On Student Mathematical Thinking

Keith Devlin's perspective on mathematical thinking, as presented in this blog post, has been influential in my own thinking about how students interact with mathematics in our classroom.

I am happy to accept this as a working definition for myself:

Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns."'

This is not to say that I think novices (students) are always thinking like experts (mathematicians). I don't. School math and professional math are not the same things. But I do think that students can adopt the identities of (young) mathematicians in the same way that they can adopt the identities of (young) scientists: by using some of the tools of the disciplines as they work at their own levels. Students of all ages are encouraged to construct scientific hypotheses about the world, without controversy, whereas I think it is far too easy for us to present all school math as axiomatic, or even universal, without giving students a chance to see into the mathematical world themselves.

Keeping the expert/novice divide in mind, then, here are some things that students will generally NOT be doing:

-constructing proofs by contradiction
-developing theorems
-proceeding from rigorous logical structures like "there exists a..."
-using set theory

Some things that students will, and I believe should, be doing:

-developing their own reasoning, and "proving" things as true as they can, for any given task
-exploring truths about arithmetic, like commutativity
-systematically exploring why things like the formula for area of a triangle work for all triangles
-making generalizations using algebra, for example about a given linear pattern
-choosing the best "tool" for the job, whether it be adding, subtracting, multiplying, dividing, or higher powered tools, as they get older
-constructing models for how things "might" work, in the context of a given problem

Here is a nice piece that argues that mathematical thinking doesn't look anything like mathematics. If you don't believe that, I give your Andrew Wiles' proof for Fermat's Last Theorem: "Modular elliptic curves and Fermat's Last Theorem". Probably only a few hundred people in the world can read that.

For those keeping track, I am absolutely not arguing that any of these things are done in the absence of content.
I do believe that the content itself is the vehicle for thinking mathematically. I am also unabashedly on the side that says students aren't given enough chances to develop their own reasoning. Mathematical tools that just sit in the mental toolbox are no better than dusty old tools in my basement. They should be used.

I also think mathematical content should be presented for what it is, in discipline-based language. Even the viral Common Core problem on "5x3" was an ideal opportunity to teach kids of any age about commutativity. You might say kids can't handle that- I myself have seen grade 2s in @MrSoClassroom use the phrase "distributive property". A lot of our coaches are also exploring arithmetic in interesting ways with number talks. And yes, they are teaching standard algorithms as well, because that's part of the journey. But number talks might be an example of what Mr. Devlin is talking about-stripping down simple arithmetic into structural elements, then putting the elements together.

Students must be given the chance to work with lots of problems, in order be able to strip them down to their mathematical elements. As they learn more mathematical content, as the grades progress, they will be more able to do so. Sometimes we do that in a more structured way, such as giving a few problems that are obviously "about" Pythagorean Theorem. Other times, we might let students play with the structure of problems (I used to like having students write their own Pythagoras problems, to see what they could do with the problem structure).

Why not? Students come up with things much more interesting than the old "ladder against a wall". 

Even better is using that structure to explain something in the world, in this case, how televisions are sold:

A few pictures I still have and like that I hope illustrate some of these points about student mathematical thinking:

 A student created their own model for the final career point tally of 3 NHL players. He had to decide what method for projecting their career points, including a calculation method and what assumptions to use.

This one represents an attempt to systematize the thousand locker problem.  The deep structure of this problem is that all square numbered lockers are the only ones touched an odd number of times. For any student to understand that, they have to first come up with a tool that will help them find the structure. 

This problem is something about numbers on horse stable stalls. I forget. They made it all the way through a systematic organized list before realizing they could start to generalize about their results. 

This one represents a students' understanding of a dot pattern. She saw the structure as adding a group of three dots to the side each time. It was a big "a-ha" moment, and led her toward the algebraic expression that models this pattern.

This picture represents a students' choice of tool for a problem involving using a certain amount of fencing to enclose as much area as possible. Students were generalizing about the meaning of "squareness" with regard to area and perimeter.

This picture is one of my favourites, It represents a kid exploring the class of polygon called a "trapezoid", in order to attempt to figure out how you might get its area. It's exciting because it led to...

