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## Whiteboards in Math

Recently, I tweeted out a link to Nat Banting’s post on whiteboards. It reminded me that I was going to write a post here about my experiences with them this year. I’ve been trying them out. I’ve done some things well. I’ve learned some things.

The three things I like most about having students use whiteboards in class, probably in order, are:

• How their work “pops” off the boards so that I can see it easily. As I circulate, it’s so easy to provide feedback.
• How comfortable students are working on a dry-erase surface. They are not afraid to try things. They are not afraid to make mistakes.
• How easy it is to group and pair students to give and receive feedback from peers when their work is on whiteboards.

The things I learned, I learned by making mistakes, and by asking you questions. Here’s one I threw out in the middle of a class in which I was making mistakes. I was having students do questions on the whiteboards, and then copy them into their notebooks. Of course, they hated it. They were doing everything twice. The high school math teacher in me is having a hard time letting go of the mentality that I must write it on the board and they must copy it down, or they haven’t learned it. I still want them to leave my class with something useful and correct written down in their notebooks for future reference. Please don’t judge me for that.

In the middle of that class, I put the call out on Twitter.

The responses were immediate and helpful. Talk about real-time formative feedback! My pocket was buzzing in the middle of the lesson. You can see all the responses here.

Christopher Danielson (@Trianglemancsd) saw right to the heart of my question and threw back at me, “Also, how much needs to?” That’s really what I was wrestling with.

Consensus was that between none and very little of what is written on whiteboards gets transferred directly elsewhere. This whiteboard work is the learning phase. It’s messy. It’s getting transferred to their brains, honest. One suggestion what to have students take a photograph of their work if they think they will need to look back at it. Frank Noschese (@fnoschese) suggested Lens Mob as a way to create an album they could all send their work to. Jacqueline (@_Cuddlefish_) uses them a lot, and I respect her experience. I wish she blogged (too busy geocaching, I suspect). She tried to convince me to let go of the need to have stuff written down.

So the question, then, becomes what do they do AFTER they have finished the messy learning on the whiteboards? Some teachers go straight into practice. Nat Banting (@NatBanting) and Bowman Dickson (@bowmanimal) talk about having students reflect on their whiteboarding at the end.

One teacher I work with this year helps her students summarize a few key things in a “Student Study Guide” that she provides. Heather Kohn (@heather_kohn) does something similar by giving her students a guided notes page for things that she feels need to be formalized. Of course, there’s always the textbook for future reference.

I’m in the middle of figuring out what I am going to do next year. If I land back in a classroom, I will make sure I end up with a set of mini whiteboards. I may still make kids write some stuff down on paper. Don’t hate me for that.

I take no credit for this idea at all. I was in a classroom this morning, and the teacher had the students play a game of Radical SNAP. The students were totally engaged, and were enthusiastically converting between mixed and entire radicals. It’s pretty simple to set up.

Materials: You need one deck of cards with the 10, J, Q and K removed for each pair of students, and one giant square root symbol per pair of students. This one should do the trick: Giant Root

Pair off the students in your class. Each pair gets a deck of cards, and should remove the 10, J, Q and K. Shuffle the remaining cards, and deal them so that each person has half the deck, face down.

Mixed to Entire

The students flip over their top cards. The student on the left puts his in front of the radical, and the student on the right puts hers under the radical. The first student to correctly convert the mixed radical to an entire radical wins the round.

Entire to Mixed

The students flip over their top cards, and put them both under the radical. The first student to correctly simplify the radical or to identify that it can’t be simplified wins the round.

## Engineering Effective Discussions

In the spring, I was working on a series of posts about formative assessment in math class. I got sidetracked by starting a new blog, and kind of let it drop. This morning, however, I read this great post from Max Ray about questioning, and it brought me back to formative assessment.

One of Dylan Wiliam‘s 5 Key Strategies is “engineering effective discussions, questions, and activities that elicit evidence of learning.” From Dylan William’s book, Embedded Formative Assessment:

There are two good reasons to ask questions in classrooms: to cause thinking and to provide the teacher with information that assists instructional decision making.

Max is right. Good questions that cause thinking in math are tricky. Most of us lean towards asking recall and simple process questions. With practice, we can learn to throw out deeper questions as easily as we ask recall questions.

Max’s post contains a number (26 to be precise) of great questions that prompt discussion. My two favourites are:

• What do you notice?
• What do you wonder about?

Questions like the two above feel safe to students. They don’t have to worry about being wrong. They can think and respond without fear.

