# Tag Archives: MTBoS

## 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

## 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

## 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.

Filed under conversations, Geometry, group work

## Factor Craze

I didn’t have the greatest teaching day today, so I thought I’d try to remind myself that I CAN do this teaching thing by describing an activity from a few days ago that was successful.

Factor Craze, which I think I found via Fawn Nguyen, is one of NCTM’s monthly “Problems to Ponder”, and it asks:
Which numbers have exactly three factors?
Which numbers have exactly four factors?
Which numbers have exactly five factors?

This problem was a great introduction to factoring with my FST (2nd half of an extended Algebra 2) kids. They saw factoring last year, but this year I wanted them to really understand how they were coming up with the equivalent expression instead of following a list of steps from the teacher. So I used Factor Craze to spark some conversations about factors.

I have my students seated in groups, but I had them think on their own for a minute before working with their group. I actually started with the question Which numbers have exactly two factors?, which may seem rather elementary for high school juniors and seniors, but as I suspected, many had very little knowledge or experience with the concept of prime numbers.

Most groups started by writing down examples of numbers that had the required number of factors, but I prompted them with, “What’s a way to describe ALL numbers that have exactly ___ factors?”. All groups eventually came up with prime for exactly two.

When they got to exactly three, most groups found out that 4 and 9 worked. I asked them if there was anything special about numbers 4 and 9. “Oh, oh! They’re perfect squares! Perfect squares have exactly three factors!”

So I respond with, “Do all perfect squares? What about 16 and 25?”

“16 doesn’t work. Oh. But 25 does!”

So I say, “Nice. So some perfect squares but not all perfect squares. What type of perfect squares work?”

And so on. Most groups figured out that squares of prime numbers have exactly three factors. Only one group in each class was able to delve into exactly four factors before we ran out of time.

I really liked how this problem posed a challenge for every student. For some, just remembering what prime numbers are like was a challenge. For others, it was recognizing a theme for exactly four factors. Either way, all students were developing an understanding of factors.

We later moved on to greatest common factors and factoring expressions, and I think laying the ground work with Factor Craze made a difference.

Filed under FST / Algebra 2, group work, productive struggle

## High and Lows (Mostly Highs!) of the First Two Weeks

I survived my first two weeks as a high school math teacher! So many things have been running through my mind, but right now I’m going to make a list of things that are going well and things that need improvement. I just want to get it all out. I hope to blog more regularly from now on!

Things that are going well

• I love my school. It’s so, so, so good. My colleagues are incredibly supportive and amazingly talented. Our students care about their school and each other. I am very fortunate to be part of such a strong community.
• There are some very effective school-wide policies in place that administrators, teachers, and students are all on the same page about. I feel like this really promotes school pride and diminishes behavior problems.
• My Geometry and FST students are awesome kids. I am so impressed by them.
• Creating a classroom that values mistake making. This is a work in progress, but I’ve got a decent start.
• Establishing a classroom community where the kids feel comfortable talking to each other. Seniors are good with this (too good, actually), and I’m still working on Geometry kids.
• I have established some classroom routines! Phew. Thank you Andrew Stadel for Estimation180. It’s been a great way to start class every day. Similarly, ending class with an exit ticket lets students know that we work until the bell, as well as provides me with some great feedback.
• Using whiteboards (both big and small) has been an effective way to get students to share their thinking and to just get some students to write something down.
• I’ve done some deep activities, problems, tasks, or whatever you wanna call ’em that have produced good results.
• I am learning every day.
• I am finding time to exercise and cook dinner. (Sleep is another matter. Looks like I might pick up drinking coffee again…)

Thank you to all the inspiring teachers who share their wonderful ideas and activities so that I can use them. I stand on the shoulders of giants.

Things to improve

• Classroom management. Can you tell I’m a first year teacher?
• Similar to the first point, I struggle with engaging every student when I’m talking to the whole class. Group work is my strength: students discussing with each other with me floating around from group to group asking questions and guiding them along. In contrast, I feel like I’m not strong enough at whole-class lecturing and encouraging note-taking. I think I just need to be more strict about it. No talking when I’m talking. Pick up a pencil and write something down.
• Kids who are absent. And the kids who are just now switching into my class. How can I get them up to speed?
• Checking homework and going over answers. What a big ol’ unproductive time sink.
• Better hand-writing. I save my Smart Notebook documents and upload them to my class website for students to use as a reference. Neater hand-writing would be easier for kids to read and follow.

Have a lovely weekend, everyone! Here’s to a great year!

Filed under culture, productive struggle

## Desmos rocks my world

Last night I took the batteries out of my TI (83 plus!) graphing calculator to use in my bike lights. Sorry not sorry. Priorities.

In case you haven’t already discovered this fantastic resource, there is an online graphing calculator (and so much more) called desmos. Use it once and you will never want to use your old TI again. It’s easy to zoom in and out. It’s IN COLOR. If you are graphing multiple functions, you can make each one a different color. You can create sliders. And it is all FREE. Also be sure to check out all of the beautiful artwork while you’re there.

I won’t be ditching the TI for good because using it is a major part of my school’s current FST (aka the second half of Algebra 2) curriculum, which is fine. Students need to learn how to use TIs because right now they are the accepted technology for tests, both in the classroom and for standardized tests like the ACT and SAT.

Desmos is definitely worth incorporating into the classroom though. I used desmos with great success last summer when teaching summer school, and I think it’s great for doing investigations and creating visuals. That tiny, pixelated TI screen seems rather clunky and out-dated next to desmos, where students can really “see” the graphs and play around with them more easily. When I personally do math, I always use desmos if my laptop is with me. (Being significantly lighter than my laptop, the TI is more likely to be in my backpack on any given day.)

So I plan to use both desmos and the TIs in my classroom this fall. The TIs will be our go-to use-every-day type of calculator, but I’ll pull out the laptops as much as possible to use desmos for graphing investigations. I also hope the kids will come to appreciate desmos and start to use it at home or when they come to the math resource room during study hall. From summer school, I already have desmos investigations made up for quadratic functions and rational functions, but they were made hastily and need some improvement and some updating to more closely match my school’s FST curriculum. I’m excited! Now I just have to do some work and make these plans actually happen.

How do you incorporate desmos into your classroom?

Here are some examples I’ve found on the MTBoS:

Fawn’s Des-man which inspired the desmos team to create this awesome version of the project

Bob Lochel’s Desmos Filing Cabinet