What is Math?

As a preassessment for my capstone course in mathematics, I asked these soon to graduate math majors what is math, and what are the big developments in math. This goes along with both my belief in preassessment, and wanting to ask questions about what I really want to know. Here are some of their responses. (See all their blogs in a urlist.)

Roz Chast
http://rozchast.com

What is math?

• Lots of "math is more than numbers" or "it is not about computation"
• Relationships
• Sara: "math is patterns."
• Jennifer: "Math is more of a way of thinking.  It is logical reasoning.  It is looking at patterns and relationships.  It is problem solving and explaining phenomena that seem unexplainable."
• Kate: "I think math is thinking logically, understanding facts and finding relationships between different mathematical concepts."
• Alex: "Math is the study of relationships that many people take for granted."
• Annette: "math is an explanation or an attempt to explain relationships we find in nature or ones that we create."
• Kristine: "math is how we can apply numerical values to how the world works."
• Basis of science
• Becky: "Math can be used to problem solve, find patterns, make predictions, provide reasoning, and much more which is all in the tool box of math – the resource for information."
• Bryce: "Math is the science of solving problems."
• Josh: "Math is the one science that every other subject has in common."
• Kenton: "The reason that I call math a "science" is because without math, no scientific studies would be able to be quantified, all studies would have to be qualitative and therefore much less precise."
• Logical system
• Kerry: "Mathematics is a fantastically broad, beautifully intricate, complexly connected concept of numbers and symbols."
• Emily: "math is thinking critically about different kinds of systems. These systems can range from the number system in algebra and calculus, to a system of shapes, such as geometry, to a system of rings as we saw in modern algebra."
• Andy: "Mathematics is a method for conveying logical principles. A proven mathematical theorem is a reality about logic; it is some organizing principle inherent in the human mind."
• Language. 
• Duncan: "mathematics is the most beautiful language in all of the universe."
• Final answers:
• Biz: "For me, math has been a source of intrigue, education, and frustration."
• Matt: "To ask what is mathematics is like asking what is life. There is no definitive answer."

Top 5
I also asked what are the biggest moments/discoveries in history of math. (Top 5 or milestones, etc.)

• Famous Stuff
• Pythagorean theorem 9
• Fibonacci Sequence 9
• Pi 4
• Fundamental Theorem of Calculus 3
• Natural logarithms and e 2
• Kepler’s laws of planetary motions
• Any of the Fundamental Theorems 
• Law of Sines
• Quadratic Formula
• I Ching
• Fractals
• ϕ
• Culture
• Numbers 9
• Calculators/Computers 8
• Abacus 3
• Discovery of zero 3
• Understanding of different formulas
• Development of measuring units
• Ancient Geometry
• Math in astronomy.
• Concepts
• Unit circle
• Function
• Pattern
• Combinations (in counting)
• Infinity
• Axioms
• Parallel postulate
• the need for complex numbers
• prime numbers
• Fields
• Calculus 7
• Algebra 4
• Calculus 3
• Non-Euclidean Geometry 3
• Geometry
• Differential Equations
• People
• Euclid 7
• Euler
• Newton
• Archimedes
• Gauss
• Karl Pearson (new to me!)

Not too different than I might expect, although the diversity of responses was pretty interesting here.

One more interesting comment:

• Danielle: "The founders of mathematics struggled and devoted their whole lives to the theorems we now take for granted when studying in our classes. I can’t imagine devoting my whole life to proving what we consider now a simple concept." 

If you have a moment, I would love to hear your responses to these prompts in the comments!

Image: I saw this at Peter Liljedahl's very worthwhile site. Even Google couldn't help me find the original. Doesn't it look like it's from the New Yorker? EDIT: Tweeps nailed this one: Roz Chast, a frequent New Yorker contributor.

Spin Square

Brief post to test GeoGebra embedding.


Inspired by a typically great Daniel Mentrard sketch.  I wanted to play around with some of the ideas. In this you can make a random or ordered mosaic of a tile you design yourself, that is spun through a number of possibilities which you can also set. Hope you make something neat!

Smith and Stein Demand

(Note: crazy semester and I didn't finish a lot of started blogposts. But this one felt important to finish. Great experience in teacher ed, and I'd love some feedback on how to make an activity for these ideas for teacher prep.)

