NPR wants to advise your pizza order…

Quoctrung Bui from NPR says that there are at least 74476 reasons that you should always get the bigger pizza. (The article has an awesome interactive graph, too!)

If we could mix the article with a math exploration, we could provide an awesome opportunity for a math-literacy activity that can combine reasoning, reading, writing, and some number-crunching all in the same experience. That’s a nice combination. Also I suspect the content hits close to home for most students. (The leadership in our district is often looking for opportunities to increase authentic reading and writing in math classes. This seems to fit the bill quite well.)

Here’s an activity:

Although without fail, the menus from a variety of local pizza joints will probably be a bit more engaging. (Look for an update coming soon…)

Shauverino Pizziano

But the big question is why?

According to Bui: “The math of why bigger pizzas are such a good deal is simple: A pizza is a circle, and the area of a circle increases with the square of the radius.” 

Yup… that’s pretty much it.

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Okay… I solved it, but how’s that math?

Its on these warm, spring days that some learners tend to start checking out. They recognize our typical reviewing and recirculating attempts to help students recapture some learning before the last tests and exams. But some of them learned the content the first time. Essentially, this time of year can bring with it a lot of down time for our students who learn the fastest.

That’s when I like to pull out my old college textbooks. I spoke before of the power of ungraded bonus problems. If interesting and placed properly, they can provide powerful opportunities for thinking simply for the sake of thinking. I like to give them a window into math that “doesn’t look like math.” After all, most of the content that K-12 mathematics includes has some commonalities that students get used to. They’ve gotten used to “what math looks like.”

Today, four groups got a problem that I adapted from one of my undergrad courses that I took at WMU with Dr. Ping Zhang who, along with Dr. Gary Chartrand co-authored this book which was the textbook for the course I took.

The problem is adapted from Example 1.1 from Chapter 1. It asks the students to create a schedule for 7 committees that share ten people. Now, the purpose of the problem in the book is to give a simple example of how a graph can be used to visualize a complex situation. Often the way a problem is mathematically modeled can change the intensity of the solution process.

 

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Figure 1.1 from Page 1

 

The reason I like to give this problem (and problems detached from the K-12 curriculum in general) is that in May, to curious students, these problems tend to hit the perplexity button just in the right spot. In fact, all the students looked at the handout at first and were unimpressed. Until I asked them to read it… then, they seemed to just want to see what the answer looked like. As one student put it, “It seems really easy at first, then you get into it and it’s actually harder than we thought.”

Graph theory is about creating visual representations. I didn’t want to ruin their experience by pigeon-holing them into trying to represent this they way I knew they could. And given the time, that didn’t stop this group from creating a similar idea. Not with a graph, but they did use crayons.

 

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To hear Kailey explain: “We just gave each person their own color. Then we knew we could have two committees meet at the same time if they didn’t have any colors in common.” That isn’t different thinking, really. Just a different representation.

I enjoy the conversations that come out it. No grade. Just doing math for the sake of thinking about something that’s interesting. It’s especially interesting to students like Katie who said, “Okay… I solved it, but how’s that math?”

 

Reference

Introduction to Graph Theory (2005) Chartrand, G., and Zhang, P., New York: McGraw Hill

My experiences with Kahoot!

Kahoot Student Front

A teacher across the hall (@nolink10) encouraged me to try Kahoot!

So, a Spanish teacher across the hall from me encouraged me to try “Kahoot!”. Kahoot! is a online quiz maker that works a lot like pub-style trivia. A teacher makes a quiz. Students log into get a chance to take the quiz. The question goes up on the screen and students try to get it right. Get it right quickly, you get many points. Get it right slowly, you get less points. Get it wrong, you get no points.

So, knowing the typical doldrums that the “last-day-before-a-unit-test” can fall into, I decided to try it as part of my test review. So, here’s how I used it: I posted 9 trig problems around the room. Students paired the students up and sent them around in 90-120 second intervals to solve each one. I encouraged them to show as much work as humanly possible to the point of being excessive. (This is an instruction that often gets ignored.)

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This part took about 20 minutes.
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Then, I fired up the ol’ projector and sent the students to kahoot.it. The quiz had a pin# they had to enter when they arrived. Then they could choose a nickname. (I’d advise some fairly clear boundaries on the nicknames. Just sayin’.)

