I have fallen in love with the function-approach to algebra, and if I were teaching algebra 1, or even below that (as the CCSSM allows), everything would be taught from the perspective of a function.

Starting with functions from day one, you can reinforce everything about the functions, domains, ranges, families, etc with each and every lesson (just about!).

operations on expressions? Yep, start with a function that has the expressions you want. You can get so much more out of it. This can lead to so much more understanding. In class right now, students are dealing with rational functions in the form f(x)=a/(x-h) +k, finding the characteristics, then rewriting it as a quotient of two polynomials and comparing the graphs. Good stuff, and can be done with every type of function. I think it gives meaning to algebraic manipulation, why do we do it, what happens when we do it, and how does it really change the original function. These are powerful questions whose answers give real depth and meaning to the algebra classes.

I went through the EMATHS training, but the classes I taught never really let me explore the shifts that strongly (namely statistics and AP Calculus). I had a feeling that the function-based approach was, in my opinion, stronger and more flexible than the other way, but since I never had a chance to really explore it, it wasn’t cemented.

It is now. Everything that can be done in a math class can be done from a function perspective and so much more. I will never go back!

There have been questions about learning procedures, the rigor of some of the algebraic skills like “simplifying” expressions, applying properties, etc. It’s all there, all of it. It’s just a different way of getting there.

I will leave this with a story. Since we were converting a rational function from one form to another, and to get the students thinking about what is going on, I took a piece of paper and told them this was a function. We can do a lot with a function in this form, and we came up with a lot of uses for it. Then I crumpled up the paper and asked how I changed the paper. Did I add anything? Did I take anything away? Is it still that original piece of paper, just dressed up differently? Then I asked about the uses of my crumpled up piece of paper. Can I do the same things? Can I do different things? I highlighted the different uses of the wad of paper by playfully tossing it around the room. When we discuss this as a group, hopefully we will get to the point that even if a function is in a different form, it really hasn’t changed (check the graphs), but the uses for it have.

Wouldn’t it be great if that idea can be reinforced almost all the time in Algebra 1 and Geometry, and not just at the end of the 3rd unit of Algebra 2?

Thankfully, the CCSSM is all about the function in high school!

Hi all,

The last couple of days have been heavy on the direct instruction. I saw that this group wasn’t getting too much out of working in groups, and since the upcoming topics (logs, log properties, solving exponential and log equations) could be approached a little more traditional, I made the command decision to focus on small chunks of information and practice. So far, it’s working pretty well.

These students are solving the equations graphically and then algebraically, which is all I can ask for right now. They are getting more and more comfortable with the tech and with the content, which will lead to next week and more group work on some exploration activities. They are making pretty good progress!

The room is filling up with a ton of chart paper: formula walls, word walls, procedure walls, Nspire shortcut walls…running out of walls!

Hi all (anyone that is taking time from their summer vacation to read this!)

I am one week in, and I have learned quite a bit of interesting things. I’ll give a brief synopsis of the fun so far:

1. Set-up
Setting up the technology was a bit of an issue. The biggest thing was getting the Ti-Navigator set up and running. I had to upgrade the operating systems first and do a lot of the hardware upgrades. Not a big deal, and not unexpected, but it took a few days to get them started. Finally, on Wednesday, I got the bugs worked out of the navigator, the class set up, and everything working. More on that later!

2. The Teaching
Even being out of the classroom for only a year made the first couple of days a bigger challenge than I anticipated. My last classroom experience was a class I taught at Macomb CC, which is a little different than teaching in high school. I found myself falling back into that routine a little bit more than the high school routine. It took a couple of days for me, but I hit a groove during the day on Thursday.

3. The Students
It took a while to get them into the swing of things as well. Summer school is an interesting experience, and can be a very dry, extremely oppressive atmosphere (in my opinion) for the students. Since it is credit recovery, they are there because of past failures, and their attitudes about math and math classes are very negative. Put them in a room that is high stakes, for five hours a day, and this could lead to a very negative atmosphere. So far, though, the students have been really good and have been participating, especially in the last couple of days. In fact, they are very engaged, and I think I’ve found the reason for it:

The Navigator System.

Once I got this thing up and running, and once I got used to it (it took me a couple of days, and I think the class slowed a bit because of it), things started to roll! For example, yesterday we were dealing with exponential functions. We had a whole class discussion of what exponential growth was (I use The Matrix as an example, when Agent Smith, the bad program, develops the power to copy himself, and we discuss how long it takes him to take over a world with 8 billion people in it). With some guidance on my part, we developed the function f(x)=1*2^x, and what all the parts of the function were. I expanded it to f(x)= initial value * (growth factor)^x. We worked through some examples of finding the parts of the function, the initial value and growth factor. ( f(x)=a*(1+r)^x )

Instead of the typical “give the students a couple of minutes, let the brave one give the answer and move on,” way of trying to tell if the students get the answer, I sent out a quick poll. It is amazing that when there is an accountability on the screen showing how many people have or haven’t answered, how fast the answers come in.

Every student had an answer, and we had a discussion about the answers, about every student’s answers. There were about 3 misconceptions that came up that we were able to address immediately. Every student saw their answer on the screen, and I was able to give instantaneous feedback, and students got instantaneous feedback.

This is a form of formative assessment (FA), and I’ve used it a lot over the last couple of days. It is amazing how effective FA’s are at clearing up misconceptions fast and moving the class forward. I was trying to use them before I left the classroom, but I don’t think I fully “got it” then. I get it now. Granted, I have a great tool to use, the navigator, and not all the math teachers have them. There are other ways to get them going. More on that in the future.

I am rambling, too much coffee. I will discuss more of my views and experiences as we go through the summer.

Highlights:
—I love the technology!
—Formative Assessments are a huge key to moving a class forward. (and you don’t need the tech to do it, you just need some creativity)
—Get the students engaged, working, and hold them to it!
—These students, who hate math, are showing improvement!
—Give them hope that they can do it, give them tasks that they can succeed on, and build on that success.

Rick is an excellent source on Differentiated Instruction. I was fortunate to see him give a few sessions at the DI conference in Chicago. Give him a try.

Here is an interesting point of view on giving 0′s on a 100 point scale.

Do you agree or disagree?

How would you explain that what you do in your classroom is the best practice for your students?

just some things to think about…