[Julie McNamara] Fractions: What’s There to Talk About?

McNamara began the presentation by comparing fractions on a number line. (See handout.) Using Cuisenaire rods, she demonstrated one way to compare the relative size of fractions. Then she used the rods to connect the fractions to their locations on a number line.

We watched a few video clips in which Marilyn Burns interviewed students about fractions. One student described whether or not 11/12 + 1/5 is greater than 1. Another compared 5/12 and 5/8. As we discussed the videos in groups, we wanted to see more students who actually struggled with the concepts. It was interesting, though, to see that the students weren’t given any tools to help them answer the questions. At one point, a student was struggling and asked for a paper and pencil.

We were then asked to make a conjecture about the location of A on the number line:

Which is a better choice, 3/5 or 7/8, for the location marked A on the number line?

Two participants in my group immediately folded the paper to look at 3/4, a strategy that no other group considered. We used and shared a variety of strategies to find the location of A. The presenter recommended that we give our students those same opportunities in the classroom.


[Jeffrey Wanko] Developing Proof Readiness With New Logic Puzzles

I’d like to believe that I didn’t act star-struck during Jeffrey Wanko’s talk on logic puzzles. I did restrain myself from asking him to pose for a picture with me, and I didn’t ask him to sign my handout (like the lady behind me did). However, I couldn’t resist going up and telling him that I gave a “very similar” talk last year at Asilormar [note: in this case "very similar" means I basically stole the idea for my talk directly from Wanko's article in Mathematics Teacher, Nov 2009, volume 103]. He was very nice about it though, so I must not have come across as too crazy.

Jeffrey Wanko started off by giving a little background to the types of logic puzzles he was presenting about, puzzles that he referred to as “language-independent.” Meaning that, once you know the rules, there are no words to translate or interpret in order to solve the puzzle.

As a group, we came up with a few examples of commonly known language-independent puzzles. No surprise that Sudoku was at the top of the list. Fun fact: Sudoku means “single number” in Japanese, but the puzzle was actually invented by an American. Okay, two facts.

Wanko talked for a little bit more about teaching proof and deductive reasoning, citing Michelle Cirillo’s article in the November 2009 issue of Mathematics Teacher, ”10 Things To Consider When Teaching Proof.” And then it was time for the puzzles.

The method that Wanko used to teach the two puzzles he presented was an exercise in inductive and deductive reasoning in and of itself. He presented the group with side-by-side examples of unsolved and solved puzzles and asked:

  • What do you think the goal of the puzzle is?
  • What do you think the rules of the puzzle are?

After taking a few minutes to notice and discuss with the person sitting next to  us, Wanko opened up the group discussion and we established the set of rules and the goal for the first puzzle, Yajilin. The rules and goal were also listed on the reverse page in our packet, which Wanko wisely did not distribute until this point in the talk.

We worked through an example puzzle as a group, sharing out things that we noticed and filling in the beginning of the puzzle together. Then, Wanko invited us to work on a puzzle of our choosing from the packet.

After a bit of fun puzzle-solving time, Wanko called us back together to repeat the process with a second puzzle, Ripple Effect. He ended by answering some general questions and talking about a pilot study he had done which showed that students who had taken a weekly logic-puzzle class showed gains on a logical reasoning inventory from the beginning to the end of the 10 week class. He also (very briefly) plugged his books.


Recaps Around the Web

[Amy Ellis] Laying a Foundation for Learning to Prove

It’s hard not to enjoy sessions when you’re already drinking the kool-aid. That said, Amy Ellis did a fantastic job of balancing research and practice around laying a foundation for proof well before Geometry class (on a side note, I still hope to lead a session called “Proof Doesn’t Start in Geomety” at some point in the future). She gave a convincing argument for the importance of introducing the idea of proof in early elementary school, and more importantly discussed structures and cultures that promote proof at any age.

Ellis started with a simple, but robust problem asking students to explore a growing rectangle in Sketchpad. She showed some student data

Student Work Growing Rectangle

She then talked about student conjectures and their explorations, including some nice proofs in pictures.

