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.

Read this news report on the controversy: http://www.cbc.ca/news/canada/manitoba/canada-s-math-teachers-should-get-back-to-basics-report-says-1.3090099

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:

http://www.youcubed.org/wp-content/uploads/2015/03/FluencyWithoutFear-2015.pdf

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

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?

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:

"Use of critical and creative thinking processes." <most under explained line in @OntarioEDU math curriculum! pic.twitter.com/WYIJWbnAwb

— Matthew Oldridge (@MatthewOldridge) May 12, 2015

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:

As I said in my #Ignite, what is math for, if not to make critical judgements like this? #OAME2015 pic.twitter.com/uXprdl2kTb

— Matthew Oldridge (@MatthewOldridge) May 12, 2015

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.

An open space for mathematical thinking. #EngageMath pic.twitter.com/H9no0tJOJy

— Matthew Oldridge (@MatthewOldridge) May 12, 2015

Roll Up The Win To Win, Odds, Luck, and the Law of Large Numbers

Pre and post ceremonial losing cup stomp. #timhortons #RollUpTheRim #odds #probability pic.twitter.com/qLz8fd9Wp4
— Matthew Oldridge (@MatthewOldridge) February 8, 2015

A Warm Cup Full of Hope?

"Roll Up the Rim to Win" season is a yearly warm cup full of hope in the otherwise dreary month of February. There is a car at stake, after all (one for the whole country), and the winner could be me.

Here is how the narrative of luck plays out in our heads:  I realized after the fact that I threw out a cup in Target (speaking of dreary Februarys).  Briefly, it crossed my mind,  "what if it was a big prize?"
We might feel the same way if we threw out a lottery ticket.

Here are some reasons why I shouldn't worry about it:

-The stated odds (from http://www.rolluptherimtowin.com/en/in-restaurant) are 1/6 to win any prize. I only had about a 17% chance of winning *any* prize. I can live with forgetting about that cup.

-there are 3 cars to be won in Ontario.  There are some 143,000,000 cups being sold in Ontario.  See here: Restaurant Rules. This is statistically zero. Actually, do some lotteries have better odds?

-there are 94 t.v.s to be won in Ontario.  Again, statistical zero.

-there ARE 23,000 $100 gift cards at stake.  I calculated a 0.16% chance of winning one. Again, I can live with not rolling that cup

-that means the 1/6 odds basically apply to food prizes. There is a 17% chance I missed out on a doughnut.

-the picture above is from the first cup i've had this year.  I didn't roll the second, because I threw it out. Expecting a streak of 5 losses in a row is how you might need to think about it.  Then again, you can sometimes roll a "six" in a dice game, when you need it...

The Law of Large Numbers, and Our Perception of Luck

I will let you read the Wikipedia article on the Law of Large Numbers. The  "house always wins" rule, you might call it.  One of the most brilliant ways to teach it is to use a coin flip simulator, combined with the kids actually flipping their own coins a number of times.  See the Virtual Manipulative here. They watch the sim go to very close to 50/50 over just a few thousand trials, and understand that the odds must come true over time.  How do you think casinos stay in business?  Because they *know* how much they will lose and gain, as an average, on every single bet.

I used to have kids record their flips like this:

e.g. HTHHHTTHHTTTHHTTT...

There is an amazing trick you can pull on them. Have them record one real trial of 50, and one fake trial.  You will be able to guess which one is fake, every single time!  Why?  Simple-our brains perceive 50/50 as truly random, so kids will make their fake trial *too* random!

Real trials of 50 will usually have a streak like:

HHHH or TTTTT.

Look for those.

How does this apply to Roll Up the Rim?

Let's say you have won 3 doughnuts in a row.  You feel lucky right?  You might even feel like you should go out and buy a lottery ticket.  Wrong- after you have had a streak of good luck is the *worst* time to go out and buy a lottery ticket.

Here is your streak:

WWW

But what cups did you have before, and how many after?

Is your true distribution something like this?

LLLLLWWWLLLLLLLWLL...

4/18, or just ahead of the odds. I suppose that would be a bit lucky.  But there's a reason for the saying "quit while you're ahead..."

We're talking about math here, folks, so this even has a name:  Poisson clumping.  Random events occur in bursts.

Now that we've talked a bit about Roll Up the Rim odds, here are your choices:
-ignore the tabs, because there is a 5/6 chance every single time that you have lost
-keep dreaming of that new car!
-just enjoy your coffee and have fun

Breakthroughs, "A-has", and "Finding the Door"

I highly recommend this article on Yitang "Tom" Zhang's breakthrough on the Twin Primes conjecture. It's readable, inspiring, and full of lessons for elementary and secondary teachers of mathematics.  Don't worry about understanding the math (although it does explain it pretty nicely), just read Tom's story, from unemployed math PhD helping out at Subway, to becoming a professor, to his big breakthrough.

