On the #LessonClose hashtag there have already been some great lists of closing structures, and some blogs that reflect more deeply on the logistics, timing, purpose, and value of closing routines.

I thought it might be nice to have an open thread for reflecting on some of the resources shared. Speaking of which, there’s a Google Drive Folder of all the resources shared on the #LessonClose hashtag.

Here’s what I’m thinking about: we’ve started to identify some different decisions that have to be made about Lesson Closes. What other decision points are out there? How do you personally weigh these? What tools do you like to use?

**Timing: **How long do your closes take? How do you make time for them? What do you do when you need a short closer? A longer one? What lessons need longer closers? Could a whole lesson be the close from a previous series of lessons?…

**Public/Private: **When do students get to see each others’ closers? When not? How do you decide? What tools/routines do you use to public closing/private closing? Are private closers always more for the teacher than the student? Are public closers always more for the students than the teachers? How do you decide whether a lesson needs a closer that’s more for your formative assessment purposes and when the whole group needs a closing experience?

**Tools: **Do you use journals? Exit slips? Poster paper? Sticky notes? Technology? Is the technology more like a quiz or a poll (e.g. Kahoot, Socrative, Poll Everywhere) or more like a place where notes/journals can be made public (e.g. blogs, Desmos Activity Builder, Google Docs, or Evernote)? How are the tools you choose informed by timing and public/privateness?

**Oral or written: **New evidence shows that the act of writing changes the learning that happens (and that handwriting and typing are cognitively different too). What are the considerations that go into choosing an oral closer like a class discussion or think-pair-share vs. a written one like a journal or exit ticket? Also, what kinds of activities straddle the space between oral and written (e.g. think-write-share or gallery walks)

**Open vs. Specific ****(New from David Wees) : **There might be several different ideas in here. One is open-ended prompts for students vs. questions with answers (which could range from solving problems and showing work to multiple choice Poll Everywhere type prompts). Another is what I think of as “reusable” prompts vs. specific ones. For example, almost any lesson could end with “How does what we learned today connect to something you already know?” Or “What new wonderings do you have based on today’s lesson?” Or “Based on today’s lesson, what do you think we will do tomorrow?”. But only one specific lesson could end with, “David solved the brownie problem using a table, and Shelly solved it using a pattern that she noticed after a few guesses. Copy some rows of David’s table into your notebook that you think show the pattern Shelly noticed. Use words and arrows to describe how Shelly’s pattern shows up in David’s table.”

**Types of Reflection Questions ****(New from David Wees) : **David started us off with some examples, and I brainstormed some more:

- Reflecting on what you do and don’t understand (e.g. “One thing I learned… One thing I am wondering”, or rating your understanding on a scale or stoplight)
- Predicting what will happen next
- Listing new questions that have come up
- Summarizing
- Taking a stance on a controversial topic (e.g. based on today’s lesson, what is your answer now to the question “Is the answer to a multiplication problem always greater than the numbers in the problem?”)
- Connecting representations
- Solving a new problem using a suggested strategy (or multiple strategies)
- Teaching/explaining a procedure
- Reflecting on personal feeling about the lesson (e.g. what was hard/what was easy)
- Connecting strategies (from
*Intentional Talk*) - Troubleshooting a common error/error analysis (from
*Intentional Talk*)- Also finding the hidden error/mistake game (from many blogs and tweets)

- Attempting to justify a conclusion that was drawn (from
*Intentional Talk*) - Attempting to clarify or put in your own words a strategy or concept or argument from the lesson (from
*Intentional Talk*)

What other dimensions are there? How do you think about them? Would these be useful dimensions for organizing lesson closing activities?

]]>There seem to be several conversations among math teacher bloggers and Tweeters about if and how they use “non-routine problems,” the role of asking vs. telling, whether it’s okay to give students hints or not, that often come down to a belief that sounds sort of like this:

*The best teachers say the least*

Math teacher hot-button terms like “productive struggle,” “high cognitive demand,” and even “problem solving” are lumped in with the two ideas or beliefs that the articles above debunk:

- It’s important to teach kids to solve novel, non-routine problems
- The best teaching is minimally guided — no hints, no lectures, no worksheets

Are you surprised that I think the two articles are basically right — not in the grand claims they make about how math should be taught, but in their specific claims of what the research shows about teaching and learning? I think that the math education research community has gotten pretty sloppy about both of these claims, which opens up a lot of good thinking and good research about how to teach to criticism it doesn’t deserve. Here’s why:

- I think, for the most part, kids don’t need to be taught how to solve novel, non-routine problems. Any generalized, transferrable, “higher-order thinking” skills that exist are so general that kids already have them, and from a young age. Research on Cognitively Guided Instruction shows that young kids are capable of reasoning about mathematical situations in order to get correct answers.
- If the best teaching were minimally guided, why would humans have organized kids into schools for centuries? The more technology, the more culture, the more stuff we humans produce, the more we need to ensure that our kids master skills, gain knowledge, and develop specific ways of communicating that let them be productive members of society — and it really is content knowledge that differentiates experts from novices.

Okay — so if kids don’t need to learn problem solving skills, and the point of school is to master content, then shouldn’t we just tell kids what we want them to know, and get on with it as quickly as possible?

NO! That’s where I differ a LOT with the authors of the articles cited above. I draw very different conclusions from the research about the importance of domain-specific content knowledge in problem solving, for several reasons:

- Experts in the field have organized their knowledge in highly useful, highly interconnected ways (schemas is one name for that, I think). There’s lots of evidence that, for generations now, most American students are not graduating with highly useful, highly interconnected deep content knowledge in mathematics. And I’m not talking about the knowledge needed to be a mathematician, just the knowledge needed to, say, reason confidently about percents to make smart decisions about spending money. It seems pretty clear to me that the worked examples and exercises based on textbooks methods that still dominate most math classrooms across the country aren’t creating experts, despite exposing students to lots of content knowledge and lots of worked examples. Maybe we’re using the wrong examples, but I think it’s deeper than that….
- There’s an interesting (and growing) body of work that indicates that learning from listening & seeing worked examples is not easy. Experts learn well from lecture, because they already have organized schemas for attending to the important information, and they already have questions in mind that they use to make sense of what they are hearing. They interrogate the material and their own understanding of it as they listen (and after they listen). Novices, on the other hand, may attend to non-salient details, they may think they understand something but actually be completely wrong, and they might misremember what they heard in the lecture to match their own misconceptions, rather than adjusting their thinking to take in new information (Derek Muller of Veritasium fame breaks down some of that research in a video, but if you love methods sections and analyses and stuff, you might check out his dissertation or a related journal article about how students learn more in settings that create confusion, rather than clarity).
- Students are perfectly capable of solving problems, and learning from their own problem solving. Books like Children’s Mathematics and Young Children Reinvent Arithmetic demonstrate how children can solve problems that they haven’t been instructed in solving or shown examples of, using everything from direct modeling to sophisticated numerical reasoning. I combine this idea that children come to us able to reason and think mathematically, with Manu Kapur’s work on productive failure and the idea that students seem to learn more from lectures/direct instruction after being put in a situation where they grapple with (and ultimately fail to solve) ill-defined problems for which the content they are about to be taught is useful.