•What do the second differences mean?
•Why are they constant?
•Will they always be constant? If so, why?


Ellis then swung over to the research side, making a number of claims, including something we teachers can always use reminding of: “Students over-rely on examples as proof (Chazan, 1993; Harel & Sowder, 1998; Healy & Hoyles, 2000; Porteous, 1990).” “Students have no problem generalizing, these generalizations just aren’t always right or productive.”  “Students tend to generalize in recursive ways before explicit ways.”

In middle school parlance, we’re talking about the difficulty in getting our students to shift from “I do 3 examples. They all work. Done.” to proof.

We talked with partners about how you know when something is correct in mathematics. My personal answer: when you can convince your skeptical peers that your reasoning is valid.

Convince myself, convince a friend, convince a skeptic.

She then shared some student responses to this same question, many of which were pure gold.

“You experiment until you find five examples to see if the answer is right.”

“When you get the test back.”

“You don’t really ever know if anything is right, you just have to hope and pray that you are right.”

Ellis then introduced a new problem, something she called Eric’s Property.

Eric came up with a new mathematical property. He thinks this property is true for every whole number.

First, pick any whole number.

Second, add this number to the number before it and the number after it.

Your answer will always equal 3 times the number you started with.

Do you think this is true for every whole number?

She shared a few examples of “proof by three examples”, and then moved on to the following abstraction.

“So here are random objects. Okay. This is a number. 3 Triangles. This one has more triangles. It has 4 triangles. And this one has 2. So if you take one of the triangles and you put it over here, it becomes 3 of the same thing.   So 3 of these would be the same thing as this plus this plus this.”

She also shared a traditional algebraic proof of this. I noticed she referred to the former as a proof and the latter as a formal proof. An audience member asked the same question I was thinking, asking why the former was not a formal proof. I was unconvinced by her answer (while I do think this student is using the triangle symbol to represent both “some number” and “one”, I don’t otherwise see how this is any different from x-1, x, and x+1).

Ellis ended the talk discussing classroom practices and habits that promote proof. It’s fine (and important) to allow incorrect conjectures to play out. Generalizing doesn’t have to be algebraic. In earlier grades this can be done visually or verbally. Proofs too often start as exercises to verify versus proofs to explain (and in my opinion, students have a much harder time proving things that are “obvious” to them). Give kids the opportunity to conjecture and prove. Extend tasks to encourage more general reasoning. Make observations, conjectures, and argument/proofs. Don’t get so caught up in the structure of proof. Provide problems for which examples fail.

Use peer critiques. Have students explain why their statement is true to a student in an earlier grade. Ask students: will it always be true, when will it be true, are there times when it won’t be true. Make a conjecture wall. Share (and I would add create as a class) criteria for a valid proof.

And to add my own final recommendation, don’t fall into the trap that proof must require algebra, two columns, and/or a less eloquent rehash of Euclid.

More at http://website.education.wisc.edu/aellis1/Home.html

[Scott Farrand] Problem Solving as PD for the CCSS

Farrand kicked this one off with a figure composed of 18 paper squares that had a perimeter of 20. “Can you come up with a figure using 18 squares that has a larger perimeter than this one?” he asked participants. “How about a smaller perimeter?”

Farrand then passed out baggies of paper squares that we cautiously manipulated on our laps. This session was begging for tables. With squares placed corner to corner, we found a maximum perimeter of 72. We agreed on a minimum perimeter of 18. One participant stacked the squares on top of each other for a perimeter of 4, but this only resulted in clarifying conditions for the shapes.

West, Farrand, and Stetson teamed up to ask a lot of questions. On the side, we were asked which questions perplexed us the most.

Through a series of these questions we were coaxed into thinking about what happens to the perimeter as we add squares. We quickly discovered that when squares are added, sometimes perimeter increases, but at other times it doesn’t. Here’s what we found:

The group’s discovery was that changing the number of contiguous squares in a row or column has no effect on the perimeter. The perimeter is equivalent to a figure with the row or column maximized. Putting a finger on why this is the case and using language to explain it pulled in some rich math practices.

