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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.

IMG_2331

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.

Before he passed away a few years ago, we would occasionally attend my father-in-law’s church. Attending someone else’s church is a strange thing. I almost always came away with a few things I agreed with, a few things I disagreed with, a few things I was curious about, and a few things that seriously offended me.

On May 1, I attended someone else’s church again. I joined (I think I was invited) a whole bunch of University Mathematics Professors from across Alberta at their 2014 Alberta Mathematics Dialogue, put on by PIMS at Augustana in Camrose, Alberta. Of particular interest to me was a series of talks on Number Theory, but I wasn’t there to engage my own mathematical interests. I was there to listen to these math professors analyze the Alberta K-12 math curriculum.

The sessions I attended were:

  • Leif Stolee (Retired Edmonton Public Schools Principal) – NEW MATH: Building the Roof without laying the foundation.
  • Vladimir Troitsky (University of Alberta) – Critique of Alberta K-4 math curriculum
  • Cornelia Bica (Northern Alberta Institute of Technology) – Grade 5-6 Mathematics Curricula Around the World
  • Christina Anton (Grant MacEwan University) – Mathematics Curriculum in Junior High School
  • John Bowman (University of Alberta) – What is Missing in High School Mathematics Education?
  • Mark Solomonovich (Grant MacEwan University) – Now and Then; Here and There: problems with what they call education.

For details beyond the titles: the full abstract of their talks.

Based on the four categories of reactions I had in my father-in-law’s church, I present, without commentary, my reactions to the sessions I attended.

Things I heard that I agreed with…

  • All of the Math Professors emphasized that basic facts and problem solving are both important.
  • Cornelia Bica wants students who persevere, solve problems, and do more than just follow recipes.
  • Cornelia Bica talked about a blend of pedagogies. She said, “Singapore and China aren’t as hardcore as they used to be.”
  • Cornelia Bica said that pedagogy should be left to the teachers.
  • John Bowman stresses that he is not criticizing K-12 teachers. He is asking for leadership in making changes to curriculum.

Things I heard that I disagreed with…

  • Troitsky insists that bar graphs, charts, patterns and shape and space are not mathematics.
  • Cornelia Bica said that you need basics before you move on to problem solving.
  • Christina Anton stated over and over again that concrete and pictorial representations need to be removed because they are a waste of time. Only the symbolic should be taught because it, alone, is real mathematics.
  • John Bowman said, “Strategies and tricks are interesting for the brightest students. However, the average student (including future professionals) will be better served by learning the time-tested algorithms for arithmetic computation that we learned as children and continue to use in our daily lives.” (This one might be better placed in the category below. My gut says, “disagree”. My head says, “think about it.”)

Things I heard that I want to think about some more…

  • Two of the speakers (Troitsky and Bowman) mentioned that they believe kids can’t develop personal strategies until they have first mastered a traditional strategy.
  • Cornelia Bica carefully analyzed at what level different topics are introduced in different curricula around the world.
  • Cornelia Bica’s analysis included comparing instructional time in mathematics as a percentage of total instructional time. Alberta was the lowest of the regions she showed.
  • John Bowman spoke about Math 31 (HS Calculus) and whether we need it.

Things I heard that really offended me…

  • Vladimir Troitsky said that kids who can’t multiply 7 x 8 or those who need 5 strategies to do it should go work at McDonalds.
  • Christina Anton said that concrete material and pictorial representations should be reserved only for special needs students.
  • Leif Stolee opened with Matthew 8:28-34. It’s a story about demons and swine and Jesus. As I understood him, Alberta Education is the demons, K-12 teachers are the swine, and he is Jesus. I not sure I followed him completely.
  • Leif Stolee told a story about working in a school in which I would later work. Two of his English teachers applied for a job at Alberta Education. One of them was a superstar (Leif’s words). The other was mediocre. The mediocre teacher got the job. He extended this thought by pointing out that good teachers like himself can no longer advance in Edmonton.

Edit: After Dr. Troitsky’s comment below, I looked back at my notes from his talk and found that he had also included patterns and shape and space in the list of things he found in the curriculum that are not mathematics. I have added them in that bullet above.

Edit: Based on Dr. Bowman’s comment below, I changed what I originally had above  (strategies and tricks should only be for our brightest students, the other students are best served by learning algorithms) to the direct quote he provided.

At a session I did on Friday, I mentioned that if anyone was interested in links to bloggers speaking in support of our revised math curriculum, I’d pass them along. Rather than email them over and over again, I thought it would be more efficient if I catalogued them here.

These are the educators I know of who are speaking in support of our revised curriculum. If I’m missing any, please let me know and I’ll add them. If you don’t have a blog and would like to add something here, let me know and I can publish it as a guest post. I can even turn off commenting so mean people don’t insult you.

Update (March 18, 2014):  A Petition For Educators to Speak Out In Support of Revised Curriculum

Deirdre Bailey

Cherra-Lynne Olthof

Geri Lorway

David Martin

Joe Bower

Me

Radical SNAP

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.

Photo 2-12-2014, 11 01 43 AM

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.

Photo 2-12-2014, 11 11 48 AM

 

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