A few quick words on a function context

Geoff Krall did us all the favor of preserving a brief Twitter conversation about a lovely applications of functions example found by Taylor Belcher.

Go have a look. Won’t take you long.

[View the story “State Hate Functional Relationships” on Storify]

Status [#MTBoS meta-commentary]

My electronic colleagues worked very hard this summer on issues of inclusiveness, welcoming and participation in our electronic staff lounge.

See, for example, the following people’s thoughts from recent months:

• Chris Robinson
• Dan Meyer
• Kate Nowak
• Hedge
• Gregory Taylor

I kept quiet during this discussion because I didn’t have much to add to the conversation.

Now I do.

I think a lot about status in my classroom. As teacher, I have a lot of power to influence the status of my students. By encouraging, valuing and using student ideas, I can help each of my students to be seen by others as a valuable member of our classroom community. By making sure I spread opportunities to speak, and by differentiating the ways that students are able to contribute, I can help to spread status more equitably.

I do this imperfectly, but I do think about it a lot. I like to think that I get better each year.

When I notice that I am doing a poor job of it, I try to change my ways.

An important point is that status is conferred on others by social agreement, and that this agreement is often implicit. While I can influence the status of individuals in my classroom, I cannot impose it and neither can individual students.

Now, it seems that all social spheres have dynamics that involve status.

In any case, our sprawling electronic community of math (and other) teachers (and other interested parties) certainly does. I think that status is part of what my colleagues were sorting out this summer.

If we are concerned with welcoming new people, and with making people feel like valued members of the community, we would do well to make conscious decisions about this.

Here are a few things I do in this spirit:

• Link to the work of others in my blog,
• Comment on others’ blog posts,
• Tweet interesting things others have said (I am more likely to tweet a quote with a link than to praise),
• Engage publicly with new voices (e.g. the dreaded “.” before an @ mention on Twitter makes a response more public, especially if the person being mentioned has fewer or quite different followers from one’s own)

And here is something I no longer do:

• Maintain a blogroll

Don’t get me wrong. If you maintain a blogroll, more power to you. It certainly is a way of conferring status on colleagues, and it certainly can be a helpful tool to those who are new to the world of blogs.

But I found that maintaining a blogroll meant that I was making more permanent judgments than I cared for. I didn’t edit it very often (too much work, and too low priority), and when I did edit, I didn’t like where my mind went as I decided which blogs to include. It felt too much like picking teams for kickball.

When I killed my blogroll a while back (a year maybe?), I pledged to more frequently link and tweet the interesting work of others. That has felt much more satisfying and effective to me.

In sum, you don’t need to do what I do. I don’t advocate that at all.

No, I advocate thinking about the ways in which all of us—high, medium and low status members of this community—can support positive social dynamics in this space. And I argue that paying attention to the ways we influence the status of our colleagues (and of ourselves) is one way to get better at this.

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Wrapping up the [#algorithmchat]

Intentions were good and initial interest was high for #algorithmchat.

And then people realized how incredibly boring algorithms are unless you’re really, really into them. And it was the end of the school year. Et cetera.

We did get a bit of Twitter back and forth, and Karl Fisch dove in twice. Which is awesome.

Let me know if I missed anything with the list below, which I believe to be the comprehensive collection of posts on the matter—in order by posting date.

Hopefully, this list will get others thinking and we’ll add to it. Find me on Twitter, or post your link in the comments.

May 6: Reading group. Overthinking My Teaching.

May 6: Algorithm nation. The Fischbowl.

May 12: Algorithms, quadcopters, and the CCSS-M. The Fischbowl.

May 15: Common numerator fraction division. Overthinking My Teaching.

May 22: What is “the standard algorithm”?. Overthinking My Teaching.

Various dates: Posts on algorithms. A collection of posts from David Wees: Thoughts from a Reflective Educator. [Technically, these were not written in response to our original article, but they certainly are on topic.]

A few thoughts on integer operations

Responding to a request here.

@Trianglemancsd Interested in your thoughts of the examples provided for neg. x neg. in @ddmeyer‘s latest: http://t.co/CrfguOW0cP

— Chris Robinson (@absvalteaching) June 20, 2013

The short version is that Dan got it right (go ahead and give it a read if you haven’t already).

I do have a bit to add to the conversation, though. Dan pointed to the difference between being a teacher who views engagement through the lens of being useful in the world outside the math classroom and one who views it through the lens of curiosity. That’s a really nice distinction.

I want to point to an additional subtlety.

The formal mathematical view says that the distributive and associative properties of multiplication and addition ought to carry over to integers, and deduces the relevant result. The exact train of thought, and where it begins depends on the grade level. Consider Hung-Hsi Wu’s thoughts on the matter:

The key step in the correct explanation lies in the proof of (-1)(-1) = 1 (as asserted in the grade 7 standard). Pictorially, what this equality says is that multiplying (-1) by (-1) flips (-1) to its mirror image 1 on the right side of 0. A more expansive treatment of this topic in accordance with the CCSMS [sic] would show that, more generally, multiplying any number by (-1) flips it to its mirror image on the other side of 0.

The approach suggested by Brian is formal as well, but it’s different from the one above. It doesn’t tell it asks.

@PaiMath @ddmeyer Somewhat. From what I know neg #s were invented for debt. Once they exist, we might ask what happens when we multiply?