...the beginning of a generalization about trapezoids. In my experience, making statements about all cases, beyond ones that they have found themselves, is tricky. A nice article about how generalization develops in early algebraic reasoning is here. 

This one represents a generalization with algebra about a version of the tables and chair problem. Gradually students realize that the "+2" stands in for the two chairs on the end, that are always there, even as more tables are added. 

In summary, I believe that students are very much capable of mathematical thinking, even if they don't always "think like mathematicians". They need lots and lots of chances to break down problems into their basic structural elements, to explore their own reasoning, and to start to make interesting generalizations on their own. 

Hope #oneword

When I think of my #oneword for 2016, I don't think of anything specifically related to teaching and learning. I don't think of "rigour", "grit", "moxie" (apparently a thing  now?), or anything to do with technology or mathematics.

I won't take a stand for something new and trendy, and I won't take a stand against something I see as stale and old. In 2016 I would rather be for things, rather than against things, anyway. I am for optimism, and I am for hope. I don't believe the world is getting worse every day. It's getting better. I am conscious now, on the planet, and my consciousness, as of all humans, goes back and forth between hope and despair, between the darkness that we stare into, and the light that comes from within (or without, depending on your spiritual beliefs).

I am for being hopeful. I am for supporting each other, when we're struggling, or feeling down, hurt, or broken-hearted. I am for the mental health of our teachers, administrators and support staff. I am for supporting our students to be hopeful, even in the face of trauma and despair. Classrooms are places of hope. (You can ignore trendy arguments against "safe spaces", classrooms must, always, be safe spaces).

This will probably be seen as another post about putting human relationships above all else in our education systems, and it is. It's about sustaining hope in each other. It will probably be seen as another mental health advocacy post, and it is. Most struggles are struggles of the mind. Look around you- are you aware of the struggles inside each and everyone around you, even if they don't show on their faces, or come through in their words?

There are people around you who:

-stare down the vicious monster called depression on a daily basis
-struggle with dependency (particularly on alcohol- if you are an adult in this world, you know someone who struggles with alcohol- you just don't KNOW it)
-are critically ill, or have loved ones who are, or who struggle with chronic pain
-have unresolved trauma in their past
-have suffered through broken relationships, heartbreak, heartache
-everyday swallow the bitter pill of disappointment over unrealized hopes, dreams, aspirations

I could go on, but I won't. Be the "light in the darkness of misery" (E. Costello) for others, but above all, take care of yourself. As much as is possible, be hopeful. Be not so fearful, and know that everything has its equal, and its opposite. Light has darkness, darkness light. If you feel despair, you can feel hope. Be ever hopeful.

Light vs. Darkness by Ritesh Man Tamrakar 

(CC, Flickr: https://www.flickr.com/photos/_rmt_/5569820904)

Multiple Entry Points for Mathematical Tasks: Lessons from Phys Ed Teachers

We had an interesting discussion about mathematical tasks this morning, specifically, rich tasks, and what makes them so.

A principal in the room brought her unique physical education background to the discussion. If you have, say, your baseball unit happening, and you have some elite level players in your class, and some who have never touched a bat and glove before, how do you structure your classes?

I think it's fair to say you can deploy what Boaler, and others, call "low floor, high ceiling" tasks. Check out the YouCubed tasks here and see what you think of them. Further, you can recognize that some students will need more time with thinking tools, or exploring of concepts, where other will need more time on specific practice elements.

This brought to mind my own experiences in grade 9 phys ed class. I was only ever good at one sport, basketball. This was because of endless hours spent shooting baskets at the school down the street. This led to being on the basketball team. That year, the basketball unit in phys ed class happened in the middle of basketball season. The games in class were too easy. Most people didn't know how to play basketball at all. Having only this one single entry point to the game of basketball was probably quite frustrating to those who were just learning how to play. That said, those at lower skills levels would have benefited more from skills drills, than from being in a game they weren't prepared for.

I think we see something similar play out in our math classes. Many of our students come with prior knowledge of the topics that will be taught, especially as math is now more "out there" in the culture than ever. (If you don't believe me, ask your students what they know about pi!) Others will struggle to even enter the mathematical content, having gaps in their skills or conceptual knowledge, or a history of struggling with math.