The ideas that mathematics must be taught in a way that enables novice learners to organize it into a highly interconnected, coherent schema or body of knowledge, that novices don’t necessarily have the skills to learn the way experts do (let alone the motivation or disposition/mindset, both of which have been shown to have huge effects of ability to learn), and that novices do have problem solving skills that they haven’t been taught but are rather part of their birthright as thinkers, suggests to me that there are different conclusion to be drawn from studies showing that minimal guidance is not effective and generalized problem solving skills aren’t what separates novice from experts.

The conclusion I draw is that problem-solving is a tool to be used in the service of learning content, and that content in turn becomes a tool to solve more and more interesting problems. I think that what students are learning in math is specific ways of solving problems (and learning what kinds of problems are interesting to try to solve, and what kinds of problems math can help with, and how to see the world as mathematically rich). I think students come to us with varying degrees of ability to solve problems in novice ways, and we help them develop their existing schemas and get more expert and efficient and solving problems by helping them see coherent “understanding stories” from basic understandings to highly mathematical, useful skills. For example, we help students move from modeling arithmetic problems with objects and counting to solve them, to making use of place value to fluently solve arithmetic problems in their head. We do this, not through minimal guidance, and not through worked example after worked example, but by carefully structured sequences of tasks that build on what we expect all students to be able to do, which we use to find out novice versions of increasingly mathematical ideas, and use tasks and worked examples and student-shared examples to scaffold students to develop more expert skills and content knowledge — knowledge that is connected and useful and feels relevant and important to the student.

Here’s why I think this carefully guided instruction works (and is hard, and is basically what everyone whose kids are learning is doing, albeit with a different style, whether their desks are in rows facing a chalkboard or their kids run around in fields in schools with no walls):

Mathematics is a coherent, highly interconnected body of knowledge, that includes both mathematical objects and (specific) mathematical ways of thinking. For example: algebra. Algebra is a set of thinking tools and heuristics (rewrite expressions to make them simpler, make strategic use of equivalence & its properties) and a set of master-able skills (use variables accurately, manipulate expressions & equations accurately, solve equations). There is a grammar and a vocabulary and a way of thinking. Students come to us with more and less novice versions of the thinking tools and heuristics (just ask them to go play with EDC’s mobiles), and not only do we need to help them master the skills and vocabulary, we need to link those skills and vocabulary to their existing schema around equivalence, while strengthening that schema.

I believe in the *learning theory *of constructivism — the idea that learners are actively engaged in trying to make sense of instruction* — AND I believe the research suggesting that they are not very good at making sense of what we want them to make sense of, whether it’s because of our instruction or whether it’s because they have unproductive beliefs about math and themselves, or a lack of motivation to learn, or they don’t know what to attend to, or they are assimilating what they are hearing into their own (wrong) ideas rather than shifting their thinking to take up new beliefs. So everything we want students to learn through listening and observation has to be presented to students who are primed to learn, in ways that support them to attend accurately, to recognize how the new material contradicts their current thinking, and to have chances to test their new (and old) beliefs against reality until they are solidified.

I also believe that students’ beliefs about things like who does math, what it means to do math, how math is learned, etc. are so important that pedagogy has to take them into account — even if that means sacrificing clarity or efficiency to support the belief that math can make sense to students, and good math ideas can come from anyone.

By now, it should be clear that while it would be cool if “minimally guided instruction that teaches students to solve any kind of problem” worked, it’s unlikely to. Carefully guided instruction, which is what I think actually works, involves thinking like:

- balancing the desire to help students master content exceedingly fast with the desire to provide experiences from which students can feel like “I can do this! I have the power to think mathematically!”
- being strategic about WHO provides worked examples — do they come from a carefully selected peer or from the teacher? Which will best support sense-making, engagement, and the learning culture the teacher believes these learners need?
- ensuring that students are ready to learn from instruction — that they have questions/curiosity to guide their listening, and a sense of what will be important to attend to and what to ignore
- sequencing the strategies that are shared from more novice to more sophisticated, to create a coherent storyline of how the formal math connects to student thinking.
- harnessing students’ existing abilities as problem solvers to create situations in which students are ready to learn math, rather than assume students aren’t problem solvers and try to teach them to be.

So… I would like math ed researchers to stop trying to prove that we can teach kids how to solve all math problems by teaching them a dozen strategies and turning them loose, and assuming that problem solving can replace learning content. And I would like them to stop contrasting minimally guided instruction with worked examples and direct instruction, and instead focus on helping teachers understand the kind of guidance and the kind of examples and the timing of examples and who should present examples so that the most learning happens.

I would like to see even more research that helps teachers answer questions like:

- what is the range of problem-solving skills that students present with at different age levels, and how can we support all students to have a basic ability to, say, understand a problem context and be willing to try stuff?
- what are best practices for supporting productive disposition and metacognition (which, it seems, can be situational or become more robust and actually transfer) — how do we help students be better math students?
- after arithmetic, where we actually have a pretty good sense of the strategies that young children invent, and how to support learners who need to see more examples and need more guidance to learn that from their peers in a culture that supports motivation and productive disposition (yes, I’m talking about CGI!), what are the conceptions that children are likely to come to use with, and what are the “understanding stories” that help move students from novice conceptions to a highly connected, more expert body of knowledge?
- what are the main factors teachers use to make teaching and learning decisions, and what is it useful to know about mathematics and learners to make better decisions? How do we help?
- how can we re-conceptualize the body of math knowledge so that it is clear to teachers and learners what the interconnections are, what kinds of problems each domain is helping to answer… essentially how do we make it clear the main, important thinking tools of math to ensure we are teaching students the most important ideas, as ideas, and making them super full of velcro-hooks for attaching more ideas to.

Wanna play?

