Confessions of a 13 Year Veteran

Ok, I admit it.  My lesson planning skills suck!  It just hit me the other day. I am reading all your blogs and seeing all the cool things you are doing in your classrooms, and all I can ask myself is, “Why don’t I think of stuff like that?” 

 I think I know why.  I cut my teeth on CPM (another post on CPM coming soon) and all of my lessons were pre packaged.  All I needed to learn was to ask good questions and get out of the way.  That suited my style very well.  By the time NCLB came around and we dropped CPM, my school was already going lock step with pacing guides and common assessments.  Neither one of these approaches allowed for much innovation nor did they require a bunch of thought.  It didn’t help that I became the varsity baseball coach in my first full year of teaching and spent way more time planning practice than I did lessons.  The real problem was that I didn’t really know how much I had bitten off until recently. 

Now that I am teaching middle school math as the high school guy brought in to handle all the GATE kids, I am not only responsible for my own classes but for setting a tone and pace for an entire department.  I feel more accountability now than ever before.  It has been in these last three years that I have really started to ask reflective questions regarding not only my practice, but good practice in general.  I also have access to tools that I didn’t even know existed when I was at the high school. 

So here is a snapshot of where my lessons were compared to where they are.  Feel free to take your shots and help me make this better.  This lesson is stolen  borrowed adapted from a lesson in the April 2009 issue of Mathematics Teaching in the Middle School. 

What I would have done before:

I don’t know if I would have even bothered to re type the lesson, I may have just photo copied it and given it like this:


I would have passed out this handout and told the students to represent each savings plan as a: graph, input/output table, algebraic expression and verbal expression.  My students would have muddled through it and most would have just jumped through another hoop. 

What I did today:

I toggled between these four slides and asked students to pick out what information they thought was important.  It was interesting that some students felt it necessary to try to copy the information while the rest of the class was willing to observe and jot down what they thought was essential info.



Questions students came up with:

  • Does the chart represent the amount of money Diana makes each week or the total she has saved?
  • Will she continue to save at this rate? “Yeah, look at the “dot, dot, dot.”


Teacher: “What can you tell me about Yoni?”

Class: “She has $300.”

Teacher: “How much will she have in 3 weeks? 2 years? 100 years?

Class: “$300. $300. $300.”


A few students were quick to point out that we only needed to focus on two or three key points in order to make a generalization of Michael’s situation.  They immediately zoomed in on (0, 30) and (10, 80).  However, I did have one group who decided to focus on the last point (20, 130). They decided that Michael averaged $6.50 per week.  But when they were asked if their trend would continue, they quickly realized that they had failed to consider the original amount of Michael’s savings. savings-plans-multiple-representations_5

Chandler, one of my 7th graders decided to ask, “Does x represent weeks?”  Beautiful, that info was on the next slide. 

The question of the day had to come from Paul: “So, if the only point we know for sure in Michael’s graph is (30, 80) wouldn’t he have more money if he actually started at (0,0)?  In his mind he was seeing this…


…which led to an interesting discussion on how slope represents the amount of money saved per week. 

My students are starting to catch on to how things work in my class because as soon as these slides came up:savings-plans-multiple-representations_7



I asked, “What is my next question?” To which they all responded:


Once they had made the decision on which represenation they prefered, I had them represent the other three savings plans the same way.

At first the justifications for “why” one representation was preferred over another were weak at best, lazy and unthoughtful at worst:

  • An input/output table is more organized.
  • It’s easier to see the data.
  • I like it better because it is better and less worse than the others.

Okay, that last one was mine, but you get the point.  Being the father of four boys, I can smell an “alright get off my back, already” answer a mile away, I continued to prod.  Eventually we had answers like this:

  • I prefer the chart because it is easier to see which numbers go together; on the graph, you have to work a little more to see which numbers are related.
  • I prefer the graph because I can actually see how the different points relate.  The slope helps me see how fast someone is saving.
  • I prefer the verbal expression because it helps me understand the overall situation better.

We eventually got to the discussion on “Who has the most money right now?”

Hands fly upright around the same time I hear: “Yoni! Danny!” 

Then Adrian pipes up, “When is now?”

Gotta love that kid!


6 Responses to “Confessions of a 13 Year Veteran”

  1. 1 Nick Hershman April 16, 2009 at 2:46 am

    I have one easy suggestion, don’t do blue type on a progressively blue-er background. It gets hard to see at the bottom. Pick colors that contrast well, it should be easy to read.

    Definitely seems more engaging. I was busting butt to do presentations at the beginning of the year, but I like GeoGebra sooooooooo much. You can have a pre-pared file, or start from scratch, it might have made your student’s point about saving from (0,0) easy to show with precision in class. I found it really useful for having kids do some graphing today – having them come up to the computer and click the two points they found on their graph and having geogebra plot the line. Having seen your farm geogebra, you know it way better than I do, why not use it instead of the graph slide?

