# An Attempt at Mistake Analysis

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First of all, if you somehow read my blog, but not Andrew Stadel’s, stop here and subscribe to his blog.  He’s been doing awesome work with 3 Act videos, his estimation site, and recently analyzing math mistakes.  Then, as Mr. Stadel said in a recent blog, you gotta check out Michael Pershan’s Math Mistakes page as well.

Ok, so here’s my contribution to the mistakes conversation.  I had students do this activity (it’s a functions matching activity) that I got from Mrs Reilly’s blog.  It went pretty well and I was pleased with the results in general.  Then, as an extension I gave each group a whiteboard with a quadratic equation on it.

That’s it. No real instructions.

They naturally decided to make a table, a graph, and a sentence for their equation.  This ability to go right to work creating different representations is something that we’ve been working hard on in physics and algebra, and I was pretty proud of them for just doing it without explicit instructions.

The results in one of my classes were not so great though.  Their boards were riddled with mistakes and we ran out of time in class to do any real work on fixing them.

But the next day (different classes), I had a spur of the moment idea.  I had a hunch that this class was going to do a little better on them in general, so I thought “Why don’t I give them the same problems and then have them compare their boards to the boards with the mistakes?” And then that turned into the students writing notes to the other class to help them out.  Here’s the boards with mistakes and the notes (some are more helpful than others):

Reflections:

I love the idea, but it didn’t work exactly as planned.  The students were not really thinking about the specific mistakes made by their peers.  They were mostly just pointing out that something was wrong.

I like that the mistakes weren’t obvious necessarily, but that did make it harder for students to spot them.  I think I can build up to this activity by using the mistakes game more frequently in class (basically, they insert a mistake on purpose and the rest of the class tries to find it).  By getting good at finding these purposeful mistakes (which should be easier to spot), they should be better prepared to spot accidental mistakes and even misconceptions.

# Why Energy Pie Charts?

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Note: This is my first year teaching freshmen physics and it’s also my first year teaching physics using modeling.  Lots to learn!  I’ve had to remind myself of this a lot:

Anyways, this post is about energy.  On with it then:

ENERGY UNIT

So after discussing the various forms of energy and then giving the students some time to play with Phet’s energy skate park (with some guiding questions), we jumped right in to a lab.  The groups were each given an inclined plane and a cart.  We discussed what we could measure, and decided on height of the cart as the independent variable and velocity as the dependent variable.  The velocity should increase as the height increased right?  Let’s see!

Problem: Why are we interested in this relationship?

They didn’t have a clue, so the lab was a total disaster.  Some groups had issues collecting good data (that’s another problem for another blog post) and the rest had no idea why they had collected good data.  I think the reason is because they didn’t start the lab with a good question in mind.  What is the velocity vs height graph supposed to tell us about energy anyways?

Back to the Drawing Board

As I told my students today, it’s just not good science to run an experiment with no purpose in mind.  So where should we start then?  As always, the blogosphere has the answer.  I found this post by Kelly O’Shea.  Starting with pie charts? Sounds good to me.  Maybe that will help us gain a better understanding of energy transfers and then we can head back to the lab.

When I discussed this with my colleague who teaches the same class, he asked why I decided to go with pie charts at all (instead of going with bar graphs).  This comment from Kelly’s blog post shows others have had the same question:

You can check the comments on Kelly’s blog post for her answer.  But my answer was that I didn’t really know where else to start and Kelly’s lesson seemed to make sense.  But after doing the activity, I feel like I’ve got a much better answer.  First, here’s what we decided on for our three pies of a ball falling from a really tall building (I used Empire State):

The discussion that came up based on these pies centered around “what should the proportion of kinetic to potential energy be in the second pie?”.   That was a question that everyone seemed to want answered and no one had a solid answer to.  “Well,” I asked, “what would we need to know in order to determine that proportion?”

BAM!!

They asked those questions right away with exactly zero prompting from me.  All I did was write them down.  Then, it was super simple for them to see that right there on the board staring them in the face were the exact questions we were attempting to answer with our inclined plane set ups.  Now we have a purpose for our experiment!! We are trying to answer the question, “In what proportion are the gravitational and kinetic energies of the ball when it’s about halfway down?” (or perhaps anywhere on the way down? :).

Other Good Arguments That Came Up

In this whiteboard, you get a visual representation of the students arguing about two things.  First, should there be thermal energy in the first pie?  Someone had drawn it in and then someone else in the group erased it (after a board meeting with another group).  The first student stood by his original pie though.  His reasoning was sound, he explained that thermal energy is internal energy and that the bunny must have some internal energy to begin with because it’s got a certain temperature.  Then argument two is in the third pie, where once again students dueled over whether a wind up toy would be left with some elastic potential energy even after it stopped moving.  It seems the side of “nope” won out on that one.  This next group agreed on the third pie, but not the first:

Sorry, purple expo marker! You will probably not last long in the hands of this group.