*or, if they aren’t trying to make sense of math, at least they are trying to be successful in or cope with school — students often come to realize that this means making sense of the teacher’s motivations, expectations, grading schemes, etc. and not the content itself; my favorite example of this is in Jean Lave’s “The Culture of Acquisition and the Practice of Understanding” on page 29, which, sadly, is not part of the Google Books preview, but is summarized here

]]>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”

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**1) “You can’t divide by zero.” Explain why not, (even though, of course, you can multiply by zero.)**

First of all, what would it mean? Is anything divided by zero equal to infinity? That’s what I thought as a kid, but I’ve since encountered experiences that suggest that 3/0 = ∞ is thinking of infinity as a number and thus not really mathematically sensible. But I got to that idea by thinking of how I understood division: How many times does 0 go into 3? Or, what times 0 gives 3? But even infinity isn’t a sensible answer to those questions. The limit of the sum of a bunch of zeros as the size of the bunch approaches infinity is not 3. Infinity zeroes is not 3 or any whole number. So those are intuitive reasons why the answer isn’t zero, it’s not even defined.

Mathematically, though, the best reason I’ve heard for why 3/0 is undefined is that if it were defined, you could make any number equal any other number! For example, if 3x = 5x, does that imply 3 = 5. Nope: you have 2 possible moves you can make solving 3x = 5x. Either use the additive property of equality to write 0 = 2x, and then the division property of equality to write x = 0. 3x = 5x -> 0 = x. Or you can try to apply the division property of equality immediately: 3x = 5x -> 3x/x = 5x/x -> 3 = 5. You have two choices for not breaking math. Either accept that 3 = 5, or say that the division property of equality only holds when you’re not dividing by 0, and since 3x = 5x is only true when x = 0, 3x = 5x only implies 3 = 5 if the division property of equality allows dividing by 0.

That’s kind of related to the fact that if 3/0 = ∞ (or even if 3/0 = some special made-up number §) then that means 3 = 0 * ∞ or 3 = 0 * §… but then doesn’t 5 = 0 * ∞ also since 5/0 = ∞? And if 3 = 0 * ∞ = 5, then 3 = 5 again. Unless you want to invent this whole shadow number system so 3/0 = §_{3} and π/0 = §_{π}… but then what the heck kind of arithmetic would you do with those numbers? How would you visualize them? Why would you do this? On the other hand, please see question 10.

**2) “Solving problems typically requires finding equivalent statements that simplify the problem” Explain – and in so doing, define the meaning of the = sign.**

Well I know an easy example of this. Say I want to find the value of x that makes this statement true: 2x + 4π + 12 = 7(x + 2π) – 5x + 10π + x

I could guess values of x and see when I got a true statement. I could plot a graph of each side and look for the point of intersection. But easiest would be if I could simplify the above statement without changing which value of x made it true.

So… if 2x + 4π + 12 = 7(x + 2π) – 5x – 10π + x

That’s equivalent to 2(x + 2π) + 12 = 7(x + 2π) – 5(x + 2π) + x, by the distributive property of equality

Which is equivalent to 2(x + 2π) + 12 = 2(x + 2π) + x, by association and the distributive property and arithmetic

Which makes it easy to quickly guess and confirm that x had better equal 12.

Of course, I made up the example so I knew just which simplifications would get me quickly to a form of the equation that was easy to solve for x. There were many points at which I could have made other decisions and written the expression in other, equivalent ways, such as by expanding any terms with parentheses, combining like terms, adding and subtracting the same thing from both sides, etc.

As for what the = sign means, it means that the expressions on either side of it are equivalent aka have the same value. Because of some properties of equivalence relationships in our math, it’s “legal” to do certain kinds of moves to equal statements and you know you haven’t changed the truth of that equals sign.

I wonder what examples people might have come up with that didn’t have to do with the = sign? Or that had to do with the = sign but not solving for x?

**3) You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.**

Here is one reason it works. a/b ÷ c/d asks, “how many c/d’s are in a/b?” Because of what fractions mean, we can think of c/d as c 1/d’s. In other words, 3/5 is three one-fifths, 10/7 is ten copies of 1/7, etc. So… if we think of a/b ÷ c/d as asking “How many c/d’s are in a/b?” an easy way to calculate that is to ask first about how many 1/d’s are in a/b.”

Example: How many 3/5 are in 10/7? First lets find out how many fifths are in 10/7. There are 5 fifths in every unit, so there are 10/7 of 5 fifths in 10/7 of a unit. How many 11/20 are in 3/2? There are twenty 1/20s in 1, so there are 20 + 10 1/20s in 1 1/2. There are 20 * 3/2 or 30 twentieths in 3/2.

Now how do we use the fact that we know how many fifths are in 10/7 to find how many 3/5 are in 10/7? How do we use the fact we know how many 20ths are in 3/2 to know how many 11/20s are in 3/2? How do we use knowing how many 1/ds are in a/b to know how many c/ds are in a/b?

Well, if we know there are 50/7 fifths in 10/7, and we want to find how many sets of 3 of those fifths are in 10/7, we just break the 50/7 into groups of 3. Aka divide 50/7 by 3. The answer is 50/21. We got that by doing 10/7 multiplied by 5 and divided by 3, which is the same as 10/7 * 5/3.

Same with the 3/2 ÷ 11/20 example.

3/2 = One whole and a half.

There are 20 twentieths in the whole, and 10 twentieths in the half, for 30 twentieths in all.

How many sets of 11 twentieths are in 30 twentieths?

30/11 of course!

And finally: There are a/b * d 1/ds in a/b. There are c groups of 1/d in c/d. So there are (a/b * d)/c c/ds in a/b, aka a/b ÷ c/d = (a/b * d)/c = a/b * d/c.

**4) Place these numbers in order of largest to smallest: .00156, 1/60, .0015, .001, .002**

At least I don’t have to write an essay for this one… First of all, 1/60 = 100/6000 = (100/6)/1000 = (16 2/3)/1000 = 16.666666…/1000 = 1.66666…/100 = .1666…/10 = .01666…./1 = 0.016666…

So, in order, we have .002 > .00166666… = 1/60 > .00156 > .00150 > .0010

I think you were trying to trick me (and it did make bells go off in my head) with what to do when there are no digits after the first ones you compare. Like is .001 bigger than or smaller than .0015, since .0015 is fifteen ten-thousandths and .001 is one thousandth. And since Alum comes before Alumna when you are alphabetizing, since a blank after a letter is prior to any other letter after a letter.

But I wasn’t fooled! .0015 is five ten-thousandths bigger than .001 which is ten ten-thousandths!

**5) “Multiplication is just repeated addition.” Explain why this statement is false, giving examples.**

Gracious! It’s a good thing you included the “just” because it’s certainly false that multiplication is just-as-in-only repeated addition. Though repeated addition can be used to calculate and conceptualize many kinds of multiplication problems. Here are two examples where repeated addition doesn’t make sense.