The challenge for us then is structuring our classes so everything can have something to do that will challenge them, as they work toward understanding the big ideas of the math curriculum, and acquiring the skills they will need to progress.

Here is an example of a task, Four 4s, that I think nicely allows students at all levels to practice their operational skills. This is less a task, and more a class of tasks.  In other words, this task could easily generate many more extension or follow up tasks involving operations. "Use all/any operations to make a certain number/integer/fraction", etc. It is possible to enter with just adding and subtracting, then multiplying and dividing, negative integers, squares, exponents, square roots, and so on. In my experience, students who have the comfort with the basic operations will try and come up with more and more elaborate expressions- nested brackets, exponents raised to exponents, fraction bars, and so on.

Another example: change "basketball class" from my example to "integer class".

If our curriculum goal is understanding and being able to add and subtract integers, consider all the actions any given student (or small group of students, for guided explicit instruction) could take:

-working with 2 colour tiles to explore the zero principle
-working with number lines to understand  integers visually
-exploring a context for integers, like golf or temperatures
-doing a worksheet on adding and subtracting integers, if they are ready

Many more activities are possible. Having flexible groupings based on readiness, and providing direct instruction as needed are the key things here. The math teacher is more like a coach, providing feedback, guidance, and explicit instruction as needed. "Sage on the stage" and "guide on the side" are outdated descriptions of what we do- perhaps "coach in the middle" is more like it. Here is a piece I wrote on being a guide on the side, versus being a "coach in the middle".

How To Be A Good Reader, or, How To Get Better At Math

For some of us, we can no more go without reading than we can go without breathing. Famously, I used up all my Hardy Boys before we even got to Europe, when I was 7. We had to find some more books in English. I can't fall asleep without reading, even if it's just those two pages, as the words swim, and the book (or tablet!) falls smack on my face as I fall asleep. I need to read my entire Twitter feed, and all of my flips in Flipboard, at least once a day. I am a subscriber to that dying industry, newspaper delivery. I keep up as much as I can with Batman, and lots of other comics.

I read a lot.

I think i'm a pretty good reader.

Schools do a fantastic job of teaching reading strategies these days. That is, they get students to pay attention to the things that are happening in their mind as they read- visualizing, connecting to our own lives, or to other texts, and making inferences about what is to come. Schools do a great job at making these things that readers do explicit, and they do a pretty good job at providing an interesting variety of texts for kids to read.  They are doing an increasingly better job with digital texts, and needed critical literacy skills for "the now".

The best way to get good at reading is to read. A lot. Schools are very good at promoting literacy outside of schools, and parents of young children all know the importance of taking their kids to the library, and reading to them as much as possible.

This makes sense, right? After all, words are all around us. Being literate in the world is being able to read, the world. Kids, from the time they are in their baby seats, are looking around, trying to decode signs, or ads, to "read the world".

Do we make the same connections with math? How do you become a numerate person? (My outstanding Peel colleagues have written about the 4 roles of the numerate learner here.) I would argue we often don't know how. Either we never figured out how to transfer mathematics out of our classroom experiences, into the world, or our school experiences with math were so dreadful, we can't even imagine the thought.

The Ontario Council of Directors of Education has a new resource to help parents help their children with math. I  saw Lynda Colgan (Queen's) present this resource. In here words, parents know more math than they think, they just call it "common sense".  That squares nicely with my favourite definition of mathematics, from Jordan Ellenberg:  "the extension of common sense by other means.

Common sense to a plumber is using Pythagoras to fit pipes in small spaces. Common sense for me this morning was scrounging change for Callum's lunch order. Common sense to many of us is searching out the best deals on Black Friday. We deal with negative integers every year, around November, as Canadians.

Being numerate in the world is finding numbers in the world, yes, is "reading"the world with math. Many of us watched the Blue Jays make a deep run this October. Did you ever think about what needs to happen for a .299 batting average to go to .300? It's hockey season. What does a .908 save percentage mean to a goalie? What about a 6% shooting rate, if you're Ovechkin? What does it mean that 39.6% of us voted for the Liberal party? What is 50% off, then an additional 20% off that shirt?

