Sunday, July 26, 2015

My Grudge with "Grudge"

I'm flying home from Twitter Math Camp near Los Angeles, and after successfully figuring out how to steal the airplane's wifi, I decided to write another post. This is what I do. I go to a conference, get inspired to contribute to the MTBoS community, and write a blog post. You must understand that once I get home, all motivation to do such a thing will be lost. That's what Netflix would like me to believe anyway.

There is one contribution I've made to the online community that has received a lot of good feedback from students and other teachers. This is a game called Grudge. I gave a survey to my students at the end of this year and asked them what were their favorite things were from my class. Grudge was near the top of the list. ("Mr. Kraft" was at the very top of the list, of course.) 

There is no question in my mind that it is a review game that engages almost all of my students almost all of the time. I also feel that I present it in such a way that students seriously consider their answers and are eager to understand their mistakes. But there is a problem with the game. On occasion, students will team up on other students, and while it is not always expressed, I do believe that feelings can be hurt. As Matt Vaudrey once expressed in a tweet, it hurts the class culture. It promotes competition instead of collaboration.

I've learned that any activity I use in my class should not only be engaging and promote academic growth, but should also encourage students to be respectful to one another.

Sunday, April 19, 2015

What the hell is mean absolute deviation?

When I first started looking at the Common Core standards for sixth grade a couple of years ago, admittedly, there was one standard I had to do a double-take on:

6.SP.B.5.C: Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

And, like many of my colleagues, I thought, "What the hell is mean absolute deviation?" My horror was confirmed when I googled it and saw how complicated it would likely be for my students.

Looking in some textbooks and online resources, I was continually left wondering why my students would even care about mean absolute deviation. I mean, you do all of these steps, you get a number, and then what? What does mean absolute deviation tell you?

I figured that the only way my students are going to have any access to this would be to compare different data sets, make a quick judgement about which one has more variability (which can be very subjective) and find some way of quantifying that variability. On top of that, I wanted my students to create their own data where the goal would be to have the least amount of variability.

I then remembered the "Best Triangle" activity I did with Dan Meyer. In this activity, Dan asked four teachers to draw their best equilateral triangles. (Notice that Andrew and I have points in our nostrils.)

Rather than having the students evaluate the teachers' triangles, I had them create their own. I started the lesson off by asking the students to draw, what was in their mind, the perfect triangle. Immediately, there were several hands that shot up from students who wanted some clarification, but I told them to just do what they thought was best. After a quick walk-around and throwing some random triangles up on the document camera, it seemed that almost everyone was trying to draw an equilateral triangle. A few students argued that a right triangle could be considered a perfect triangle and I admitted that my instructions were very vague and their interpretations were justified.

We then brainstormed all the things we should look for in the perfect equilateral triangle. Students agreed that we needed three equal sides and three equal angles. They then made a second attempt on the whiteboards to draw perfect equilateral triangles. I asked everyone to make a quick judgement about which triangles they thought were the best, but soon ran out of time for the day. After the students left, I quickly took pictures of their triangles and took measurements in millimeters. (Admittedly, this is something I would have preferred having the students do on their own, but my class time is unbelievably short...37 minutes.)

The next day, I told my students that I took those measurements and found a way to rank all of the triangles from all of my classes. Next, I showed them the five triangles which represent the minimum (best), first quartile, second quartile, third quartile, and maximum (worst) of the data (in order below). This was a nice way to show a sample of the triangles as my students had just finished learning about box-and-whisker plots.

When I first showed them these triangles, I asked them to figure out which triangle represented the maximum and the third quartile. The other three triangles were not easily identified, however, we noticed that if you reorient the triangles so that one of the other two sides was on the bottom, the inferior triangles no longer looked equilateral (leaning to the left or right).

I explained that ranking these five different triangles didn't provide too much difficulty, but I was confused how to rank triangles that looked very similar. I gave the three following triangles as an example and had students vote on which one they believed looked the best:

In each class, there was a lot of disagreement about which triangle was the best, and more often than not, the majority picked the wrong one. I then provided the side lengths of each triangle (above in millimeters) and asked the students, "how can we use these measurements to rank these three triangles?"

