Figuring out How to Present a Lesson

      I often know what topic I’m going to teach and have an idea of how to present it, then before class starts I’ll change my mind a couple of times about how I want to present it. For example, today I am teaching function operations with 7^th hour Algebra II. I made these function dice so we could have some fun while adding and subtracting functions. But today I’m feeling a little low on energy and don’t want to stand up in front of the class doing all the work at the board, so I thought I would get them to present. I’m sitting here thinking about it, and I’m not sure how I would get that to work. So then I thought I might split them up into four groups and have them each study an operation and present it to the class. But then we don’t get to use the function dice, and if they used them in their groups it wouldn’t be as exciting as the whole class using them together. Then I thought about Mr. Turner’s idea of showing them what happens to the combined function on a graphing calculator. How do I incorporate that? Now I’m thinking I’ll have students roll the dice to get the functions, have one student scribe or solve the combined function on the board, and then have them all enter the three functions into the calculators and talk about what is happening. I have to make sure I do some functions with different domains, though, so we can talk about how domains affect function operations. (The function dice are all quadratic or linear functions.) Then with dividing, we can really talk about how domain matters when you have a fraction, and especially when you cancel expression on the top and the bottom. I think this could work, but it’s a couple of elements going on at once, and I’m not sure if it will all come together or get confusing. I still have a hard time remembering what to do when and how to hold it all together. I think the function dice and the student writing the combined function on the board will be the minor parts, and us talking about what’s going on in the calculator will be the important part, especially in terms of domain. With all of this, I don’t do any of my original presentation, which was to show that if you solve each function for a number, say f(3) and g(3) and add these two together, you get the same thing as if you add the functions first and then solve for (f+g)(3). This is also interesting. I had a worksheet on it, but I don’t think it got across what I really wanted it to. It’s interesting that there are a couple of ways of presenting the topic, and I really don’t know which is best. It would be great to touch on all of them so students could see some connections, but there’s not time for that. And sometimes it gets confusing when you present multiple representations at once. (Although it is important to try to use different modes so that students have different access points to the lesson.) I am not planning on using the calculators with 4^th period. We will definitely do the function dice, and I might try to show them how plugging in a number works for both the two original functions and the combined one.

Update:
We did the lesson as planned, and it went pretty smoothly. The kids definitely liked throwing the dice. I figured they would roll them on their desk, but they're a little big for that, so they threw them at the board and let them bounce off. We went pretty quickly through how to add, subtract, multiply, and divide functions. I asked them if they felt comfortable with what we had done, and they said yes. I hope that they got it. I feel like they did, but I wonder how well it really sunk in and how much of it will stick. Once we finished the basic functions, we had more time, so I went ahead with the graphing calculators. Although we didn't go in depth with what was happening, I think it was good for them to see the two original functions and the composite function, whether it was added or subtracted. So overall, everything I had planned fit into the lesson, which was great! And we had fun with the dice, too.


On a side note, two of my kids were so excited to come to math club today that they skipped chess club! I am amazed that they are so excited by math, and I love blowing their minds with things they've never thought of before. The looks on their faces are priceless.

Assessments

Ever since Daniel Schneider brought up the issue of assessing students in this blog post and subsequent posts, it's been big news in the Blogotwittersphere. This is definitely an interesting topic. I have only created one quiz so far while student teaching, and while I do formative (informal) assessments along the way, I don't have much experience with creating summative assessments (those are like quizzes and tests, anything that sums up what students have learned in a formal way). I just wrote my first test for my Algebra II class which they will take next week and we will see how that goes.

The thing is that summative assessments are super important! They send a strong message to students about what you think is important and what you value. You may have been doing cool projects all along, and then on the test you expect students to do procedural type problems. That would be a confusing message. I really have no clue how to align what I value in learning math with my assessments. In fact, I'm still struggling to communicate on a day-to-day basis in class what I want students to learn. There are many aspects of math I want to spend more time on -- non-traditional things that I think will pull in some students who up to this point have hated math and think they're terrible at it. Exploration and discovery-type things. However, I am finding there is a huge time crunch -- too much material to get through in too little time. I've had the discussion with one of my CT's about how the state is pushing down so much more information on students than in the past and there's no way to cover in all -- except in a speedy and sloppy fashion. I could go on here, but I'd like to get back to the question of assessments.

Having to write a test for the Algebra II class is really making me reflect on how I have taught the chapter, and how well I have taught it. There are many many things I would like to improve. It's too late to get into more of this, I just wanted to write down some quick thoughts, which maybe I can expand on later.

Park Math

I posted a couple of days ago about a problem I used to introduce the idea of converting from radical expressions to expressions with rational exponents. I found the question posted on Michael Pershan's blog, and he referenced Park Math. When I looked up Park Math I found that they have a whole curriculum that they developed and are willing to share. I emailed them and received the curriculum today. I'm excited to take a look at it and see what I can use in my own classroom!

Student Engagement

I’m starting to become aware of different ways of engaging students. At Purdue, the math education department encourages us to follow a method called Launch Explore Summary (LES) where the teacher launches the lesson with a hook or something that will lead into the topic. Then the teacher gives students an activity to explore to the topic. It could be a worksheet, a hands-on activity, work on a computer or calculator, or something else. The point is for students to “get messy with the math.” The teacher’s job is to circulate around the room and answer questions and guide students. At the end of the lesson, the class summarizes what they’ve learned. Often students will present their work to the class, or the teacher will help the students summarize their learning. As a student teacher I’ve found it difficult to implement this method for a number of reasons. One of them is that I can’t come up with a way for students to explore certain topics, like simplifying radicals. Another is that there just isn’t enough time. When I do have students thinking about more conceptual topics or playing around with a topic, it takes much longer than my CT’s lesson on the same subject. I don’t know how I could get through all of the standards using LES most days. Perhaps it is possible, but I just haven’t found a way. 

