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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.

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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.

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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’ve been blogging about my experience at the Alberta Mathematics Dialogue last week, in which a group of university mathematics professors offered a critique of the K-12 math program in Alberta. My colleague, Pat, attended as well. Pat has more than 30 years experience as a teacher and consultant in Alberta. She has a BSc (math major), BEd and MEd. As a high school teacher, I can’t pretend to know a whole lot about how young children learn mathematics. Pat, however, is truly an expert in this area. I asked her if she would be willing to share a few words here, and she agreed. What follows are her words.

 

I’ve been an elementary teacher since 1979. It’s a designation I’ve always been proud of, even though it seems the complexity of the work is poorly understood and not always respected. For most of the past 4 years I’ve been out of my classroom, supporting Alberta teachers in the areas of mathematics and assessment. I attended the 2014 Alberta Mathematics Dialogue in Camrose on May 1.

In addition to attending the presentations examining the Alberta K-12 mathematics curriculum, I was able to join a round-table discussion at the end of the day. The presenters from the earlier sessions were there, along with other interested participants. The discussion focused again on the math curriculum – past, present and future – and its impact on mathematics learning in Alberta classrooms.

There was overwhelming agreement among the post-secondary faculty in attendance that the math skills of their students have significantly declined over past 10 or more years. This is not an area I have expertise in, but I’m willing to work under the assumption that they know what they’re talking about, and are not guilty of looking to the past with rose coloured glasses. However, almost no one in the room seemed prepared to question the causes of this perceived decline. It seemed accepted as a truth that changes to the Alberta curriculum caused the problem, and that reversing those changes would fix it.

Alberta teachers (as well as teachers in many jurisdictions around the world) have been asked to teach math through more of an inquiry approach – teaching math through problem-solving rather than for problem-solving, if you will. Teachers present problems for students to explore, and then help them use this exploration to develop an understanding of math concepts and strategies they need to move their learning forward. Personal strategies for operations are part of the equation, and a mastery of basic facts is still critical. (Even as I try to explain this in a nutshell, I sense the eye-rolling of the masses of critics who see this approach as so much hogwash. Please accept for a moment that I have some serious experience to back up my opinions.)

In my classes I have mathematically talented students who need to be challenged, as well as students whose past experiences have made them fragile, uncooperative, discouraged and hard to motivate. I need to find a way to interest all my students, sometimes almost against their will, in the problems I’m asking them to explore so they can begin to grapple with the ideas that might be useful to solve them. Once students have worked to solve a problem, sometimes unsuccessfully, they are far more likely to be interested in thinking about an approach (mine or another student’s) that might do the trick. I try to give them a need for the math I want them to learn. A hard lesson I’ve learned after many years of teaching math to elementary students: as much as I’d like to, I can’t do the understanding for my students. All I can do is my best to engage them in thinking about what I need them to think about. I have to rely on them to do the hard work of making sense of it.

It is unbelievably complex work, but an inquiry approach in my math classroom helped me and my diverse students function as a mathematics community. Without a doubt, I was a better and more successful math teacher using the current math curriculum, as well as the one before it, than I was using the 1975 mathematics curriculum (which, according to Anna Stokke of the University of Manitoba, was the last excellent math curriculum in Alberta). My students thrived under an inquiry approach.

I’m pretty sure I don’t need to lecture the mathematicians in the crowd about the difference between “correlation” and “cause and effect.” The perceived decline in math abilities is correlated with an enormous number of changes and challenges that have impacted students and teachers in Alberta schools in the past years, and the curriculum is just one of them. I find it fascinating and disturbing that critics, particularly in the media, seem so unwilling to consider the possibility that the task of improving math achievement is far more complex than it might seem at first glance (and, in my opinion, impossible to measure using a single standardized test). An easy fix like making the curriculum more rigorous or traditional or focused on basics almost certainly does not exist.

Recently, when I polled a roomful of university educated adults about their opinion of math as students, about a third of them admitted to having hated it. I fail to see this as evidence of the great success we had back in the “good old days.” Instead of  blindly charging back in that direction, why don’t we take a deep breath, set aside the destructive, combative nature of the current debate, and support the work of our teachers and curriculum developers (who, believe it or not, bring essential skills and expertise to the table) in whatever way we can. The challenges we face are more than failure to memorize times tables. The world we live in is changing at a dizzying rate. Preparing our students to navigate it successfully is the most important work I can imagine.

