Monday, November 21, 2016

Overcoming "Test Mystique"- My Principles For Mathematics Assessment

"Test mystique", to use Bennett, Dworet, and Weber's (2008) term, is persistent and pervasive in our dealings with our students, as tests are sometimes seen to have a "mystical capacity to open a window into a student's inner being and the workings of his or her mind". 

Some teachers are reluctant to trust their own informed judgements and classroom assessments, deferring instead to testing results, many of them standardized, for the "final word" about the strengths and needs of their own students.  As the authors note, teachers will often "defer to test results even when those results contradict their own observations and conclusions arrived at over months of on-site observations and analyses". 

I believe this is wrong. We need to trust our eyes and ears, most of all. When I stopped giving math tests, I felt I only knew my students better. That said, short quizzes and tests are appropriate in some circumstances- I liked them as "checking in" on basic skills- for example, can you add and subtract integers correctly?

Wiggins and McTighe (1998), note that using tests as the mark and measure of student achievement is a "long-standing habit".  Test mystique still prevails-perhaps in math classrooms more than any other subject.

Here are some assessment principles that I think are effective and fruitful for the mathematics classroom.

1.  Engage students in purposeful talk about what they are learning as they work on a classroom task. Walking and talking is assessment. Record and use tracking sheets with anecdotes from this purposeful talk, and treat it as valuable data, part of the bigger picture of how and what the student is learning.

2.  Don't be afraid to not give a test.  Tomlinson (2008) talks of her early years, when she only knew she was "supposed to give tests and grades", although she didn't like them. 

3.  Find a system to document observations and anecdotes from conversations.  It is often hard to stop in the "flow" and write things down, but techniques like using checklists will help.  I have experimented with using tape recorders (old school), and now apps, to record observations, as my handwriting is rather difficult to read. Remember the power of the camera roll- take pictures of work in progress frequently.

4.  Use frequent checkpoints in larger assignments.  This relates to the concept of chunking-allow students to check-in frequently in their learning.  Give students checklists which will aid in task completion.  If, for example, a student is building an electric car in Science class, give dates to bring materials, dates to complete sketches, and detailed lists of how the task is to be completed. In the math classroom, allow students to check in as they work on longer problems or assessments. Help their thinking develop.

5.  Give frequent feedback to improve student learning.  A state of helplessness often sets in when an exceptional student does not know if he or she is doing the work "right"; feedback helps to redirect a student positively, and giving feedback helps bring a task closer to completion.  I have read, and refer to quite frequently, the work of Black and William (1998) on feedback for learning.  It is quite a long work, but boils down to simply this:  feedback works!  One single piece of directed feedback is sometimes all it takes! 

6.  Trust your own judgement! 

Sunday, August 14, 2016

The question is not, “how best to teach mathematics?” The question, educator, is “how best for YOU to teach mathematics?”

The question is not, “how best to teach mathematics?” The question, educator, is “how best for YOU to teach mathematics?”

The debate over what constitutes “good” math instruction flares up over and over again. We are over 25 years into “reform” mathematics curricula, which started with the publication of the National Council of Teachers of Mathematics standards (1989), and has its deeper roots in the “‘new math” after Sputnik was launched in 1957.  In recent years, in Canada you may recall the 2015 CD Howe institute report, which gives recommendations for educators on how best to teach mathematics, including the idea that teachers should consider a balance of 80% direct instruction, versus just 20% what they call “discovery” instruction. The debate is usually characterized as being between “back to basics” advocates, and “reform” or “discovery” mathematics advocates, and that’s partially true, although the real truth of modern math classrooms is by no means as binary as newspaper articles on the topic would have you believe.

Here’s an image that might depress you.


This little pamphlet was written in 199 (the former Peel Board of Educaton)! It would seem that, 25 years on, we are still having some of the same debates. It might seem like we aren’t moving forward, but I would argue we are leaps and bounds ahead of where we were a generation ago.

But 25 years later, the question still remains, for North American educators, how best do we teach mathematics in K-12 classrooms? What methods workk best? How do we balance “basic skills” and concepts”? Should they be set opposite to each other? Are these even opposites at all? The intent of this short blog piece is not to attempt to unravel terms like “discovery”, “inquiry”, or even “direct”’ or “explicit” instruction, or even “procedural” and “conceptual” instruction.  (On that last pair, fifteen years of work by Bethany Rittle-Johnson and her co-researchers is showing that procedural and conceptual understanding iterate on each other, in a constant back and forth. In short: it may not matter which type of instruction comes first).