Farrand then challenged us to use this understanding to calculate the perimeter of a figure without counting:

Throughout the session Stetson facilitated “Teacher Time-Out” moments, highlighting approaches that were used to get us thinking more deeply about the problem. “Did you hear the question Rick just asked? What sort of thinking would that require you to do? What mathematical practices are in play here?” Participants reflected on the importance of capitalizing on student curiosity and facilitating student sharing.

We then explored figures composed of equilateral triangles with the guiding question, “How does adding triangles affect perimeter?” One big surprise of the session came at this point: adding an equilateral triangle to a figure of five triangles can actually decrease the perimeter.

I found this problem to be refreshingly devoid of context. There was no talk of minimizing fence material for a dog’s pen, for example. It was a great opportunity to reason and justify conclusions. Discoveries were made and a few surprises propelled participants into further questioning. West, Farrand and Stetson reminded us that problem solving tasks are key sources for the use of mathematical practices in our classrooms.

[Andres Marti] Make Math Move: Modeling Algebra and Geometry with Sketchpad

Basically the premise of this “talk” was to get some hands-on experience with Geometer’s Sketchpad. We started off by downloading the free trial from the Key Curriculum website (you get a 20 minute preview, which you can restart as many times as you want for another 20 minute session).

Once everyone was up and running, Andres Marti cracked a joke about how we’ve all been trained by Microsoft Office to “avoid the Help menu at all costs,” but had us confront our demons and click through to the website support. Which is awesome. We looked at a couple of the tutorials on our own (Marti gave recommendations for varied levels of experience with GSP and turned us loose). I played around with making various types of transformations.


Marti stopped us midway through to show us some other features available on the Sketchpad resource page, such as Dynagraphs. We played around a bit more before it was time to go.

I had been looking forward to learning more about creating animations, which we did not do. Marti said that such a lesson was not feasible in the time allotted. Given the varying levels of experience around the room, from people who had had none at all, to the man sitting next to me who had a couple of circles swaying back and forth before I’d finished downloading my free trial, Marti did a good job of differentiating and allowing for lots of individual exploration. The workshop certainly whetted my appetite for doing more with Sketchpad, though now I have to convince my school to spring for the upgrade to version 5.

BTW. More evidence that the support for GSP is pretty phenomenal:

[Michael Shaughnessy] Reasoning and Sense-Making: Keys to Student Engagement

Shaughnessy’s morning keynote covered Reasoning and Sensemaking (RSM). He spent the first ten minutes reviewing the different resources NCTM offers. Then he illustrated RSM with four tasks he had us work on and debrief. You can find all the problems in his slidedeck but the fourth one interests me most.

Problem 4

A “data detective” exploration. What do you notice in these tables? What do you wonder about?

As we noticed sums between rows and columns and speculated what they could be describing, he progressively added column and row headings, which in turn sent our reasoning down new, different paths.

His point was that RSM can and does occur in “naked number” contexts.


[Karen Arth] Mathematical Modeling

This is the second session I attended from Karen Arth at this conference and I’ll re-iterate again that I appreciate her communication skills. She also has a lot of good ideas, even though I disagreed with one of her premises in this particular session.

She started by asking for our definitions of “modeling” and I thought the audience’s suggestions were astonishingly perceptive. Two well-turned phrases:

  1. “Start with life, do the math, check back with life.”
  2. “Approximating the world.”

“So what I’m hearing from you guys is ‘application’,” Arth said, and I made a note to ask her a question about that later.

She gave the following Algebra II problem, which she called “traditional”:

Write the equation of a parabola going through the points (0,0), (20,50), and (40,0).

a) Use the form y = a(x-h)^2 + k.
b) Solve for y when x = 25.