After a few unproductive guesses, someone usually asked to find the differences between the measurements, which lead to someone else asking to find the sum of those differences or the range. They notice that the ranges for each triangle are all 20 mm. Someone usually calls me out for doing this intentionally...which I did.

Next, somebody will ask about the mean of the numbers. I act dumb, as I do with every suggestion, and we explore that possibility. We find the means, and it would seem that we have again hit a dead end.

I have say that at this point, some classes were completely stuck, and some kept going with it. For those that were stuck, I told them that to me, the mean (157 mm for the first triangle) represented the side length that the triangle drawer had intended for each side, but sometimes he or she fell a little short of that goal (149 mm), or overshot it (169 mm). I then asked them to compare each drawn side to "the perfect side length". We found the differences of each length and the mean, and soon after, someone suggested finding the sum of those differences.

At this point, most of my classes were satisfied that we found a method of comparing the triangles. We just had to look at the sum of the differences from the mean. The best triangle was the triangle that had the lowest sum. A couple of classes even went one step further to find the mean of those differences. In reality, there was nothing wrong with either of those methods. However, the second method WAS THE MEAN ABSOLUTE DEVIATION!!! When I first started planning this lesson, never did I think my students would intuitively come up with this concept.

This was the first time I've taught this lesson and I realize that there was a lot more I could have done with it. Given more time, I could have had students work in groups to come up with their own methods for determining the best triangle (similar to Dan's lesson plan) and we could have compared the methods later.

Side note: Dan says that "the best solution is to use the fact that an equilateral triangle is the triangle that encloses the most area for a given perimeter". Sixth graders are not at a point yet where they can find the area of a triangle just given the side lengths, so some other solution was necessary. Technically, my method is flawed because it favors smaller triangles. If you double or triple the size of a triangle, it doubles or triples the mean absolute deviation. This is noticeable in the data as smaller triangles were preferred. A better method would have been to compute the percent differences from the mean, but this would have greatly complicated an idea I was just trying to introduce for the first time.

Wednesday, November 19, 2014

Minecraft and The Coordinate Plane

I explained to my students today that my son forces me to play a game called Minecraft and sometimes we bury treasure chests for each other to find. I pulled up the map below and asked my class how they would describe the location of the treasure.

Students suggested a bunch of very vague directions:

  • It's in the desert.
  • It's where the snow and the desert meet.
  • It's next to the large pond.
  • No, I didn't mean that pond. The other pond.
  • Go northeast, then dig.
None of these directions were that helpful. While some of the more detailed ones could have gotten me closer to the treasure, it's still difficult to find it unless you have the exact location.

Enter the coordinate plane. Some students were familiar enough with the game to know that x-, y-, and z-coordinates are given to you on the map. (They were cut off on my original picture.)
Of course, my students weren't exactly sure what those numbers meant, but it didn't take long for them to see that these values were simply directions from the origin of the map (white crosshairs) and they would provide the exact location. 

I particularly liked this introduction because it created a need for the coordinate plane (Dan Meyer did something similar here).

Sunday, September 28, 2014

I'm Crushing Your Head

Yesterday, I e-mailed my favorite estimation guru, Andrew Stadel, a question about estimating and collecting data. He said I should share my insights with the rest of the world. So, for the dozens of you who read my blog, enjoy!

The other day, I wanted to start easing my sixth graders into estimation (before diving into Andrew's, so I put this up as a warm-up:

For most of my students, this problem caught them off guard. It seemed as if no one has ever asked them to guess the length of something. Some were confused about what I was asking and it was apparent in their answers. I made a line plot for each class and noticed that about 80% of each class thought that side B was 24 if I was referring to some archaic property of rectangles that says that the longer side of a rectangle is twice the length of the shorter side. Only a few students in each class even got close to the right answer (which I've put at the bottom of this post).

