Monday, December 14, 2015

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".


Wednesday, November 25, 2015

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?




Friday, September 25, 2015

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:


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.


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).

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.


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.

Thursday, September 10, 2015

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:

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






Thursday, August 20, 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!



Wednesday, August 19, 2015

Critical and Creative Thinking in the Math Classroom (Updated)

I believe critical and creative thinking are both essential to doing math.  Yet I believe both are relatively unexplored areas with our young student mathematicians.


Here is the lone reference to critical and creative thinking in the Ontario curriculum:



The star below the achievement chart is a footnote explaining that critical and creative thinking are present in some, but not all, math processes. It does not elaborate which! Obviously, this is not helpful- if the math processes are the actions of doing math, it makes sense then that these actions will, at times, encompass critical and creative thinking. Further compounding the problem, critical and creative thinking are, at best, ill-defined. The role of teachers in teaching critical thinking is debated- see Daniel Willingham’s piece here.


What is critical and creative thinking in the math classroom? What does it look like in the math classroom? I am starting from the presumption that all kids are capable of critical and creative thinking. My second presumption is that mathematical knowledge and skill gained as children go older allows them to think creatively and critically. Third, I don’t buy the typical (and somewhat ill-defined) notion that creativity and critical thinking are only typical of “higher order thinkers”. It depressed me to no end when I did my literature review on these two topics and found that much of the work on these two types of thinking were done with gifted learners.


The other common line of thinking is that critical and creative thinking are somehow opposite, or at odds or competing with each other. I don't buy this false binary. Typically this binary is set up as “making” versus “assessing” or “judging”. I believe that both are intrinsically tied together.

Here is an example I like to come back to. A student came up with his own method for predicting the career points scored of several hockey players. In his judgement, here they are:
Here's a nice quotation on critical and creative thinking:


"These two ways of thinking are complementary and equally important. They need to work together in harmony to address perceived dilemmas, paradoxes, opportunities, challenges, or concerns (Treffinger, Isaksen, & Stead-Dorval, 2006).


Further, Poincare said something to the effect that mathematical creativity is simply discernment, or choice. Our young mathematicians will make judgements as they are solving problems, deciding which path to follow, and when. They will pick the best representations for their mathematical work, and their own idiosyncratic mathematical voice will come out. (Given a classroom culture of math talk, our students will find their voices. “Voice” is not just for the English classroom) Doesn't that sound like critical and creative thinking, combined in one neat mathematical package?


I have a dislike for overly complicated frameworks and definitions that clutter and obscure important concepts.  Einstein may have said something about how if you understand something, you can explain it to a child. If we can explain the quantum world without jargon, we can explain educational concepts without jargon, so here goes. Here are my personal working definitions of each:


Creative thinking: making something new.
Critical thinking:  making sound judgements.


Yes, these are deliberately economical. Yes, you could add to these definitions if you wanted to. But if you are a student, and you are doing a mathematical problem or task, you are making something new every single time. There will be patterns and trends in the strategies and tools that individual students use that further differentiate more “unique” or “divergent” work which will perhaps “more” creative.  I also maintain that, provided we don’t oversimplify our mathematical tasks to take students’ judgements away, they will be constantly hypothesizing, choosing, testing, and revising their work.  


How does this happen in the math classroom? How can we harness these two powerful types of thinking?


In the first case, if we don't see math as a generative process, a creative process, then we will not find creative thinking. Look closely at the picture I started this post with: both problem-solving and inquiry are mentioned.  To the former: problem-solving classrooms will always have an element of creativity, unless we force our own methods, techniques and processes on our students. It will always be our job to consolidate purposefully, and to offer suggestions as to more efficient or effective solutions. The range and variety of the student work, with all its understandings and misunderstandings will lead us to that point. A balanced math program with strong foundations and and a spirit of questioning will always lead to interesting lines of inquiry-questions leading to more questions.

The beautiful diversity of student work:
Here is a video where we analyze the student work in our LearnTeachLead project, "Loving the Math, Living the Math." 