Sometimes, questions can be improved by turning your lesson around. I spoke to a teacher last year who was working on 3-D shapes with his class. He had the nets all copied and ready to have the students cut out, fold, and tape. It seemed more like a lesson on cutting, folding and taping, so he scrapped it. Instead, he brought out models of the 3-D shapes, and asked the students to create the nets that could be folded up to make the shapes. It ended up being an incredibly rich discussion.

One of my favourite conversation-extenders comes from Cathy Fosnot. When a student responds to a traditional question, extend the conversation by simply stating, “convince me.”

The more we can engage students in conversation with each other through effective questioning and planned activities, the more likely they are to come to their own understanding of the topics.

## Two by Four

The first strategy I’d like to share for embedding formative assessment (Remember, formative assessment isn’t always a thing. It’s feedback for learning) is referred to by the consultanty-types as “two by four”. While generally working in the field of consulting, I should point out that I’m not much of a consultanty-type. I’m not a fan of fancy processes and the equally silly names we give them. So, for classification purposes only, I present to you the two by four.

One of Dylan Wiliam’s 5 key strategies is to activate students as instructional resources for one another. This strategy is my go-to one, mostly because I’m trying hard to embrace the philosophy of the revised program of studies, which suggests I should orchestrate experiences from which my students can extract their own meaning. When I active them as instructional resources for one another, they learn from each other, rather than from me writing notes on the board.

This strategy is really simple, and can be used to teach just about anything I can think of in our curriculum. Two students learn something through some kind of structured activity. They check in with two more students who learned the same thing and compare strategies and results. When they agree, they move on.

Let me tell you how this worked in a class I tried recently. I was invited to model formative assessment in a Math 20-1 class (Pre-Calculus 11 elsewhere in the WNCP). The topic of the day was multiplying radicals. In my classroom days, I would have written some rules on the board, done some examples, and then given an assignment for practice. The students would not have been activated much at all.

For this lesson, I built a structured activity that I hoped the students would work through and learn what I needed them to learn about multiplying radicals. That activity is posted below.

Some notes about how it went:

• First of all, I should point out that this does not need to be photocopied and handed out. It would work just fine as a set of questions that the teacher poses to the class. I ran it off, because I was going into a stranger’s classroom and I wanted to make it as simple as possible for the students.
• As the students worked, I circulated and listened in. I helped when needed and as little as I could. I regrouped them at the appropriate times and with the appropriate people. By the end of class, almost all of the students taught themselves how to multiply radicals. I gave exit slips at the end (another formative assessment strategy) and the exit slips showed me that most students had met or exceeded my expectations. Two students didn’t manage to learn the material. This next part is huge. Wait for it…
• It wasn’t my class. I didn’t know many of the student names. Despite that fact, when I saw the two exit slips that showed little or no understanding, I described to the classroom teacher exactly who I thought those two students were in terms of where they were sitting. I nailed it. Because I was circulating and working closely with the class, I knew exactly who was getting it and who wasn’t. That’s feedback, folks. That’s formative assessment, folks. The old me (remember, the one who stood at the board writing example after example) would have had no idea who was getting it and who wasn’t. Unit tests were frequently sources of great surprise for me.
• I was fascinated by the strategies they were coming up with. On the first page of the activity, all students concluded that $\sqrt{a}\times\sqrt{b}$ was equal to $\sqrt{ab}$. What they did when I introduced coefficients on the next page was really interesting. The room was split into two camps when they came across $3\sqrt{2}\times6\sqrt{3}$. Many of the pairs of students got right to it and multiplied the coefficients, then multiplied the radicals, and came up with $18 \sqrt{6}$. They confirmed their answers using their calculators, and moved on. The other camp wanted to rely on the only rule they knew at that point ($\sqrt{a}\times\sqrt{b}= \sqrt{ab}$). Their solutions looked like this: $3\sqrt{2}\times6\sqrt{3}= \sqrt{18}\times\sqrt{108}=\sqrt{1944}=18\sqrt{6}$. They were converting to entire radicals, multiplying, and then simplifying. It worked, but it was highly inefficient. So when it became time to let the pairs meet with other pairs to see what they had learned, I made sure that each group of four contained one pair with the efficient strategy and one pair with the inefficient strategy. Students were taught the more efficient strategy by other students, rather than by me. That’s formative assessment.

I did something similar in a Math 30-1 (Pre-calculus 12) class earlier this year. The description and resources are available here.