-1 * -1 = 1.

Does adding -1 to itself -1 times make any sense? And if it does, would it really make sense that the result would be 1? Nah…

sqrt(2) * sqrt(2) = 2

What does it mean to add something to itself a irrational number of times? If you really get deep with it, you could define this using the distributive property as a limit: sqrt(2) * 1 + sqrt(2) * 4/10 + sqrt(2) * 1/100 + sqrt(2) * 4/1000 + … which (assuming you’re comfortable with adding something to itself part of a time) *approaches* 2. But then you have to be comfortable with defining sqrt(2) * sqrt(2) as a limit that approaches 2, and not just plain old 2, which seems kind of sad. Even though these days the limit and the thing itself are thought to be the same, such as 0.9999999999… = 1, I am still more comfortable with having a concept of multiplication such that sqrt(2) * sqrt(2) just is 2, no limits, based on, say, an area model.

**6) A catering company rents out tables for big parties. 8 people can sit around a table. A school is giving a party for parents, siblings, students and teachers. The guest list totals 243. How many tables should the school rent?**

Piece of cake. You think I’m going to be dumb about remainders or rounding or give an exact answer, but I know this trick!

243/8 = 30 tables with 3 people having no seats.

The school could rent 31 tables to be on the safe side, or they could take a risk and just rent 30 tables figuring that if 243 people say they’re coming it would be crazy to expect exactly all 243 to show up. All it would take is one family feeling under the weather and you’d have a whole unused table.

**7) Most teachers assign final grades by using the mathematical mean (the “average”) to determine them. Give at least 2 reasons why the mean may not be the best measure of achievement by explaining what the mean hides.**

Umm, is this a math question or an assessment question? One thing that the mean hides is whether you know some things not some different things, or whether you kind of know a lot of things. For example, what if your grade was based on 2 assessments, one on each of 2 major topics? What if you were taking a class on how to be nice and clean and you had a test on cleanliness and a test on niceness. Say that you are a really sweet, stinky person and you learned a lot about niceness and nothing about cleanliness, and so you got a 100% on your niceness test and a 0% on your cleanliness test. Your classmate is an okay person and kinda messy about personal hygiene, and got a 50% on both tests. You both have the same final grade (50%) in the class even though you have really different profiles as nice, clean people, and really different needs for remediation.

Another thing the mean hides is change over time. Let’s say you’re taking a class on learning to ride a bike, and when you start the class you can’t stay on the bike for even 1 meter. By the time you’re done, you can stay on the bike without putting a foot down for balance for 1000 meters. And say you developed that skill all of a sudden late in the class, so your distance assessments looked something like:

1, 1, 1, 1, 2, 3, 5, 5, 3, 4, 2, 3, 1, 3, 5, 2, 5, 5, 200, 500, 800, 800, 1000, 800, 1000. Your mean would be about 206 meters. But what about your friend who started out the class as a fine biker but developed vertigo towards the end of the course: 1000, 1000, 1000, 1000, 1000, 0, 0, 0, 0, 0, 0, 0. Her mean distance is about 416 meters. Does that mean she’s a better biker than you are at the end of the course?

**8) Construct a mathematical equation that describes the mathematical relationship between feet and yards. HINT: all you need as parts of the equation are F, Y, =, and 3.**

The length of 3 feet is as long as the length of 1 yard. That does not mean 3F = Y though because F usually refers to the number of feet and Y usually refers to the number of yards, not the length of feet and yards. Since the length of 3 feet is as long as the length of 1 yard, the number of feet in a given distance is three times the number of yards in a given distance and therefore F = 3Y. If we defined F as the length of a foot and Y as the length of a yard, it might make sense to say 3F = Y but that would not help us know how many yards long a distance we had measured as a number of feet was.

**9) As you know, PEMDAS is shorthand for the order of operations for evaluating complex expressions (Parentheses, then Exponents, etc.). The order of operations is a convention. X(A + B) = XA + XB is the distributive property. It is a law. What is the difference between a convention and a law, then? Give another example of each.**

Shoot… I’m surprised that X(A + B) = XA + XB is a law. I thought it was a property of our arithmetic system. Like maybe an axiom? Or a property of a ring or a field or some such. Aren’t there arithmetics without distributive properties? I think, though, that I could talk some about what is different between PEMDAS and the distributive property (although it would have been helpful to have said “the distributive property of multiplication over addition in the real numbers” or something more official like that).

So… what’s the difference between PEMDAS and the distributive property (aka the d.p.o.m.o.a.i.t.r.n-or-something?)

Well, I think it’s that changing PEMDAS wouldn’t fundamentally change the way numbers relate to each other, how we solve problems, how we do math… just how we write it. Changing the distributive property would make it really hard to solve equation, it would change what was equivalent to what, it would make it really hard to know how to think about arithmetic with integers or fractions, etc.

Back in the day, there wasn’t really PEMDAS because we didn’t use symbols much. If someone wanted to write 3x + 7 = 10, he (usually, though sometimes she) would write something like “A quantity is tripled and seven is added to the result. The sum is ten.” I think there are good reasons that mathematicians chose to invent a symbol system where multiplication and division are done before addition and subtraction, exponentiation is done before multiplication and division, precisely because of the distributive property, and I have a hard time believing that there are clearer, more compact ways to express “A quantity is tripled and seven is added to the result. The sum is ten.” than using our PEMDAS, but I imagine that just like there are many languages and ways to express the same thought, there can be many mathematical ways to express the same fundamental relationship. And there are different ways that we express calculations. There’s a computer notation called “Reverse Polish Notation” in which you press 3,5,+ to get the computer to add three and five. If you want to do 3 – 4 + 5, you press 3 4 – 5 +. It makes it so you don’t really need parentheses. The math is exactly the same though, it’s just how you tell the computer what to do that’s different.

