Make Math Memorable (A response)

I've had the opportunity to engage in further conversation about math education with Dr. Robert Craigen, Assoc. Math Professor and co-founder of WISE Math. My response to Dr. Craigen's most recent comment wouldn't fit in comments so I've included it, along with the initial response, here.

Monkeys or Mathematicians (Math is More Than Memorization)

Pedagogy trumps curriculum every time.

Dylan Wiliam

The recent parent-driven push for a “return to basics” shift in math curriculum in Alberta is not unexpected. Our post-industrial society remains regrettably focused on relaying and assessing content over process. The deeply embedded desire to quantify student thinking for the sake of a neat, uni-dimensional continuum that claims to represent student potential results in the inevitable association of learning with factual and procedural recall. Quite simply, we've designed schools to train and measure our children. We group them by age, divide their days into standardized units and test them at regular intervals in order to compare them to their peers. Memorization is easy to measure in math so we convince ourselves we’re holding kids accountable by measuring their recall. This also allows us to rank and sort students effectively without actually engaging them in conversation, something PISA has effectively mastered. However, making a judgement about the quality of an entire math curriculum based on data snapshots from a moment in time is not only irresponsible it's ridiculous. Advocating that because memorization scores have dropped, an entire curriculum should re-focus on memory work is incredibly shortsighted. We've already been there. It wasn't awesome.

Math is beautiful

Deirdre Bailey

It has been a bit of a battle this year to convince our students that mathematics is not disconnected. They seemed to arrive in our classroom at 9 years old with the conviction that the discipline exists sequentially, layered based on varying degrees of difficulty, some of which will remain inaccessible to the more artistically minded for most of their lives. We have been working hard to share that math is in fact a wonderfully complex web of recurring concepts, ideas, and patterns accessible through many different points from a variety of perspectives, and consisting of infinite possibilities awaiting discovery.

Our year began again with conversations about what we call multiples, what it means to be a multiple, and what a multiple of the number one is. We wouldn't let our students dismiss multiples of one as "obvious" or "easy," insisting that they consider what it means to be the number one. For example, how the number one can be manipulated without losing its integrity and how it is a part of other larger numbers. Before the fall break, we had explored multiples of one to nine, discovered patterns, noticed which ones fall into columns on hundreds chart, and noticed which multiples connected to others and how.

On Monday of this week we shared Perry the Platypus' birthday dilemma....

at this point in the year, students were quick to glue the problem in their journals and begin documenting their thinking as they worked through answering Perry's question. Some flipped back through their journals to remind themselves of previous discoveries they had made about multiples. Some organized their work in charts and some in diagrams or bullet points. Each was now familiar with the idea of writing down every thought or "a-ha!" that resulted from their considering the problem. 

Almost all of the students jumped right to identifying that in one year Perry would be a multiple of 2, 4 and 8 because every multiple of 8 is also a multiple of 2 and 4. Many wrote notes to remind themselves that the smallest multiple of any number is that number itself. One cool observation that resulted from this problem was that Perry would be a multiple of every single one of these numbers by the time he was 12 BUT that when he was 11, he would not be a multiple of any of these numbers! One student wrote... 'It seemed important to notice that both 7 and 11 are only multiples of one and themselves...' PRIME! Another student noticed that when Perry was an odd-numbered age, then he was only a multiple of odd numbers and he expanded to state that odd numbers can only have odd factors!

The coolest part of the week however, was the conversations that resulted at the tables who had begun working through an extension to the problem which asked..

How long will Perry have to wait to be a multiple of 2, 3, 4, 5 and 6 all at the same time?

At first the question seemed overwhelming and intimidating to many students. They had been preconditioned to focus on finding a solution. We suggested that they look instead at which numbers on a hundreds chart were definitely NOT a solution. For example, which numbers on a hundreds chart were NOT multiples of 5... Right away we were met with excitement.. 

"WE JUST ELIMINATED 80 NUMBERS! A multiple of 2, 3, 4, 5 and 6 HAS to end in 5 or 0." 

Students excitedly crossed 8 columns of numbers off their list when one double takes again... 

"Wait... the number we're looking for also has to be a multiple of 2.. it CAN'T be odd... FIVES ARE OUT!"

"So we've got ten numbers left and we're looking to see which one of these ten is a multiple of 3, 4 and 6.."

"But if it's a multiple of 3 it HAS to be a multiple of 6.."

"Right so just 4 and 6.."

Looking at multiples of 6 students immediately eliminated everything but 30, 60 and 90. One noticed that multiples of 6 only end in zero if they are multiplied by a multiple of 5. The last step was to eliminate numbers that were not multiples of 4. As 30 and 90 were eliminated, another student noticed that multiples of 4 which end in zero HAVE to have an even number in the tens spot.

As we wrapped things up for the day on Thursday, one student commented as he reluctantly closed his journal; "Mrs. Bailey, I never knew math was so exciting. It twists and turns and loops and connects all over the place and it just seems to go on forever!" 

Math is beautiful.

The Candy Problem

I mentioned the Candy Problem in a previous post in which I alluded to having provided the kids with a challenging math problem that even teachers had been taking a significant chunk of time to solve. We presented the problem as part of an end-of-semester formative assessment. We had had many conversations previous to presenting them with this problem about multiples, factors, common factors, number patterns and multiplicative relationships. Our goal was not to evaluate the kids based on whether or not they could complete the problem but to provide them with a means of generating conjectures and demonstrating their thinking process.