I was kvetching on Twitter - I mean, consulting my PLN - about preservice teachers whose desire for fun, shiny lessons can distract them from finding lessons of substance. Nicora Placa, a math ed prof and teacher educator (with a fabulous blog),  suggested Smith and Stein's framework for cognitive demand.

That's Margaret Smith, an all-star researcher at University of Pittsburgh, and Mary Kay Stein, who seems like her partner in crime at Pitt. Their paper(s) and book on the 5 Practices for Orchestrating Discussions has been very influential here at GVSU this fall. Several profs are using them to organize our first math for elementary teachers course. [Lengthy aside: book, NCTM's number one seller; MTMS article, very readable for preservice and inservice teachers; Christopher Danielson's 5 practices posts, start here; it synthesizes a lot of great research into worthwhile mathematical tasks and instructional practice that they have done with other partners.]

The cognitive demand article (full issue MTMS Feb 98; also available on JSTOR; Reflections on Practice: Selecting and Creating Mathematical Tasks: from Research to Practice, Smith and Stein, Mathematics Teaching in the Middle School, February 1998, v.3, n.5.) is great for a professional development discussion. They have several highlighted discussion questions that get to the heart of the matter. The article boils it down to four levels of tasks, with detailed breakout at each level.

The reflection questions are very theoretical for preservice teachers, though. (The article relates a professional development discussion with teachers, though, which is excellent reading for them.) But it also includes tasks to analyze. So I had my students try them for a while and discuss them at the table. Without exposure to the article or to Smith and Stein's cognitive demand framework.

(Click to see full size)

Luckily, there's a lot of overlap with out content. The fractions, not so much, but there's a lot of fraction language in the common core geometry and statistics, so it's been discussed.

Immediately, they went into test mode. Which is especially interesting as we don't take tests as such. (SBG, rather.) It took several urgings to get them to try the problems cooperatively, ask each other questions and share methods. I noted that they got very engaged - maybe I need more individual problem solving or even practice in class. (I'm reminded of our pre-assessment and how that's a tension for me.)

Afterwards, we talked about what did they notice about these problems. They focused on the subject of the content and the story problem aspect. Except they were impressed by the realism of these story problems. Many of them spent a significant amount of time on task C, and it was the most discussed at the tables.

Next I asked them to, in their groups, rank the problems on the amount of thinking required by students working these tasks.

We talked about the tasks that had significant agreement or disagreement or were at the least or most thinking ranks.

• Task A was easy. The visual made it easy to extend the pattern. (Many skipped over the description.)
• The group that rated Task B the hardest had trouble making sense of the text.
• Task C had so much computation, calculators were required and the problem had extra unneeded numbers in it.
• Task E started to bring out some task analysis. It was easy with the picture, but the content was hard. People who had difficulty understanding the picture shared that, too.
• Task F was ranked easy because it was an easy pattern compared to the ones they had been doing in class.
• Task H was "just a rule"

Now we looked at the Smith & Stein exposition:

 

It was astounding to me how ready they were for this discussion. They read for themselves, discussed in their groups, then rated the tasks by this framework within their groups.  In particular I was pleased at how they discussed non-routine vs a difficult-to-execute algorithm in terms of difficulty, and the value they placed on connections and representation. They were able to make connections from their own experience to challenges students might face. For example, the symbolic rule for the train pattern (Task F) and going from the observation of 2 more each time to what that means in the rule. Unfortunately, some of them feel the solution to this will be in what to tell the students, but the majority saw the value in presenting cognitively demanding tasks.

The activity is obviously raw. I'd like to formulate some questions to support them finding some of these task characteristics themselves - but that's challenging without students to observe. 

How might you teach a lesson on these ideas? Surely that's a teacher task with a high-level demand!

Activate Prompt

Looking at the blog it looks like I retired from teaching! Disappointing oreo lesson and I'm out. I think this has been my most extended blogging absence...

.. and this is only a brief return. But there should be more posts coming if I get a chance to write up all the stuff I was doing.

Matt Enlow tweeted: One of my favorite Geo lessons of the year: Put down the iPads. Let's talk. How could we define the "center" of a triangle?

I responded: @CmonMattTHINK wish @ddmeyer would put together one of his cool electronic samplers for that.

