Ethan, Fran, and Gloria have summer jobs at the local Dairy Freeze. They collect their own tips and then share them equally. One week Ethan collected $25 in tips, Fran collected $48, and Gloria collected $41.

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For some time I’ve been thinking about ways to gain minutes in classrooms so that students have more time to talk (and learn). Think-pair-share, turn-and-talk and other partner or group conversations help but invariably there seems to be a need to pull the whole class together for the whole discussion. Once we do that, though, we’re again having the “one person talk” mode — maybe it’s not the teacher doing the talking but it’s still just one voice.

BUT, here is the nugget that was suggested in the discussion post.

“**I’ve started to write down what I hear while I’m monitoring the class during their discussions and project them for all to see. Then students add any other thoughts that were discussed that I didn’t hear.**”

What a brilliant idea!

I responded,

“*As you record those comments and project them, do you find that your students refer to them? Do you still take whole class time to review those comments or might it (maybe with time and practice and suggestion) not need to be discussed as a whole? *”

Her response was,

“*I am noticing that as I post things and continue to monitor, others will say that they have the same thing. It actually seems to be encouraging discussion in groups and may be adding other ideas to continue their group discussions. I am not spending as much time on whole group discussion when I use this format.*”

I’m sure there are still moments where a whole class discussion is a good idea but this idea of projecting a compilation of ideas generated by the variety of pair/group discussants is a powerful idea!

If you try it or have already used this technique, we’d love to hear your stories!

]]>A regular hexagon and an equilateral triangle have the same perimeter.

]]>Did you know that there is a famous illusion associated with Lincoln’s “stove pipe” hat? The hat looks like it’s taller than it is wide, even when the height and width (including the brim) are the same. Let’s make a hat:

- Cut a circle with a radius of 5 1/2 inches.
- Cut a circle from the center of the first circle with a radius of 3 1/2 inches.

The smaller circle will form the tip of the hat, and what’s left of the larger circle will form the hat brim. If we had a rectangular sheet of paper of the right size, we could make the cylindrical part of the top hat.

]]>Mia drew a shape with exactly 4 sides.

It has 4 lines of symmetry.

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- Write 23 X’s on a piece of paper.
- On your turn you can erase or take away 1, 2, or 3 of the X’s.
- Turns alternate. You cannot skip your turn.
- The person who erases or takes away the last X wins.

I’ve always felt a connection to Punxsutawney Phil because my birthday is on February 2nd, the same day folks watch to see if Punxsutawney Phil sees his shadow! One year, I noticed:

- Phil’s height = 51.2 cm
- Suzanne’s height = 5′ 4″
- Suzanne’s shadow length at Gobbler’s Knob = 76.2 cm

And then the hypothesis that culturally competent teaching is one that explicitly leverages students funds of knowledge is useful — it lets me think (as a teacher) about one clear(ish), specific, recognizable goal that I can point my self-assessment towards. And the other skills I need to become culturally competent fall in line with that goal. I need to recognize my students as having knowledge, recognize that knowledge even when it doesn’t look like mine, and connect to students authentically so that I can leverage their knowledge in meaningful ways.

How I choose to answer your 2 important questions will have a big influence on how I use the hypothesis to assess my own teaching. I’m wondering if there are more and less powerful ways to answer the two questions.

For the both questions I turn mostly to the work of Lisa Delpit, especially her article “Skills and Other Dilemmas of a Progressive Black Educator,” an article that even on a third reading I still catch myself skimming over certain parts of defensively because it’s too scary to confront deep racial biases in my own practice as a progressive white educator. There’s a copy at this link: https://www.ebooksclan.com/reading/skills-and-other-dilemmas-of-a-progressive-black-educator-EG7b.html

