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.