Then she gave us the following revised problem, which she implied was the same task only with modeling:

McDougal’s Restaurant has a play area for children under and around their giant arch (in the shape of a parabola with negative orientation). They plan to set up a new activity that allows children to bungee jump from the arch. The manager, upon hearing of your team’s expertise, hires you to calculate the maximum stretch of the rope that will keep the kids safe. The arch is 50 feet high and 40 feet wide at the base. The jumping location will be 5 horizontal feet away from the axis of symmetry of the arch.

a) Write an equation to model the shape of the arch.
b) What’s the maximum length to which the cord could stretch to keep McDougal’s safe from lawsuits?

She had us create a graphic organizer for our work that included sections for “Notes,” “Labeled Picture,” “Table,” “Estimates,” “Assumptions,” “Calculations,” and “Recommendations.”

“If I look at your picture, if I look at your table, if I look at your notes, I want to be able to tell the whole story,” she said.

We worked on the task in groups and presented it. Then I asked my question. “Earlier you equated modeling with applications. Is there ever an applied task that doesn’t involve modeling?”

She told us about several SBAC performance tasks in the modeling strand. Bruce Grip asked if my question had been answered. I said I was still unclear on what features of those performance tasks made them modeling. Is it enough to have “real world” objects in your problem?

Our own definitions of “modeling” are interesting, of course, but the CCSS and other documents have offered their own, very specific definition, which includes, among other skills:

  • Identifying essential variables.
  • Formulating a model that uses those essential variables.

Arth had us talk about our model’s assumptions, which is a core component of modeling, but her task also gave us the essential variables (the height of the arch, the width of the base) and it gave us the model also (“the arch is in the shape of a parabola with negative orientation”). This seemed to make the task something other than modeling.

I said all of that and she replied that, given her student population, she had to offer them some of that information because the task might be too foreign and unmanageable otherwise. I’m sure she knows her students’ capabilities better than I do. Whatever our students’ capabilities right now, though, I hope we’re all moving in the same direction, towards the same definition of modeling.

[Karen Arth] Making Functions in Algebra Active and Interesting

Karen Arth is a strong communicator, full stop, and for that reason alone I’ll be sure to look for her byline in future conference schedules.

She also works for CPM, so if that’s curricula you use, it’s likely you’re already familiar with the function activities she described in her talk. I’ll explain them briefly and share her resource [PDF]:

Function Machines

You give student four functions on cards. Ours were:

  • f(x) = sqrt(x)
  • k(x) = -x/2-1
  • g(x) = -(x-2)^2
  • h(x) = 2^x-7

Then you give them an input and tell them what the output should be after you compose all those functions.

For example, Arth gave us 6 as the input and told us that 11 had to be the output of the last function.

We shuffled cards around and finally determined that f(h(k(g(6)))) = 11.

These functions aren’t chosen arbitrarily, though, and that’s where you take your small-group instruction and whole-group debrief. k(x) changes the sign of the input and creates non-integer outputs if the input is odd, for instance. So Arth asked us why we made the choices we made, and our responses revealed a lot of mathematical understanding.

Silent Board Games

“Teachers love these,” Arth said. “Because students love them, because they’re silent, and because the hand that goes up is not necessarily what you consider the A student.”

Arth drew a horizontal x-y table on the board. There were two complete x-y pairs. The other pairs were only half complete.

Students were to raise a hand when they thought they knew a missing part of the table. Arth would then invite the student up to write the missing part. She’d do nothing if the number was correct and she’d erase it if it was incorrect.

Beneath that table on the board was a blank for “English Sentence” and then “Math Sentence.” It felt a bit like Wheel of Fortune, where you know the missing pieces and then decide to solve the puzzle.

Here, again, Arth emphasized process. “What was your strategy? What were key moments for you?”

One participant said that once someone else completed the x-y pair (0, -1) she knew an important fact.

Personally, I was confused throughout much of this activity. If Arth said the function was linear, I didn’t hear it, which meant I had to figure out what function family we were even looking at all while my fellow participants were having no trouble completing the puzzle.

I’m interested in hearing from people who have tried this activity in a CPM class. Do struggling students find it as easy to participate as Arth suggested?

