I have a small calculus class this year (only 6 students), and to be honest, I wasn’t sure what to expect at the beginning of the year. I have taught these kids for the previous two years and so I knew that none of them were exceptional math students. Great kids, who mostly don’t seem to care too much about math. Boy, have they surprised me. The kid who last year could hardly keep his eyes open in class is this year coming up with some of the best input of all. The one I would have considered the weakest student got the highest grade on the quarter test. I’ve been having a lot of fun with this group. Today we learned the trig rules for derivatives. This might have been my favorite day this year. I started by asking them to graph y=sin(x) and label all important points. Then we used a calculator to find the numerical derivative at several different points. We plotted these points on the same graph as y=sin(x) until they were all certain that the derivative was cos(x). Some people may disagree with me on this, but I didn’t attempt to prove this fact, we just accepted it based on our results.
Then, without any prompting on my part, someone asked “so, is the derivative of cos(x) = sin(x)?” And they all set off on figuring out if that was true. It didn’t take them long to figure out the correct rule. And now they had a graph of this derivative as well.
Next question, again without me doing anything, was “so does that mean that the fourth derivative of sin(x) is just sin(x)?” I asked why he thought that would be true and he explained that each successive derivative of sin(x) was a graph that was shifted left pi/2 units. Once you shift it 4 times, you will be back to a sin(x) graph. I have to admit, I had never thought of it that way. That’s one of the things that is so much fun about this class. They see things so differently than most. Naturally we had to verify that this student was right in his reasoning.
Then the next comment came. “Where’s tangent? Tangent has to be in there somewhere.” Okay, let’s talk about tangent. And so they derived the rule using trig identities. Then they wanted to know if the derivatives of tangent followed a pattern like the derivatives of sin and cos. Which naturally brought up the need to know the derivative of sec(x). And so it went for the entire hour. I got the ball rolling by asking them to graph sin(x) and find it’s derivative at several places. They took it from there, and went exactly where I was planning to take them. All I had to do was facilitate their discussion and ideas. It just may have been my favorite lesson so far.
On a related note. This year, as I have tried to step back and let kids take more ownership of their own learning, it has been a lot of fun for me to see the creative ways they approach things. Sometimes it is exactly how I would do it, but many times they come up with a process or a way of thinking that I was not expecting. I’m loving the opportunity to see how they are thinking.
Last week I missed two days of school to attend training on the new common core state standards. The training was largely a waste of time, but that is another story. This week I missed part of yesterday and all of today because I was sick. The means that I have had a sub for more than three days out of the past 6 days of school. Unfortunately, our district does not have any subs that know anything about math. So when I am gone I have to try to come up with stuff that my classes can do without any help at all. Most of the time I am largely unsuccessful at this. For today I had a short youtube video that my Algebra 2 classes were supposed to watch about graphing absolute value equations. Lame, I know, and they probably hated it, but what else can I do when the book doesn’t explain it well and the sub won’t know anything about it? Is it a sign of a good teacher or a bad teacher when my classes don’t get much accomplished when I am gone? I need some ideas about what I can do when I have a sub besides just review worksheets.
Really, isn’t a lot of what we teach in a typical high school math class just a lot of “shortcuts” to solving problems? Take graphing for example. In Algebra 1 kids learn about slope-intercept form and point-slope form and all the other little things that can make graphing a line quicker. They spend so much time on these shortcuts that by the end of it they don’t even remember what they are doing or how a graph connects to an equation (and maybe they never knew that to begin with). An even simpler example is absolute value. I’m not sure when students are first introduced to absolute value, but I’m going to assume that they are initially taught that the absolute value of a number is its distance from zero on the number line. Before too long they just remember the shortcut that “absolute value just means make the number positive.” I was shocked last week when I asked one of my pre-calculus classes what absolute value means and no one could tell me. All they knew was that you make the number positive. So we talked about the meaning of absolute value and then extended that meaning to distances between any two numbers. Once they understood absolute value in terms of distance, they had no trouble figuring out how to graph inequalities such as . (Yay, I just typed my first latex equation!)
One thing that I have been thinking about a lot lately is the fact that we so often teach kids the shortcuts or the quick way to figure out the answers. I don’t think there is anything wrong with doing this, but I am realizing that I need to make sure that they understand the concepts before they start learning the quick ways of doing it. For example, kids will have all sorts of problems solving absolute value equations and inequalities if they just see absolute value as “make the number positive.” On the other hand, once they understand what absolute value means, solve equations with absolute value seems obvious to them. So my challenge to myself for this year is to slow down and not be so quick to show them the “shortcuts.” Instead, give them time to understand the big ideas and concepts first.