The first question, about school math vs. home knowledge makes me wonder if there’s a parallel to be drawn between what Delpit sees as black children’s hidden literacy and fluency, and the need to focus explicitly and immediately and critically on conventions and the language of power. The first time I read the article I thought, “it’s different for math — there aren’t strong, inventive, and different math cultures.” But that was a long time ago. Now I do think it’s very likely that there are math cultures out there, ways of knowing and thinking about and arguing with and about number and quantity and pattern and shape and relationship and fairness and proportionality that are cultural, and that children come to school fluent in. Not all children, just like not all black children come to school able to play fluently with language and rhyme, and not all white children come to school fluent in how to do middle-class American school talk. But I do think it’s likely that there are (mostly unexplored) quantitative ways of knowing out there, just like there are different literacies. I’d love to be part of the research team exploring that question! A quick example is that when it comes to financial literacy, the latest research tends to show that even when people with not much money are making what seem to be irrational decisions (not having a bank account, cashing checks at check cashing places), when you ask them about it, they’ve always considered their options and come up with the best option available. Financial literacy folks are coming to realize that lots of people are good at quantitative financial reasoning, they are just taking a whole nother set of constraints into account.

And, clearly, there are a lot of ways that people have been explicitly locked out of getting to make sense of mathematical symbols and tools for thinking quantitatively — think of most Americans’ struggle to know when 20% off or $40 off is the better deal. That kind of skill to unlocking the culture of power totally exists in math. Unlike in literacy I’m not sure there’s a direct way to say: “here’s how our home math thinks about percents” and “here’s how school math thinks about percents” because I’m not sure if the concept of percent is so universal — but I wonder if there are “home math” ways of expressing proportional relationships that teachers could be more accountable to?

So that’s one way I think about the distinction between home knowledge and school math — that kids clearly have quantitative reasoning skills, as do their parents, neighbors, grandparents, mentors, older cousins/friends/siblings. That knowledge needs to be brought into the classroom and used as resource. I usually try to do it by listening to kids in play situations, or when they’re talking about other stuff, or by putting them in settings in class that elicit that home-knowledge way of thinking, and then finding ways to bring those experiences and conversations into the classroom, with an almost math-ropological lens. “What were you doing there in that conversation? What was so powerful about it? How can we apply it to more situations? How can we all get good at that kind of thinking? How do mathematicians notate that thinking? Can you think about this the mathematician way?” I often think of Lesh and Doerr’s idea of model building as my guide for this (http://www.amazon.com/Beyond-Constructivism-Modeling-Perspectives-Mathematics/dp/0805838228).

But there’s another piece here too. Home knowledge in most homes is usually concerned with quantitative reasoning in context, about stuff we care about. Number play and quantitative imagination in most homes doesn’t go much beyond some wondering about really, really big numbers (this is totally me making stuff up from anecdotal experience and watching people shamelessly on public transit). And way more so that word play and literary imagination, wondering about number (especially about non-whole numbers) is discouraged because it’s seen as too hard and too confusing for parents, peers, older listeners, etc. to engage with.

The exception to this seems to be a certain culture of nerdiness (mostly white boys) who like to take ideas apart and consider them in the pure imagination world — who like numbers because “they don’t lie” or “they aren’t messy” — there’s a cultural value put on mathematical wondering and playfulness that it transcends social and emotional realities. I think that appeals to white male cultural norms (the culture of power!) in particular for all the reasons from elevating “rationality” to having the privilege to decontextualize and believe in a pure, unvarnished, rational truth. I’ve got a source on this one: http://nataliacecire.blogspot.com/2012/11/the-passion-of-nate-silver-sort-of.html

There’s a connection between the nerd culture that elevates math as “un-messy” and school maths. It seems likely to me that for students (of all races and genders, but particularly non-male and non-white) who don’t connect to or see the value of something that sees itself as transcending messiness, then there needs to be both an explicit conversation about things like:

- the conventions of word problems

- why people bother with such “dry” stuff, as, say, proving the Pythagorean theorem

- an appreciation of math that does grapple with messiness (probability & statistics, mathematical modeling, financial math)

- an appreciation of what students bring to math, both in bringing the real-life to math, and also recognizing their power to engage in conversations about abstract mathematical ideas. It’s damaging to assume that because kids cultural background foregrounds different things that they can’t also enjoy and do backgrounded stuff.