One of the best parts of really getting to know your students is starting to see inside their idiosyncratic mathematical thinking. For a long time, I felt like creativity was that certain "je ne sais quoi" of the math classroom, a "know it when I see it" type of thing. When I thought this, I probably didn't have a broad enough definition of creative thinking. I was waiting to be bowled over by stunningly divergent solution paths. That does happen, but not always.


Since, I have been watching for more subtle evidence of creativity.  Students using new thinking tools, or subtly tweaking a solution path or process they may have got from talking with their classmates. Creativity is there to be found in the math classroom.

Here is an example of a student finding a new use for Minecraft as a thinking tool to represent data:



Inquiry is also hidden in that little line in the picture from the curriculum above. Inquiry to me means: asking good questions. Are our students question askers? There are some astounding numbers floating around about the ratio of students asking questions, to teachers asking questions, in a typical math classroom. Question askers are typically critical thinkers.  Once your classroom is an open space for wonder, your students don't stop wondering! Questions lead to answers, leading to more questions (I once called this the "inquiry tumbleweed").


The key thing is that students are becoming more confident in their judgements as young mathematicians.  I want them to be able to use their mathematical thinking tools to decide "what's best", or "what's fair". I want them to justify their thinking. I want them always probing the mathematical world around them with their confident judgements.


This is one of my favourite things to tweet now and again:

This work came out of our LearnTeachLead project involving proportional reasoning: http://learnteachlead.ca/projects/loving-the-math-living-the-math-part-1/.  I found some very precision judgements happening, like students telling me a cup of pop was worth exactly $1.26. Not $1.25, not $1.27- $1.26. The power of their thinking led them to this conclusion.


There a nice quote in this book excerpt about how the "best way to think critically is to think critically". We are risking circular logic there, but think about it: the best way to learn to think, is to think. That is why our classrooms should be open thinking spaces. If they are, our students will be constantly making judgments, testing them, revising them, and drawing meaningful conclusions about the important mathematical work of the classroom.
See:  

Treffinger, D. J., Isaksen, S. G., & Stead-Dorval, K. B. (2006). Creative problem solving: An introduction (4th ed.). Waco, TX: Prufrock Press.

"Loving the Math, Living the Math" at LearnTeachLead is here. 




Monday, June 22, 2015

Meet the "New Math"...Same as the "Old Math"?

The title is partly flippant, but mostly a music geek's Who reference ("Won't Get Fooled Again"). We spend a lot of time debating "new" math reforms (that are in some cases over 50 years old), and we spend a lot of time thinking about the things that are "old" (time-tested, essential, or that truly matter). Where we stand on these issues says a lot about our own beliefs as educators, and things we learned and internalized in our own schooling.

I honestly thing we need to focus on the common ground that we all share in our beliefs about math teaching and learning. Polarities like "new"/"old" and "discovery"/"traditional" don't tell the truth about how things actually are.

What follows is a short module designed to get us examining our own assumptions about math teaching and learning. Its purpose is not to inflame. Its intended audience is anyone who cares about math teaching. If that is you, please feel free to comment.

Meta moment: the text below is also in this Google doc if you want to add to it:  https://docs.google.com/document/d/1imNVRX01EBHA3NKu9RkOz0UkBb_f0e-JgqLztuRrxrQ/edit?usp=sharing

On “Back To Basics” vs. Reform Mathematics in Schools

The debate over what constitutes “good” math instruction has flared up again recently in Alberta, with the publication of a recent CD Howe institute report, which gives recommendations.


Questions we all must answer for ourselves, as mathematics educators:  

  1. What do we mean by “basics”?
  2. What factors have influenced your own personal definition of the things that are “basic” to math teaching and learning?

Further to #2, reflect on your own personal experiences with school math. To a large extent, your answer to the question “what is math?” was informed at a young age by your school experiences.

3.  Did you see math as creative, vibrant, beautiful, interesting and alive as a student?

4.  Does it matter whatsoever if you experienced math as creative, vibrant, beautiful, interesting and alive as a student?

“Discovery” Learning: What does that mean?