First, our students have these kinds of breakthroughs each and every day.  They don't have to be Archimedes style "eureka" and "jump from the bathtub" breakthroughs.  They aren't Einstein imagining himself on a beam of light, and conceptualizing relativity.  But they are breakthroughs, nonetheless.  Rather than bolts from the blue, these are often subtle and sudden realizations that they are closer to solving a problem than before.  You know, the "a-ha" moments.

These are the moments when things become clear, when they can see their way through a problem. They can begin to articulate a solution through the complex interaction of math processes, content and background knowledge, and classroom culture that we call problem solving.

There is a lovely description in the article of Mr. Zhang walking around and thinking about the problem, until one day, he "found a door".  He realized what tool he needed to solve the problem, and how he would do it.  The unbelievably complex math he was working on began with selecting tools he would use, and finding a way to represent the problem.  Sound familiar?  They should- they're two of our math processes.  If the math processes are the "actions of doing" math, doesn't it make sense that a professional mathematician would use the same processes as our young learners?  I think so- practice, experience, and knowledge background being the main difference.

The growth mindset math learner might say, "I can't solve this problem...yet."  We might change that to "I haven't found the right tool...yet."  Or:  "I don't know how to represent this situation with math...yet."

We could also say, "I haven't found the door...yet."  After you find it, you can walk right through, after all.

10 Good Things

I was challenged by Brian Aspinall to do a #10goodthings blog post, so here it is.

So many good things happened in 2014- reflecting back makes me very happy.   Through good fortune a lot of things fell into place.  These are my 10 good things from 2014.

1)  Minecraft and Math happened.  See video clip about the work of our classrom here.  The real hard work of using Minecraft as a consistent classroom tool is being done by people like @GumbyBlockhead and @zbpipe, and all the schools around the world on the MinecraftEDU servers.  

For me, this was less about technology than about an opportunity for differentiation and student voice.  What will happen if I let them try the math task (or even a textbook question!) in the Minecraft environment.  The results far surpassed anything I would have thought possible.  I kept asking:  "are you sure this will work in Minecraft?", and kept having my skepticism proven wrong each time. 

My blog post about the journey is here. 

2)  The journey into inquiry math.  Pictured is a student's projections using their mathematical model for how many points these players will score in their careers.  We use "wonder as the fuel" in our #GeniusHour math, and saw how questions spark more questions, which spark more learning.  Letting students loose to explore a math topic of their choice was amazing.  Their work exceeded all expectations!  My account of #GeniusHourMath  is here.  

3)  Starting #PeelMathChat with @raspberryberet. Here is a Storify of our very first one.  It is always inspiring to see educators talking math in the evenings, when they could be binging on "Fresh Prince" on Netflix, or knitting a scarf.  But seriously- our PLN is strong, and this is one way we share and collaborate.  

4)  Social time on Twitter (and IRL) with friends like Carla Pereira, Jonathan So, Tina Zita, Helen Chapman, Debbie Axiak, Melanie Essex, Jay Richea, Donald Campbell, Aviva Dunsinger.  Because, you know, it's not all work, all the time.  #Peelnomnomchat and other fun chats often bring a certain needed levity to life.  Now let's all help #Peelsockchallenge take off as a hashtag!

@Carla_Pereira2 these #noms are back! pic.twitter.com/j81eNIsMKm
— Matthew Oldridge (@MatthewOldridge) January 20, 2015

5)  Presenting about Math and apps at #TLDWpeel in the summer (which led to a further 2 part workshop with @tina_zita, and my #BIT14 presentation).  Sitting on the expert gaming panel about Minecraft was amazing too. 

6)  Watching @MathletePearse at #BIT14, and having to sit in the hallway because it was so packed.  Doing my session right after, 3 doors down, and having the room totally packed.  

Also at #BIT14, the power of the people in creating change in education:  the informal yet amazing discussion with @MrSoClassroom and @MrAspinall at #BIT14 on assessment and math (in turn, sparked by a Twitter conversation) spoke strongly to the power of informal networks of information sharing and professional collaboration. I also turned around and @avivaloca was right behind me!

7) Harnessing the power of the #MTBoS to improve my teaching and learning.  When you have a worldwide network of people like Dan Meyer and many others sharing daily, you are never short of ideas.  