Now what about the distributive property? Does it have to do with PEMDAS? Or is it independent of that? Can you use the distributive property in Reverse Polish Notation or does it require a system with parentheses? Answer: Yes, the distributive property is independent of how you write it, and has to do with two things being equivalent. The distributive property of multiplication over addition tells you that X(A + B) is equivalent to XA + XB. It doesn’t matter if you write it X(A + B) = XA + XB or A,B+X* = X,A*X,B*+. (A way better explanation of this is here: http://mathforum.org/kb/message.jspa?messageID=4538619)

One reason arithmetic would be different without the distributive property is that the distributive property lets you know that -1 * -1 has to be equal to 1. Because we know -1 * 0 has to be equal to -1 * (anything that adds up to zero if you do the adding first). Whatever notation you use to tell you to do the adding first, just make it clear to people. So without using PEMDAS I could write -1 * 0 is equivalent to adding 1 and -1 and then multiplying that sum by -1, since -1 + 1 = 0. Using PEMDAS I could write -1 * 0 = -1 * (1 + -1). Now, without the distributive property, I’d be done. I wouldn’t know if -1 * 1 + -1 * -1 was equivalent to -1 (1 + -1) and so I couldn’t simplify or further calculate anything about -1 (1 + -1) other than to go back and forth between -1 (1 + -1) and -1 * 0. But with the distributive property I can keep going. If I use words and not PEMDAS I can write -1 * 0 is equivalent to adding 1 and -1 and then multiplying that sum by -1, which is equivalent, by the distributive property, to writing doing -1 * 1 and -1 * -1 and adding the results. So I know now that -1 * -1 + -1 * 1 = 1 * 0 = 0. And I know that -1 * 1 = -1 because anything times 1 is itself. So -1 * -1 + -1 = 0. The only thing I can add to -1 to get 0 is 1, so -1 * -1 must equal 1. But without the distributive property, how would I have know that!?! Without the distributive property, we can’t prove a lot of the arithmetic we take for granted (like combining like terms, using place value-based algorithms, etc.)

**10) Why were imaginary numbers invented? [EXTRA CREDIT for 12th graders: Why was the calculus invented?]**

I actually know the history on this one. I’m skipping the extra credit. My understanding is that a LOOOONG time ago Italian guys wanted to come up with rules for finding intersections of cubic functions and lines. Every line intersects every cubic function, but sometimes their formula that usually worked spit out values for intersections that involved adding up expressions involving roots of negative numbers. “Yuck, our formula is broken,” many of them thought, but others thought, “Well, but what if there *were* a reasonable way to do arithmetic on these puppies?” They knew what their answers should come out to by carefully graphing and finding the intersections, and they saw ways to manipulate their answers using the usual rules of multiplication with just a light suspension of disbelief, and get to the known intersection. But… this line of reasoning was pretty well abandoned because even though it mostly didn’t break math (the distributive property, commutative and associative properties, and all the usual results of multiplication on the reals still held), having two positive numbers which, when multiplied, made a negative result, did seem to break math. And breaking math is generally frowned upon by mathematicians and math students alike.

So… why do we still use them? For some reason, someone wanted to think more about these roots, and had a reason to think of them on a Cartesian plane. They were able to show that these numbers could be thought of as vectors on a plane, and the algebraic operations on these numbers not only made sense visually on the plane but also made a hard problem (finding the results of rotating and stretching a vector) easier. So then people gradually came to accept that it was okay to break the idea of “a positive times a positive is a positive” because of thinking of multiplication in a new way, since this new way didn’t break any other ideas about multiplication, had a great visual analogy, and made life easier in an important new field of vectors.

**11) What’s the difference between an “accurate” answer and “an appropriately precise” answer? (HINT: when is the answer on your calculator inappropriate?)**

If my teacher gives a daily subjective participation grade out of 3 and a daily subjective homework completion grade out of 5 and I had some quizzes out of 10 and 20 points and some tests out of 100 points and my final point total is 538/635 then it is inappropriately precise to say that my grade is 89.449% and to quibble with me about whether that rounds to an A- or a B+. If we only went to the 1s place in our point calculations, then any decimal after the 1s place is meaningless, and maybe even after the 10s place. Best to round to the nearest 10 and say I got an A-.

**12) “In geometry, we begin with undefined terms.” Here’s what’s odd, though: every Geometry textbook always draw points, lines, and planes in exactly the same familiar and obvious way – as if we CAN define them, at least visually. So: define “undefined term” and explain why it doesn’t mean that points and lines have to be drawn the way we draw them; nor does it mean, on the other hand, that math chaos will ensue if there are no definitions or familiar images for the basic elements.**

I have no idea! I have hung out in Geometries where “planes” were not flat, they were hyperbolic or lived on spheres. I’ve met Geometries where circles and lines were equivalent, and called clines, and Geometries that included a “point at infinity” among other points. Geometry was still meaningful, that’s for sure, and the points, lines, and planes weren’t like good old Euclidean ones.

What I don’t know is why these different kinds of points, lines, and planes are “undefined”? How do we differentiate between this kind of plane and that kind of plane if they are undefined? Is the undefined part the part that makes 2 very different planes still planar in some sense?

Maybe points, lines, and planes (and other undefined terms) are like pornography, and we don’t have to define them because we know them when we see them? http://en.wikipedia.org/wiki/I_know_it_when_I_see_it

**13) “In geometry we assume many axioms.” What’s the difference between valid and goofy axioms – in other words, what gives us the right to assume the axioms we do in Euclidean geometry?**

I don’t know. I think that’s an open question. For a long time, didn’t we think that it would be goofy to imagine a geometry without Euclid’s 5th postulate? And then it turns out it wasn’t? Didn’t most smart people agree it was goofy to violate the “positive times a positive equals a positive” until they finally agreed that complex numbers were useful and meaningful? Isn’t negotiating whether an axiom will turn out to be goofy and valid part of what makes mathematics a living, breathing, fuzzy, human, creative domain? And what makes math so hard? You kind of are walking a line between madness and sanity when you make rigorous truly new mathematics. How do we know there will never be a mathematics in which dividing by zero is useful and meaningful? A geometry which angles can be trisected using only valid construction tools? I’m not sure we do!

Update:

David Radcliffe’s answers: https://docs.google.com/document/d/1FK8m_oVaS_UWS4grETMAfAhByN_vrQC0TR85ld5FXj8/edit

Erik Johnson’s answers: http://step1trysomething.wordpress.com/2014/04/25/answering-the-conceptual-questions/

Welcome:

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**problem-solving**or**communication?** - Anything else you’re looking/hoping for?

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I was guest-teaching a class that had been studying exponential growth (doubling and tripling, mostly), writing recursive and explicit rules, making tables, and graphing. Today, we were introducing the concept of compound interest as an exponential scenario, and the teacher’s goal was to have the kids recognize just that — compound interest can be modeled with exponential functions (and is cool and important, because free money!).

I was expecting that the hard bits would be the vocabulary of compound interest, the concept of investing, and the concept of the bank giving you more money each month because you have more money each month. And all of those things were hard for some subset of kids, but what was really hard for the majority of kids was doing mathematical operations with the quantity 8%.