— Bryan Meyer (@doingmath) June 17, 2013

I cannot overemphasize the importance of this difference.

Of course it is appropriate to tell kids stuff sometimes. Of course it is. But there is far too much of that going on in classrooms already. Wu is concerned with steering this telling in a mathematically correct direction. That’s fine.

But I don’t want Griffin and Tabitha‘s mathematical educations to depend on better telling. I want them to explore and to wonder. I want them to commit to their ideas and see what the consequences of those ideas are, and to revise their thinking when their present ideas are not good enough to explain what’s going on in the world.

And what I want for my own children is no different from what I want for my students, and no different from what I want for all children.

There is a place for good, mathematically correct explanations. I want kids to experience those when they’re the right move.

More importantly, I want them to learn to think for themselves.

Appendix 1

I am working this summer on an article about integer operations that I’ll submit for publication. If you have an interest in such things, keep an eye on Twitter; I’ll be looking for a couple of critical readers in a few weeks.

appendix 2

I attended a lovely session on integer operations at the Minnesota Council of Teachers of Mathematics spring conference back in April. Two University of Minnesota grad students, Christy Pettis and Aran Glancy, presented a useful framework for characteristics of good integer contexts. I now pass it along to you.

Clear and logical opposites. Integer contexts should have clear and logical opposites. An important point here is that money and debt are not clear and logical opposites for many kids. If I have 3 dollars and owe 2 dollars, it is not obvious to kids that this is the same as having 1 dollar. Indeed in many respects it is not the same. Credit and debt are logical opposites, but more abstract. This may be inherent in working with integer operations.

Net value. The context needs to be able to support the idea of numbers as being net values. Kids should be able to reason in the context about 2 as 3—1, or 1+1, or 3+(–1), etc. Not all contexts support this (cf: chip boards).

Zero is not empty. This follows from the net value idea, but emphasizes the special role of zero in the integer system. In particular, the context needs to support seeing zero as the state of the existence of an exact set of opposites.

Help some Minnesotans think something through, won’t you?

I spent some time on a conference call today as part of my duties as VP of math for MCTM. On the line were (mostly) elementary teachers and coaches. The fundamental problem at hand was beautifully stated by a participant:

If we want to grow wonderful, fabulous teachers, we have to make it easy for them to stay connected.

As the conversation proceeded, I found myself asking How does having organizations that are fixed in their physical location make sense in fostering these connections?

Several teachers described the kinds of connections and resources they would like to have access to, but that they do not (and hence why they are reaching out to MCTM and to the state department of education). These include (in no particular order):

1. Both occasional synchronous and ongoing asynchronous communication among teachers and policy and content experts;
2. Concrete resources for helping teachers move from a textbook-focused classroom to a standards-based classroom (especially a Minnesota standards-based classroom; we are not a Common Core state);
3. A place to ask about something that happened in the classroom today and to get help with what to do tomorrow as a result;
4. A way of promoting and organizing useful resources and filtering out the junk;
5. A place to have longer conversations about bigger issues such as children’s long-term development of place value concepts or current research and curricular innovations relating to subtraction; and
6. Video of classrooms in action—there seems to be a real hunger for seeing over reading classroom scenarios.

As I listened and thought about my own teaching, I became less and less convinced that a centralized resource is the answer to these needs. Indeed, the only need related to our particular state is number 2 above. But all of those other needs (and many of the general ideas underlying number 2) can be met by folks who have never set foot in our state.

I have built my own version of this support system online. Here’s how:

Blogs. Writing my own and reading those of others has generated tons of useful ideas for every aspect of my work. Just today, I took something from Chris Hunter’s blog to use with my College Algebra students, and I adapted Fawn Nguyen’s Google Form for my standards-based grading reassessment in all of my courses.

Twitter. Pretty much everyone whose blog I regularly read I also follow on Twitter. The reverse is not true, so Twitter expands my network. In the past week, I put out a general request with organizing my students’ reassessment requests and got reminded of Fawn’s system.

You know what I need? A system for organizing S reassessments. Esp. helping them focus on what they'll do to improve. Ideas? Links? #sbar

—
Christopher (@Trianglemancsd) November 05, 2012

Also I have had several back-and-forth conversations with teachers on a range of topics. For example, I replied to the following…

Classic math mistake: Adding Fractions. mathmistakes.org/?p=519

—
Michael Pershan (@mpershan) November 06, 2012

Twitter facilitates a lovely combination of asynchronous resource-hunting and synchronous problem-solving. It can also be a tremendous time sink, so time management skills are essential.

Email. Back to asynchronous communication. Email for me remains a medium for having extended private conversations. I am not a fan of listservs, mailing lists or email conversations that attempt to involve more than about three people.

Experience. Having spent a couple of years in this space, I have developed a working knowledge of what’s available. I know who to ask about what’s happening at middle school, who to ask/read/recommend on assessment issues. I know who will give me a new perspective on social media use. I know who to talk to for questions about college teaching.

All of this reinforces the idea-for me-that the role MCTM could play in meeting this request from our members isn’t so much creating a space for interaction, as it could be facilitating teachers’ entry into this much larger space that already exists.

I have no idea what that would look like either, though.

How say, math-o-blog-o-Twitter-sphere?