8)  Being the host classroom for the learnteachlead.ca project "Loving the Math, Living the Math." This was the most inspiring 3 days of teaching I have ever had.  I'm not even sure what to say  about it.  I've been tweeted from across the province ("i'm watching your video") and even had someone say, "i've watched all your videos."  I can't even put into words how many things I learned about math teaching and learning through this project (or rather, it would be a very very long essay, and a topic for another day.)

It's not me though- it's our amazing Peel students who are willing to grow their brains in math, take an inquiry stance every day, and #EngageMath with us. Together we are strong. Believe in the power of kids, let them be kids, let them talk, and engage their natural curiosity about math and willingness to learn, and they will do great things!

9)  Being hired as an Effective Math Resource Teacher in @PeelSchools. This was a goal i'd had for over 5 years, and included a handful of unsuccessful interviews.  For my personal journey, I think of the Japanese proverb (and personal mindset motto), "fall seven times, stand up eight."

It is the greatest privilege to be able to engage math every single day with our amazing school board-students, teachers, principals, superintendents, everyone.  

10) Trying to live as ethically as possible, and being the best father, husband, and human being I can.  

 Is your hourglass 25% or 75% full in 2015? Do we focus on what we have, or what's missing? 

Slow Down...and Do the Math (Part 1)

The full text of the problem is missing, but you get the idea- it's a fairly typical seeming growing dot pattern.  Many of these patterns are deceptively simple- it's fairly easy to describe how it grows, to construct a table, and to come up with an additive relationship for how it grows.

From the Ontario curriculum, you might expect any student from even grade 2 and up to be able to say SOMETHING about this pattern.  The problem strongly fits explicit teaching of the math process "Representing"- tables, graphs, explanations, and algebraic representations all work.  The patterning/algebra curriculum is ideal for showing the variety of mathematical representations that we can use.

This problem worked very well for co-teaching in a grade 7 class.  After about 15 minutes of paired work, all student pairs had something down on the page.  They had made tables, extended the patterns, and made conjectures about how the pattern grew.

Math process of the day. How do we create representations of something so abstract as mathematics? pic.twitter.com/tUx71Rvw8r
— Matthew Oldridge (@MatthewOldridge) November 28, 2014

This is where I had a conversation with one of the teachers in the room about how the students "should" have been done 5 minutes prior. On the surface, that seemed like it might be true-there was something on every page, and the students' initial line of thinking was tapped out.

Having a bunch of experience seeing students work with growing patterns, I thought that many were "ready" to make the leap from additive to multiplicative reasoning, and quite possibly to generalizing using algebra for the nth term. Asking one or two key questions here would be the key to getting further with our thinking.

Here is an example of a pair that was ready to jump to generalizing:

The proof is the circling of the groups of 3 on the right.  Groups of 3 leads to the idea of multiplication by 3, which can get us to the ideas of variables and constants.

Another teacher suggested asking for the 15th term.  What typically happens here is that those students who need to keep adding to show the constant growth will mechanically extend the pattern, and get the correct answer.  But those who are ready to take the conceptual leap will often jump in with insights, like lightning out of a blue sky.

Here is where the differentiation happens- some students realize they understand the problem right down, all the way to the bottom.  They have broken it down to its mathematical elements, and can use those elements to construct their own explanations for what's happening.

Given an extra 5 minutes, here is what one pair had to say about the 15th term:

-it has 15+16+17 dots.

How did they know?  Term 1 has 1 +2 +3, term 2 has 2+3+4, etc. This was quite an efficient way of thinking about the pattern.  It would allow them to give the number of dots in the nth term with simple addition.

In the consolidation, some students revealed that they knew 3n + 3 would work for any term.  They  had little to no prior experience with algebra, but were able to make that generalization.  I think further work in this classroom (on other situations/patterns) could focus on generalizing through multiplying, and generalizing for any term.  Here is where precise explicit teaching through mini lessons, and purposeful practice work would come in.  The next week or two of classes could all be planned from what we learned that day. Such is the power of consolidating our thinking.

This is just the sort of insight that students will come up with, given more time.  Where we can, let's slow down, and the let the thinking happen.  Curriculum can be a rush, yes, but sometimes that extra 5 minutes is the difference between getting somewhere with our thinking, and really getting to a more mathematical place with our thinking.

Jo Boaler and others have written eloquently about the problem of speed in working with students on their number fluency.  Slower but deeper thinkers can often be turned off of mathematics by our constant rush-to get from topic to topic, strand to strand, report card to report card.  Let's slow down and do the math!