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…

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.)

A Note on Equity

Yesterday one of the guidance counselors sent out an email asking for nominations of a sophomore student to send to a leadership conference. This student was supposed to demonstrate leadership and contribute positively to the school community.

I thought about who to nominate. A student who immediately jumped to the front of my mind was a black male who fits the criteria and is an awesome person in general. He’s kind of quiet though, so I wasn’t sure whether to nominate him or not. I ended up not nominating anyone and decided to let other teachers make the nominations.

Today the guidance counselor sent out the list of nominated students. The black male mentioned above was on the list. I immediately thought, oh, good , I’ll vote for him.

Then I saw another name on the list, a white male, who I have in class this year. I didn’t think of him yesterday, but he is a great leader, and I am so thankful to have his positive influence in my classroom.

So, of course, the typical internal debate ensued. Do I vote for the black kid or the white kid? Does the black kid “need” my vote more? Is this an opportunity that he might not get elsewhere?

I went back and forth for quite awhile, and then started scanning other names on the list. Suddenly it occurred to me that perhaps I was having the wrong debate. I was so stuck on racial equity, but what about gender equity? How come I immediately focused on two males for a leadership conference?

I am regularly in disbelief (and sometimes in shock) about the lack of women in leadership positions, yet here I was debating between two male students to send to a leadership conference. Although I immediately considered racial inequity, I almost didn’t even acknowledge gender inequity. Weird. This made me wonder if I am somehow influenced by societal norms about men assuming leadership positions.

Once I came to this realization, I completely changed tactics. I recognized many female students on the list and quickly settled on one who I think is, and will continue to be, a great leader and positive role model.

She happens to be white, but I’m completely satisfied with my choice.

 

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.

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.

‘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.

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.

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.

Keep Yo Head Up

I came home from school on Tuesday and told my boyfriend that I wasn’t going back. Of course, I was overreacting to one bad day, and after a good night’s rest, I went to school as normal the next morning. The rest of the week went great, so there you go. It’s so easy to focus on the negative and forget the positive that I have to continually remind myself of what’s going well in my classroom, or I’d never make it.

Last week, my administration granted me the privilege to watch an experienced colleague teach a class. I watched a Biology class that was so well taught that I left feeling inspired. I used to think classroom management was my biggest issue, but after watching that lesson, I realized that time management might actually be my biggest problem. I wasn’t managing time at all. (That sounds crazy, but it really shouldn’t surprise me. I won “Most Likely to be Late to their Own Funeral” in high school. The clock in my car is always set to a random time. I never rush anywhere or feel a sense of urgency.) For example, I used to set the kids off on a task with absolutely no time-frame. I guess I thought that every kid had to finish before moving on.

To avoid wasting so much time, I need to keep things moving fairly quickly. I need to set time limits on activities. I can’t have kids messing around because they think they have the rest of the class period to accomplish something that really only needs 10 minutes of concentrated effort. I need to have tasks for kids who finish early, and I need to recognize that there’s not enough time to wait for 25 kids to each discover something on their own. One thing that has been effective for me is to have a list of two things that students need to accomplish within a given time-frame, with the second item being one that doesn’t require all students to get to before moving on.

Lessons learned there for sure, and that’s a good thing. Other good things: Had a good 1-1 conversation with a student who acts out in my class and likes to challenge me. Two students I’ve been trying to get a hold of for awhile finally came in after school to take missing tests and even waited 10 minutes extra for me because I was in a meeting that went long. Did a a fun investigation in FST (blog post coming soon, hopefully). Taught some algebra-heavy topics in Geometry and no one threw a fit, including me.

Did a long problem in Geometry, and at the end a student pointed out “that answer doesn’t make sense”. She was right, it didn’t, but I couldn’t see a mistake in my work. I stepped back and told everyone to try and find the mistake. Finally, one of my favorite delinquents (he’s not really) piped up, and said “The point is labelled wrong in the diagram. It should be (1,0) not (0,1).” Dang it, a typo on the handout. But all was well because I got to model mistake-making in front of my students. Then I had to go through and basically re-do the entire problem which was also good because the students got to see it again and hopefully understand better.

Oh yeah, and I’m the advisor of the newly formed snowboard and ski club. Held a brief meeting in my classroom after school just to see how many people were interested, and within a few minutes, my room was packed. It was awesome. One of my Geom kids has stepped up as the student leader of the club, which is exactly what I needed.

Always, Sometimes, Never

I debated some Always, Sometimes, Never statements with my Geometry kids today. In groups, they had to choose the word that they thought went in the blank, as well as draw a picture to explain their choice.

Some example statements (taken straight out of our textbook):
Two planes ________ intersect in a line.
Lines ________ have endpoints.
Lines that are not parallel ________ intersect.
Two points _________ determine a line.

That last one created some interesting discussions, particularly in my last period. Many students wanted to put Sometimes in the blank. I didn’t look at the textbook’s answers, but I assume the authors wanted Always in the blank.

Why did so many students think Sometimes? Well, I think the statement was kind of confusing to them. What does it mean to “determine” a line? Does “a” line mean one line or does it many any line? I tried to resolve the matter by putting two random dots on the board and drawing a line through them. “Look, I can draw a line connecting any two points.” Not particularly convincing.

The students then told me to draw a line going through each of the points (parallel lines, for example). “See,” they told me, “there’s two lines, not a line.” I didn’t really know how to respond to that. I told them yes, I can draw different lines through each point, but only one line will connect them.

Well, I think I convinced them that any two points could be connected with a line, but we just left the Always, Sometimes, Never question unanswered. Which is okay. Of course, some kids insisted, “But what’s the answer?” and I replied, “Well, I think it’s Always, but I don’t think it’s totally clear.”

Perhaps the answer would have been less ambiguous if the original statement was Two points can _________ be connected with a line. But that statement seems way less powerful. So now I am intrigued by the word “determine”. I definitely think it’s important. It’s hard to explain to the kids what is meant by “determine” though.

One instructional difference I would have made during the activity was to require new people to be the writer and the speaker for each statement. In a couple groups, it was very obvious that two or three students were doing all the work while the others checked out, so some sort of rotation would have been smart.

I want to start the next class by playing Sarah Rubin’s Draw It game because some of the drawings I saw today were definitely off the mark, but that’s okay. Visualizing lines and planes and space can be tricky. I love seeing their eyes widen when they begin to “see” it.