and the MG community. I just read this article, which was a good reminder to finally attend a basketball game, as well as organize another MG SNOWBOARD AND SKI CLUB!!!!!!! trip. I agreed to be the advisor of the new snowboard and ski team, and it has been mildly hectic, but fun, so far. The other day I realized I have a more experienced background in sports and recreation teaching than I do classroom teaching because I started teaching sailing lessons when I was 14.

Ok, that looks like a pretty good list. Now I just have to work on the enormous pile of grading that I have to do.

Filed under planning

## ‘Twas the day before break…

My goal for my Geometry classes today was to be as mathematically productive as possible, given that it was the last day of classes before break. The plan was to review the last assignment, take the quizlet (what my department calls formative assessment), do an extension problem, then make a Koch Snowflake if there was still time.

The extension problem was the “Shortest Path Problem” which I highly recommend.

It turned out to fit perfectly with what we’re learning right now. It also sparked some rich conversations and good reasoning, and everyone could at least venture a guess, even if they didn’t really know what to do to figure out the exact shortest path.

This plan was carried out differently in each of my three Geometry classes. In the first class, I reviewed several problems and concepts, kids followed along, asked questions, the usual. They took the quizlet. I passed out a half-sheet with the scenario typed out on it. I didn’t include a diagram, thinking that the kids should make the diagram. That was a mistake because the wording isn’t exactly clear, so some kids drew the tent and camper on opposite sides of the river and all sorts of random things. So I had to draw the diagram up on the board for everyone, which slightly killed the magic, but at least we were all on the same page.

A few kids calculated the distance of a path, but not the shortest, and then wanted to be done. I needed a way to motivate them to keep working. In a rare moment of brilliance, I decided to keep score. I announced “Jesse found a path that’s 1,518 feet, can anyone beat that?”. Then I’d write the student’s name and their shortest path on the board. It became a competition to see who could find the shortest path. I let things linger too long in my first class because a few students were really getting into it and asking wonderful questions like, “how do you know that’s 450 feet” and “can you show me how you got that”. So unfortunately several kids had checked out, but at least everyone did something with the problem.

In my other two classes, I skipped the homework review and went straight to the quizlet because there was no way they were going to sit and listen to me blah blah blah about their homework problems on dilations and scale factors. In my first afternoon class, student behavior dictated that decision. In my second afternoon class, I asked them what they wanted to do, and almost everyone said, “let’s just take the quizlet”. So in those classes, there was plenty of time to do both the shortest path problem and the Koch Snowflake.

This time I just asked them to read the problem on their own, and then I read it aloud and drew the diagram as I read so everyone started out with the correct diagram. In one class several kids said they didn’t know what to do to get started, so I said “guess and check” or “if you were the camper, where would you go if you wanted the shortest distance”. This was an excellent starting point for those kids.

The snowflakes were fun too. We saw the Sierpinski Triangle this year, so I brought that up again as a reminder of what a fractal is, but then said that the fractal they were about to make was going to be more holiday-themed. I gave everyone some triangle graph paper to help them with their triangles. At first I thought maybe I’d have them construct the equilateral triangle, but using the graph paper was a good call.

Filed under fun, Geometry

## The teacher learns

I’m getting better at making my expectations clear. Giving quick, short directions right away and repeating them until all students are with me sounds obvious, but it’s easy to move on without some kids and then you never really get them back.

Always be one step ahead of the kids. Pass out and explain the next task to the kids before they start their quizlet so that kids who finish early have something to do.

I freaking love warm ups. Haven’t figured out a system for them yet though. Should I preprint the questions on a half sheet? Should I grade it? I think the answer is probably yes to both of those questions, but I don’t love the idea of using more paper or having more things to grade.

Graphic organizers are great. A few phrases in a few boxes is more writing than we usually do in math class. They work on it individually, then in groups, then I solicit answers and go over it as a class.

That reminds me: cold-calling = awesome. I have cards with student names on them that I use. Open ended questions or questions with more than one right answer (give me one of the transformations we’ve talked about) are best.

I’m almost half-way done with my first year! The lows have been low, but the highs have been high, and I keep reminding myself just to be better than I was yesterday. Always learning.

Filed under classroom management, productive struggle

## 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.

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.