# Real World Factoring

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I got asked today “is there a real world connection to factoring?”. I hadn’t heard that one in a while and I think it’s because I’ve been doing a good job of “starting at the end” like this awesome teacher. Factoring though, that one had me thinking I’d better just stick with what I know. So I showed them the area model for multiplying two binomials, then we worked that backwards and called it factoring. We even did some whiteboards earlier today. I told them a new student was coming and they would need to teach them the skills they just learned using only one whiteboard. Here’s what they came up with:

Not too bad. I also asked them plenty of questions to make sure each group member could explain what was on their board. And there were a few mistakes (some still in there), but that was ok because we fixed them as a group! So, clearly they’ve got this skill now. But the question still came up “is there a connection?”. Dammit.

I think that my techniques here were good because they can factor a trinomial now. And we also have made connections between quadratic equations and real world projectiles:

The constructed response portion of our last exam even had a real world connection:

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But where I failed was connecting these events.

So I think what I should’ve done was start the unit with a video of a projectile.  Then get the students asking questions like “how high will it go” or “where will it land”. Then when I show them factoring (using the same method I just did) they won’t ask “where’s the connection?”, they’ll just proceed to answer their own questions using this handy new skill they learned (plus maybe a few others).

# Algebra Physics Connections

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I just remembered today that I wrote a post about how the constant velocity lab in physics matched up nicely with the linear equations stuff in algebra, but then I never followed up on the results.  Here goes:

I started the unit on systems of equations with this problem.  It was awesome that they took to it so quickly.  With very little help from me, each group came up with a testable solution to the problem.  Some used the chart, some took it a step further and graphed the lines.  “Interesting that they were able to come up with such specific solutions then” you might say.  Well, what they did was approximate the solution with the graph or chart.  Then some groups used guess and check to find the number, and some groups used technology to find the intersection (Desmos or TI-84).

Next time, I need to make sure to let them test their solution with the actual cars.  That was the plan, but I never went back and did it, d’oh!

Anyways, in the past the sequence of learning happened this way:

1. Learn different techniques to solve a system of equations.  Graphing first, then substitution, then elimination.

2. Practice those techniques on a bunch of problems.

3. Apply the techniques to a “real world” problem.

Even as I type this, it seems like a logical way to do it.  You can’t apply a skill until you master the skill right?  I think that’s why this continues to be the way that so many math teachers teach this unit.  But my experience starting with the real world context showed me that they can and will find solutions to problems even if they didn’t start out with all the right skills.  There seemed to be two types of students in general:

1. Students who understood the problem already and came up with a solution fairly quickly.  These students should be allowed to use their class time for extending the problem into different contexts and doing whatever assessment I’ve planned for this skill so that they can move on or tutor other students.

2. Students who don’t really understand the problem or how to solve it.  I think that the standard way of teaching this unit tries to cater to these kids.  But this way is better for them too.  First, they benefit from just watching a student (#1) solve the problem (as they work in groups to create the whiteboards).  They also are (hopefully) intrinsically motivated to learn the skills because they see how they can be used to solve the problem. These students can use their class time learning the basic skills of using an equation to generate an input/output table, and then plotting those points to make a graph.

Note:  They were the ones that wrote those equations on the boards we started with.  In physics class (also with me), they started with just the cars, some plastic markers, and some questions.  The ladder of abstraction was climbed from there.

# Let’s Contextualize!

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In my Algebra class, I’ve made a big push this year to more often start with a context and then work our way to an equation, graph, or whatever more abstract representation of that context.  But for this activity, I decided to start with the abstraction and have the students invent the context.  Here’s the set up:

1.) Students are set up in groups of two or three.  Each group starts with either a table of values or a function:

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2.) The groups are told to create a whiteboard with the following:

• a table
• a graph
• a function
• a context

What I liked:

The students are asked to create multiple representations for the same function.  Sometimes I ask things like “make a graph from this table” or “write an equation from this graph” but stop short of having students create more than one new representation.  It’s nice for them to make the connections between the different representations in an activity like this.

Writing the context was something new for most of my students and they were confused by it.  This confusion led to good questions being asked and good conversations between students.  One group that started with the function wanted it to represent the growth of a plant over time.  But it’s got a negative slope, so that didn’t really make sense.  It was nice that they had the graph already on their whiteboard to help visualize this.