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Then the questions came up on the screen and they could use their laptops, tablets, phones, or wi-fi enabled tech to answer. After each question, the correct answer is revealed and they got a chance to ask clarifying questions. It is possible to set multiple answers correct. The next question doesn’t appear until the teacher clicks “next”. The standings are updated and displayed after each question, too.

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So, what did I think? Well, the students sure enjoyed it. Although, I am curious how much learning got done. I suppose we’ll have the the test to offer some insight into that question. Also, one negative is that if the student device goes to sleep, Kahoot! kicks them out of the quiz. In one larger class (32 students), students were having trouble reconnecting and only about 17 students finished all nine questions. That wasn’t the case in my other class (of 22).

The students get to rate their experience after the quiz is done. Ratings were generally (but not all) positive. Also, the teacher gets an opportunity to download an Excel file that reports out all the data including the answers for each student (whether they finished or not), the breakdown of answers for each question and the student survey results. That is a nice piece.

I would encourage you to chime in if you have experiences with Kahoot! or something like it. I feel like tools like this can be useful, especially in BYOD schools.

Trig Curiosity

There is something about a question not being graded that makes the students aggressive and risky. That can create the conditions for some of the best thinking. There are many days when I think grades and points and the division between problems that I will “collect and grade” versus the ones that I will not.

In the system in which I exist, sometimes bonus questions on formative assessments are the only way to really perplex a student – to push them at the risk of pushing each student beyond their current ability to reason, but still get a solid effort.

On today’s quiz, I added the following question as a bonus:

If you type “Tan 90″ into a calculator, you will get an error message. Knowing what you know about trig, discuss the possible reasons that taking the tangent of a right angle in a triangle would make your calculator show an error message.”

This isn’t something that has come up in any of our discussions. I would like to share with you some of my student’s answers.

From James: “It has an opposite which is the hypotenuse, but it has two adjacents so you wouldn’t know which one to use unless you put it in the calculator.”

From Tyler: “There is no such thing because when you plug in Cos 90 you get 0 and when you plug in Sin 90 you get 1. Maybe it is because since Tangent is TOA, it tries to add up to 90, so like opposite is 30 degrees and adjacent is 60 degrees.”

From Brianna: “Because Tan 90 would be opposite/adjacent, but the opposite side of the 90-degree angle is the hypotenuse and you can’t have the hypotenuse on top.”

From Jeremy: “It shows an error message because the right angle on a triangle doesn’t have a defined opposite or adjacent side length because the angle is touching both legs.”

From Lauren: “With tangent, you are finding opposite/adjacent. Those are the legs, and that 90-degree angle is being made by the legs.”

From Dayna: ” There could be an error because the opposite of the right angle is also the hypotenuse of the triangle.”

Quiz Bonus 3

From Ally: It’s not clear where the negative idea comes from, but it is curious that in a Trig world of decimals and fractions, 90 in the other functions gives 1 and 0.

Quiz Bonus 2

From Victor

 

Quiz Bonus 1

Perhaps Josh’s picture says it all.

 

Now, the next question: If the “two-adjacent-sides-so-the-calculator-doesn’t-know-which-you-mean…” explanation wins out…

…then why don’t we get an error message for Cos 90?

New Blog Design

I decided it was time for a new blog theme.

I felt like the previous theme I used (the blue and gray) was optimized for tablets, which was fine, but led to some HUGE margins on a desktop screen. Longer posts and comments took a lot of scrolling to read and I couldn’t find any way to fill that space. I think this design fixes it. Plus it’s a bit brighter.

Anyway, just thought I’d post a bit of a notification if you are a frequent reader (thank you so much, by the way!). Wouldn’t want you thinking you’ve ended up in the wrong place.

The Odd Logic of Achievement Gaps

My goal in this piece is not to suppose that the spirit of closing achievement gaps is wrong. It’s not. It’s wonderful. The idea that we, as a culture, have decided to turn our attention to improving the situation for those who are struggling the most is a blessed endeavor indeed.

However, we need to be careful that what we are doing is going to produce the results that we want.

Just so we are all on the same page, I understand achievement gaps to be the difference in achievement between one group and a different, higher-achieving group. They could be groups based on demographics or location, or age, or any number of identifying characteristics. In Michigan, right now, the focus tends to be “Top 30-Bottom 30.” The means that we will create groups of the highest 30% of student scores on any given assessment and the bottom 30% of student scores and compare them. The difference in average score from those two groups would be named “the achievement gap.”