I would hope that all educators make up their own minds. There are very personal choices to be made, based on one’s own strengths as a teacher, the strengths and weaknesses of your students, and the type of math being taught. As a general rule of thumb, there are very few math concepts that I believe can’t be introduced in an intriguing and interesting way. No, I don’t believe algorithms should be taught first, either. But algorithms are one piece of the puzzle, one we shouldn’t discard or scoff at. A lesson on how the standard multiplication algorithm works doesn’t take much time, and shouldn’t be avoided. Show your students how the algorithm pulls together partial products, adding them together, and giving one neat number. It’s not magic- it’s mathematics. I think students should get an Algorithm Licence- show how it works, and you can use it forever after, no questions asked.

Many big ideas in mathematics can be introduced through simple and interesting investigations. Think Pythagorean theorem, or “finding” a value for pi, for example. Of course, good investigations are nothing without powerful teacher talk- deciding when to intervene, when to give whole class instruction, and how to consolidate the activities. Kids should never be left to just “discover” math on their own-teachers have an important role to play in constructing understanding of powerful math concepts. Kids need us.

unguided ins.png
You can probably agree that, for the most part, your instruction is somewhere in the middle of this line, depending on the topic and lesson. Both fully guided and unguided instruction are equally disastrous. (Fully guided instruction is less disastrous, in my opinion, but I don’t believe it’s optimal in this day and age). The real trick is knowing where and when to let students “discover”, and where and when to tell or explain (yes, telling and explaining is still part of our jobs!)

We must make up our own minds about how best to teach mathematics. My friend Tim Boudreau said the following of me in a tweet:

personal stance.png

Even then, it depends on the lesson. For myself, I believe at times I swung out way too far toward unguided instruction. The evolution of my practice was realizing that those precise and short and specific 5-10 minute mini-lessons to the whole class went such a long way. But my big evolution over time was realizing how kids talking to other kids about mathematics needed to be the mainstay of my classroom. Both are needed!

Others will have a different optimal balance. Some years your class will be more suited to exploring more, or will need more direct instruction, and that’s fine. Professionals need to know their own strengths and weaknesses, and teach to their strengths, while tempering their weaknesses. Classroom practice is a daily grind. Know yourself, educator, and know your kids. Make purposeful instructional choices. Explore powerful and interesting mathematical ideas with your kids. Don’t be afraid to explain concepts to them when they need it. There is no one single “best practice”- there are better and worse practices, though. More importantly, there are better and worse practices for YOU. Choose wisely and well!

Thursday, June 23, 2016

Partial Metrication/Partial Understanding

Source:; Wikimedia Commons

This post was inspired, as always, by talking to a fellow teacher, in this case about the numeracy that kids need to navigate in the world.

I will begin with a confession: I am relatively innumerate, with regard to Imperial and US Customary units. Not coincidentally, I was born the year Canada went Metric.

If I am cooking a steak, and it's supposed to be 1" thick, I have no solid conceptual anchor for what that looks like, in my mind. I know I am a bit over 6 feet tall, but I only  know I am 183 cm tall because my driver's license says. So, when my sons were born, they were around 9 lb, so I guess that made them about the weight of medium size turkeys, at that time. I never learned mental math tricks for converting between pounds and kilograms. When I travel to the United States, temperatures in Fahrenheit make no sense. Right, except on my house thermostat, on which I know 70 feels cold, if using air conditioning in the summer.

Home Depot is a strange world of things I barely know about. Don't even ask me about that. I know fractions well, but not fractions of inches in the screw section. Is a 2 by 4 really a 2 by 4? I hear not, but how would I know?

The sordid history of metrication in Canada is covered nicely in the Wikipedia article. Basically, it stalled, due to our historical relationship with the UK, and our current relationship with the United States. We are stuck in the middle, seemingly permanently.

Why does this matter?