## Peter Liljedahl on Problem Solving

Last week, I had the pleasure of attending a workshop with Peter Liljedahl from Simon Fraser University. The session was on assessment, but he opened with his take on problem solving. I’m going to discuss the assessment stuff in a later post, but I really wanted to talk about his problem solving process while it was fresh in my mind.

Peter grouped us randomly by having us select cards, and then gave us the following problem.

1001 pennies are lined up in a row. Every second penny is replaced with a nickel. Then every third coin (might be a penny or a nickel) is replaced with a dime. Then every fourth coin is replaced with a quarter. What is the total value of the coins in the row?

He gave each group a whiteboard marker, and a window or small whiteboard to work on. We were to solve the problem on the space we were given. Peter circulated and encouraged us to look at what was written on the other windows if we were stuck. In the end, most of us were satisfied that we were right. Peter never told us the answer or revealed it in any way.

Here’s what interested me about Peter’s take on problem solving, which is very similar to what I have advocated as Learning Through Problem Solving.

1. The problem wasn’t written down any where. He just told us the story, clarified, and put us to work. This isn’t a textbook problem.
2. He grouped us randomly, rather than letting us sit with the people we came with. Throughout the day, he kept on re-grouping us. His feeling is that this frequent changing of the groups leads to more productive classes, and helps include all students. Kids are more willing to work with someone they don’t like when they know that it won’t be for long. This process helps include even the shy and “outcast” students in the classes. These students are included because the people working with them know that the groups will change again very soon, and so they can be patient for that long. Peter also has a way to assess group processes, which I will discuss in the subsequent post on his assessment material.
3. Writing on non-permanent surfaces is critical to Peter’s process. He has observed that when kids work on paper, they take a long time to start. When they work on something like a window or a whiteboard, they start right away, because they know they can erase any mistakes easily. Groups that are stuck can step back and see what the other groups are doing as a way to help them get started. When finished, each group’s work  is displayed prominently around the room so that students can share solutions with each other in a non-threatening way. Students could do a gallery walk and discuss the different solutions, or sit down and write down one that works for them individually.
4. This is the third time I have had the pleasure of working with Peter. He has never once told us the right answer. This drives teachers nuts. I’ve tried it in classrooms, and it drives kids nuts. The message, though, is that they have to figure out the answer. The teacher isn’t going to let them off the hook. An even more important message is that the teacher is less concerned about the answer than she is about the process. These are two pretty nice messages.

Peter’s problem solving process fits really nicely with the Learning Through Problem Solving discussed frequently on this blog. There’s always a “yeah, but” when I share some of this with teachers. They have one whiteboard, no windows, and nothing else to write on in their classrooms. Let me solve that problem for you.

Superstore has $20 whiteboards that are just the right size. Staples has tons of different sizes available for less than$30 each. I’m confident you could get enough whiteboard space in your classroom for 15 groups to work for less than $300. Your principal will find that kind of money in a hurry, especially when you explain that you are going to use it to enhance mathematics instruction by engaging students in group processes that will lead to better problem solving skills. I’ll write you a letter if you need one. Read Full Post » ## Name Blanks When I was in high school, I had a teacher who used something other than boring old “Name: __________” at the top of his tests. He’d put things like, “Hello, my name is: _______” or “_____________ is going to ace this test.” When I started teaching, I did the same thing. The kids really liked it, and any time it left room for creativity, they did some neat stuff. Over the years, I acquired a pretty long list of them. Whenever I created a new test or handout, I would open up the document with all of them in it, and pick one. Sometimes I asked students to write their own. There is nothing pedagogically brilliant about this. It just amused me. Here’s the document. Read Full Post » ## Let the Kids Ask the Questions October’s issue of the NCTM Mathematics Teacher magazine included an excerpt from an interesting article in their Media Clips section. They used a Forbes article that dealt with tween earnings, and the portion they included is: Hollywood’s 10 top-earning tweens collectively pulled down$107 million between June 1, 2007 and June 1, 2008.   According to Alloy Media and Marketing, it’s a rabid and often indulgent fan base: America’s 20 million kids aged 8 to 12 spend $51 billion of their own money annually and influence$150 billion more in spending by their doting parents. Tied for the top spot: Hannah Montana star Miley Cyrus and Harry Potter lead Daniel Radcliffe. The young talents each banked a cool \$25 million in the last year.

I love this article for the questions it could elicit.  The problem with the Mathematics Teacher Magazine treatment of this is that it follows the article with 8 questions we are supposed to ask kids.  Teachers might see this, and defer to the questions provided, thereby missing a great opportunity.  I’d much rather show students the article, and have the kids decide what would be interesting to explore further.  There are lots of potential rate and ratio questions that could come from student exploration here.