Dan asked: “Electronic samplers”? (I meant like his Ice Cream Stand Future Text)

Matt explained: Maybe Ss are given a series of ∆s and asked to click where they believe the "center" to be? Then show all marks at once

But Christopher  saved the day: That is easy in activeprompt. No need to involve Mr. M

Oh, yeah! ActivePrompt. Riley Lark's cool programming venture for making interactive polling diagrams that I always meant to do but never got around to doing. Pretty quickly - with a slight distraction of making a GeoGebra sketch with tools to find four of the centers - I put together three prompts. Want to try them? (If you're an early reader, don't expect much of the results!) (EDIT: turns out the responses are only visible if you're logged into Active Prompt. After a few responses accumulate I'll post the pictures. Try the polls though!)

Poll and Results

Poll and Results

Poll and Results

I'm pretty curious to see the results! Want to compare to the geometric centers? Here are those images.

Thanks, Matt, Christopher, Dan and Riley. Now that I've blogged about it, I'm sure I'll remember ActivePrompt next time.

My Oreo Lesson

Finally... my chance to do the oreo lesson!

I'm teaching one of our math for elementary education courses and the content includes measurement and statistics. I love measurement as a context which needs statistical understanding. Measurement introduces variability, and has a strong need for producing a number to represent typical. If the question is, "How tall is the ceiling?" then 2.60, 2.7, 2.725, 2.73, 2.735, 2.735, 2.74, or 2.76 meters is not a satisfying answer.

The oreo lesson, if you are unimaginably unfamiliar with it, is the brainchild of Christopher Danielson, aka @Trianglemancsd, the purveyor of much fine snack food mathematics. (All the oreo posts; this one is sort of a wrap up.)

Previous to this lesson, we investigated measurement, did an introduction to statistical typicals, and worked on statistical displays. (Two of those covered in a previous blogpost.) On the day before oreo day, I brought three packages of oreos to class: regular, double stuf, and mega stuf. Their interest was definitely piqued; it was like they could smell the sugar. Not much mathematical interest, though. So I prompted - what might a mathematician wonder about this? They immediately jumped to the idea of is it really double, and what is mega. Then we brainstormed together - what do we need to gather data on for the next day?

Their list:

OREO: data to collect

weight of the whole cookie

weight of white stuf in each cookie

height of each cookie (mm)

diameter of each cookie

weight of cookie/black sides

height of black cookie

height of white stuf

diameter of white stuf

how many of each size fit in a specific container/height

volume by displacement

compare deliciousness of different types

nutrition information

stuffing v serving size

calorie content (burning)

Not bad. Calorie burning turned out not to be viable with that short of a notice... but I'd like to see it! I made a data sheet, so that we'd have a whole class worth of data, and a google spreadsheet to share.


The points about measuring like a scientist (half of the smallest unit) and recording to show the accuracy measured are obviously still in progress. Also the statistical thinking of gathering and using data need more development - most were happy to answer the main question with just their measurement. "It's double." "It's more than double." "It's less than double." No one used the measurements, they went entirely by weight.

That wasn't what bothered me. I expect those kind of goals to take time.

What bothered me was that they weren't into it.

They were excited about the cookies, and figuring how many each person got, and eating the cookies afterward. But they weren't into the math.

Dave Coffey sometimes recounts (or makes fun of me for) how I want to be obsolete. Sitting back and watching students direct themselves at the end of the semester. I always want to hand off to the students. Have them make it their lesson. Look, here's a pile of data! On something interesting! What can you do with it? What else could we look into? How many ways can we come at the question?

But on this day, they said no.

My personal metaphor for this is a Smothers Brothers routine. (That's how old I am.)


(The whole brilliant bit... the show was amazing. Steve Martin got his start there as a writer, for example. They used their folk singing to make the show safe for sharp political commentary. Like we use math class as a ruse to get students problem solving and thinking critically. They were cancelled and replaced with Hee Haw.)

So this lesson felt like, "Take it, class!"
"No."

My response was to ask them to make sure they got all their group's data, and to write about the measuring and their answer to the question for a standards based grading assessment. And this is a compliant class, so they did, and did a good job on it. But that's far from the peak experience for which I was hoping with this lesson.

Part of the problem, I think, was in my desire for efficiency. By introducing the problem in the previous class and then making a record sheet, I took the initiative from them. They went into fill in the blank mode, from long habit in math class. Another part of the problem was lack of a focus, in the workshop sense. I think I should have discussed statistical thinking with them, and how that's different from single measurement thinking. It's all about the data! This is very reminiscent of the Barbie Bust. It was my problem and my lesson. "My" doesn't help me be a better teacher. (Gollum.)