In Ontario, there is no mention of Imperial units until grade 10, and then only in the Applied course:

 -perform everyday conversions between the imperial system and the metric system (e.g., millilitres to cups, centimetres to inches) and within these systems (e.g., cubic metres to cubic centimetres, square feet to square yards), as necessary to solve problems involving measurement (Sample problem: A vertical post is to be supported by a wooden pole, secured on the ground at an angle of elevation of 60°, and reaching 3 m up the post from its base. If wood is sold by the foot, how many feet of wood are needed to make the pole?)

By this point, it's too late. Numeracy starts at birth. There is also an unbelievably patronizing aspect here: the Applied students are more likely to go into the trades, and therefore are the only ones who "need" to know Imperial units, while the students in the Academic stream march steadily into more abstract territory, with Algebra, and Calculus, as always, the pinnacle of K-12 mathematics education. 

Meanwhile, the goal of the school system is to graduate educated, literate and numerate adults. Being caught between two systems of measurement is not doing our students any favours. 

Wednesday, June 8, 2016

Representations-Tools for Mental Activity, or End Results of Tasks?

I have worked with a lot of teachers who tell stories of being pushed toward one single correct way of representing their math work by their teachers. If we teach that way, we are probably teaching our students to be too inflexible, too rigid with their mathematical representations. Typically, we focus on the product (written) of the representation, rather than the entire mental process that leads to representation.

The Pape/Tchoshanov paper is instructive for teachers in getting kids to develop their own mental representations, and to get them out into the physical world.

In their words:

"We use the term representation(s) to refer to both the internal and external manifestations of mathematical concepts."

This is a huge argument for paying attention to your students' thinking, in order to make it visible. We need to help kids visualize representations of interesting mathematical concepts, in order to imagine them into being.

So we have two complementary aspects of mathematical representations:
-the act of representing (verb)
-the representation itself (noun)

This quite neatly corresponds to both NCTM and Ontario curriculum definitions of representation: as both process and product.

The authors propose a "'zone" of interaction between mental and physical interactions.

The nice example used is of "six"-a kid develops an internal mental image of "'six', then connects it to sets of six objects, the numeral 6, the written word "six", the spoken word "six", and so on.

As kids advance in math, we should encourage representational thinking. That is, building and having a repertoire of ways to externalize math concepts. Tables, graphs, tree diagrams, area models, arrays- these are all examples.

The authors put it nicely:

"the development of students' thinking skills requires a multiple representational approach."

Put simply: representations are tools for reasoning.

Two things to think about.

Kids often mathematize math tasks by drawing pictures with little or no mathematical content. Do these count as representations?

"Students often produce representations that lack meaning."

Manipulatives are generally a good thing, as long as they are accurate mathematical representations of the concept, the mathematics you are working on. Daniel Willingham presents this caveat here.

Manipulatives, if not chosen wisely, can get in the way of mathematical representation. Let the tool fit the task (and the math).

The end goal of representation is good communication. Can the student communicate their thinking, in words, or on paper, using mathematical representations?

Even better, can they start to see how different representations are connected? Do they see a table of values, a pattern made with cubes, a graph, and an algebraic expression as fundamentally the same thing?

How do they show the same math in different ways?

Fluency with multiple representations should be a goal of thinking math classrooms; we can help kids to visualize, conceptualize, and make real their understanding of powerful mathematics.

Sunday, May 29, 2016

Make Reasoning A Routine

I have written before about how talking changes everything in math classrooms. Kids will surprise you with the power of their thinking. They will conjecture, and wonder. They will back up their reasoning with mathematics.

Reasoning shouldn't be an event in math classrooms. Reasoning should be a routine, or even- the default state of the classroom. I have a strong belief in defining terms in plain language, so I will define reasoning in math classrooms as, "providing mathematical reasons to support our answers, verbally, or on paper." If reasoning is pushed further, we get into the concept of providing airtight "proof", and generalizing for all cases. In K-12 education, you will some breakthroughs with proof, and a growing ability to generalize (we particularly see that in the progression of algebraic concepts and thought from K-12). The ability to generalize develops the more kids are given the chance to reason through their ideas, and, of course, as their mathematical toolkit develops over time (learning all four operations, and then powers and roots, fraction sense and arithmetic, algebraic reasoning and working with equations, and so on.)