Is infinity a number? (My own children want to know). 1+1=2 (That's Alec's favourite "number" right now). Are we there yet? (Not long now!)

Math is everywhere for us to find. To get back to the point of this blog post, the best way to get better at math is to do lots of math. Do your school math, if you are in school. Get curious and watch that Numberphile video about prime number encryption. "Play" with the calculator on your phone. Join a MOOC. (Who knows, you might be among the 6% of all people who actually finish the MOOC.)

If you have small children, let them play with money. Explore the meaning of the equal sign. Count a set of objects. Count again. And again. And again. Explore different ways of making 5. 10. Play with a hundreds square. Look at the cost of toys in flyers. Give them $10 to spend at the dollar store, and see what they do with it. Do math. Everyday. Just do math.

                                                     Baseball: the mathiest of all sports

How much wall space do you need for this t.v.? How much would the tax be?

Read. Read lots. 

Whole lot o' Costco mayonnaise! What's the unit rate?

What percent bigger is Olympic ice, compared to NHL ice? Why does that matter?

Connective Tissue and Bone: On Multiplication Facts

Jo Boaler is making the news again for her stance on teaching times tables. I don't wish to wade into that debate, per se, but to consider how personal and idiosyncractic our knowledge of multiplication facts can be.

I wrote this tweet because I am very curious about what mental models or imagery others may have when presented with multiplication facts:

Is there anyone out there who *hasn't* memorized their times tables, and is willing to describe what they "see", or their workarounds?

— Matthew Oldridge (@MatthewOldridge) September 19, 2015

The "see" part was inspired by an incredible Twitter conversation started a while back by @Sue_Cowley and @surreallyno.

The Storify of that discussion is well worth your time. I know it blew my mind.

An amazing array of responses https://t.co/Uaoq69oagr generated by @Sue_Cowley 's question! #mathchat Constructivism at its best.

— Cristina Milos (@surreallyno) July 6, 2014

I don't personally have access to any of the interesting mental models, visuals, strategies or schema that others report having. I just don't I can't wind back the clock to how I learned to multiply. I just have the schema I have. 7x8 ("fifty-six"). 8x9 ("seventy-two).

@surreallyno @Sue_Cowley I can honestly say I have no imagery whatsoever when I think of a multiplication fact. I just hear the number.

— Matthew Oldridge (@MatthewOldridge) July 6, 2014

My working hypothesis is that I learned solely by rote (flashcards and worksheets).  I don't fully remember. If my hypothesis is true, my ability to break multiplication facts down, and find the connective tissue between them, came later, through repeated exposure to them, and lots of practice.

So if you say, 8x8, I think "64". I "see" nothing. But I can break down this fact many ways.Learned rote, then made sense later? @Madame_JB

— Matthew Oldridge (@MatthewOldridge) September 24, 2015

I am not willing to wade into the knowing vs. understanding debate here. I know this fact, and I understand what "64" means. But my 7 year old self didn't have the benefit of the more than 30 years of experience I was later to get with multiplication facts. Did my seven year old self "know", or "understand" that 8x8=64? I learned up to the 12 times tables, but I stumble on some of the 11 and 12 times table facts.

Children should have the opportunity to make sense of numbers, to play with them, and find what I call the connective tissue, the very fabric that makes up all numbers. I won't take a stand against automaticity, as an end goal, but I think the means are more important.* If my school aged self simply memorized the times tables, then didn't do any further work making sense of them, I probably wouldn't have the rich schema for numbers that I have. This is the worst case scenario described by Boaler and others. In my case, I had many more years of math to come, and I use a fair amount of math in my job.

Further, I would have liked things like playing dice or card games to practice facts, or listening to others share their thoughts and strategies during number talks. There's a lot of nice and balanced ideas in this Reddit thread about learning times tables.

I think that, especially in K-8, children should be encouraged to "play" with numbers-find patterns, break them apart, make connections between them. I used to do a lot of lessons with the hundreds square, do you know how many interesting tasks you can build from that?  Again, I am calling this the connective tissue of number (or lack of connective tissue, when it comes to primes, but that's another story). Maybe just learning multiplication facts on their own is all bone, no tissue.