So in an effort to engage students more in the lesson, I am trying to use different methods of teaching and learning to mix things up. For example, yesterday I presented a new topic and demonstrated an example on the board. Then I asked them to get into groups of three or four and I gave each group one problem to work out. Once they worked it out, they had to write it up on the board and present it. Other methods I’ve used it having students use mini white boards to practice procedural problems; for some reason they are more engaged using white boards than pencil and paper. Another of my CT’s uses centers or stations for review days. She’ll set up the desks in groups and have a set of problems at each station. She places students in groups and they work the problems at each statement for about 7 minutes, and then they move to the next station. It’s more social and gets kids moving around. I’ve also had students practice simplifying radicals in a Row Game. (See earlier post for the worksheet.) 

Using an LES, discovery, inquiry, Three-Act Math, or similar approach not only engages students but get them thinking more deeply about math and in different ways. They learn critical thinking and problem solving. They learn to see and utilize patterns, to make connections, and to utilize resources. These are all good mathematical habits. However, I can’t always find a way to implement these approaches with every topic I’m teaching, and I don’t know how to make the time for them and still cover all the standards. I was starting to worry that I’m turning into a teacher that only lectures. Then I realized that there are other ways to engage students and make things interesting without having to plan an exciting and deep lesson every day. While I would like to be doing more non-traditional lessons, I can still help students learn math and keep them interested by just mixing up my presentation styles and the way they practice problems. Realizing that has taken off some of the pressure I was feeling about not being an innovative and creative teacher every day. I can do my best every day with the time and energy I have, and then try to making small changes one at a time. I’m hoping that there will be more opportunity for interesting math problems in the next chapter on exponential and logarithmic functions.

 

Losing Steam

I’ve only been student teaching for a little over three weeks and I’ve already reverted to traditional teaching methods. It’s not like I did anything crazy at first, but I did try different strategies. I used powerpoint, then worksheets, I had students work on mini white boards in pairs, I used think-pair-share, and I made them answer conceptual questions and think about problems without much background information. Since they’ve taken their quiz in the middle of last week and I realized how little they learned, and since they had a requiz that took up two days, I feel like I have been spending all my time trying to get them back on track. I have an expectation that I will get them through one section every two days. That is the guideline that my CT (cooperating teacher) gave to me at the beginning of student teaching when I asked him about pacing. While he has not held me to it or said anything critical about my pacing, I feel pressure to get through as much material as possible. In order to catch them up to my CT’s class (which is really only one day ahead), I’m lecturing, assigning more homework, and letting them work on homework in class so they get practice while there is a teacher on hand to answer questions. I find it extremely boring and life-sucking. It’s often what students expect in math, and they respond to it well enough, but I don’t enjoy it at all.

Writing about it makes me realize that the main thing keeping me from trying new teaching strategies is that I’m afraid my students will get behind. I want to prove myself, and to me that means keeping my students on the same time schedule as my CT’s class. I don’t have any solutions, but noticing that I’m in the way of delivering creative lessons is a big lesson in itself.

Radical Expressions to Rational Exponents

My first unit during student teaching has been on radical expressions. I am following the textbook closely, but am trying to challenge the students with conceptual questions in each section, not just teach them how to do the procedures.

I found a reference to a Park Math problem (Book 5, p 66; and I will reference the blogger who mentioned it when I find him/her: Michael Pershan) that helps students think about the relationship between radical expressions and expressions with rational exponents. I tweaked it and wrote up the following two questions that I posed to my Algebra II classes: 

If x ∙ x = x^N, what is N?
If x^a ∙ x^a = x, what is a?

I let them sit on it for a couple of minutes, and I asked them not to shout out the answer. Within about three minutes in one of my classes a student had it. Still the other students sat thinking. I gave them another minute, and then I asked the student to tell the rest of the class what he got and how. He said a = .5 because .5 + .5 = 1. The thing that surprised me about it all is that the student who first got it normally struggles in my class. He sits in the back by himself and pays attention, but it takes him a little while to catch on. I know he can get it because when I work with him individually he'll get it after I explain it. I love that he the one to figure this problem out. It gave him a change to shine.

Funny enough, the same thing happened in my other Algebra II class. A girl who struggles with math even more than the student mentioned above figured out that a = 1/2. Her explanation: "It's common sense."

This little event reminded me that one of my goals for teaching math is to show students that there is more than one kind of mathematical thinking. There is doing all the procedures - and those kids know if they're good at this because they've been doing it their whole math career - and then there are a whole slew of other skills, like reasoning, estimating, thinking abstractly, puzzling, finding resources, visualizing, solving a simpler problem to help you figure out a harder one, etc. I believe that every person has a mathematical sense, it's just not utilized or made known to them. Most students think math is a set of rules because that's how it's often presented, and that's how I've been presenting it, too. How can I make math seem coherent, logical, and useful to students? That's a challenge I expect will take me a career to figure out.

Radicals Row Games

I created a Radicals Row Games based off of Kate Nowak's original idea.

This one includes a Sum column where students add their answers together to get what's in the Sums column. It was too hard to come up with radical expressions that simplify to the same thing, so I just added up the two answers and created a column for it. I like having the Sum column because it allows students to work independently and check their answers without coming to me to ask if they got it right. 

Radicals Worksheet

Here's a worksheet I created on Multiplying Binomial Radical Expressions, Conjugates, and Rationalizing Binomial Radical Denominators

I like it because it leads students through the steps of multiplying binomial radical expressions (reminds them of FOIL) and then uses the idea of conjugates to get them to think about how to rationalize a binomial radical denominator. Most of the students filled it out without my help until they got to the last problem.