In the current culture of accountability, small achievement gaps are generally considered to be a good thing. Big gaps are a bad thing. I’ve been in multiple data meetings where the goal is to make a plan as to how we can “shrink our gaps”.

But achievement gaps are curious little creatures. Big ones suggest that our highest achievers (on any given assessment) performed a lot better, on average, than our lowest achievers. Given this environment, this is seen as bad news for a lot of teachers. But in reality, this is quite good news for at least 30% of the students.

Small gaps are an equally mixed blessing. A small gap simply means that there wasn’t much difference between the two groups. It doesn’t mean that either group did well. In fact, some of my smallest achievement gaps on unit assessments have come when the whole group struggled.

Once again, I want to reemphasize the goal is noble. Ultimately, we want to keep the high achievers achieving highly, and elevate the achievers of the low achievers. Here are some thoughts as I ponder that idea.

1. As a former professor of mine, Joe Kretovics, once said, “If you are going to close the achievement gaps, those lowest achievers have to learn more each year than the highest achievers.”

Think about that for a minute. That is a simple statement to make and a terribly difficult task to pull off. First, you have to consider a complete role reversal. Closing achievement gaps doesn’t mean getting the “bottom 30” (as the lower-achieving students have come to be known in Michigan) to keep up with the Top 30. It means that they need to outperform the Top 30. This is because, in general, in addition to a ton of other factors playing into their achievement, low achieving students also tend to be less proficient in the content than the higher-achieving students. Shrinking gaps means catching them up and moving them on at the same time. This is going to be tricky because…

2. Most academic supports are utilized better by the higher-achieving students than lower-achieving students.

There is a reason we are talking about the “achievement” of the students. This isn’t about brains. Practically all of the students in “the bottom 30” are as smart as the students in “the top 30”. In general, though, the circumstances surrounding the students have contributed to whether or not the ability has manifested itself into academic achievement. In general, the higher-achieving students have a better track record of seeking out support, and taking advantage of resources that are available to make them more successful at school. Email and twitter reviews aren’t a lot of help to students without internet. After school programs and Saturday school aren’t a lot of help to students with transportation issues. Extra practice problems and test retakes aren’t a lot of help for students whose away-from-school lives contribute to a practice of not doing homework.

But those are all fantastic helps to students who are used to taking advantage of them. This is why a lot of these programs tend to grow achievement gaps instead of shrink them. I give test retakes. Each student gets one. Each test, it seems, I have an awful lot of students who are turning C+’s into A’s instead of the target audience which are the people turning E’s into B’s. All this leads me to believe that…

3. Perhaps “shrinking the gaps” is a goal that, while noble, might not be particularly useful.

Perhaps we are in a brave new world of school data. We are only now getting a chance to explore all that we can collect, disaggregate, and report. We are getting excited about all it can tell us and the potential for what it would all look like in a perfect world.

It seems that we haven’t yet made it to the stage where we are able to discern which metrics tell us the most, which metrics are useful, but only peripherally, and which metrics are really pretty useless.

My feeling is that achievement gaps belong in that last group. Not because of the spirit, but because of the effect. The fact that students in Michigan are being labeled by personnel in their schools as a member of “the bottom 30” should make us nervous. (No kids should go through school being called “a bottom 30 kid”) The fact that we might skip out on a useful academic experience for the students because the gaps might grow as a result should make us nervous. The fact that we might consider leaving our “top 30” to sit bored because we don’t want to push them for fear of a growing gap should make us nervous.

There are ways to support lower-achieving students. Perhaps focusing on achievement gaps isn’t one of them.

Mosaic Math

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I found some more math in Lansing, MI. Right downtown, actually. The Lansing Art Gallery has a rather artsy front entrance. It looks like this.

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See the green circle? See it? Looks like Pac Man. This one:

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Closer inspection reveals that it is a green mosaic circle partially covered by a red mosaic rectangle. This grabbed my attention while I was strolling waiting to pick up a Cottage Inn pizza. I wondered how many green mosaic pieces were “underneath” the rectangle. I also wondered what percentage of the circle was covered up by the red rectangle.

Let me know if you get a solution. I’ll let you know when I get one.