Here’s the lesson plan:

- Pair kids and ask them to discuss, “You won the lottery and there are two prize options: would you rather have $10,000 now or $20,000 in ten years? And what would you do with the money?”
- Individuals answer the above question by making a graph of the money they’d have over the next 10 years based on their choice.
- Share a few different graphs on the document cam and discuss the story they tell.
- Pretend to get a call from a banker offering to hold the $10,000 in a CD earning 8% interest, compounded annually.
- Ask kids “What would you do now? Take $10,000 now, $20,000 in ten years, or invest the $10,000 in the 8% interest rate account for 10 years?”
- Define the question: will earning 8% interest on $10,000 over 10 years earn you more money than just waiting and letting the lottery folks give you $20,000 in ten years.
- Turn students loose to use a table (strongly encouraged), graph, or rule to help them answer that question, using their knowledge of percents.
- Come together as a class to compare & discuss tables, and convert tables to graphs and tables and graphs to rules to help us understand what’s going on in compound interest situations.
- Practice on another compound interest situation.

So it turned out that the students right away thought of investing. On their warm-up graphs, some saw investment as a linear scenario (I’ll get $1000 a year from investing in the stock market) and others saw their growth rate increasing as their money grew (exponential graphs).There was engaged discussion about spending vs. saving vs. investing, and more than one student knew what compounded annually meant when the banker called.

Here was the part I wasn’t quite prepared for. When 8% was mentioned, I right away heard fear about the use of percents, and students asking tentatively, “doesn’t that mean we have to divide something by 8 or zero-point-eight?” Maybe half the students knew that 8% was the decimal 0.08 (most went right away to 0.8, which sounds a lot like point-zero-eight but is a lot different!), and only one could state clearly that 8 percent was 8/100 which was 0.08, since 0.8 was 8/10 or 80/100. And worse, even once we’d established that 8% could be typed on a calculator as 0.08, more students wanted to divide than multiply (I guess because they wanted a smaller number)?

So the conversation about making the table wasn’t around what I was hoping — identifying those students who’d found 8% of the principal but forgot to add it to the principal and so who got 10,000, 800, 64, for the balance, and identifying those students who’d found that she made $800 in interest the first year and so reasoned she’d get $10,000; $10,800; $11,600; $12,400; etc. Then we could have talked about linear vs. exponential growth and what that has to do with compounding, and we could have talked about 8% vs. 108%, and gotten into some key ideas that would let us make sense of an exponential rule.

Instead we got bogged down in 8% vs. 80%, and number that didn’t make any sense, like $10,000 / 8 or .8 or .08. Even though we talked through those issues, I lost a lot of kids in talking through those issues, and we didn’t have very good conversations about the other, more connected-to-the-concept-of-exponential-growth conversations.

So, what would you have done, with 20/20 hindsight? Here are my questions:

- Can a class of students, the majority of whom don’t understand how to calculate with percents, learn compound interest? Should they be asked to? If so, how? If not, what should we do instead?
- Would there have been a way to differentiate this lesson so that more students got through the right amount of it at their own pace? What could I have expected everyone to have gotten? What could I have done to support the most struggling? The most advanced?
- Could technology have helped students focus on the concepts, rather than calculations? How? What technology?
- Is there a way to present this concept more conceptually, so we avoided getting bogged down in the calculations? Were the students conceptually ready but lacked fluency, or were there underlying conceptual issues I needed to address?
- How long does it take a good understanding of percents to sink in? Did I need to stop and teach a conceptual foundation for percents and then come back to compound interest? What would that do to my pacing guide (already pretty far behind with snow and 4 rounds of state testing instead of the 1-2 we used to do)?
- And finally, does thinking about this particular story give us any more insight into the general question of teaching kids who come in with gaps in their understanding, fluency, or both?

I think it’s really important to be aware of and responsive to students’ outside-of-math-class cultures and experiences. Here’s why. I want all my students to work as mathematicians in my class. I believe that all students have a playground in their brain for shapes and quantities, that all students can wonder, observe, conjecture, explore, refine, reason, explain… And I also believe that all my students are pretty novice about these things. I have to be able to listen to them and hear them as mathematical thinkers, even when their expressions are very, very novice — and therefore different from what I’m used to hearing.

Students’ outside-of-math-class experiences change the “accent” in which they speak math. Not necessarily their actual accent (though some of our students have accents or speak math in a different language than we do!) but more the way they think and express their thinking.

We don’t realize we all have accents. We think of people with accents that are strongly different than our own as being “weird” “different” “cool” “special” — and rarely think of ourselves as being people with accents (unless we have an accent that’s different from the people around us so it gets pointed out to us a lot).

I have a hard time recognizing my students doing math and being mathematicians when I am distracted by their “accent,” and I’m more distracted by it the more different it is from how I conjecture about, reason about, and explain quantities and shapes. In fact, I don’t always realize I have an accent — I believe I’m fluent in math and speak the universal language of math. Which makes what my students are saying and thinking sound “weird” “special” and even wrong.

Here are some concrete examples that I’ve read about or experienced:

- Dealing with uncertainty and the issue of right/wrong — different cultures, and in the US, different economic classes, have different ways of thinking about uncertainty, especially when it comes to uncertainty in school. In some cultures (for example, the latte-sipping liberal elite!) school is a place to learn to argue, to poke holes in accepted ideas, to get troubled and cause trouble. In other cultures school is a place to learn the right way to behave, talk, etc. School is your ticket into proper society… so encountering uncertainty, challenging authority, poking holes, etc. is bad news. This doesn’t mean that students whose cultures relate differently to authority and knowing should be told what to do and made to just memorize procedures… but it does mean that they’ll engage differently with the challenge of checking on their own whether a solution is correct, that they will need explicit coaching & parameters around times that they are debating the relative merits of different arguments when there is a teacher in the room who could be the voice of correctness but is choosing not to be.
- Written vs. oral communication — like everything, there is as much variation among individuals of the same “culture” as there is between cultures on this… but… that said: the role of writing vs. talking is different in different cultures. In some cultures the word on the page has a different kind of authority than the spoken word. In some cultures reading and writing is more of a communal, oral exercise where people talk about what they’re reading or writing as they read and write it. In some cultures girls are brought up to be more reserved, to not blurt — and so they may find it easier to use writing to marshall their thoughts before speaking. In some cultures, you talk things out before you put anything down on paper. So a “think-pair-share” task where students write silently and then share orally may go awesome in one classroom and bomb in another, because the students in the other classroom would have to “talk-think-write” to be successful on the same task.
- Consensus building & arguments — this one I’m more out on a limb about. I don’t know the research and have only read anecdotes but… it seems to me like one thing that happens in math class is there’s a tension between knowing things because you’ve deduced them through rigorous logic, knowing things because you have faith in the person telling them to you, and knowing things because they seem right inductively. Back in the Dolciani days, in theory everything was deduced through rigorous logic, and in practice a lot was taken on faith from the teacher. Student-led investigations these days in theory begin with intuition and inductive reasoning (wow, in all these examples, making a triangle just knowing the side lengths guaranteed that my triangle was exactly the same as yours) and end with rigorous deduction (that will always be true because I can prove there will always be a rigid transformation that maps the three sides onto their corresponding sides and since rigid transformations preserve shape and angle measure, all the corresponding parts will be the same). In practice, I think a fair number of investigations build up students intuition and inductive reasoning, but then the conclusion that this always works is taken on faith again. And it seems to me that there’s a cultural piece at work here. What is the way to win an argument in students’ out-of-math culture? Convincing others? How? By logic? By coercion? By force of personality? By showing lots of examples? By offering counter-arguments? Which is more important — being right or everyone agreeing? If students’ out-of-school experience is that agreeing is important, and that it’s not polite to bring a counter-argument if everyone else agrees, then where does that leave them in math class? How do we honor their experience of argument and also help them participate in doing math, as mathematicians?

