Friday, October 31, 2014

Will it fit?

I used this lesson in MTH 221 (Mathematics for Elementary Teachers) to address Common Core State Standard 7.G.B.4. It seems to have a lot of potential, but there are still some elements that I think need to be tightened up. These are written in red - along with some other thoughts. I would appreciate any feedback on how to improve this lesson.

[Schema Activation]
How many of you know your fitted hat size? For example, I wear a seven-and-seven-eighths. Today, you are going to find your hat size and what that number means. 
I like the DC cap for a couple of reasons. First, obviously, DC are my initials. Second, it foreshadows the circumference and diameter relationship we will explore in the lesson.

In the past, I have had students measure a bunch of circles to find the ratio between circumference and diameter. It has been a struggle to make it an interesting lesson. This connection to something personal (hat size) seemed like it might be an improvement.

According to, there are a couple of ways to determine your hat size.

We will use both a flexible tape measure and their printable ruler in order to...

... consider the following questions:
  • How are hat sizes and head-measures related? (In other words, if we didn't have access to their table, can we determine a hat size given a head-measure?)
  • We are supposed to be working on CCSSM 7.G.B.4. Is this connected in some way to circles? Too obvious?
  • Why don't hat makers just use the head-measurement as the size?
Measure your head using both the flexible tape measure and the printable ruler.

Feel free to wear the printable ruler as a stylish headband as you work. Optional

Place both your head-measure and hat size on a sticky note and place it at the proper coordinates on our graph.

What does our graph show? Is there a relationship? If so, what do you predict the relationship to be? (If we input 22 inches for head-measurement, what hat size is the output? What if we input C inches?)

Here are a couple of tables for hat sizes from Let's use them to see if we can determine the input-output rule they are using to find hat size from head-measurement. 

There are other hat size tables (for example), but I like that this one makes 22 inches a hat size of 7 because 22/7 is often used as an approximation of pi.


What did you find? What is the rule?

Des found the following: y = 0.333x - 0.333. Does it work for our table? If it's correct then what hat size does a person need for a head-measurement of 24?

Okay, I wanted to play with the new Desmos linear regression feature - sue me. Is it a problem that this line doesn't go through the origin? That the slope actually represent an approximation of 1/pi? During the lesson, it seemed like this portion required a lot of scaffolding.

So what does the hat size mean? Let's take our headband and place it on the table. Notice that it is nearly the same shape as a circle. Now measure the distance (diameter) across your headband (circle).

In my case, if I measure what is approximately the diameter of my headband, I find the length is close to my hat size (seven-and-seven-eighths). How about you? What does that suggest our hat size means?

The students were most impressed by this portion of the reflection. They liked that the hat size number was not some arbitrary value - that it was actually connected to something mathematical. I looked for some history of hat sizes to explain why this value is used instead of circumference, but Google failed me.

Now what fitted hat size should I buy from if my head measure is 24 inches? Seven-and-two-thirds doesn't look like it's an option.

In an earlier unit, students struggled with the idea of independent and dependent variables and creating graphs that accurately represent a real-life situation. Because a hat maker does not make all possible diameters, we decided it didn't make sense to connect the dots. Instead, we came up with the graph shown above.

One of the reasons I like this activity is because it does connect with so many other standards, like 6.EE.C.9 and 6.SP.B.4. What do you think? Does this lesson have merit - is it worth saving? If so, how? Please add your thoughts in the comments.

Updated: As much as I loathe Pi Day, this piece on Stormy Kromers (hats made in the Upper Peninsula of Michigan) might make a nice connection.

Friday, October 17, 2014

At what level is his thinking?

Pierre van Hiele

My colleague, Jon Hasenbank, and I have been discussing the van Hiele Levels of Geometric Thinking and what they mean for teaching and learning in mathematics. I am particularly interested in finding videos of people sharing their geometric thinking so that we can apply the Levels and evaluate their thinking. If you are not familiar with the Levels, here's how Pierre van Hiele, the architect of the Levels, described the first three Levels in Developing Geometric Thinking through Activities That Begin with Play [PDF].

In my levels of geometric thinking, the "lowest" is the visual level, which begins with nonverbal thinking. At the visual level of thinking, figures are judged by their appearance. We say, "It's a square. I know that it is on because I see it is." Children might say, "It is a rectangle because it looks like a box."
At the next level, the descriptive level, figures are the bearers of their properties. A figure is no longer judged because "it looks like one" but rather because it has certain properties. For example, an equilateral triangle has such properties as three sides; all sides equal; three equal angles; and symmetry, both about a line and rotational. At this level, language is important for describing shapes. However, at the descriptive level, properties are not yet logically ordered, so a triangle with equal sides is not necessarily one with equal angles.
At the next level, the informal deduction level, properties are logically ordered. They are deduced from one another; one property precedes or follows from another property. Students use properties that they already know to formulate definitions, for example, for squares, rectangles, and equilateral triangles, and use them to justify relationships, such as explaining why all squares are rectangles or why the sum of the angle measures of the angles of any triangle must by 180. 

