Wednesday, July 30, 2014

Planning for Personal Change

How do teachers change?

I've been reading a lot of great talk lately about teachers who want to change the way they teach. Some people don't know how to change. I don't know either, but I do know that any real self-directed learning benefits from a plan. Whether you're learning new things about teaching, about math or about basketball, a bit of structure can help keep us disciplined and focused on the important things. Consider what follows an attempt to start a conversation about what structures might support a teacher's personal learning efforts.

How do you learn to write good web activities?

One of my goals this summer was to improve my computer programming skills. Early this summer I decided that I needed to make sure that I was making solid progress, so I decided that it was important to have a self-study plan. After a few different versions, this is what I've settled on:

I always have a project that I'm working on. I try to pick projects that will be useful to me, but that will also force me to learn some new skills. I don't try to just read about programming or take lessons unless I know what I need. I think people learn best by doing something, and then trying to reflect and generalize lessons from that experience. In the absence of a class or a teacher, I think projects are a decent way to consistently land on good experiences to reflect on.

I write down some questions that I have. I have a project notebook. When I sit down, the first thing I do is scribble down some things that I don't know. Below are this afternoon's questions:


When I'm stuck, I search broadly for ideas. Usually what happens is that I'm stuck, but don't even have the language or concepts to get myself unstuck. For example, today I knew that I wanted to make something that had a bunch of pages as part of one activity, but I had no idea what you'd even call this. So I had to search the internet and do research to even understand what exactly it was I needed to figure out.

I try to take breaks every 30 minutes or so. I didn't do a good job with this rule today, and it showed. I ended up spending too much time searching and tinkering, and not enough time incorporating the new knowledge. Since I think that powerful learning comes from reflecting on experiences, it's important that I take frequent breaks to give myself reflection time. 

At the end of my session, I write down my next steps. This is a helpful way to reflect on the last couple of hours and think about what I've learned and to focus on what I need to do next. I rarely need to consult my "Next Steps" list when I'm starting a new work session, but it's there in case I need a reminder. This list often includes questions or reflections on what I don't yet know. Here's what my "Next Steps" looked like after today:


I used this same process to make sure that I was making steady progress on my "Why do kids struggle with proof?" project. It worked pretty well there too.

Supporting self-study about teaching

Could the above structure help me get better at teaching? What's great about the above routine is that it keeps me focused on questions that matter instead of diving off into an endless timesuck of resources. That's why it also kept me on target while I was studying proof -- there is an immense body of literature about proof, and I needed to keep myself from getting distracted by interesting, but non-crucial resources.

(Incidentally, whenever I take one of these tangents I end up feeling massively overwhelmed. I always feel good when I'm following my central questions.)

I'll be trying to use this structure over the next month as I dive deeper into feedback, an area that I I know I want to get better at before the school year starts.

Does a plan help you change?

Learning is really hard. Change is really hard. Change is learning, so it's exactly the same sort of hard. I've found that adding a bit of structure to my self-study really helps me learn/change.

I think that this might be a good direction to take the "change" conversation -- deep down into the nitty-gritty details. What are you trying to do, and how are you planning on doing it? Answering these questions for myself has helped me be more efficient and less anxious about learning new skills. 

An added benefit of this sort of planning is that it helps me really develop my theory of how learning works. Making and improving a self-study plan gives me a chance to think through my assumptions about how people learn, in general.

Thursday, July 24, 2014

The Top 4 Reasons Kids Struggle With Proof, According To Teachers of Proof

I asked 37 math teachers why they think kids find proof so hard in geometry. Here were there top four responses:
For a moment, let's take a close look at the idea that kids aren't logical thinkers. It's a common view among teachers, and it's also an assumption that's built in to a lot of textbooks, activities and curricula. It's why the second chapter of your book might involve a careful analysis of "if...then" statements in non-mathematical contexts.

A concern about the logical capacity of kids is also why you might design an activity around the idea that "children are fuzzy thinkers."

This morning at TMC14 I lead a workshop on proof with a bunch of really thoughtful geometry teachers. They had some really good points:
  • Kids use logical reasoning to argue with their parents, right? So kids sometimes have the capacity to reason logically.
  • When we dug into some math and tried to prove some ideas that were new to us, we found it difficult to offer justifications and proofs. So our adult ability to articulate our reasoning was being stretched beyond comfort.
In short, this means that thinking logically isn't an always thing. It comes and goes, depending on the context. Kids can't reason about school geometry, we math teachers struggled to reason about unfamiliar math. 

The instructional implications of this are that we can't hope to improve our students' ability to reason about geometry by improving their ability to reason logically in general. Kids can use logic, sometimes. We need to help them reason in geometry.

Tuesday, July 22, 2014

Rebecca Talks Logic With Her Kids!