  2. 2 David Cox April 16, 2009 at 3:10 am

    I noticed the blue problem when I posted the slides, but for some reason there is plenty of contrast in my classroom. I have skylights right over my SMART Board so using a white background is pretty glaring.

    I just started using GeoGebra this year and Dr. Fahlberg-Stojanovska has been a great resource. I love that program as well. You might be right about having the GeoGebra fired up and ready to go. Do you think that it would hurt the flow if I went from Smart to GeoGebra?

  3. 3 Nick April 16, 2009 at 5:16 am

    I do it quite a bit. But it does tie you to the computer. I’ve started sending kids up to do stuff on GeoGebra. Anything they did on paper if it involves graphing should be pretty easy for them to transfer to GeoGebra with just some basic instructions from you on the tool etc. I was almost even wondering why not make a static graph image in geogebra. Cheers.

  4. 4 David Cox April 17, 2009 at 1:33 am

    Last year, I started using a wireless tablet with my SMARTBoard and I really enjoy that. It allows me to do a lot of the things at the board, but I can move around. It is kind of the best of both worlds.

  5. 5 samjshah April 18, 2009 at 6:57 pm


    Three points.

    One. I asked my tech people about getting a tablet computer hookedup to the smartboard, and they said that it wasn’t possible. Is this what you have… not a real computer tablet but this board:

    Two. Thanks for putting your work out there for discussion.

    Three. It sounds like this was a really productive lesson. My favorite part is the discussion at the end of which representation is better, and why. I’m still in my second year of teaching and I’m at the point where I am making handouts like your first handout. And I think that a really good discussion can be had from your handout, honestly. Do you think your students wouldn’t have left class with an understanding that info can be represented in 4 ways and some ways are better than others depending on the situation? I guess my question to you is: what do you think they gained by this new way of presenting, that would wouldn’t have been able to have from the handout?

    My thoughts.

    I don’t think I get Yoni’s slide at all. Is that a chart, graph, statement, or expression? How are the kids supposed to know it represents that she has $300 in savings for all time. I was like “Yoni spent $300 at the grocery store” and “Yoni found $300 in her jeans” and “Yoni has a salary of $300.” Do they just invent the situation and then you as a class go with it? I had the same problem with Diana’s slide. Is this how much Diana spent on lottery tickets each week? How much she saved?

    Building off of that, it is this lack of context which I suspect you like — because it gets kids to ask questions about what information they need in order to interpret — which makes me wonder if that’s the best way to get the points across that: (1) information can be represented in four different ways and (2) sometimes one way is better than the others. It seems like a distraction. I know dy/dan will probably read this and cry out “LESS IS MORE!” But at least to me it seems like a distraction.

    Writing “Amount in bank” on Diana’s chart wouldn’t mean that you couldn’t have a discussion about how the numbers are reasonable for the amount she might have in her bank (it isn’t in the trillions), the fact that each week she’s putting in the same amount, etc.

    I think something to consider, after the class discussion is over, is to either have a bunch of crazy data/situations represented in all four ways, and then have students discuss for those data which would be better. They’ll find that the way we like to represent information in one situation won’t be the most useful in others. (Or have your students try to think of some data/situation which is better represented in a way different from the way they preferred in the class example.)

    Again, thanks for sharing! As I said, I’m still working on the handout/simple smartboard phase, so take my thoughts/questions with a mountain of salt.

  6. 6 David Cox April 20, 2009 at 3:59 am

    1) I don’t have an AirLiner, but I have something similar. It is a tablet made by Qomo and I can’t recommend the wireless tablets high enough. The nice thing about the AirLiner is that it is a Smart product so you can legally use the SMART software with it. IMO, the strength is in the software, not the IWB.

    2) It is my pleasure to put the things I am doing out there. I appreciate you for the same thing. My only regret is that I didn’t find all of these blogs sooner.

    3)I think all of your points are good ones. Yeah, my students probably would have been fine if I left the worksheet alone. But one of the things I have been working on lately is getting students to actually engage in the problem and not just jump through the hoop. But, you are right, the primary objective of recognizing multiple representations and judging the “best way” to represent a situation would probably have been met using the worksheet. I think Yoni’s slide would have been a problem, if I would have put it up and left the room. I like the fact that I can deconstruct a “word” problem and offer information as they ask and/or earn it. Too often, kids take information like the $300 on Yoni’s slide and impose their point of view and assume they know the context when they really don’t. You see it in Geometry when a kid wants to assume a right angle just because “it looks like it” in the picture. I am trying to blast away at that nasty habit. So I figured that I could either give them all the info at once or a little at a time and have them wrestle with what it all means.

    The question I ultimately have to answer is whether or not this deconstruction was more effective than it would have been had I just offered the worksheet and led them through it. I would have to say yes–if for no other reason than the fact that I didn’t have students jumping ahead and trying to get the problem over with because I held the information they needed to get the job done.

    Hey, don’t feel like you have to offer a mound of salt with your comments. You are asking questions in your second year that I didn’t know existed until a couple of years ago. Keep ’em coming.

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