It’s so, so important to me to meet mathematicians and math teachers from many cultures, and hear them talk about their experiences with school mathematics, their children’s experiences in school mathematics, and their students’ experiences with school mathematics, to help me hear my own accent.

As a white, American, mostly East Coast, mostly male, mostly middle-class person, I am so used to argumentation and picking things apart and valuing logic and trying to see “why does that work?” and “will that always work?”, and doing that kind of argument everywhere from the dinner table to friends’ houses that when I don’t see those same “accents” in my students I think they aren’t mathematical. I don’t see my students’ use of different kinds of mathematical knowing as they use their spatial intuition (not my strength) and can just “see” why something works, let alone when they decide to agree on a conclusion that doesn’t follow deductively because it preserves the feelings of everyone in the group (even if some of them “know” it doesn’t work). I don’t hear the mathematics in my students’ accents so I don’t know how to support them to be accountable to the math even as they’re being accountable to their group, or accountable to their lived experience of the world.

So… if you have a different perspective about your students’ math, about your own mathematical habits of mind, if you come from a culture where school math is a little different, where arguments happen differently, where authority and rightness are interpreted differently… I want to talk to you at TMC14, and I want to talk to you explicitly about your background and experience and how it’s made a difference in your math classroom!

]]>- Divide 63 by 3 and use that as the middle number
- Guess and check, generally starting in the teens or low 20s
- Write an equation: x + (x + 1) + (x + 2) = 63*
- Think of 63 and 60 + 3 and find three consecutive digits that add to 3 (o, 1, 2) and then divide the 60 into three 20s, yielding 20+0, 20+1, 20+2. (This was the rarest strategy).

I’m always trying to define for myself what the heck Mathematical Practice #7, Look for and Make Use of Structure, means in actual kids’ thinking. I think strategy #4 is a particularly good example of a middle- or elementary-school version of making use of structure, taking advantage of place value.

I wonder what would have happened if the three consecutive numbers had added to 60 or 61.

I wonder if the fact that the structure they found doesn’t make every type of problem easier invalidates it as a strategy somehow?

Hmm….

Update: an upper-elementary student used a strategy that combined algebraic representation and looking for structure:

Y + (Y+1) + (Y+2) = 63.

Y + (Y+1-1) + (Y+2-2) = 63 – 1 – 2

Y + Y + Y = 60

3Y = 60

Y = 60 ÷ 3 = 20

Perhaps surprisingly, that’s a controversial statement. First of all, there are folks who want to make political statements about teaching and argue that the only way to improve learning in schools is to end poverty or extend the school day or to write harder assessments or to fire more teachers or to pay teachers based on test scores or… And then another kind of controversy is among teachers themselves, as they talk about what their jobs are. Should teachers be writing standards? Writing curriculum? Writing lessons? Writing assessments? And if so, what kind and how many and when?

I’ve been working in two pretty different school districts. Both have high-needs populations and would like to see improvement in their test scores. Both have a lot of initiatives going on, and a lot invested in those initiatives. PLCs, data-driven instruction, computerized formative assessment benchmarks, literacy across the curriculum, students in career-based academies, lesson-planning and curriculum mapping packages, and more.

Each district gives teachers an average of 70 minutes of teacher planning time per day (not counting lunch, because people should get to eat lunch without multi-tasking). There’s an expectation that some of that time will be in structured collaboration, and some of that time will be individual work-time. What the teachers I talk to *want* to be doing, first and foremost, and *don’t ever* have time to do, is:

**The basics**(before the school year, perhaps): Looking closely at the existing curriculum (i.e. with the the book open in front of them, reading carefully and talking over the details) and figuring out:- What is being taught/emphasized
- How kids are expected to figure it out based on the experiences
- Why it works that way
- How to tell if it’s going well

**Implementation details:**Working with the curriculum they’ve got to figure out:- How to engage
*their*kids in the experiences they’re supposed to be learning from - What scaffolding
*their*kids will need to get from the initial experience to the expected outcome - Anticipate multiple methods students might come up with for making sense of the experience in the curriculum
- Anticipate and structure activities that incorporate multiple approaches and multiple representations that kids can/will use to make sense of their experience
- What will be hardest for the kids and what kids might do/say to show they’re struggling
- How to meet kids’ struggle so it’s productive
- Planning concise interventions for when struggle gets unproductive
- Avoiding sloppy language that gets in the way of precise understanding
- Good questions to engage students, focus them, redirect them, etc.