In this picture, I stood beside two kids while they explored various trapezoids. They were very close to finding something out about the area of all trapezoids. With a bit of a push, they could have gone from examining specific cases, to generalizing for all cases. They were tantalizingly close. I didn't get to see if they got there in the end. I do know this: letting those kids play with trapezoids is a lot more interesting than just throwing out "the" trapezoid area formula and having them do exercises. There is time for that, later.

There is a lot of stuff out there that is user friendly, repeatable on a day to day basis, and supports getting kids to share their reasoning. With the exception of number talks, these are all things that have sprung up from the dynamic and amazing #MTBoS. I suspect lots has been written about using these things as routines, so this is just a brief summary and survey.

Here are a few things you can do.

Number Talks. Depending on how you choose to do your number talks, you may be more focused on specific strategies for mental math, than reasoning. but number talks are portable, short, and get kids talking.

Estimation180. We used one involving a piece of a pie with some of our adult learners, and I was thinking it could go to fractions, or well, pi, if you wanted to actually take the picture and carry out some calculations. I think these are mostly good for developing that horse sense, or intuition about things like quantities. How many? How much? Why do you think so?

Fraction Talks. This site (and Twitter, @FractionTalks), has a wonderfully diverse selection of pictures that can inspire reasoning about fractions. It is so, so important that kids don't see fractions as a strange and new species of number, when they first encounter them. Kids should be able to reason with linear, area, volume, and set models of fractions.

In this picture, a teacher is sharing her reasoning about the lovely "Quarter the Cross" tasks.

Flags make lovely fraction provocations:

Hat tip to @Madame_JB for this one!

WouldYouRather Math.  This one presents two options, and has kids chose which one is best. The instructions include the lovely formulation, "justify your reasoning with mathematics." I love it. The best part about it- you could make your own. You just need two options that inspire interesting reasoning. I am fond of pizza tasks- what's better, at what price, a medium or large, that sort of thing. 

I am tagging @MathManAnusic to talk about Which One Doesn't Belong and Visual Patterns.

Given what's out there, it's pretty easy these days to make reasoning a daily routine, and you should. You could pick any one of these websites, find something matching the curriculum you are working on, and use it as part of your routine. 

Get kids talking, exploring interesting tasks, and let them amaze you with the power of their reasoning. 

Wednesday, May 18, 2016

Letter to the Editor of a Major National Newspaper on the Ontario Revised Math, Which Wasn't Good Enough To Print, Because It's Not Negative At All, In Its 200 Words, And Refuses To Indulge In The "We're Terrible" Narrative About Ontario Education

What’s missing from the Globe and Mail’s articles on the new Ontario math strategy is the voices of actual math teachers-who teach every day, in classrooms from Windsor to Moose Factory.

The pervading media narrative about math education is far too negative.

My personal perspective is that the Ontario curriculum is strong, with an appropriate balance of skills and concepts. What is needed is not a complete course correction. With subtle tweaks, and reshaping of the curriculum to focus on bigger mathematical ideas, and making sure skills don’t go overlooked, I think it could be even better.

We already have world-class teachers, who with the new focus on math, are getting even better. I have seen massive breakthroughs these past two years. I have seen teachers learning more new and interesting math content than ever before, and they are excited by it! .

$60 million is nothing to sneer at. Together, we can go from good to great, if we keep working together. Change and improvement are not sudden. There is no “silver bullet”, only careful, patient work.  

Wednesday, May 11, 2016

Math is...What Parents and Teachers Make It

My OAME 2016 Ignite speech, "Math is...What Parents and Teachers Make It," is here, in PDF form.

I think a lot about what math "is", and "is not". I am involved in professional learning for practicing teachers, I am a teacher myself, and, I have two young children.So it is that I have conversations like this, in my house.

I thought about just letting that drop but naw, teachers need to teach, so we explored it, with both boys. That's a pretty decent provisional definition of infinity for a 5 year old, but can we push his thinking further? I asked him "what's the biggest number you know?" He started with 12. I said that if I add one, I get 13. The play continued. I got Callum to write a 1 and a string of zeroes down, and showed him that you can always add another zero. He doesn't really know place value yet, but I was thinking that playing with ever bigger numbers would help him. 

