# Category Archives: Uncategorized

## Here’s my plan (subject to change)

Yes. I understand that there are conflicts in the list below. And yes, I understand that I will change my mind, or get engaged in a conversation and miss a session. I understand all of that. Nonetheless, here’s my plan for late next week in Denver.

And yes, I will gladly accept recommendations for sessions I may have overlooked. I have highlighted (in red type) people whose work you need to check out if you have not already (with an emphasis on people who my blog readership may not have run across—I’m not gonna highlight Dan Meyer ’cause you know who he is.)

# Thursday

8:00: Teaching and Learning of Algebraic Thinking: Research Insights, Daniel I Chazan, Mark Driscoll, Megan Franke

8:00: Continual Formative Assessment Using the Common Core Mathematical Practices, Karen Fuson

9:30 Algebraic Thinking When Solving Equations and Doing Word Problems, Daniel I Chazan

11:00 Measuring Length and Time in the Common Core, Grades K–1, Constance Kamii

12:30 Keeping It Real: Teaching Math through Real-World TopicsKarim Kai Ani. (Note: I would like to point out that this session will be painful for me to attend, as he mopped the floor with my sorry-ass pony tail in our Chicago debate. Also, I could not get into his session in Philadelphia last year, so SHOW UP EARLY!)

12:30 Meeting the Challenges of the Common Core Standards, Alan H. Schoenfeld

1:00 Research in Algebraic Thinking: Continuing the ConversationMegan Franke, Mark Driscoll, Daniel I Chazan

2:45 mARTh: Using Creative Expression to Connect Students to Mathematical Concepts, Hannah McNeill

# Friday

8:00 Teaching and Learning of Proof: Research InsightsE. Paul Goldenberg, Patricio Herbst, Eric Knuth

9:30 Essential Mindsets for Tilling the Soil for the Common Core, Steven Leinwand

11:00 Keeping Our Eyes on the Prize, Philip Uri Treisman

02:00 Creating Opportunities for Students to Engage in Reasoning and ProofMargaret Schwann Smith

2:45 Powerful Online Tools Promote Powerful Mathematics, Eli Luberoff, Patrick Vennebush

# Saturday

8:00 They’ll Need It for Calculus, Christopher Danielson

11:00 Tools and Technology for Modern Math Teaching, Dan Meyer

12:30 Viral Math Videos: A Hart-to-Hart Conversation, Vi Hart, George Hart

## What do we need functions for?

In this online functions course I have been teaching, and which is nearly halfway complete, a question has been kicking around in various forms. What do we need functions for?

There are many versions of this question. What do we need functions for (in middle school math)? What do we need functions for (in solving visual patterns tasks)? Et cetera.

One permutation of this question comes from the blogosphere’s own Michael Pershan. What do we need functions for (in mathematics more broadly)?

Here was my response to that question in class this morning. I do not consider this to be the definitive answer to it. Instead, it is an example. One of many, no doubt.

Apologies for the imperfect typesetting on the mathematics below. It’s good enough to get the point across and I don’t have time to look up all the relevant TEX commands to get it looking just right.

You know what blew my mind about functions on returning to higher math in graduate school? That we can make them our servants. We can make functions work for us. My introduction to the idea that we can create functions that—like robots—we can program to make something useful happen was the characteristic function. I am going to assume most of you have not taken graduate Real Analysis, so let me explain quickly.

The characteristic function of a set is defined to have the value 1 when the input is in the set, and to have the value 0 otherwise. Formally, that looks like this:

That capital Xish thing is the Greek letter “Chi”. Here is the graph of .

So what, right? Well, the characteristic function links sets with functions. We can begin to talk about the size of a set based on the results of operations on the set’s characteristic function.

One question that mathematicians have been interested in dealing with over the years is how big is the set of rational numbers? By using the characteristic function , we can rephrase that question in terms of functions. Without giving away the first three weeks of graduate analysis, you can imagine the graph of this function as being a whole bunch of points on the x-axis and a whole bunch of points at y=1. Indeed, there are so many of each that you can’t see that the function is a function. It looks like two functions (Note; I have cheated and just plotted y=1 and y=0 below—I do not believe Desmos can work with the condition that x must be rational).

The question about the characteristic function on the rational numbers that corresponds to asking How big is the set of rational numbers? is this: What is the value of the integral of the characteristic function? That is, we ask what is the value of the following expression?

$\int_{[0,1]} \chi_{(Q)} (x)\,d \mu\$

Geometrically, we are asking how much area is shaded below (given that the points at y=1 correspond ONLY to the rational values of x in the interval 0≤x≤1).