Around about 4th grade in the early 1980’s, I was walking home from school and saw the new kid in my class following the same path. We got talking and realized that he lived just around the corner from me. The proximity law that governs childhood friendships dictated that Andrew and I would become best friends.

Andrew and I had a lot in common. We were both smart (although too modest to tell people that, until today). We both loved reading trashy horror novels (Stephen King and Dean Koontz were our favourites). As a team, we rocked the junior high debating circuit in Alberta for a couple of years.

Andrew and I grew apart a bit in high school, and then he moved to Ontario. I saw him once in the early 1990’s when I drove across Canada. A year or two later, he showed up at my parents’ house one day when I just happened to be there. Five or six years ago, one of us managed to track down the other, despite Andrew’s aversion to Facebook. We correspond infrequently by email, and occasionally update each other about our families. Last weekend, I got an email from Andrew. He is not a math educator (I think he does government training), but he reads my blog. He sent me this:

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Source: http://imgur.com/JDZ1WDk

That’s the thing about curiosity and perplexity in mathematics. You don’t need to be a math teacher to know a good 101qs when you see one. Everybody wonders about things that can be explored mathematically.

I wonder if Justin lost or gained followers when he got arrested.

I wonder if this growth is linear. I wonder if I could have a couple more data points.

I wonder when and if Kim will pass Justin.

I wonder why Andrew knows more about what I do than I know about what he does. I’m a terrible friend.

Recently, I was sent a copy of a letter from a mathematics professor to a science journal. The mathematics professor opens his letter by explaining that mathematics professors are the most qualified individuals “to speak with authority on the subject of mathematics and the pedagogy of mathematics.”

Below are some quotes from the letter.

Having had considerable opportunity of late to observe the preparation of students entering college from this community as compared with that preparation some fifteen years ago, I can only deplore the modern tendency to give at most a superficial attention to fundamental subjects.

The teachers in the high schools and the elementary schools are working just as hard as ever, are just as efficient as ever, but they can not obtain as good results under the handicap of present-day curricula. The student can not be trained to think in as effective a manner as he was fifteen or twenty years ago.

But those who have been most responsible for this unfortunate state of affairs in the high schools and the elementary schools, far from realizing the work of destruction that they have already done, are now endeavoring to complete it by attacking what is left of valuable educational training in the curricula of today.

It sounds exactly like what we are hearing in Alberta right now. Here’s the punchline, though. This letter was written in 1914. Full text of the letter.

Despite this mathematician’s prediction of impending doom, I submit for your debate and discussion that we have still managed to have a pretty good 100 year run.

Further, this letter was sent to me by an Alberta mathematics professor who indicates frustration with this back-to-basics push in our province.

In my post on the Alberta Math Dialogue, in which a group of Alberta university professors got together and offered their critique of our current curriculum, I mentioned that I heard some things that really offended me. Two of them aren’t even worth elaborating on (and for the record, were not uttered by a university math professor). Two of them had to do with math and I addressed one in a previous post, in which I discussed how I sometimes wonder if university math professors truly understand who we teach in K-12 math. It has some relevance here. It’s ridiculously long. Perhaps you should read it first. The second comment that offended me had to do with the use of concrete (hands on materials) and pictorial (drawing) representations.

In her critique of the Junior High curriculum, Christina Anton from Grant MacEwan University, talked about visiting a junior high math classroom and seeing the students colouring and using fabrics. From the context she described (polynomials), I suspect she saw a frugal teacher who had made algebra tiles out of old fabric, rather than spending sparse school money on a commercial set. Because she got a good laugh out of this, Christina kept coming back to it, and it became the running joke of the day. The Edmonton Journal even published the joke.

It may come as a surprise to you, as it did to me, but Grade 9 students here are required to use sticks, tiles, swatches of cloth and colouring to do complex math operations such as multiplying polynomials with monomials.

Here’s the thing, though. It’s not funny. After her session, I offered to show her how algebra tiles connect to base 10 blocks and make a nice bridge to symbolic algebra in grade 10. Christina dismissed me, and stated emphatically that concrete and pictorial representations are not real mathematics and have no place in the junior high curriculum. Only symbolic representations (the x’s and y’s and so on) are real mathematics and they are the only things that should be taught.

Such statements show the true naiveté of (some, not all) mathematics professors about who we teach in K-12 schools, and how those students learn. Concrete and pictorial representations help students make the jump to symbolic. For many students, they help form a critical bridge to understanding.