This is still a work in progress. It's also just one example of how you can model math talk at home. I really do think parents have enormous power to expose their kids to everyday math. Tackling infinity is not for everyone. The new resource Inspiring Your Child To Learn and Love Math has all kinds of practical activities parents can do with their kids, from cooking, to shopping, to all kinds of games and fun things you can do to get kids seeing that math is a normal, interesting, inspiring, and fun part of their lives. 
And from the teacher''s side, I think that tweet says it all. We have such an enormous power to shape what mathematics IS, in students' minds. Is it disconnected topics, strands, and facts? Or is it a powerfully connected and interesting body of knowledge? Is it a way of thinking you only do in math class? Is it a subject where you can do powerful critical and creative thinking?

Do we expose our kids to both the awe-inspiring mathematical world, and, math used in the real world?

I hope so.

Here are some of Melissa Dean's (@Dean_of_math) amazing students' responses.

From Kindergarten kids: 

1 + 2
10 - 1
Playing thinking games

From Primary Kids:

The language to explain certain happenings.
Fun for me. Math is all around us. 
I love to do math because, first, it makes your mind smarter. Next, it just makes me happy. Lastly, I love my teacher every day in math.  

From Junior kids: 

Important, since, well, it’s used everywhere for everything, and so it’s basically necessary in our lives. 
Awesome and it helps you learn. Math is also cool.
Everywhere. This is actually true, considering it covers all strands of life. For example in gym, strategic games, and science! You also use it when you don’t know it. That’s how math is everywhere. 

And from this grade 6 philosopher: 

Everything. Everywhere you go, everyone you meet, all have some connection with math. It’s logic, common sense, and thinking out of the box, not only the seemingly tedious arithmetic and problem solving. Math has grown into the world, and so the world wouldn’t be the world without it.

From Intermediate kids:

A way of solving equations/problem. Math is everything, everything is math.
A combination of numbers and symbols that is useful for every day life and can help you in the future (finding a job.)
Needed in life. Without Math, we’re idiots! Yeah!

And from a grade 8 philosopher:

A metaphor of life. Asks you to solve the problems it creates. It’s simple. Its just us creating ways to explain things we don’t fully understand.

Thursday, March 24, 2016

Talking Changes Everything In The Math Classroom

Talking changes everything. 

That's been my biggest revelation over the past 14 years of teaching.

When I first starting teaching, I didn’t know that math classrooms could (and should) be talking classrooms. I saw them as quiet spaces, where you could often hear a pin drop, and students silently approached the teacher’s desk, but then only when they had a question.

Often we treat math solely as an individual activity.I myself have memories of sitting in a desk, in a quiet room, textbook open, doing problems in a workbook. We didn't talk about our math work, and we didn't collaborate with each other very often.

Assessment happened between us and our teacher, usually at the end of the unit, or on quizzes in the middle of a unit. We made our corrections if need be, but by that time, the class had moved on to the next topic.

When I first started out, I sometimes had to cover for another, very, very experienced math teacher (30+ years). She was very, very good at what she did. Homework was taken up, lessons were given, and students were set back to work again. Kids never talked. I remember how surprised I was by that. They expected the classroom to always be a silent working space. There is a time and place for that, but I think we are better served by talking about their mathematical ideas.

Talking changes everything, and when kids start to talk in the math classroom, they don’t stop. They will talk about the problem they are working on with others. They will come in the next day still talking about the previous day’s work. They will come in eager to talk about prime  numbers, or videos about math they watched on Numberphile. They will wonder aloud about things like infinity, and if pi never ends.

Math classrooms should be talking spaces, because talking spaces are collaborative thinking spaces. Lucy West identifies five types of classroom talk: voicing, repeating, adding on, waiting, reasoning. You can find a number of nice videos here. Discourse, Reasoning, and Thinking (LearnTeachLead).

Of these, reasoning is integral to math class. Is a negative added to a positive always negative? Positive? Why does a triangle have an angle sum of 180 degrees? Which is bigger- 3/4 or 5/8? Why?

I have previously written that assessment shouldn't be an event. Neither should talking. If you haven't already, start with something concrete and manageable, like having kids turn and talk to the person behind them at crucial junctures of the lesson. Get used to having kids explain their own arguments, using white board, or document camera, or good old fashioned chart paper.