- Key vocabulary that will help students organize their thinking and communicate better with others

- How to engage
**Classroom management and routines:**Looking at the curriculum experience and content to decide on & find or invent:- Figuring out how to make complex experiences (Barbie Bungee! Centers! Algebra Tiles!) go smoothly
- The right blend of whole-group, small-group, and individual work for
*their*kids on*this*experience - Classroom routines for engaging kids in the work & the discourse to most efficiently experience and harvest the fruits of the curriculum tasks

**Looking at student work:**Differentiating instruction, adjusting whole-group instruction, attending to the social-emotional needs, etc:- Reflecting back on student talk, written work, etc. to figure out if the kids learned from the experiences
- Making decisions about revisiting topics, moving ahead, doing small-group interventions, asking kids to stay after school, etc. based on kids’ work
- Noticing individual students and how they’re feeling, engaging, thinking, and learning
- Noticing the group and group norms, behaviors, etc
- Planning social/emotional interventions as well as mathematical interventions
- Making notes about what to do differently next year

And sometimes, because teachers are creative and wise and fun:

- Identifying gaps in student understanding, numeracy, etc. and figuring out if and whether to address them, and if so, finding additional resources to do that
- Bringing in an extra activity that they found online, heard about at a conference, dreamed up, etc. because it seems like it would swap in or supplement nicely something in the curriculum
- Making time for students to pursue projects or topics that they are interested in — sparking mathematical passions

Here’s what the teachers do instead during the structured collaboration time, which eats up most of their “planning” time (leaving about 0 minutes for making copies, grading, using the restroom, jotting notes on what just happened in class that you want to remember for next period, tomorrow, or next year, posting the homework online, responding to parent calls and emails, attending IEP meetings, making handouts/visuals, organizing manipulatives, setting up the classroom for today’s experience, and all the other tasks teachers need to do as individuals):

- Teachers writing or aligning curriculum maps, scope and sequence, etc.
*Unless teachers are getting extra money and extra release time and extra professional development and working with experts because they like this sort of thing, it’s the wrong level of granularity. It’s like asking physicians to be medical researchers, too. People who write curriculum are smart enough that teachers shouldn’t have to rewrite it to make it usable (one hopes, and if not, buy a different curriculum) and it’s not*that*hard to make sure the stuff on the end-of-year test is covered before that test by re-ordering chapters with a sparing hand. Teachers should be focused on implementing curriculum to meet the needs of their specific group of students, which is NOT trivial!* - Teachers writing tests, called formative assessment tests, but which aren’t used to figure out why kids are struggling and what to do about it.
*If all the time is spent writing traditional tests and quizzes that won’t be recorded in the gradebook, and not talking about*why*students are making mistakes and what can be done to help them*the next day*then writing paper-and-pencil formative quizzes is not so useful.* - Teachers meeting to learn about new school climate, routines, rules, initiatives, etc.
*This stuff is important, but good instruction is probably the main contribution to school climate. If classrooms aren’t places where kids are invited to work hard and supported to succeed, then they won’t want to be there, which impacts climate. Of course, like eggs and chickens, it’s hard to get kids to work hard and succeed if their overall experience of school is chaotic, unsafe, etc. I’d caution administrators to make sure that teachers have LOTS of time to plan & reflect on instruction, and be frugal in the amount of time spent engaging teachers in face-to-face meetings about school climate.* - Teachers going over statistical data about which broad topics students don’t know.
*Data-driven instruction is only as good as the data. I think most teachers already know*what*their students don’t know. I think it’s harder for teachers to know what their students*do*know, and why their students can’t do the things they can’t do. Do students have “buggy algorithms” that they think work? Are they giving up before they begin? Do students know they don’t know but are trying something anyway? An exercise I love to do is ask, “What way of thinking about the math makes this student’s work coherent to them?” I can’t easily do that if all I know is that they chose C instead of B on this problem, and I definitely can’t do that if all I know is that 25% of them missed 80% of the “measurement items”* - Teachers filling out paperwork, like lesson planning templates.
*Teachers are often good at following instructions and crossing t’s and dotting i’s because they’re told to. It’s how you survive a job in a bureaucracy! Lesson-planning forms are only useful if they’re part of a focus on good instruction, and every nuance of the form is pointed towards encouraging dialogue about good instruction, and grounded in teachers’ concepts of good instruction. I’d lean towards introducing forms the way I’d introduce any procedure: start with the concept of the good instruction, solicit teachers’ current lesson-planning methods, talk about what works well and what we could improve, and generalize to a form that works for our community and our definition of good instruction.* - Teachers meeting in cross-discipline groups to discuss student behavior problems or school-wide, non-instructional or supplemental initiatives.
*Again, fine stuff, if there’s time for it. But right now, there’s not enough great math instruction happening in so many high-needs schools that I’d prioritize instructional planning and reflection over all this stuff. Squeeze in the conversations about tricky kids, how the 4th grade is doing overall, or cool cross-curricular projects after planning and reflecting on good instruction. Otherwise, planning and reflecting on good instruction (which is kinda the whole point of being a teacher)**is what gets pushed to the side.* - Teachers doing work for extra initiatives such as finding readings for the reading across the curriculum initiative.
*Again, cool stuff, but not as important as good instruction.*

(Almost) none of the above is a total waste of time, but none of it is the stuff that research on teacher collaboration actually suggests is useful. Considering how little time teachers get to plan, let alone collaborate, it’s frustrating to me how much of that time is wasted time. It’s also frustrating to me how much time teachers spend writing lessons or searching the web for lessons. I’m certainly very guilty of that — I tend to start with a vague sense of the topic, Google for it, and then try to do what I find on Google. My main focus is on what to do, not how to do it.

What I should be doing is starting with the curriculum, figuring out what kids are supposed to be doing & learning from that doing, and then applying my particular expertise to how to get my kids from A to B. It’s really concrete — should they be reading the questions and writing answers, or is this piece best done orally? Should students work in small groups on the main task and then share out, or should I lead the group through a demo and then have them try on their own? If small groups are right for this task, what will it take to keep them working productively, and how long can I expect each group to stay on task? Will it work if the groups are random? Will I need to make sure Keion doesn’t work with Marvin, or can they get it together today? If a group finishes early what should they do? If a group gets stuck, should I tell them what to do, ask a question, have another group share out, or something else? How many different “kinds of stuck” can I anticipate and how many different responses can I prepare? How long can I wait for a group to catch up before moving on? Will students’ struggles with integers get in the way? Will I let them use calculators? If students use Guess and Check to solve all the problems on this page, what are three things I could say/do to help them look for and find a more efficient approach? The list goes on and on… And I haven’t even scratched the surface of student-work questions like, based on what I asked them in their exit ticket yesterday, who needs more time grappling with yesterday’s concept? How can I help Deshaun get more fluent with division facts so he can make sense of one-step equations? What does this class believe counts as a right answer to an inequality — do they get that it’s a range of numbers that make the inequality true? If not, is there something in my language that could help them access that, or is there some challenge I could engage them in that would bring that point home? How does my curriculum make that idea “pop” and why did they miss it the first time through? Why does Emily consistently evaluate 2x when x = 5 as 25, but can solve 2x = 10?

If teachers don’t have enough time to ask, answer, reflect on, and revise their thoughts about the questions above, then we shouldn’t be filling their time with things other people get paid to do, like writing curriculum, writing fancy-schmancy benchmark tests, looking at data that’s not useful on the individual student level, or discussing which minutes of the day the bathrooms will be open to students.

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