After we talked about some estimation strategies such as using your hand as a guide (see picture below) and identifying lower and upper limits of reasonable answers, many were eager to try another problem. As each of my classes is only 37 minutes in length (crazy, right?), I told them that we could try another one the next day.

"I'm Crushing Your Head!"

So, here's the problem I gave them the next day...

And sure enough, their guesses were much more informed. As with yesterday's estimation, I made line plots for each class's data and we could see that many more students were closer to the right answer. As a class, we felt that progress was made.

And then came the beauty of the line plot itself. For every class, I asked: what do you notice? In one particular class, we noticed that the data points were spread out. In another class, we saw that we had outliers. In another class, we saw that somebody guessed 18 inches, so they really must have been thinking that the rectangle was a square. In another class, we noticed that the data was skewed to the left or closer to a bell curve. In many of the classes, we noticed that students typically underestimate (which I'm very interested in understanding why, but I'm not going to delve into that here).

Later in the day, I noticed that the data from one class was very similar to a previous class. So I put both data sets up, and all of a sudden, we weren't just evaluating different students' guesses, but two different data sets. Finally, I added a third set, and we started having discussions about which class guessed the best. And the kids were really into it and coming up with some interesting ideas about how to determine the best class.

And I thought, this is awesome. Not only are my students driven to become better at estimating, but now they're looking at using math to help figure out if they're getting better at it and if they're better than somebody else. (They're downright vicious when you throw a little competition their way.)

By the way, the answers to the two estimation challenges are: The first rectangle is 12 inches by 32 inches. The second rectangle is 18 inches by 26 inches.

Thursday, September 18, 2014

Every Math Teacher in the World Should Do This...Right Now!

Yesterday, I was teaching students how to find the greatest common factor of two numbers. We start this lesson by using easy numbers to work with (like 10 and 14), list all of the factors, circle the common factors, then determine which of these common factors is the greatest. No big deal.

Next, we moved on to bigger numbers (48 and 84), and it became much more challenging. Some students just don't know their times tables that well, especially past ten. 3×16 equals 48? Even I'm a bit sketchy on that one.

I showed the students how to write the prime factorization of 48 and 84 using factor trees (which they've already learned), how to identify the common prime factors, and finally, to multiply them to find the greatest common factor. I then immediately sent these students to the whiteboards surrounding my room, so that they could practice finding the GCF for a different set of numbers. As you can see in the picture below, every student has their own space to work.

What happened next? Only the greatest damn thing ever! When students are working on the whiteboards, I can see everything happening at once. It's like I'm looking at the freaking Matrix. With a quick glance, I can see which students got it, which students are making minor mistakes, and which students have no idea what's going on. I can quickly identify errors for students. I can ask a stronger student to help a struggling one. Once a student has the correct answer, I yell, "Great! Erase it! Next problem!"

And the kids love it. As soon as the kids walk into my classroom each day, they ask "are we working on the whiteboards?" As soon as I say, "Go to the boards!", they rush out of their seats potentially harming each other as they make their way there. As soon as I put a problem up, they quickly get to work, Even the students that I know would typically struggle in math class, love the whiteboards and are learning much more because of them.

Now imagine what would happen if these same students were doing this work in their notebooks at their desks. Would they be enthusiastic? No. Would I know how much my students understood about the lesson? No. Would I be able to help students in a timely manner? No. Would they learn as much? Probably not.

I cannot stress enough how much these whiteboards have transformed my students' growth. If you do not have enough whiteboard space on the walls in your classroom, install them as soon as possible. It is the most important thing you could possibly do.

If you'd like more information about this, visit Alex Overwijk's blog post on it (who I give credit for teaching me about Vertical Non-Permanent Surfaces). I believe Peter Liljedahl deserves credit for bring the research on VNPS's to light.

Saturday, September 13, 2014

Real World Math

My school recently instituted an end-of-the-day program where almost all of our students partake in different activities. There's a journal day, a current event day, a homework day, a silent reading day, and a real world math day.