What, if anything, do we mean by “discovery” learning?

See if the Wikipedia article helps us here.

Here is an article that seems to be the main one used as fuel against “pure” discovery learning. (Or, learning that is done with minimal teacher guidance).

In your opinion, what are some math concepts that can be “discovered” through exploration, and careful teacher guidance.  

I will give your the relationship between any circle’s circumference and its diameter. What can you give me? (Examine Ontario TIPS unit here.)

A lot of time has been spent unravelling what we mean by “direct instruction”, “explicit instruction”, and “inquiry learning”.

Is “guided discovery” the same as “direct instruction”?

As a teacher, what is your own personal definition of direct instruction?

Here is John Mighton, founder of JUMP Math, on what he sees as “guided discovery”: http://www.theglobeandmail.com/news/national/education/kids-cant-figure-out-math-by-themselves/article15087557/

Fluency, Automaticity, Memorization:  3 Sides of the Same Coin?

My assumptions:
-more practice with more number facts will lead to being fluent with them
-encountering number facts frequently through meaningful work, problems, and games will lead to them being committed to long term memory
-flexibility with facts, gained through things like number talks, will lead to more connections being made between numbers, and facts
-K-3 is a time to build a deep base with number sense
-teaching through problem solving, as specified by the Ontario Curriculum, is the best way to build procedural and conceptual understanding
-direct instruction does not always mean talking to the whole class at the same time
-mini lessons on key mathematical concepts, given at a key time in the instructional sequence, pack the most punch

Assumptions that are often made by “back to basics” advocates:

-number facts should be committed to long term memory ASAP, to “get them out of the way”. Grade 4-5 for memorizing times tables, for example, is seen as far too late
-standard algorithms should be taught as early as possible
-encountering problems without prior scaffolding is too much of a cognitive load on students
-forcing students to explain their thinking gets in the way of actually doing math
-direct instruction should take up most of our time.

Here is a Jo Boaler article on fluency that is popular right now:



Develop your own list of assumptions. What common ground do you see? What things do we all agree on?

Thursday, May 14, 2015

Surprised By Their Mathematical Thinking

I have been thinking a lot about mathematical surprise these days.  Specifically, I have been thinking about all the different ways we can be surprised in our math classrooms.

We have worked a lot at creating mathematical thinking spaces for our students. The lovely monograph by Dr. Chris Suurtamm Making Space for Students to Think Mathematically nicely formulates how our classrooms can be mathematical thinking spaces.




The above image is some student work on a proportional reasoning problem. (I was going to say unit rates, but leading in that direction might already be taking some of the thinking away, don't you think?) 

The video clip linked here shows 3 of us analyzing the chart paper you see in the picture:  analyzing the thinking. Our students used many surprising strategies. Some were radically different, using completely different thinking tools, or perspectives on the problem.  Some were subtly different. There were huge mathematical implications on the rates they chose to use, for just one example.

I had a chat with a brilliant teacher the other day, and he told me he identified 26 distinct solution paths for one problem (the classic "tug of war") problem. It may be this Marilyn Burns one, or another- it's a classic context for a problem. One this one you may rightly note: we as experienced thinkers might choose to model it with algebra. This is only one type of solution! Don't underestimate the flexible thinking of the novice, with less experience and context to hem in their thinking! (Dr. Brent Davis out of Calgary calls this the mistake of the experienced thinker)

Once we remove our preconceived notions of what the problem "should" be, we can focus on what sort of thinking our students are doing. They will surprise us!  I haven't been involved in one collaborative inquiry where we haven't been surprised by at least some element of one solution! We must be open to surprise, in our classrooms.

One minor caveat: I am not saying here we are "surprised" because we don't know the math at hand. We must know the fundamental big idea in the math, and the connections to curriculum (both content and process). Many of our teachers and coaches are working from the "Five Practices for Orchestrating Productive Mathematical Discussions" book. The key practice for me is anticipation: we must know the math, do the math, and anticipate student responses to the best of our ability.