## Golden Ratio and the Human Face

This is an activity a group of teachers I was working with fleshed out based on ideas on Mark Wahl’s website, and here.  This activity is appropriate for Math 10-3 (Measurement) and Math 10C (Measurement or Real Numbers).

Statues of human bodies that the ancient Greeks considered most “perfect” embodied many Golden Ratios. It turns out that the “perfect” human face also contains many Golden Ratios.  This task allows students to take measurements of an ancient statue, and check whether these measurements approach the Golden Ratio.  Then two celebrity photos are provided and the students check how beautiful the celebrities are based on these calculations.  Finally, students can use pictures of their own faces and GeoGebra to check how they match the Golden Ratio.

Part 1 Students perform the measurements on a photo of a statue, and calculate the ratios requested.  These measurements could be made by printing the picture and physically measuring, or by inserting the picture into GeoGebra and using the software to measure.

Part 2 Students complete the same measurements, either on a provided celebrity photo, or one of their own choice.

Part 3 Students then take their own picture and insert it into GeoGebra.  They can then measure, and check how closely the ratios in their own faces match the Golden Ratio.  Students who are not comfortable using their own pictures can choose other celebrities from the internet. (I didn’t teach Cameron Diaz, but her face is much more likely to be beautiful using this test than my own…)

The complete student package in Word format is available for download here.

## Prepare Kids for a Contract Job, Not an Hourly Wage Job

I had a long drive home from an assessment session yesterday, and I was reflecting on some of the conversations we had. We talked about how some teachers justify strange assessment practices because they sincerely believe that they are helping prepare their students for the real world.

It occurred to me that this view of our role in preparing students for life after school treats the real world like a minimum wage job at a fast food restaurant.  This view of the real world values things like showing up on time, having a great attitude, and working hard. Those are admirable qualities in the guy who serves me my curly fries. They should have nothing to do with my assessment of students’ mathematical abilities.

A better view of our role in preparing kids for the real world would be to treat the real world like a contract job.  This view of the real world values things like taking responsibility to finish what you start, taking as long as you need to do the job right, seeking help from others when needed, and producing good work every time. These are great qualities in the guy who designs the bridge I drive across every day. These are the things I should be encouraging in  my students, and should be reflected in my assessment practices.

This morning, as I was making my toast and about to put peanut butter on it, I was reminded of something I did early in my career in a Math 31 (Calculus) class.  It was at least 15 years ago, and way before we had digital cameras or Dan Meyer, but I think it definitely qualified as a WCYDWT.

One morning in the early 1990s, I was having peanut butter and toast, and had gotten near the bottom of the jar.  While scooping peanut butter out, I got it all over the handle of the knife, my hand, and then of course, my pants.

It occurred to me that the jar really was taller and narrower than it should be, and in a rare moment where everything came together for me, I realized that the shape of the jar was probably significant and we just happened to be working on Maximization and Minimization problems in Math 31.  Since digital cameras hadn’t been invented yet, the only multi-media artifact I could take into my classroom was the jar itself.

So I took my jar of peanut butter into class, slammed it down on the table in front of the class, ranted for about 5 minutes about how mad I was at Mr. Kraft for getting peanut butter on my hand and pants, and then told the class that we wouldn’t be doing Calculus that day, because I had to write a complaint letter to Mr. Kraft.  I sat down at my desk, and pretended to write, and hoped.  It took a couple minutes, but then it happened.  One student said something like, “Mr. Scammell, there must be a reason it is made in that shape.”  Another jumped in with, “It’s probably the shape that uses the least material but still has 1 kg of peanut butter in it.”  Then they argued for a bit, and decided on their own to figure it all out.  They measured, calculated, argued some more, and then told me to write my letter, because you could hold 1 kg of peanut butter in a container that was wider and shorter so my hand wouldn’t get peanut butter on it.

Again, I pretended to write while I waited and hoped.  And then they extended.  One kid stopped the whole thing, and said that there must be a reason it was the shape it was.  He observed that the lid was thicker than the other plastic, and therefore must cost more.  They re-ran their calculations based on an estimate that the lid cost twice as much as the other plastic, and then concluded that the shape was logical if we assumed the lid cost more.  It’s too bad that this was all done years before the internet hit our schools, or perhaps we could have quickly researched further to find out costs and contact numbers for Kraft.

I tried the Peanut Butter Rant several more times over the years, with varying degrees of success.  Some classes got right to the one important question, while others took some prompting.