This year I have just 6 kids in Calculus. This is my third year teaching these kids so I feel like I know them pretty well. So I don’t feel like I need to spend much time on the first day of school going over rules, procedures, etc. I just wanted to jump right into doing math. So I gave my students some of my kids’ toys and had them build something with them.
If you’re not sure what this is, it’s a collection of plastic track pieces that can be put together lots of different ways. Then you put a marble at the top and let it roll to the bottom. After the kids had built a simple track, I asked them how fast the marble rolled down the track. One kid immediately said that the speed was different at each point on the track and that we could find the marble’s average speed by finding the velocity at every point along the track and then averaging them all together. After a little discussion we decided that the ole distance/time equation might be easier. So they set to work on that. Then I asked them to find out how fast the marble was moving at a particular point on the track. They didn’t every hesitate, just immediately knew that they needed to find the average speed on an interval near that point. (I thought they might have to think about it for a little bit at least.) Before they started, I asked them to guess what it would be. After we had talked about the difference between average and instantaneous velocity (something that we will revisit more tomorrow), I asked them to graph the marbles height vs. time and the velocity vs. time. What I hoped they took away from this was a beginning understanding of using limits to find rates of change and some practice in graphing motion. The graphs that they came up with were not totally accurate but they generated some great discussion about the motion of the marble that I think will be useful later on in the year. Overall I was happy with the way the lesson turned out. Tomorrow we will be going into a little more detail about the limit process and looking at examples of numerical data as well as equations. But this activity gave me something that I can refer back to as we move into more abstract stuff.
When you make a blog, of course you have to have a name for it. Now I am not a creative/funny/witty person, so I started trying to put in a lot of simple, unoriginal names. Everything I tried was already taken. So I started thinking about all the math jokes and puns I have heard in the past. That, naturally, reminded me of a particular student I had in class last year. He could come up with a pun at the drop of a hat and he frequently gave us all a chuckle with his puns related to whatever we were learning that day. On one particular day another student asked “What is a pun?” (Yes, I know this has nothing to do with Pre-Calculus, but something we take a brief detour). The entire class proceeded to try to explain to this young woman what a pun is. After a few minutes everyone got back to work. Then a little later in class this punny student asked if she now understood what a pun is. She replied yes and he responded with “so you can secant you, what a pun is?” There were a lot of other puns during the year, some were probably better than this, but this one sticks out in my mind the most. So that’s what I went with for my blog name.
P.S. – If any Pre-Calc students from the 11-12 school year happen to read this, you should know that I also considered a name like ISawTheSign or ChangeOfAceOfBase 🙂
So I took a class this summer called Making Mathematics Reasoning Explicit (MMRE). The main idea behind the course is that students should be actively involved in learning math and in explaining (justifying) the things that they learn. Shortly after the class ended I was with my 5-year old son Ethan. We were getting tires put on my car, and while we waited he asked to play the “plus game.” Basically the plus game means that I give him a simple addition problem and he tells me the answer. Now I just want to say that I had nothing to do with inventing the plus game. Ethan and his older brother Connor came up with it and for some reason they love to play it whenever we need to pass some time. But playing the game with them has given me some fascinating insight into how young kids begin to develop math concepts. It has also made me realize that it is totally natural for kids to justify and give explanations for what they know. Here is how the game played out that day at the tire shop:
Me: What is 9 + 5?
Ethan: 14. You know how I knew that?
Ethan: Well, I know that 10 + 5 is 15. So 9 + 5 would have to be 14 because 9 comes right before 10.
Maybe I’m just a proud parent that likes to brag about my kids, but I was impressed that he was able to demonstrate that level of reasoning with addition (I might mention that Ethan has not even started kindergarten yet). But what really stuck me was the fact that he wanted to explain to me how he knew the answer. I have seen similar situations with Connor also. It seems like they just naturally want to explain and make sense of what they are learning. Contrast that with the typical high school student who just wants their teacher to tell them the answer so they can get their work done. Somewhere along the line they have lost that sense of excitement that comes from learning and figuring things out on their own. Somehow I would like to try to get them back to that place. I’m hoping that some of the ideas from the MMRE Institute will help me accomplish that.
Well this is the first post on my new blog. I was finally motivated to start my own blog after following the blogs of other math teachers who have a lot of wonderful ideas. One of them, Sam Shah, challenge people who were on the fence to start a blog and sign up for new bloggers initiation. Basically, the idea is that he will email those of us who sign up a prompt once a week for a month to help us get started on our blogs. So here I am, I we will see where I go from here.