I have since heard stories about professional mathematicians who stumble when presented with certain facts (like 7x8), or fellow teachers who use what they call "workarounds" with things like the 6 times table.  I'm not sure at this point though, what's a "workaround", and what's a "strategy". My mind is my mind, and yours is yours. We need tissue, and we need bone.

*I am also leaving aside the thorny, knotty issues of what constitutes "fluency", "automaticity", and "memorization". Others have covered that quite well.

What Things In Our Math Curricula Do You Think Have Become Outdated (...Or Even Obsolete?)

The always interesting Keith Devlin writes here about the future of math. As a mathematician, he has a good perspective on the ways in which math education has changed. For Devlin, our education systems are lagging behind the world of math. Math has changed, but has how we teach it?

I will quote a whole paragraph here:

"To most people, mathematics means applying standard techniques to solve well defined problems with unique right answers. They have good reason to think that. Until the end of the 19th Century, that’s exactly what it did mean! But with the rise of the modern science and technology era, the need for mathematics started to change. By and large, most people outside mathematics did not experience the change until the rapid growth of the digital age in the last twenty years. With cheap, ubiquitous computing devices that can do all of the procedural mathematics faster and more accurate than any human, no one who wants – or wants to keep – a good job can now ignore that shift from the old “application of known procedures” to new emphasis on creative problem solving."

Jordan Ellenberg is also quite persuasive on this topic, and if you haven't read his book, How Not To Be Wrong, you must. Machines can do the laborious work of long calculations, so our brains are freed up to do what we do best-think.  Conrad Wolfram is another who sees computing as a large part of the future of math.

We may not need to do long laborious calculations by hand, but we do, of course, need basic number sense, combined with our intuitions and our operational skills. One example from my own schooling is calculating approximations of square roots by hand- I remember doing it in school, but I am not aware of anyone who teaches it now. Of course, if you are a certain age, you remember having to use slide rules and log tables. Probably nobody is nostalgic for that!

I got to thinking today, after an interesting conversation with @MathiesUnite. We talked about how, with the dawn of Desmos, drawing graphs by hand is a less useful skill than ever. Have you ever planned a lesson that involved drawing a graph, and looked around 20 minutes later, and some kids hadn't even put the scales on the axes yet?  I have. I have seen terrible hand drawn graphs. The worst. I have seen graphs so scary bad they  didn't even look like any more than a bunch of squiggles on a page. Why  should we not use the tools at hand to draw graphs more effectively?

But Desmos (and others) are so much more than that-you can compare two data sets, at a glance, within minutes, freeing up time for discussion, interpretation, and analysis. Nobody would argue that we don't need good data interpretation skills these days.  

@BrianPenfound offers some pushback on the idea of digital tools replacing graphing by hand completely:

@MatthewOldridge I am on the fence about that one. I think similarly to calc. If introduced too early, can become crutch @MathiesUnite

— Bryan Penfound (@BryanPenfound) September 10, 2015

The discussion got me thinking, and I posed this question, later:

Instead of always just, "graph the data", could we instead ask our students to "visualize the data"? Would that work? #MTBoS #EngageMath

— Matthew Oldridge (@MatthewOldridge) September 10, 2015

That's not to say our students will do FiveThirtyEight level data visualizations, nor that they don't need to know how to use conventional graphs. I just wondered, in the age of pictographs, incredible fan-made baseball data visualizations, and so on, if that might be another "new" area of math we should explore with our students.

The definition of data visualization offered by Wikipedia makes me wonder if it might not be its own literacy, on its own.

So graphing by hand might be one, do you have any other ideas for items from math curricula that you suspect could be obsolete, or, at the least, outdated in their conception?

Had a discussion about graphing by hand today with @MathiesUnite. What other "traditional" math curriculum items do you see as obsolete?

— Matthew Oldridge (@MatthewOldridge) September 10, 2015

Zooming in and Zooming Out: Considerations for Math Curriculum Planning

Use the scale of the universe website. Zoom in, all the way to the Planck Length. Zoom out, to the edge of the observable universe. Now consider your own perspective on curriculum (a very personal thing). How much detail is too much? Do we see more, or less, when we zoom all the way in  on the curriculum particulars, or do we see more when we zoom all the way out and try to find the big ideas underpinning the topic or subject matter? 