And now for some practice. Given these descriptions, how would you categorize the thinking of the individual in this video?

What evidence do you have to support your categorization of his geometric thinking? (Perhaps there are clues in the task's questions.) If more evidence is required, what questions might you ask to get a better sense of his Level of Geometric Thinking?

As always, your thoughts are welcome in the comments - as long as they are civil.

Wednesday, October 8, 2014

Whose fault is it that you aren't good at math?

Yesterday, Seth Godin posted Good at math on his blog. There was a lot that I agree with in this short piece. For example, the second paragraph begins with:
I'll grant you that it might take a gift to be great at math, but if you're not good at math, it's not because of your genes.
Unfortunately, this is followed up with:
It's because you haven't had a math teacher who cared enough to teach you math. They've probably been teaching you to memorize formulas and to be good at math tests instead.
I am not surprised that Godin employs the "blame the teacher" canard. Our nation loves finding easy explanations to complex problems and, therefore, falls back on the "bad teacher" narrative on a regular basis whenever it comes to problems in education. However, this explanation of why you aren't good at math misses an important point. A point Richard Skemp makes in Relational Understanding and Instrumental Understanding.
I used to think that maths teachers were all teaching the same subject, some doing it better than others. 
I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’.
Skemp's realization can help us to make an important distinction. It isn't that your teachers didn't care. In fact, probably the problem was that your teachers, like the rest of our society, cared too much - cared too much about you being good at math tests. And this is the crux of the problem (and another place where Godin and I can find some agreement). In the third paragraph, he writes:
Being good at standardized math tests is useless. These tests measure nothing of real value, and they amplify a broken system.
So here is what I wish Godin had written in those two paragraphs (my edits in blue):
I'll grant you that it might take a gift to be great at math, but if you're not good at math, it's no because of your genes. It's because of your experiences. You did not encounter in math class the experiences you needed to be good at math. What you received, because of our broken system's obsession with test scores, were experiences meant to prepare you to be successful in schoolmath - memorization of facts and formulas that can be easily assessed using standardized-tests.
What can we do about this disconnect between math and schoolmath? We can begin by recognizing that being good at standardized math tests is useless. These tests measure nothing of real value, and they amplify a broken system.
What do you wish Godin had written? Because until we can understand the problem, to be able to put it into our own words, it will be nearly impossible to solve it. 

Friday, October 3, 2014

Why do I make these things so hard?

During a Math Institute put on by the Public Education & Business Coalition (PEBC), we were asked to monitor our thinking while working on the Coffee Break problem from (warning, this link includes the answer) Discovery Education. Essentially, we were creating a Metacognitive Memoir that would make our thinking, not just our work, visible for others to see. Being familiar with this form of math writing, I quickly began working on solving the problem.

I started by reading the problem - the entire problem. In the past, there were times when I began working on a problem without understanding it. This lead to "wasting" time on unproductive approaches. I did not want something like that to happen on this problem.

The very first step, "Beginning with a full cup of coffee, drink one-sixth of it," got me thinking about similar fraction problems I have worked on. The second step, narrowed the list of similar problems to those involving mixtures. Fortunately, I kept reading because if I had started solving this problem with those previous approaches in mind I would have gotten off to a "bad" start. This problem was quite different from those I had worked on before.

As I was reading, mental images of coffee cups began coming to mind. At first these were simply generic cups of coffee. But pretty soon I realized that I was going to need a model that could represent the quantities presented in the problem.
In order to make the steps of my thinking visible, I decided to make a series of images, like a story board, that showed the ways I was picturing the different steps. First there was the whole cup of coffee. Then there was the cup filled with five-sixths coffee and one-sixth milk. Next I drank a third of the mixture ...
While I initially took the top third, I quickly realized that the milk would need to be mixed with the coffee. I decided to show this by splitting the cup up vertically. But what was I going to do to show drinking a half of the mixture?

As I was debating how to represent this step, and writing down my thinking about it (I could make a three-dimensional model or split each of the eighteenths in half), the facilitator asked us to stop working on the problem so we could share our results. RESULTS?!?! I felt like I had barely gotten started.

Sure enough, some of the other participants had answered the questions:
  • Have you had more milk or more coffee?
  • How much of each have you had?
What happened next got me wondering, "Why do I make these things so hard?"

Anyone want to guess what happened (what one of the participants said) and why I made this problem so hard? If you'll put your guesses in the comments, I will update this post with the answers next week.