I asked you to talk logic with your kids, and Rebecca came through in a big way! What follows is the transcript from her conversation with her kids. I've got some follow-up questions -- see them at the bottom of the post.
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Three questions were posed, and I presented them to my children in order. My kids are K (8th grade girl), M (5th grade girl), and J (2nd grade boy).

1. Ask your kid: Merds laugh when they're happy. Animals that laugh don't like mushrooms. Do merds like mushrooms? Why?

J: Of course they do!

Me: Why?

J: Because they’re not real animals. They like whatever I say they like.

M: Nu-unh. You didn’t make them up. She gave us the facts, and the facts say they laugh so they don’t like mushrooms.

K: they wouldn't like mushrooms because they laugh when they are happy, and animals who laugh don't like mushrooms.

I thought the bit about Merds eating cheese with every meal might’ve thrown J off about food preferences, so I forged ahead.

2. Ask your kid: Every banga is purple. Purple animals always sneeze at people. Do bangas sneeze at people? Why?

J: No way. Bangas don’t sneeze at people.

Me: Why not?

J: Because they’re imaginary. They can only sneeze at people if they’re sneezing at imaginary people.

K: Yes, because they are purple and you told us that purple animals sneeze at people.

M: And they certainly have enough noses to get it done.


3. "Are glasses made of rubber?" Tell your kids: Glasses bounce when they fall. Everything that bounces is made of rubber. Ask your kid: Are glasses made of rubber? Why?

All 3 of them, resoundingly, “no. Glasses aren’t made of rubber.”

J: glasses aren’t made of rubber.

M: and they don’t bounce. I’d prove it but that’s dangerous.

K: nonsense. That one’s nonsense.

---

Questions
  1. Why did the kids answer so confidently to the third question?
  2. Why did the littlest one have trouble with the "make-believe" questions?
  3. What's the difference between the youngest kid and his older sisters? What do they get that he doesn't?
  4. Would an even younger child have an easier or harder time with these questions?
  5. When can children reason deductively? When do they fail to reason deductively?
One thing is clear: we need more evidence! If you have some kids near you, ask them these questions and send the transcript along to me (either via michael@mathmistakes.org or @mpershan on twitter). Let's figure this out!

Monday, July 21, 2014

Talk Logic With Your Kids For FABULOUS Prizes!

You know all about Talking Math With Your Kids, right? Well I want to reward your handsomely for your conversations!

I've got three chats with your kids that I'd like to commission. If you send me a report of your conversation, you will win fabulous prizes. Did you hear that? FABULOUS PRIZES!!!!!!!!!

OK, here are the three conversations I'm willing to bribe you for:

1. "Do merds like mushrooms?" Explain to your children that merds are a type of animal. They're a make-believe animal, not a real one. They're blue and eat cheese for every meal.

Ask your kid: Merds laugh when they're happy. Animals that laugh don't like mushrooms. Do merds like mushrooms? Why?

2. "Do bangas sneeze at people?" Bangas are another type of make-believe animal. Tell your kids that they have fifteen feet and six noses.

Ask your kid: Every banga is purple. Purple animals always sneeze at people. Do bangas sneeze at people? Why?

3. "Are glasses made of rubber?" Tell your kids: Glasses bounce when they fall. Everything that bounces is made of rubber.

Ask your kid: Are glasses made of rubber? Why?

---

If you submit one of these conversations you will get to choose from one of three amazing prizes!

  • Prize 1: I will write you a bad sonnet. You decide on whether it's Italian or Shakespearean or Wisconsinian or whatever.
  • Prize 2: Mystery internet prize! I will send you a cheapo item from Amazon to your door. It'll be less than $10, but it will be awesome.
  • Prize 3: I will paint a bad watercolor in your praise. I have no painting talent, repeat no painting talent, but I'll do my best to honor your magnanimity. 
So run, don't walk, to your nearest child and let's have those conversations!

(Where is this coming from? From here.)

Sunday, July 6, 2014

How To Read "This Is Not A Test"

Over the past few weeks of the Global Math Department discussion of Jose Vilson's This Is Not A Test, I've realized that I'm reading the book differently than others are. I'd like to take a moment to lay out the case that this book should be read, essentially, as directed toward policy-makers and policy-advocates. In other words, this book wasn't written primarily with teachers in mind.

My main support from the text comes from this passage:
"After coming along on this journey with me, I hope you've gotten a sense of what it is like to teach--not just in urban schools, but within the parameters of any space in which we are beholden to a certain set of children, a certain set of adults, and a certain set of conditions." (p.211)
Doesn't this passage imply that Vilson has not been addressing teachers in the past 210 pages? After all, teachers don't need to get a sense of what it's like to teach.