And there it is. Real World Math. I see it over and over again. It's almost comical. People love hearing that students are learning real world math (as opposed to all of that other crap that is typically taught in math class). The other day, a colleague was writing some horrible thing called a Student Learning Objective, and he was asking me what he should write. I told him to just throw a couple of buzz words in there like "real world". People eat that stuff up.

Why does all of this bother me? Because when we keep putting real world math on a pedestal, it marginalizes everything else I try to do in the classroom. It says that there are really only a few things worth learning in school, so when something doesn't sound like it's "real world", go ahead and give up. Tune out.

Yesterday, my students and I watched Vi Hart's video about doodling stars.

The kids were entranced by this. I stopped the video to show them what Vi was saying, because, let's be honest, she does talk way too fast. I then showed them how to make one of these stars by picking a random number of points (P) and a random skip number (Q). And they thought it was awesome. At this point, I pointed out that they will probably never use this in life. But that doesn't make it any less relevant. It is beautiful and fun. And if you get any kind of reaction out of it, then it was worth your time. Not everything has to be "real world".

Side note: Throughout the day, I worked on my own star in the back of the room. Pretty damn cool, right?

Sunday, August 31, 2014

When am I ever going to use this?

As a teacher, I hate this question. For years, I would stumble with the answer, especially when I taught Algebra 1. As a former engineer, I could typically think of ways that I used math, but how does a lawyer, a nurse, or an animal shelter worker use algebra? I have no clue. And like an idiot, I would always try to construct some kind of answer that would never be satisfying to the student.

The real issue with this question is that the student only wants one answer. They want to hear you say, "You know what? You're right. You'll never use this. I've been wasting your time with this nonsense. Maybe I should just teach you how to pay your bills and call it a day."

Students don't want to hear about how every single profession uses box-and-whisker plots. The reason they ask this question in the first place is because they are frustrated. They don't get what you're trying to teach. And they're just looking for an excuse to give up. If the same students were learning FOIL, and could produce a right answer every time, they probably won't complain about never using it (even though they probably never will).

If they don't have to struggle very much to learn something, then they don't need excuses not to learn it.


Neil DeGrasse Tyson is a hero of mine. I even got a print of him to hang in my classroom. You can buy it here.

I've heard him talk about how students will often lament about how they will never use some of the things they've learned in school. Here is a panel discussion where he talks about this. In this video, he goes on to say how working on problems in physics (and math) helps rewire the brain and prepares it to solve other problems. And understanding how things work will lay the groundwork for innovation. This is exactly what most business owners want from their employees. They need problem-solvers. They need innovators.

On the first day of school, I talk about this with my students. I explain that the jobs of the future require us to be innovators and inventors. I then show them this newspaper clipping from the local newspaper:

Each year, kindergartners are asked what they'd like to be when they grow up. I read each of these responses with my students. I don't hesitate to tell them that I also want to be Elsa from Frozen. And then I point out Emmett's entry. Emmett wants to be an inventor. I explain to them that I'm really excited about this because Emmett happens to be my son (which would help explain this child's fascination with Back to the Future). I tell them I'm excited because, even at an early age, Emmett wants to learn about math and science. The motivation is there. He is already inventing things and experimenting with electronics sets.

There is only one problem with Emmett. This summer, we went to Disney World and one of his favorite attractions was the Jedi Training Academy. Basically, they give you a light saber, throw a brown robe on you, and after some light saber "training", you face off against Darth Vader. When it was Emmett's turn to fight Darth Vader, he seemed very reluctant to fight. I wasn't sure what was wrong, but after it was over, he explained that he didn't want to fight Darth Vader. He always sympathizes with the villains in movies. He wants to join the Dark Side. What's troubling is, I fear that some day he will take his love of invention, and use it for evil.

So, while I would like all of my students to be intrinsically motivated to learn about math and science, I am really worried that Emmett may someday destroy the Earth. We need smart people to stop him. And that's my new rationale for why students need to learn everything in math class.