But, I maintain: the element of surprise will still remain. Sometimes it's a powerful conjecture from a usually quiet student. Other times it's a subtle variation on the math you expected. It could be a question that arises from a student, and spurs them to create more math.  Sometimes it's a lightning flash of insight- an amazing and new solution path that you have never seen before.


Last, what surprises will emerge from the murk and (seeming) mess, when students are allowed open spaces for mathematical thinking?

Tuesday, May 12, 2015

Critical and Creative Thinking in the Math Classroom (Outtake from OAME Ignite)

Critical and creative thinking are both essential to doing math.  Yet both are relatively unexplored areas with our young student mathematicians.

Here is the lone reference to critical and creative thinking in the Ontario curriculum:

The star below is a footnote below the achievement chart explaining that critical and creative thinking are present in some, but not all, math processes. It does not elaborate which! Obviously, this is not helpful- if the math processes are the actions of doing math, it makes sense then that these actions will, at times, encompass critical and creative thinking.

But what is critical and creative thinking in the math classroom?  I'm leaving aside here the debate over "traditional" and "new" methods in math teaching and learning. I am starting from the presumption that all kids are capable of critical and creative thinking. It depressed me to no end when I did my literature review and found that much of the work on these two types of thinking were done with gifted learners.

I also don't buy the false binary that critical and creative thinking are somehow "opposite" or "at odds" with each other. Typically this binary is set up as making versus assessing or judging. But I believe that both are intrinsically tied together.

Here's a nice quotation on the matter:

"These two ways of thinking are complementary and equally important. They need to work together in harmony to address perceived dilemmas, paradoxes, opportunities, challenges, or concerns (Treffinger, Isaksen, & Stead-Dorval, 2006).

Further, Poincare said something to the effect that mathematical creativity is simply discernment, or choice. Doesn't that sound like critical and creative thinking?

I have an intense dislike for overly complicated frameworks and definitions that clutter and obscure important concepts.  So here are my personal working definitions of each:

Creative thinking: making something new.
Critical thinking:  making sound judgements.

How does this happen in the math classrom? How can we harness these two powerful types of thinking?

In the first case, if we don't see math as a generative process, a creative process, then we will not find creative thinking. Look closely at the picture: problem-solving and inquiry are mentioned.  To the former: problem-solving classrooms will always have an element of creativity, unless we force our own methods, techniques and processes on our students.

One of the best parts of really getting to know your students is starting to see inside their idiosyncratic mathematical thinking. For a long time, I felt like creativity was that certain "je ne sais quoi" of the math classroom, a "know it when I see it" type of thing. When I thought this, I probably didn't have a broad enough definition of creative thinking. I was waiting to be bowled over by stunningly divergent solution paths (and that does happen!)

Since, I have been watching for more subtle evidence of creativity.  Students using new thinking tools, or subtly tweaking a solution path or process they may have got from talking with their classmates. Creativity is there to be found in the math classroom.

Inquiry is also hidden in that little line in the picture from the curriculum above. Inquiry to me means: asking good questions. Are our students question askers? There are some astounding numbers floating around about the ratio of students asking, to teachers asking, in a typical math classroom. Question askers are typically critical thinkers.  Once your classroom is an open space for wonder, your students don't stop wondering! Questions lead to answers, leading to more questions (I once called this the "inquiry tumbleweed").

The key thing is that students are becoming more confident in their judgements as young mathematicians.  I want them to be able to use their mathematical thinking tools to decide "what's best", or "what's fair". I want them to justify their thinking. I want them always probing the mathematical world around them with their confident judgements.

This is one of my favourite things to tweet now and again:

This work came out of our LearnTeachLead project involving proportional reasoning: http://learnteachlead.ca/projects/loving-the-math-living-the-math-part-1/.  I found some very precision judgements happening, like students telling me a cup of pop was worth exactly $1.26. Not $1.25, not $1.27- $1.26. The power of their thinking led them to this conclusion.

There a nice quote in this book excerpt about how the "best way to think critically is to think critically". We are risking circular logic there, but think about it: the best way to learn to think, is to think. That is why our classrooms should be open thinking spaces.