What I didn’t like:

I have been struggling with getting everyone to participate in group activities like this.  I think sometimes the students who don’t participate are thinking that they are just skipping something “extra” and therefore not missing anything important.  But really they’re missing the most important part of the learning process.

# 9th Graders Using Excel Makes Me Happy

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First, a few flip class reflections:

Keeping up with the videos is hard!  Sometimes I’m pressed for time and the video isn’t as great as I would like it to be.  Other times they kick ass, though.  I think it’ll take a few years, but eventually I’ll have a good collection and won’t have to make many new ones.

Videos don’t have to be lectures!  Check me out y’all.  I told the kids to work out these problems and only use the videos if they needed help getting started or to check their answers.  I think part of making this work is the fact that they know they’ll be responsible for the work during class.  It’s never just about the answer.  They have to present their work to the class (or small groups) pretty much every single day.

The assignment was simple.  I gave each group a different problem that could be solved using a system of equations.  They solve the problem, then whiteboard it (the solution and how they found it).  The results were awesome!  Some groups graphed the equations, some groups used substitution, and some groups used elimination.  If any of them were wondering why we need to learn different ways to solve these problems…

But something interesting happened when several groups used tables to solve their problems.  Here’s what they looked like:

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So, even though it wasn’t in my lesson plan for the day, I decided to show them how to generate a list like theirs using excel.  I’ve been itchin to teach some excel and this was a perfect opportunity.  We generated two lists for the cost of different cell phone plans based on the number of minutes used.  Here’s what it looked like:

Anyone have any good lessons using excel in Algebra?

# Algebra and Physics are like Red Beans and Rice

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This is my first year getting to teach Algebra I and (freshman) Physics classes to basically the same group of kids.  Seriously, I thought it would take at least three years or so before the curriculum would be properly synced up.  But I don’t think I could separate them if I tried!  Everything just seems to be matching up.  For example, we just finished up our constant velocity lab in physics.  They had some little cars and they modeled the motion:

Well, in Algebra our next unit is systems of equations.  So, I want to start off with a real world problem that can be solved using a system.  How about: If we put group 1’s car on the same path as group 3’s car, when will they crash? (We can also do some where one car will catch up with another or they will never meet, etc)

The real awesomeness here is that each group already made a graph and an equation to model their car’s motion.  We still get to start with a great context, and the prep work is already done.

I’m not sure which is red beans and which is rice on this one.  (I’m thinking physics is the red beans this time!)

# Intro to Whiteboarding!

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Ok, so I’ll be using whiteboards extensively this year in Algebra and Physics (9th grade).  Not the little whiteboards (I been usin’ those man), but 24″ x 32″ poster size ones.  The idea came from a kickass workshop I attended this summer on teaching using models (not that kind of models).  Go here for info on the modeling method!  Here’s an example of one I did at the workshop:

Ours were the best boards there! In your face other groups!

So, to get the students used to the process of whiteboarding their learnings, I figured I would need a good intro activity.  Here’s what I was thinking (for my Algebra class, I already have one for Physics):

set-up = one board per three students, everyone has a marker

1. “Think of two numbers that add up to 20, everyone write down a set”

2. I’ll call for a few random ones and write em on the board

3. “Am I missing any”  “Is there an easier way to complete this list?” GO! (as in “go make a complete list”)

4. circle up whiteboards: (at least one group should have a complete list with two columns)
-“Why did you set it up that way?”
-“Do you agree?”
-“Whose labels are the best?”
-“Do we need labels? Why?”
-“Do you agree?”

5. “Is anything missing now?”  (maybe prompt for (0, 20) or (10.5, 9.5)

6. “Describe the relationship in words. Does one column depend on the other?” GO!

7. circle up whiteboards:
-“Are all the descriptions the same?”
-“Are any wrong?”
-“What might be a simpler way to write this?” GO! (the hope is they will come up with x + y = 20 or something similar)

8.  “Now we’ve got a table, a verbal description, and an algebraic description.  What other ways can we represent this?”  (prompt for graph if they don’t say it, then create graphs in groups) GO!

9.  circle up whiteboards:
-“Are all the graphs the same?”
-“Are any wrong?”

10.  “Let’s put it all together” GO! (table, verbal, algebraic, graph)

11. circle up whiteboards:
-“Where did we begin?”
-“What was the point?”
-“What’s the title of this activity?”   BTW: A bunch of my questions were inspired by watching the teacher in this Dan Meyer post.

My thinking is that since this is such a simple example, we should be able to come up with these four representations fairly quickly.  For now, the point is fitting them on the whiteboard, we’ll get to doing some real math soon enough.

What do y’all think?