Students defending their mathematical arguments:

Drop by the quiet kid's desk, and see if she has anything to say about her math. Kids will surprise you with their thoughts, their strategies. When they get used to it, they will be eager to share with you.

This kid couldn't wait to share his thinking about this pattern:

Talking spaces are spaces for wondering aloud about interesting mathematical ideas. Our classrooms are those wondering spaces, those talking spaces. Let kids start talking in math class, and they won't want to stop. We are used to thinking about "voice" in the writing classroom. How about in the math classroom? Are ready to hear students' own original and unique mathematical voices, as they talk and reason their way through interesting mathematical problems and ideas? 

A student wondered aloud about the value of this part of the $2 coin, so we found out!

Monday, March 7, 2016

Costco-the Proportional Reasoning Store

Check your local curriculum: if it's like ours, sometime around grade 6 students are asked to deal with the interrelated concepts of rates, ratios, and percents. This is part of the growth of thought in the area we call proportional reasoning. In grade 4 (Ontario), students start to think in more relative, and less absolute terms. This often includes starting to see relationships as multiplicative, not additive, as they have been doing since Kindergarten.

I tend to think that unit rates, in particular, used to be more mysterious before Costco became ubiquitous. Their entire business model is built on an economy of scale, or what we recognize as "buying big stuff, so they can sell it to us for cheaper."
The entire store is filled with very interesting possibilities for proportional reasoning problems, particularly due to how they do their price labels.

In this example, you see that the unit sold is actually 2 bottles. Each has 830 mL. Their go-to rate for liquid capacity is price per 100 mL, in this case $0.409.  And of course, how much  you actually pay.

I see lots of opportunities for interesting tasks here. Comparing the price per Litre of various liquids in Costco, for example.

How much is Coca Cola per litre?  How much should it be?

A general question you can ask about Costco prices is: "are they fair?"  You could then compare the price of water, or other liquids at various grocery stores.

As far as Costco goes, fair, and cheap, for me depend on this: can I store the item in my house?  If so, I often calculate the unit rate in my head and decided to make a purchase.

I do think other grocery stores would be cheaper on some items, so it would be interesting to compare, say using the Flipp app or other stores' websites.

This one uses the price per bar as their unit rate, in this case $0.15. The price per kg is also interesting though.

Compare to these granola bars:

Which one is cheaper? Which one would you buy? Why?

There are nearly endless possibilities for interesting math tasks in Costco. We thought it would be interesting to take a bunch of teachers there, and snap away with our phones, interesting materials for math tasks. Would you come?

Wednesday, February 17, 2016

The Three Part Lesson Mindset

We are often taught that there is a "thing"', a lesson plan, a framework, perhaps, called "the three part lesson". I discovered this video here, from the project I was involved in, offers some plain talk and background on the three part lesson, from Lucy West, Marian Small, and others.

It's mentioned in the clip that perhaps it's an Ontario thing, but it has its own Wikipedia entry, and John van de Walle is credited with coining the term. 

When people in jobs like mine get pushback from teachers about how difficult, unwieldy, and time consuming it is to teach through the three part lesson, I often think, "they''re going about it wrong." We do get this kind of pushback-I have seen it myself in social media groups, comments along the lines of, "don't even get me started on the three part lesson!" 

I don't blame them. Classroom teachers do a lot, and to be fair to teachers, designing a minds on/working on it/and consolidating the learning sequence for EVERY single lesson is probably impossible. That said, as the Ontario curriculum has it, "problem-solving is the mainstay of mathematical instruction"", and we are obliged to structure our teaching through interesting problems.  

It's probably more useful and humane to think of the three part lesson as a mindset in itself. 
Our math coordinator, Mary Fiore, has been making this point for a long time. 

As she says:
Here is how we like to talk about this mindset:
Don't think in terms of a set lesson plan, with time-bound sections. Activate student thinking with something interesting-something that inspires math to happen. If they explore this problem for an hour, a day, or even three days, so be it. Go where they take you. Watch and listen, and plan where you want to go next in your instructional sequence. Take note of any misconceptions you need to give lessons on.

Above all, a lesson is only as good as its consolidation. Think about how you want to discuss the math at the end of the task.  Consider whether your students will be sharing their work out loud, if you will be directing the consolidation, or if they will. Be purposeful and responsive. Teach through the three part lesson mindset!