It is true that many of our students can make the jump to symbolic representations fairly quickly. But even those students still benefit from the bridge that concrete and pictorial representations make to that symbolic notation. We could probably even leave out the concrete and pictorial for our strongest students and they would be able to replicate the algebra without too much difficulty. The manipulatives will deepen their understanding, though.

For our visual and tactile learners, though, these concrete and pictorial representations are absolutely critical pieces. That’s no joke.

Would I force a student who can do it symbolically to draw it for me on an assignment or test? No. Would I let a student who can’t do it symbolically show me concretely or pictorially instead? Certainly. Would I expect a student bound for university calculus to be able to do it symbolically? Absolutely.

Do we still like what Singapore is doing? To those who speak derisively about concrete and pictorial representations, I leave you with the Singapore Bar Model. (Sorry, that was the best video I could find quickly with a google search.) The Singapore Bar Model creates lovely pictorial representations that help students make the bridge to symbolic notation. These representations work, even for high school algebra.

SUM2014

Last week, I had the privilege of being allowed to take my road show to Saskatoon, SK. I presented at, and more importantly, I got to attend SUM2014, the annual conference for math educators in Saskatchewan.

It came at a time when I really needed it. It was a great two days hanging out with math educators. I learned a lot, and had a lot affirmed.

Steve Leinwand‘s keynote was a joy. If you haven’t had the pleasure of hearing him speak, check out this presentation (from another conference). As Dan Meyer says, “this guy breathes fire.”

Other highlights for me were:

  • Reconnecting with David Coffey and Kathryn Coffey, who I first met two years ago in Edmonton.
  • David and Kathryn’s session on literacy and math. We’re into that here in Alberta, too, so it was timely.
  • Meeting Nat Banting in person. Watch this kid. He’s a rising star in math education.
  • Reconnecting with Park Star. I now know (and remember) her real name, but it’s more fun to pretend I don’t.
  • Meeting Lisa Lunney Borden. We only had a few moments to chat over breakfast and before the conference started, but now I know about CMESG, which I think I will attend.
  • Briefly disengaging from David Coffey’s session on engagement and convincing the woman beside me to join Twitter.
  • I enjoyed my sessions. Some of the participants were kind enough to let me know they did too. That kind of feedback is always appreciated. Keep in touch.
  • Having supper with Steve, Anne, David, Kathryn, Nat, Michelle, Jacquie and Allison. It was a great meal, and I got Leinwand’s ear to myself for a bit. His bloggable advice on the current math debate is to build bridges. Connect with math professors. Listen to each other. He’s wise. He’s been through this before. I appreciated him listening to me.
  • Steve Leinwand referred to me in the closing session as “That dude from Alberta.” I felt like I had arrived.
  • A panel discussion with (L-R) Kathryn, David, Me, Steve. Terry Johanson was also on the panel, but we took this before we started (notice Steve’s engagement level), and she wasn’t there yet.

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We ended with Steve Leinwand modelling practice. One of the panel questions was “How do you coach or teach subversively?” We all answered (except me – my voice was gone). Then Michelle was wrapping up. She asked the audience if they had any questions. They didn’t. There was time left. Steve jumped up, asked the audience to take 2 minutes to share their conference “take-aways” with a neighbour. Then he asked them to share back with the whole group. One person shared back something he learned in my session, thereby earning a beer on me next time I’m in town. Other people shared what they learned. Steve took over the wrap up and modelled a large group reflection. That’s subversive coaching, right there, folks.

In my post on the Alberta Math Dialogue, in which a group of Alberta university professors got together and offered their critique of our current curriculum, I mentioned that I heard some things that really offended me. Two of them aren’t even worth elaborating on (and for the record, were not uttered by a university math professor). Two of them had to do with math and I’d like to address one of them here. I’ll get to the other in my next post.

In his critique of the K-4 curriculum,  Vladimir Troitsky from the University of Alberta stated

When I teach my students in calculus classes I expect them to know multiplication tables by heart. I don’t expect them to use a calculator to calculate 7 times 8 and I don’t expect them to use five strategies to calculate 7 by 8. I expect them to know 7 by 8. If they don’t they can go to work at McDonald’s and focus 21st century skill but that’s not what’s needed for university education. Maybe university education is not 21st century skill.

