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.

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.


  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?

Friday, June 27, 2014

Questioning Wiggins' Definition of Feedback

Is this feedback or evaluation? From MathMistakes.org

Feedback vs. Evaluation

Imagine a baseball coach. The second basemen just struck out...again. After his at bat, the coach heads over to the player and puts her arm around his shoulder:
"Tommy, you haven't been hitting as well as you could've lately, amiright?"
Is this feedback? In a few places, Grant Wiggins argues that this is not feedback. Instead, this coach is offering an evaluation of the player's hitting. (See thisthat, and the other thing.)

So what would feedback to the struggling player look like? Feedback would be judgement-free and informative. It would call attention to facts of the matter that the hitter herself likely didn't notice. It would look like this:
"Each time you swung and missed, you raised your head as you swung so you didn't really have your eye on the ball. On the one you hit hard, you kept your head down and saw the ball."
Wiggins thinks that this is judgement-free. And there certainly is no explicit language indicating evaluation of the hitter. The coach could've lead with "That wasn't a great at-bat," but she didn't.  It's tempting to say that the coach is only providing the hitter with plain, value-neutral information. So where's the evaluation in the coach's feedback?

I think the evaluation is there, lurking in the background. After all, imagine what the batter is thinking when the coach comes up. He just struck out, and he settles back on the bench. He knows his job is to get on base, and he knows that swinging and missing is not what you're supposed to do in the game of baseball. He knows how his teammates are hitting, he knows what sort of team they're playing. He knows whether he's the only one who's striking out and what people are expecting of him. And then the coach provides him with all this information on why he swung and missed. Here's probably how he's hearing this:
"Coach is saying my technique isn't good."
And is he wrong? Everybody involved knows that you're supposed to hit the ball in baseball. That's the point of the game! The kid and the coach both know what's expected of the hitter, and that shared context colors the feedback accordingly.


Brief Linguistic Interlude

The point here is that you can say things without actually saying them. In philosophy of language or linguistics, this is called the pragmatics of speech. The context in which you say something contributes to its meaning. As a professor of mine put it, "I love cheese" can mean vastly different things depending on the conversational context.

  • A: "Don't you love cheese?" B: "I love cheese." 
  • A: "Do you love me?" B: "I love cheese."
In the first case, "I love cheese" means roughly that the respondent enjoys cheese. In the second case, B is saying something else: "No, I don't love you. I do love cheese."


Can Evaluation Be Avoided?

Let's bring this back to math class. What your students hear when you give them information isn't just what you say. There's often a whole world of unspoken ideas communicated to a kid that depends on the classroom context.

Say that you give a kid the sort of feedback that Wiggins suggests is judgement-free:
On the first three problems you distributed the exponent to the terms inside the parentheses and ended up with a non-equivalent expression. On the fourth question you FOILed and ended up with an expression equivalent to the original.
If the task was to discover equivalent expressions, the kid is likely hearing something like an evaluation:
I screwed up the first three because I used a bad math-move. I should have FOILed for all of the problems. 
What Wiggins' calls "feedback" still contains plenty of evaluation, it's just happening silently as a result of the larger context. I think that it's much, much harder to give non-evaluative feedback than Wiggins suggests it is. (See his response here.)

What are the implications of all of this for giving responses to kids? I can imagine a few reactions:
  • Evaluation is inevitable: "There's no way to give kids judgement-free feedback. This shows that really there's nothing wrong with evaluating kids, and actually it's often helpful, as long as its done in a respectful way."
  • Purge evaluation from feedback: "Given how hard it is to give kids judgement-free feedback, we need to work especially hard to remove judgement from the feedback that we give kids."
  • That's not what feedback is: "Feedback can't be conceived as judgement-free information, since context always infuses information with judgement. We need a new definition of feedback."
  • Just do your best: "Yes, context can insert judgement into judgement-free language. You can't avoid it, but you can try to minimize how prominent that evaluation is in your feedback."
Part of this discussion needs to include research on the way evaluation can ruin feedback, and it'll also drive us to understand the how it is that feedback is supposed to help kids learn stuff. Wiggins sees feedback as helping learning by giving students information that they're missing. Maybe there are other purposes for feedback, though. 

Tell me, readers: what did I get wrong here? Did I get anything right?

Monday, June 23, 2014

Two Textbooks' Handshake Problems

Like so many classic math problems, the handshake problem is easy to state: "If everybody here has to shake everyone else's hand, how many handshakes do we need?" Because it's such a well-known problem, the way that a teacher or a textbook uses it can offer an interesting window into their perspective. The problem is common, so how you choose to use it speaks a great deal.

I'm going to share how my two favorite geometry texts -- CME Geometry and Discovering Geometry -- use the handshake problem. The question is: what do their approaches to the problem say about their takes on geometry?

Discovering Geometry's Handshake Problem:

CME Geometry's Handshake Problem:

Some Observations:
  • Discovering Geometry breaks down the modeling process into eight steps, starting with completing a handshake table. CME doesn't offer any of those steps or supports.
  • Discovering Geometry asks the student to model the handshake process with polygons and segments. CME presents the handshake problem and the diagonals problem separately, and then asks students to consider connections between the two questions.
  • This isn't in the pictures, but the handshake problem is the first problem that appears in CME Geometry, while the handshake problem appears as subsection 2.4 in the Discovering text, titled "Mathematical Modeling."

I'm very curious to know what you folks out there think about these two presentations. My suspicion is that these two presentations speak to two different assumptions about how students learn to reason in mathematics. One assumption is that students need lots of informal experience that, with feedback and opportunity, will slowly shape their reasoning habits. The other assumption is that students need an explicit model of the reasoning process, either in addition to or at the beginning of their opportunities to reason on their own.

Your thoughts and criticisms are always appreciated, but I'll amplify the invitation for this post. I also know that authors of both of these texts hang out on the internet, and they could likely school us all a bit on their work.

Comments from the Bullpen:

Fawn and mrdardy want to see Discovering Geometry take away all that support for the kids, instead put it into a teacher's guide. They'd like to see CME get rid of the explicit connection between diagonals and handshakes, preferring texts that offer radically little upfront support for students.

l hodge finds CME's explicit call for a connection between diagonals and handshakes patronizing. Is there another way to push kids towards making that connection?

fivetwelvethirteen connects this with the CCSS standards of mathematical practice. How do you help kids get better at big skills like solving hard problems or mathematical modeling? My take: let's look to other fields, like literacy, because "problem solving" is as big a skill as "reading" or "writing."