The standard definition of integrals (the Riemann integral) cannot answer this question. One way of computing gives 0, another way gives 1, so we have to say that the integral does not exist. This function is not integrable in standard calculus.

So we invent the Lebesgue integral, which is defined in terms of sets rather than areas of rectangles. And we learn that the value of this integral is zero.

That is, the set of rational numbers is so small that we can say—relative to the size of the set of real numbers—that there are essentially none of them. The rational numbers are so few that we say the size of the measure of the set is zero.

They are infinite yet insignificant.

The machinery to prove all of that is the characteristic function.

That’s one thing that functions are good for.

Dig.

## I didn’t really realize how much we need this…

This booklet, from Early Childhood Family Education, came into our home over the weekend.

Forgive the decorations, please. They are not mine.

Naturally, I was curious to see what it had to say about early math development.

Nothing.

Zero, zip, zilch. Not one word.

There are entries on Joining libraries (“They make literacy fun!”), Literacy (A full two-page spread), Writing, and Words (“Words are powerful!”)

Nothing on Number, Shape, Pattern, Counting, Math or Numeracy. I hereby humbly submit the following entries.

Counting. Children learn counting language before they learn about numbers. They learn this language through example and repetition. You and your children can count together soon after they begin to talk. Model correct counting and don’t worry about your child’s incorrect counting. Have fun with numbers. And when a child asks, “Want to see how high I can count?” the correct answer is always Yes!

Shape. Children find shapes fun. While you are at the library picking up alphabet books, grab a couple of shapes books, too. Noticing shapes in the world is the first step to success in geometry and measurement, important parts of the math your children will learn in elementary school.

Children love to get their hands on interesting shapes, too. Shape puzzles and toys can help young children to notice and investigate properties of shapes such as size, symmetry and angles.

Before children enter school, they do not need memorized number facts. They do need to have lots of experience counting and talking about numbers.

## Triscuit squares

After chatting with the Mathalicious crew about a lesson on square roots and irrational numbers, I was inspired to talk math with Tabitha (5 years old).

Me: [With a box of Triscuits at the dining room table] Tabitha! Come here, please! I want to talk to you about something.

Tabitha: I know what this is gonna be about.

Me: What?

T: Math.

Me: Right. I want to know whether you can arrange those in a square.

I hand her four Triscuits.

She quickly forms them into a square.

T: Done. Now can I eat them?

Me: Not yet. Can you do it with these?

I give her nine Triscuits and scramble them up.

She is again successful.

Me: How many more are in this square than in the last one you made?

T: I am really tempted to eat them.

Me: Right. But how many more are there this time?

T: There were 4 before. And now there are 9.

Me: Yes. So how many more in the big one?

Some elaborate Triscuit shuffling goes on, lasting about a minute.

T: Five. Wanna know how I did it?

Do you see the beauty of doing this on a regular basis? Children learn discourse patterns through exposure. Not only can she explain her thinking, she expects to do so.

Me: Yes.

T: I took 4 away, then there were five left.

Me: Nice. One more.

T: I really want to eat these.

Me: I know. Soon. Can you make a square with these?

I give her 7 Triscuits.

She moves them around. She is not especially systematic in the order she places them.

She ends with this arrangement.

T: If you took this one [i.e. the one in the upper left] away, you’d have a square.

Me: Is that a square?

T: Oh! No. It’s a rectangle.

You do it.

Me: I can’t. See I can do a square with 1 Triscuit.

T: Of course.

Me: Then I can do 4 like you did, and 9 like you did. Four had two Triscuits on a side. Nine has three Triscuits on a side…

But she can no longer hear me over the sound in her head of the crunching of Triscuits.

## We have to call this stuff out

Strange start to the day.

First was Kate Nowak noticing the hate Vi Hart routinely heaps on math teachers. As happens at 2:04 in the video below. Note especially her tone here.

Then the following comes across my desk (click on it to see the full size version).

The first line reads, “Statistics show that more students are coming from high school with weaker mathematical backgrounds than ever before.”

No citation. No argumentation. Just Statistics show.

I have replied with the following:

Wow. Gotta admit that first line is provocative.

What evidence are you basing this claim upon?
Thanks,
Christopher Danielson
I’ll keep you posted.
In the meantime, we need someone with some mad video skills to make a “Sal and Vi Hate Math Teachers” video in the spirit of “Hollywood Hates Math“. I’ll help with the archival work. Who’s in?