Reflecting afterward, I think my high expectations helped create the sense of disappointment, like an overhyped movie.  And it led me to rush into a lesson instead of building suspense and anticipation.  I think this kind of experience contributes to teachers who "tried that once" and that was enough to turn them off of inquiry-based learning.  In the end it is the learning that needs to be the center of engagement, not the cookie.

Good Data Day

Nothing revolutionary here. Just what turned out to be a nice sequence of experiences with a group of preservice elementary teachers. No handouts, just a story and pictures of what happened.

Previously... we measured some distances, from a whiteboard marker to an outdoor commons. We discussed technique and accuracy, and I shared a bit on with what elementary students struggle. They went off and measured, and then we shared data in a Google doc. (New to some, somehow.) Choice 1 we agreed on as a class. Choice 2 was up to their group. Distance to the bathroom, trash bins, and one group was obsessed with how do they measure a bus. Then we talked about how do we go from all those different numbers to a specific measurement. How tall is a desk? We want a single number! Of course, they first go to the average.



I always push on that. One, because you need to justify, man! Always. The other because the mean is so inappropriate for low sample size because of the power of outliers.

Then last class was a day to explore  measures of central tendency. I love human statistical displays and it was a gorgeous day. (October in Michigan.)

First sort: height.

They naturally lined up this way. We found the median height, with some good discussion about 19 students, which was the middle and what if it was between two people. When the middle height person stepped forward, we talked about if she was a good representation of the typical height of student in our class. People were a little uncomfortable about that. Especially at the tall end of the line.

Second sort: age.
First take, they lined up like they did for height. We found the median and discussed how well she represented typical. People were curious about the twins, who were gracious to answer age questions. I asked if it was true that boys tended to be older than girls, which led to a comparison of the significance of their position in the height chart vs their position in the age chart.

Then I asked them to line up spaced for their age. Start at 19, each sidewalk section was 4 months. "What if we're the same?" "Just line up." "Oh, it's a line plot!"   Students sorted themselves within a month by age. So when we determined the median, we talked about whether to count from the front or back of that month's line.  They thought that this showed the data much better because of the gaps, and were surprised how old it made that ancient 22 year old look.

Third sort: name length.

Finally split up the twins! For this one, we added unifix cubes. They made a stack of cubes the height of their name, first and last as it is on their birth certificate, and used that for comparison. The new question - could we use the cubes to figure out the mean? Would the mean be higher or lower than the median? "We could lay them all down in a big line and divide it by 19!" (Literal adaption of the arithmetic rule is pretty typical.) "What does that do?" "Could we all just share to have the same?" "We have these left over... what do we do with them?" Then we discussed if that meant the mean was closer to 13 or 14. Why was it higher than the median?

Fourth sort: shoe size.

I gave them the option of what to sort, but they were too shy to say anything. When I say shoe size, they jumped at it, and several said they were thinking about that. I like the ambiguity between men and women's, and the apparent diversity from shoe size not varying perfectly with height. I wanted to use the cubes again, but how do we do that with half sizes? Two cubes for each half size... that works! This time they sorted themselves into a line plot without being asked, working out their own spacing.

Again the mean was higher than the median. Will that always happen? How would it be different if we added 4 cubes to the men's shoes?

Starbursts
As they came back inside, I distributed Starbursts, very unfairly. Enter your data on the board before you sit down. DON'T EAT THEM YET. (Only one person did. Not bad.)

We discussed the challenges of using candy in class. Sugar, red dye, communicated values. (Why are prunes the only individually wrapped healthy snack?)

I asked them to make a line plot and pie chart of this data. The group that shared had a very nice way to make their pie chart:take the 80 starbursts, and think about 20 per quarter. Then half that is 10, half that is 5. "But there's 81?" someone asked. The class agreed that this wouldn't change the pie chart much.

Next was the hard discussion. If we want to be fair, how many should each person get? 2, since that's what most people got? But if people contributed, then there would be enough for 3 each. Wait a second, there's enough for four for each of us. With some left over! I tried to make the connection between this idea of equal distribution and the mean. Here's where we want to take outliers into account.