Friday, February 5, 2016

Have a Goal for Each and Every Math Class #assessment #curriculum #intentionalteaching

I once laminated a few cards that I thought I would put up at the start of each class. I had thought that there were only a certain few types of things we would be doing in any given class: explore or investigate, consolidate learning on a concept, or practice with a skill or concept.

The reality of the middle school teacher is often 40 or 45 minute periods. You need a clear learning intention for each class. Broadly speaking, you might be said to be in one phase of a three part lesson too, although we can classify those three actions as:

-activating thinking
-developing thinking
-consolidating thinking

By that measure, you might argue you are always in one phase of the three part lesson (primary, junior, or high school teachers might find you can make it through an entire iteration of a lesson on a given day).  I like the plain talk on the three part lesson from Lucy West, Marian Small, and others, in this clip.  It's a mindset, not a lesson plan, as my friend Mary always says!

We have spent a lot of time thinking about how best to use learning goals in the math classroom, and I think, regardless of how you do it, you must have clear intentions for each class. Whether you post the learning goals at the beginning, articulate them verbally, or develop them as you go, be intentional.

We must always be accountable to the math that the curriculum requires of us. Further, Growing Success asks those of us in Ontario to be accountable to the overall expectations in the curriculum. 

Further, we must have thought about the progression of learning. Is it a brand new concept? Building on something in previous grades? Is it best suited to investigation? Problem-solving? Is direct teaching needed, perhaps a mini-lesson, or small group guided instruction?

Here is an example of an overall expectation from grade 7, on integers, and some thoughts about it.

-represent, compare, and order numbers, including integers
-demonstrate an understanding of addition and subtraction of integers

What does being accountable to this expectation mean? What instructional actions could  you take? Depends on your class culture and context, but here are some general ideas, and one possible lesson sequence. Let's assume each of these is one class, of whatever length that is.

-explore contexts for integers, such as temperature, sea level, elevator floors, etc.
-make a human number line to explore counting in the "opposite direction"
-explore the concept of plus/minus in hockey (*only if hockey is a good and meaningful context for your class*)
-explore the idea that each integer has an opposite by using two colour chips to make zeroes (the zero principle)
-use number lines to add integers
-use two colour chips to add integers
-explore contexts for subtracting integers-  how does it work, to take away a negative?
-use two colour chips and the zero principle to subtract
-do some practice work on adding and subtracting (games, worksheets, etc.)
-play Integer war with cards to practice
-explore some interesting problems using contexts for integers
-attempt to reason through some generalization problems- is a negative subtract a negative always negative?, eg.

Implicit here is that lots of observations, conversations, and paper and pencil checks for understanding will happen. That's a possible 12 classes, barring the fact that some could expand past one day, and also barring any quizzes, tests, larger projects, or evaluation pieces.

Each of these could be turned into a nice neat learning goal, and you could certainly develop success criteria for this sequence with your students. What matters is approaching each and every class with an intentional goal, staying accountable to curriculum, and knowing your students.

Thursday, February 4, 2016

Do They Understand It? Just Ask! Assessing Math Through Conversations

In doing some reading about teacher efficacy, I found the above quote, from a teacher who participated in mathematics professional learning.

This is a recurring theme for our teachers. It comes up, again, and again, and again.

We recently facilitated professional learning which included looking at student work to try and figure out where the students were at in their mathematical thinking. I talked to one teacher for almost 10 minutes, trying to figure out what a student was thinking with a piece of work. We had our inferences, but really, we need the kid there to ask him. It would have been cleared up, just like that.

If you want to know what a student is thinking? Ask him. Ask her. As I went around the room that day, a big theme was: don't you just want to go back and ask some clarifying questions?

Gone are the days when the paper artifact of the work was assumed to tell us all we need to know about student mathematical learning. Look for the absences, look for the gaps- and ask. Ask the specific questions you need to ask to get at their mathematical understanding. I would have to say we are obliged to do so- Growing Success tells us to assess through conversations, observations, and products. I have been less and less interested in products over time. In the moment and current thinking in the math classroom is far more interesting to me.

