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.

Friday, September 19, 2014

Who would you want to work with?

We are in the process of making teacher-groups for Family Math Night. The teachers (MTH 221 students) will work together to develop an activity related to specific standards, try out the activity with K-6 students, and reflect on the activity's effectiveness. Throughout the project, teachers use frameworks from the 5 Practices and the Principles to Actions to inform their efforts. This is one of the ways I try to embed the work of teaching into the course.

Because I also want to prepare pre-service teachers to be your future colleagues, I am soliciting your help in identifying norms for collaboration. What are some things you look for in colleagues with whom you choose to work? I am trying to come up with five criteria that the teachers could consider as they evaluate their interactions with their peers.

I have a compulsion to use acronyms, so I made the checklist on the right using some suggestions shared on Twitter. Does this list work for you? If not, how would you adjust it? Please do not be limited by this format as you offer suggestions in the comments.

Thank you in advance for your contributions to the development of these future educators.

Friday, August 8, 2014

Where is the value in play?

This week, Kathy and I presented at the Michigan Council of Teachers of Mathematics Conference. We adapted our previous workshop on games to focus specifically on the Common Core Standards for Mathematical Practice. (Here is a PDF of the session PowerPoint.)

We used the grouping by one of the Common Core authors, William McCallum, to make the Practices more manageable for the participants. Then we concentrated our attention on Standards 7 and 8 (what McCallum refers to as "seeing structure and generalizing"). I shared how some preservice teachers had synthesized this pair into three key elements to look for while doing math:
  • Noticing: recognizing patterns by breaking things down and identifying basic structures;
  • Building: creating new knowledge by connecting ideas to what is already known; and
  • Generalizing: identifying ways to create general methods/formulas
By intentionally narrowing our focus in this way, we hoped to model the importance of highlighting learning opportunities that occur during play.

Participants were given the opportunity to play three games. Two of the games, Race to 100 and Roll a Square, provide opportunities to examine the structure of our place value system and how the structure can be used to create methods for solving double-digit combining and separating problem. We asked participants to explore the games as teachers - keeping in mind scenarios that might be used to highlight Standards 7 and 8.

I provided the following as a model scenario:
While playing Race to 100, I saw Alyssa start on 14 and roll a 10. She ended on 24. What if she rolled 3 more tens in a row? What would the Rekenrek look like at the end of each roll?
While rolling four tens in a row is unlikely, we can use the shared experience of playing the game to provide learners with a chance to notice how our place value structure can be used to build a strategy for adding 10 to a number.

By playing the games before using them with learners, the teachers can be intentional about looking for opportunities to highlight "learnable moments." Then, teachers can use reflection time to talk about scenarios they observed during  game play (during planning, during the lesson, or even imagined). This can be as simple as sharing the scenario and adding one of the questions (from this PDF) associated with the Practice Standard(s) the teacher has decided is the focus of the lesson.

Too often we do not take the time to debrief around games and make explicit some of the mathematical practices that occurred. Is it any wonder that our students respond, "Nothing," when asked by their parents or guardians what they learned in math class today. Let's not leave learning to chance and assume learners will use the skills that will help them to improve their mathematical practice.

Saturday, August 2, 2014

Am I Enough?

I am a perfectionist. Somewhere in my past, I internalized the message that my worth was tied up in being perfect. I do not blame the adults in my life for this message. Chances are they were dealing with their own perfectionism. The fact is that I thought I had to do everything perfectly, which has had a negative impact on my life.

In Daring Greatly, Dr. Brené Brown explains that perfectionism is one of the ways we try to protect ourselves from expressing vulnerability. The problem is, it is through vulnerability that we connect with others and access our ability to change and grow. My own struggle with perfectionism has at times left me feeling isolated and kept me from trying new things. As educators, if we are not able to appropriately express our vulnerability, then we risk passing along "gifts" like perfectionism to another generation of learners. (If you have not watched Dr. Brown's TED Talks on Vulnerability and Shame, they are worth your time - especially as you think about setting a classroom culture for the coming school year.)

Part of the problem, is that we are fighting against a culture of perfectionism in education. Here are two examples from this past week. (Be forewarned: one of the problems with being a perfectionist is that I read perfectionism into things where it might not exist. I will readily engage in a conversation in the comments if you think I am wrong about these examples.) First, I read a review of the book, Building a Better Teacher. The reviewer ends his piece with, "Learning on the job just shouldn’t cut it anymore." This seems to suggest that teachers ought to be perfect from day one. Then, I watched Campbell Brown on The Colbert Report. Her new project, Partnership for Educational Justice, is working to help "students fight laws that keep poorly performing teachers in their classrooms." While this sounds "common sense," it ends up creating expectations that teachers be perfect: perfect in their teaching; perfect in their implementation of district plans (even if they are pedagogically unsound); and perfect in student learning (even though they have little control over this aspect of education). Therefore, teachers are expected to be perfect from their first day to the day they retire. As Dr. Brown's research has shown, this push for perfectionism can inhibit teachers' ability to collaborate with peers, connect with students, and be innovative in their teaching. Is this really what we are after in education reform?

The woman who performed our wedding was the campus minister at Western Michigan University when I did my doctoral research. I think it is safe to say that my individual work with her helped me to overcome some of the perfectionistic tendencies that threatened to interfere with me completing my dissertation. She helped me to see that I am not called to be perfect but gracefully imperfect. This is the message I try to pass along to the teachers I work with. For me, graceful imperfection entails: (1) awareness when things go wrong; (2) acceptance so I don't resort to blame; and (3) adjustment so I can grow from the experience.

I still struggle with perfectionism. Now, however, I try to give myself permission to handle the struggle with grace. Hopefully, this post fits into that category.

Friday, July 25, 2014

What are (and aren't) the CCSS?

There is some real confusion about the Common Core State Standards in Mathematics [CCSSM]. For example, check out the #CCSStime Twitter feed from last night. Maybe this will help.

This is NOT an example of a CCSSM:
from Liberty Unyielding
It is a curricular resource selected by a district, school, or teacher to support the development of some objective. That objective might be aligned with the CCSSM, but there is no evidence on the sheet that this is the case.

Here is a curricular resource that claims to be aligned with the CCSSM:
from Create * Teach * Share
The 3.NBT.3 notation in the upper-righthand corner indicates the standard this worksheet is meant to address. It is up to the district, school, or teacher to determine if this resource does indeed meet the standard. Still, this is NOT an example of a CCSSM.

This IS a CCSSM:

You can be sure of its authenticity because it comes from this document, Common Core State Standards for Mathematics. Anything not found in this document (like the curricular resources provided above) are not a part of the CCSSM.

When it comes to development and selection of curricular resources, the CCSSM is quite explicit; it leaves these decisions to the teachers. It does not endorse any set of resources or even a sequence of topics. From the Introduction:
These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.
I am not defending the CCSSM. We can, and should, have a debate about something as important as a national set of mathematics standards. But let us have an informed debate, which starts with actually reading the document.