(Unfortunately, I showed them a graph of US income distribution at this point. Probably should have skipped that. It is one of the reasons I think statistical education is important - we can't understand our world and society without it. What do you think?)

Finally we looked at one more way to make a pie chart.

You can see two wrappers there. We compared this to the pie chart on the board, and they said that the lines really helped and they would add them. But they agreed that the drawn chart was very close.

After that we discussed how to share them fairly. I shared that this was real problem solving. The textbook problem is 81 starbursts, 19 students, how many do each get. The real life problem includes messy things like people wanting particular colors.   We did a quick check of color preferences by going to the corners of the room. One corner each for red (30%), pink (25%), orange (15%) and yellow (5%).  Center of the room for no preference (25%). A few methods were suggested, then someone said we should just do rounds of taking one. People agreed and we started. They wanted to make the pie chart for each one and who am I to argue.

So it was a good data day. Statistics and sunshine, who could ask for more?

Next class: starting the glyphs project!

Explore #MTBoS - What Makes It My Classroom

This is an interesting question. For Exploring the Math-Twitter-Blog-o-Sphere, we had two choices for Mission 1:

• What makes my classroom mine?
• What's one of my most open-ended or rich problems?

I try to write about my open ended and/or rich problems all the time.  The knots come to mind, the creative patterns lesson, a lesson that uses GeoGebra well, like the GGB Quadratics, or one of the games I love, such as Decimal Point Pickle. There's a dead simple triangle area lesson that has untold riches I should write up and I could do for this...

But the first question is the kind of reflection I only do if pushed.

<nudge>

It could be planning. Because I teach at a university, I have the priviledge and responsibility to plan each lesson, with time for revision and lots of original lessons. I try to share freely with colleagues and they give me feedback that there's a certain style that those have, and they seem to be able to guess which ones are mine. Of course, the style is probably just being goofy.

So it could be that I'm a goofball. I look in some respects like a serious, big, old guy, but wearing shorts, sandals and any funny math shirt I can get a hold of.  My first teaching emulation was David Letterman when I was a lecturer, and there are still traces of that. I use comics whenever possible, and it's a rare day that doesn't have at least one on an activity. I am encourage jokes and funny outbursts from students - it's not a raise-your-hand kind of classroom.

It could be technology. I have a Bring Your Own Device classroom and computer days where we look at math using free and open source tools like Desmos or GeoGebra or Wolfram|Alpha. I use Facebook and Google docs instead of Blackboard. I support students in using tech to capture and document their math and demonstrate it in the classroom. I read blogs with interest and use and adapt the ideas in my classroom; I make some of those posts required reading instead of stuffy journal articles for my preservice teachers. I do everything I can to encourage their blogging and use of twitter.

But I think what is essential, what is increasing more and more, and what represents what I hope to be as a teacher, is choice. The deeper I get into trusting students, and transferring responsibility to them, the more choice I give them. In class time, and certainly at home. In a pure content class, where the assessments are all about mastery of math skills and ideas, this means the choice of how to practice skills and apply ideas are up to them. Projects move from investigating my question to investigating their own. In teacher preparation classes, this semester I'm leaving everything up to them.

The always edgy Frog Applause

I give suggestions for homework and many take those, but they also in great percentage are choosing things of which I never would have thought. They are doing much more finding math in the world around them than I would have imagined. Doing more voluntary reading of good math ed articles. And definitely more creative work, though it's hard to think how to quantify it.  Students are uncomfortable with it, and frequently ask, "but what do you want?"  It feels like they're leaning away from that now.

This extends to assessments, where I provide more problems than needed, and they chooose their problems. Combined with Standards Based Grading, and they're choosing which standards they're trying to demonstrate, too. I often feel like our goals are just talk to students until it's connected to assessment. For more creative work, the choice in assessment comes from choosing exemplars of what constitutes their best work. 

thischarmingcharlie.tumblr.com

This goes against what Dan Ariely says about choice architecture. But I don't want to direct them here, I really want them to know freedom.

You can wander through the preservice elementary teachers' blogs for proof, if you're interested. No way would I have assigned some of that stuff. But it's theirs and they own it. More than in the past, anyway.

I hope to read  what makes your clasroom your own. Post your blogpost link at Exploring MTBoS. Don't have a blog? Maybe this is your <nudge>.