Number Walks

Arth took us outside and laid out a large x-y axis on the ground. Several of us stood from -4 to +4 on the x-axis. Arth then said, “Take your number and add two.” As we all formed the line y = x + 2, she had us turn and link shoulders, creating a visual of the line.

Other people took their turn. f(x) = 2^x was a real treat to watch, as the people standing at negative numbers struggled, first, to figure out if they should step backwards, after which they did their best to shuffle forward just an inch.

Treasure Hunt

You can find this resource on page ten of the handout but my group never managed to make sense of the instructions, which probably means they’ll need to be clarified or simplified for students.


Featured Comment

Matt Vaudrey:

I’ve done the Silent Function Game with my students (though I didn’t have the English Phrase and Math Phrase, which I will add). I added a few components to make it work well for any class:

–Each person can only get up once.
–You can erase or write, but not both.
–The last person writes the function rule (y=2x-7).

Then I sit in the back and make menacing looks at students who make noise.

[Lisa Nussdorfer] Using the iPad in the Mathematics Classroom

I try to attend every iPad session I can. The technology interests me itself but the massive interest of teachers at these conferences interests me even more. Lisa Nussdorfer’s session would have broken through the fire marshall’s maximum three times over if she had let everyone in the room who wanted to attend. What accounts for that interest?

These are expensive tools, after all, and the question in the back of my mind at all times is, “What problem are they meant to solve?” If “iPad” is the answer, then what is the question?

Is the question:

  • “How can we incorporate technology students love into our own practice?”
  • “How can we lighten the administrative load in our classrooms?”
  • “How can we enhance and expand access to the very math our students learn?”
  • All of the above.

In every iPad session I’ve attended, I tend to feel like the rest of the participants and the speaker have all settled on the question or questions, while for me that question is still open.

These sessions must be hell on presenters, who have to differentiate a population as diverse as any you’ve had in your classroom. We had participants with every level of experience from those who didn’t know that double-tapping the home button gets you a list of open apps to those who were reliably using box.net to distribute and receive class assignments. There were teachers managing one iPad, six iPads, and thirty iPads per classroom.

So Lisa Nussdorfer covered some of her favorite apps and fielded questions for the remainder.

She highlighted ShowMe, Educreations, and Explain Everything saying, “You’ve heard of Khan Academy. These give you the ability to be Khan.” (A participant called out Doceri as another alternative.) Later she said, “I wanted to be Khan briefly but whenever I show a video to a kid they’re kinda like ‘Eh.’ It didn’t quite have the response I was expecting.”

Another participant said, “I don’t use [those screencasting apps] because they don’t look good and I don’t sound good on them.”

Nussdorfer called Algebra Touch the coolest math app out of all of them. It’s a useful illustration of algebra skills, but a participant pointed out that “Algebra Touch does the work for you.”

She pointed at another app and said, “This is free but you have advertisements so that’s kind of a problem.” I’m hoping we’re all on Nussdorfer’s page with this one. Advertisements in front of kids should be a non-starter.

The questions from the crowd took us in some interesting directions. One asked Nussdorfer why she wasn’t recommending Fuse, Houghton Mifflin Harcourt’s Algebra textbook for the iPad.

Nussdorfer replied: “I got the sample of it and thought, ‘It’s a textbook.’ I thought it was like a textbook. If you’re going to spend money on an iPad, you should get something more than just a textbook. And it’s sixty dollars.”

Another participant called out the low resolution of writing on an iPad, relative to writing on a whiteboard or paper (see above, where the word “Doceri” takes up half the width of the screen):

I’ve found the iPad device to be not a very good device to write on, at least with math. You just don’t have a fine enough stylus or control over your writing. It’s like one of those big pencils for first graders.

As I said, I always find these sessions interesting and I’m glad I attended. At the moment, the question that interests me is, “How can we enhance and expand access to the very math our students learn?” But not just on an app-by-app basis. I’m not necessarily in the market for a list of apps mapped to content standards. I’m mainly curious how an investment of tens of thousands of dollars to put digital, networked devices in every student’s hand will result in more interesting math being taught to more students. That question, for me, is still open.