I've zoomed in often enough. My 2005 Ontario math curriculum is missing the cover, and it's spattered with coffee and highlighting, scraps of sticky notes, random notes, and scribbles. "Know thy curriculum", should be our mantra.

It takes a long time to understand how it works, longer still to figure out how to make the language of of overall and specific expectations come alive. It takes years to know it well enough to get past the idea of specific curriculum expectations as a kind of shopping list for your year or course.

Here are some planning considerations that I think make the job of working with curriculum easier. 

My first key point:  know the front matter. In the case of the math curriculum, the mathematical processes come first, and as it says:

-the mathematical processes cannot be separated from the knowledge and skills that students acquire throughout the year.

These five specific processes (reasoning and proving, reflecting, connecting, selecting computational tools and strategies, and representing) are tied in to the larger processes, which are: problem-solving and communicating. The curriculum characterizes these as the actions of math. We need to think about what sorts of things we want our students to be doing in the math classroom.

Your second concern is the overall expectations for each grade. We know from our assessment and evaluation document, Growing Success, that we must base our evaluations on the overall expectations. Using the specific expectations (many that they are) as a checklist for teaching won't do-this is a recipe for frustration. Been there, and done that, when I didn't know any better.

Consider any connections you can make between the overall expectations from each strand. Any connections you can find can only help you. Try and bust out of the "strand by strand" thinking. We do have to report on each strand, but cross strand tasks or problems will help students to see the connective tissue of math, and the underlying structure underneath.

Here is an attempt at using proportional reasoning to tie together a few strands and concepts:

Students will use multiplicative relationships to compare rates, ratios, fractions, and patterns in expressions, graphs.

Here is another bigger idea that I believe fits grades 4-8:

Percent, rate, ratio, and fractions are connected.

Try and consider the bigger ideas behind the curriculum expectations. This is difficult work, but it is the "'zooming out" part.

One version of the mathematical big ideas (by Randall Charles), are here:
http://www.authenticeducation.org/bigideas/sample_units/math_samples/BigIdeas_NCSM_Spr05v7.pdf

Many people prefer the very user friendly "Big Ideas from Dr. Small:  http://www.nelsonschoolcentral.com/cgi-bin/lansaweb?webapp=WBOOKSITE+webrtn=booksite+F(LW3ITEMCD)=9780176110789

Another consideration is our new "Paying Attention to Math" series, which includes proportional, spatial, algebraic reasoning, and fractions. Each of these monographs allows us to focus on the reasoning underneath the math curriculum. In so doing, we find that proportional and spatial reasoning, for example, can be found "underneath" a lot of our curriculum expectations.

Here is a successful planning sequence I have seen in a number of collaborative inquiries:

-use the proportional, algebraic, or spatial reasoning monograph to frame the initial discussion
-find a matching mathematical big idea (e.g. from Dr. Small)
-consider what actions students will be taking (sometimes you want to target a specific math process(es)
-match your thoughts to an overall expectation from the curriculum (and specific ones, if  you are designing something across more than one strand)

The concept web from Paying Attention to Proportional Reasoning:

Here is the concept web showing the  surprisingly large number of things that can be linked to spatial reasoning:

Here is some work on a version of the classic "tables and chairs" problem:

A possible planning sequence for using this problem:
-look at Paying Attention to Algebraic Reasoning, and see what concepts are at play (will your students just make a table, or can they generalize to the nth term?)
-consider the Representation process (you could us a table, graph, expression, or picture here)
-a big idea could be: Algebra helps us to model situations efficiently. An extension concerns a hypothetical dinner party with 1000 tables. Does algebra help us as a "shortcut"?
-grade 7 overall expectation:  "represent linear growing patterns...using concrete materials, graphs, and algebraic expressions.

I want to be totally clear here that know thy curriculum is the number one principle at work here. You must know the specifics, and be clear on the particulars, before you can zoom all the way out and focus on the big ideas. But don't be afraid to "play", and find what works for you!