To me, the whole thrust of the book is a critique of present-day education reform. I think all the short narratives he tells start gelling together when seen as part of a case that accountability reform and high-stakes testing have created at atmosphere that's bad for learning, teaching, and especially bad for Black and Hispanic students and teachers. (This was the framework through which I wrote this piece.)

I mean, he named the book "This Is Not A Test," right? And the book is full of policy suggestions, but has relatively few teaching tips. A book written for teachers would look very different, I think.

None of this means that teachers shouldn't read Vilson's book, or that reading This Is Not A Test won't help your teaching. The conversations we've been having as part of the book discussion show that this rich memoir is helping teachers continue to think through how they can better help their Black and Hispanic students. Teachers are obviously finding this book valuable, and that's great.

But I don't think that this book was written with the primary goal of helping teachers better teach their Black and Hispanic students.

I know that many of you disagree -- for all I know, Jose Vilson himself disagrees! -- and I happily invite your disagreement. Do the conversation a favor, though, and come with evidence from the book that teachers are the intended audience.

Friday, July 4, 2014

Scaffolding Proof Writing

I'm reading a lot of papers on proof and geometry class right now, and I came across one that offers some sensible, manageable scaffolds for standardish proof problems. The title is Moving Toward More Authentic Proof Practices in Geometry, and it's an interesting read.

Here's the first two problems they offer in the paper:



In Problem 1, the "Given" and "Prove" are missing. In Problem 2, the diagram is missing. They seem to have made a big list of things that can happen in geometry proofs and designed problems by excluding some combination of these things. It's the job of kids to provide those missing things.

In Problem 3 the missing thing is the "Given" and "Prove."
In Problem 4 it's the actual given statements.
In Problem 5 it's the theorems.
In Problem 6 it's the auxiliary line.





The last three problems are a bit different. In Problem 7 they explicitly ask kids to make conjectures -- something I know that I should be more systematic about than I have been in class. In Problem 8 they ask kids to find mistakes in a given proof. Problem 9 is another "missing information" problem, where this time they left out the theorem but gave you the entire proof.




This past year I really didn't push my classes to do much written proving -- though "how do we know this?" was practically a mantra in my teaching -- but I was often disappointed by my students' ability to write down logical arguments when I did ask them to explain their thinking on paper. I think that these scaffolds could be an important part of what I do next year.

Monday, June 30, 2014

Beyond "Justify"

From Discovering Geometry

Explain, Why?, Justify and Prove

Grab whatever geometry textbook happens to be nearby and scan the reasoning-and-proving exercises. (You can generally find two them at the end of the section after all the practice problems...) Take careful note of the language that's used in these questions. What exactly is the kid being asked to do when they're asked to defend their answers?

There's a variety of language that can be used for these exercises. Kids are asked to elaborate on their thinking using several different -- but apparently interchangeable -- prompts.  




In the exercises above we get a few common directives: "Explain." "Why?" "Justify your answer." If we poke around our nearby geometry text we'll pick up a few other phrases, like "How do you know?" and  "Explain your reasoning." 

Do all of these prompts sound the same to kids? Should they all? Do we want kids to think of an explanation as being roughly identical to a justification? Is answering "why?" the same thing as offering a justification? And how does all of this relate to that other core prompt, "prove"?

Reasoning Problem Makeover

A few weeks ago I wrote about something called the Hexagon of Proof, and that post was half-joke and half-serious. The half-joke part was the idea of making a catchy image that played off Bloom's Taxonomy. The half-serious part was the idea that we can teach proof more effectively if our classes have a healthy and varied diet of proof-like activities. There are natural bridges to be built between everyday discourse and the unnatural act of mathematical proof. 


We can do better than just asking kids to "justify" their thinking. There are lots of ways to provoke kids into expressing their reasoning, and there are some prompts that ought to see wider use. As an exercise, I rewrote one of the above problems in five different ways. As you read each problem, think about the different sort of student responses that each bit of prompting language might yield.

Exhibit A: Debating


Exhibit B: Disagreeing


Exhibit C: Convincing


Exhibit D: Explaining


Exhibit E: Teaching


Exhibit F: Proving

Bonus: Justifying


Does justifying have a different meaning to students than proving? I have no idea. Thoughts?

Exercises for the Reader
  1. Which is your favored version of  the problem? Would you use different versions in different situations? Explain your answer.
  2. Are there other versions of this problem that you can imagine? Construct an example.
  3. "The language used in presenting reasoning problem significantly impacts the sorts of responses that a teacher can expect to receive." Do you agree with this claim? Disagree? Justify your response.
  4. Challenge Problem! Samuel Otten (and colleagues) wrote a paper called "Reasoning-And-Proving in Geometry Textbooks." In it they analyze the types of reasoning-and-proving activities assigned in popular geometry texts. How does their analysis compare to the one given in this post? How would Otten respond to this post?