Don't be content to give grades based on the absences, gaps, or lacks. Ask questions. Listen for the answers. Seek understanding. Seek it, and, usually, you will find it.

Sunday, January 31, 2016

The Tool Must Fit The Task- or, Physical vs. Digital Manipulation #SAMR

You might know that I love snapcubes. I think they are the most versatile type of thinking tool out there. I love how you aren't closed in to a single function, like many commercial manipulatives, and how tactile they are, how satisfying it is to snap them together and make something.They can be used in algebra, geometry, 2D and 3D measurement, playing with numbers, basically anything,

Cube tasks are their own genre of task, I think. There are lots just in the Ontario Spatial Reasoning monograph. 

Here is a version of the 5 cube task you can use with most grades, with interesting results:

Here's the thing with that picture: cubes are something Minecraft does really well, seeing as how they are the basic unit in Minecraft (1 m^3, supposedly, in the actual game!)

So you might think, as per the SAMR model, that Minecraft is automatically better than using snapcubes. Isn't it fresher, newer, shinier, more "21st century"?  

SAMR Model Figure

Not necessarily. Sometimes the original tools are better. 

I came to @MrSoClassroom with this task in mind:  

I thought, sure, we should offer Minecraft as an option, as well as snapcubes. I was persuaded otherwise?  Why? One tool was enough, and it was the one built exactly for the job. Sometimes (actually, many times), physical manipulation is better than digital manipulation. Figuring out how to do the task in Minecraft (for non-players) would have slowed them down too much.

Further, one face of each cube placed on the ground would be hidden from view. That is a big problem with trying to do this task in Minecraft.

But you know what, about the Minecraft picture above? It just might be easier in Minecraft. Why? You will never run out of cubes, while trying to build your structures, in an infinite world...

Saturday, January 16, 2016

Behold, The School of the Future!

One of the best stories you can read about schooling was written in 1951, by Isaac Asimov. Called "The Fun They Had", it's about a generation that knows nothing other than teaching machines, and reminiscences about a physical gathering place called "school". Asimov knew futurism (and partly invented science fiction), but he, so far, has proven to be way off with this fiction.

You can read it, and see what you think. Certainly we all have our opinions about what the "school of the future" will or should look like, based on our own experiences, biases, and interests. A large part of what we think should happen in schools in the future is, I think, dictated by how disposed we are to see the past as better, or worse, than the present time. (This is mirrored in the larger debates over traditional/progressive schooling, and probably conservatism/progressivism in politics).

Here is a current Australian vision for the "school of the future". It frankly doesn't seem much different than the type of schooling predicted by Asimov: "electronic with a capital E!" But then you actually read the article, and it''s talking about things like" VCRs, laserdiscs, and telephones. Yep, the "school of the future" is already out of date.

I used to teach science fiction to kids, and I like to use images of what people thought the future would look like. This Reddit has some nice posts that you can look at. The human mind can always leap farther with its dreams and ideas than technology can. Filmmakers like Stanley Kubrick never thought to go past 2001 with his space odyssey. The Back to the Future movies had their own ideas about what 2015 would look like (and, strangely, hoverboards are here now).

I think technological progress is slow, relentless, and sometimes, paradigm shifting. Iteration upon iteration seems to be the way a lot of technologies progress. A lot of things in my house would be unrecognizable to my great-grandparents, but many would not.

The question to ask yourself is this: would Dewey in 1915, recognize your classroom as a place of learning, in 2015? My guess is yes. Yes, he would also recognize "21st century skills" such as creativity and critical thinking. You could teach Mr. Time Traveller Dewey how to use an iPad, and chances are, he would enjoy your class. I suspect he would quite like the opportunity to learn with modern tools.

Here are my predictions for the school of the future:

  • kids will still learn in a room with teachers
  • schools will continue to use the technology of their day (hopefully very effectively)
I am comfortable saying very little else about the school of the future.

As for "22nd century teaching and learning", i'm sure it will look a lot like "21st century teaching and learning". 21st century teaching and learning is really teaching for "the now", as my friend Mary says. What I am trying to say here is that human interactions and relationships will still be paramount. The "'school of the future" will "Human with a capital H".

File:School of the Future street facade.jpg

The actual "School of the Future" in Manhattan

Sunday, January 10, 2016

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.