So to me, culturally responsive mathematical content recognizes:

- that (almost all) students are already quantitative reasoners

- that most math CONCEPTS kids already are grappling with and can grapple with on their own, and that there are METHODS and PROCEDURES kids will sometimes discover and sometimes by taught by peers or teachers

- that certain kinds of doing math and talking math are valued more in some cultures than others, and that kids have to learn school math but they have a right to experience, discuss, and know how it is similar to and different from the quantitative talk they do at home. Kids know what is valued more by society — knowing that they don’t lack their own math knowledge and culture lets them value their home and school ways of knowing.

And then question 2, how do we honor that kids expect learning math in school to look, sound, and feel a certain way? How do we honor their cultures around authority, teaching, etc? Delpit’s article (and her follow-up, even harder to read, The Silenced Dialogue: Power and Pedagogy in Educating Other Peoples’ Children: http://lmcreadinglist.pbworks.com/f/Delpit+%281988%29.pdf) hit home for me here too. I tend (less so, but still) to act like I don’t have authority in the classroom, neither mathematical nor social. That’s how I was raised to interact with authority — good authority figures softened their authority, good subjects knew how they were supposed to act and did it based on gentle suggestions — both sides got to use questions to politely and carefully negotiate.

Not surprisingly, that is also how questions are used in academic circles — they negotiate rightness and authority, and authority is based in lots of things, but reasonableness is a big one.

Having been in lots of kindergarten and pre-school classrooms, I wonder if the “advantage” white kids have coming into school (besides not having to deal with racism, not getting disproportionately suspended, that kinda stuff) is that they get the way meaning is negotiated in academic circles. I don’t actually believe that most white kids come in with more number sense or better counting or more names of shapes or better one-to-one correspondence (or that if they do, that that accounts for much of the racialized math achievement gap). I believe that white teachers and middle- and upper-class white students already know how to construct shared meaning out of a particular kind of argument structure that is the life-blood of classrooms and academia. And it’s a thing that takes practice, and that everyone is perfectly capable of.

The search for consensus based on logical conclusions, reasoning from definitions, and paring things down to the barest of assumptions is kind of normal in a small subset of households, and emulated in many others who want to be like those households. Kids learn it by negotiating with their parents over bedtimes and allowances and whether Bert is funnier than Ernie and whether a million billion is the biggest number and a bunch of other negotiations that happen because authority comes in large part from being good at that kind of reasoning. These same kids suck at negotiating “disses” or physical conflict because they don’t learn any skills at home for negotiating with authority that deals in smackdowns… because when they encounter cops and the state the cops and the state don’t smack them down, they negotiate. It makes a ton of sense that people treated really differently by state power would teach their kids different ways to deal with authority, and arm their kids with different skills.

Learning how and when it’s safe, and even required, to negotiate with power figures, and learning explicitly how to argue in a way that middle-class white people, and academics of all races, argue, is important. I think it should be an early and explicit and critical part of math class from Day 1. I think culturally responsive teaching draws on kids’ home knowledge and ways of thinking outside of school, explicitly and critically connects it to school knowledge, and also explicitly instructs kids in the cultures of power and schooling. Even if it means showing & analyzing videos of white kids in math class arguing with their teacher about all kinds of things, from whether rectangles are squares to whether it’s fair to give a test on new material the day after it was taught.

See Grace, I told you this post was going to be really long… everything I wrote made me want to write more, and then it got late and I didn’t edit it. I hope it’s not too confusing — maybe I’ll try for a tl;dr* version in the morning.

*tl;dr = “too long; didn’t read”

]]>Kelly was watching her favorite Winter Olympic event – the four-man bobsled. On each team a driver sits in front, followed by two teammates and the brakeman in the back.

The numbers 1 through 20 were assigned to the 20 athletes. The drivers wore numbers 1, 2, 3, 4, and 5.

Before the race Kelly studied the teams. Drivers 2, 3, and 4 were riding with brakemen 18, 15, and 20 respectively in three of the sleds. In another sled were 6 and 12, and in the remaining sled were 10 and 17.

Suddenly Kelly realized that if you added up the four numbers on each team, all five sums were the same!

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