A math professor that I know and respect contacted me on Twitter and suggested that this McDonald’s quote is distracting and reflected only the thoughts of one person. Perhaps the discussion about precisely how it was worded is distracting, but I think the statement itself is important. Given the response it got in the room full of mathematics professors (laughter and nods of agreement), I believe it is indicative of one of the main barriers to truly productive collaboration among math professors and math educators in our current climate of mistrust.

My point in this post is that I’m not sure that all university math professors understand who it is that we teach on a day-to-day basis in our classrooms in Alberta. I’m not sure that all university math professors understand that not all of our kids learn math as quickly and easily as I assume all of them did.

There are kids in school right now across the province and across the grade levels who can not multiply 7 and 8. Twenty years (and two curricula) ago there were kids across the province and across the grade levels who could not multiply 7 and 8. Twenty years (and x curricula) from now, there will be kids across the province and across the grade levels who can not multiply 7 and 8. It’s not always the curriculum. It’s not always the teacher. Sometimes it’s the kid (and the kid’s circumstances that have nothing to do with the curriculum or teacher). Teachers can’t send them to work at McDonald’s when they’re not learning fast enough for us.

Anecdote #1: I started teaching in 1992. In one memorable exchange that year, I threw a 12th grade student’s brand new TI-81 calculator out the window (ground floor, onto grass) when he grabbed it to do a simple multiplication.

I’ve had several conversations with math professors recently and head them say things like, “In my country, everyone knew their multiplication facts.” Variations include changing “In my country” to “When I was in school” or “In 1982″ and so on. Let’s think about the peer groups of our mathematics professors, who we all agree are very smart people.

How many children in a current grade 3 class of 25 students in Alberta will go on to take post-secondary math courses? Given that roughly 24% of Canadians go to university, but that not all of those students will take math there, let’s put it at a generous 5. Of those students who take post-secondary math courses, how many go on to earn Ph.D.s in mathematics? I have no idea, but it’s got to be far less than one student per current grade 3 class. Most of these very smart math professors probably spent their grade school days in like-minded peer groups. I’m sure it’s true that everyone in their peer groups could multiply efficiently at a young age. I’m not so sure that everyone else in their class, country, grade, year or school program could multiply efficiently.

Anecdote #2: This is getting long, but let me share another anecdote. If you hate my anecdotes, skip ahead to the next paragraph. During my own grade school days, I had one math teacher who made new seating charts after every test and arranged us by grade, from highest to lowest. I loved this practice. Sheldon (not Cooper) and I competed all year for that top seat. We thought it was great. I never thought to look across the room at the bottom seats. I bet the kids sitting over there didn’t love it. I bet those kids hadn’t mastered their basic facts. I bet the seating chart had something to do with why they always offered to beat me up at the bike racks after school. I don’t know, though, because those kids weren’t in my peer group.

Back to my point, so I can wrap up. Many of our kids face challenges that are far greater than some of these math professors seem to understand. We can’t just send kids away when they struggle. We can’t blame the curriculum or their previous teachers and let them off the hook. We just do what we do. We teach the kids in front of us. We try desperately to get them through. We need to remember the words of one of my favorite principals, “The parents are sending us their best kids. They are not keeping better ones at home. Teach the ones you’ve got.” There are challenges in Alberta’s education system that I believe are more critical to address than our math curriculum. Those thoughts are for another day. Yes, we changed our curriculum. Yes, the PISA results seem to be slipping. Correlation does not imply causation.

I wish that the math professors I speak to would indicate that they understand that they see only the best and brightest graduates. When people like Dr. Bowman share with us what deficiencies he sees in those best and brightest, I listen intently and think about what I can do. These are productive and valuable conversations to me. At two different schools in which I worked, we had a math professor from the University of Alberta come to talk to the high school math department. At one of those schools, I had a math professor come and speak to my calculus students. During a session I facilitated with grade 7-12 teachers this year, I had two math professors from the local college come and talk to us. As school teachers, we listened. We changed practices that we were able to change.

I have enough respect the work that these math professors do that I actively seek out their input to help my teaching and to help my students. Comments like the McDonald’s one make me feel like I do not get the same respect back.

The vast majority of our K-12 students will not take university math. A large number of our students legitimately struggle with math. My biggest wish is that the math professors I speak to would acknowledge that I, as a math educator, might be capable of using certain pedagogical strategies to help those strugglers learn. In my experience, those strategies, which include the use of concrete and pictorial representations (the next post) are crucial to many of my students, and enhance the understanding of the strongest ones.

I’m a teacher. I like kids. I can’t write them off and tell them to go work at McDonald’s. I have to be better than that.

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