Having started the year with a diverse group from very different mathematical backgrounds, we were inspired by their concentration, perseverance, and the unique strategies they developed in order to solve the problem. What most impressed us was the number of students that approached us to share tentative solutions at the end of the period. They each had individual interpretations of how the problem should be solved and were eager to share the conjectures that had led them to their solutions. Since first introducing the problem, we have taken it up more in depth as a larger group. Kids have been working in teams based on original ideas they had in common for how a solution might be reached. They are excited, motivated, and eager for our next math class so that they can continue to develop and connect their understanding. Our Grade 7 and Grade 9 students have also since taken up the problem, inspired by the mathematical reasoning skills of their younger peers. If there was ever any doubt about the power of inquiry in a math classroom, these last few weeks have put it to rest for our team. Check out our video!! We'd love your feedback...

Why Math?

To recognize the crucial features of a problem, uncover latent assumptions at play, think carefully, devise symbols/diagrams that aid such thinking, and to communicate clearly and precisely...        Sam Otten

by Dan Meyer

Teaching Math: Knowing vs. Understanding

Deirdre Bailey

Each day in this process, I get a clearer idea of what powerful learning looks like. I have started to recognize what is becoming a blatant difference between kids who 'understand' and kids who 'know'. We have told them in class, we don't want 'parrots'. Parrots can recite anything we ask them to. It is not what we're looking for. Yesterday I observed a conversation in which teachers described their frustration with the time it was taking their students to 'uncover' mathematical solutions. They called it a road block and then admitted that to overcome the 'road block' they removed the problem and, 'gave them the answer', or in this case, the skills necessary to arrive at the solution.

What we, based on our prior experiences and education and even based on the way education is 'taught', assume, is that when somebody is told how to do something, they 'understand' how to do it. They don't. There are people all over the world that are capable of doing things without understanding how they're doing it, or why. As a simple analogy, how many skiers understand the physics involved in executing turns of a shorter radius than their ski? How many skiers understand that their skis have radius? As a more frightening example, how many first world home owners understand how their light switches work or understand the electrical circuits in their homes? How many adults have had mild shocks attempting to change a lightbulb because they were confused by these circuits. They can operate switches, but could they fix them? Could they pull them apart and put them back together?

Do we want schools to provide students with the necessary skills to survive or do we want schools to provide students with the ability to think, to create, to expand on what they know through critical consideration of what is possible? We believe the latter. Here is documentation of our first attempt to give power to students' own ideas instead of imparting our own.

Revelations

Amy Park

Wow, we have had two days of powerful math happening in our classes. As per our last discussion, we gave each student a 100s chart. We began by asking students to colour in all of the multiples of one. To our surprise…most kids had no idea what we were talking about. In fact, many confused looks were exchanged among the students - and us, as we realized that few kids knew where to start. Then the magic began…students started making conjectures, they began supporting and refuting others' conjectures, they were talking about abstract concepts and began developing an in-depth understanding of the concept being discussed. The light bulbs were going off. Misconceptions were discussed and partial truths were reworded to become complete truths. We discovered new definitions and new ways of thinking. It was powerful. We captured a great deal on video and we took several photos.

The discussion today was based on the question "what is a multiple of 2?". The first picture is the kids initial ideas, the second picture is their refined/reworded conjectures. The last 2 pictures are of students individual journals, where we asked them to define "even" and "odd".  This is evidence of powerful mathematical learning happening in grade 4 :) It was powerful for teachers, admin (who popped by) and for our students. Although it seems rudimentary, we believe we are setting the foundation for solid mathematical reasoning, establishing conjectures, working through the conjectures to find truths, and discovering that their is not just one way of "doing" math.

Grade 4 Math turns abstract...

Deirdre Bailey

I spent Sunday trying to combine the best of Mighton's mathematical understanding with Fosnot's teaching style in order to develop a way of reinforcing times tables in way that would be engaging, meaningful and memorable. I re-wrote the whole lesson about 8 times. I started with the idea of trying to teach number patterns but a conversation with my husband reinforced that these should arise as personal strategies. We all have different ways thinking through multiples of nine in our head...

I learned more about math yesterday than I think I might have in all of Elementary. It was fascinating. I was particularly intrigued by a series of "Rightbrainmath" videos by Mr. Numbers which visually demonstrated the patterns which result from multiples. I felt pretty excited about Math Conference today. We started by handing out number sheets and asking the kids to color in multiples of one. It became apparent immediately that none knew where to start, so rather than instruct we introduced the idea of "mathematical conjectures", ideas about what something might mean which might prove to be incorrect, partially true or completely correct. The kids volunteered conjectures on definitions for multiples of one. Initially, we had suggestions which included "any number with a one in it" and "every second number from one to nine". Eventually, we visually demonstrated that a multiple of one is any original number that can be split into groups of one and then asked the kids to support or refute the conjectures on the board based on our explanation. Partial agreements about some conjectures lead to conversation about whole numbers. Could one student and one half of a second student be split into groups of one? Why not?

The kids, on their own, debated whether zero and negative numbers should be included in a definition of multiples of one. We looked at whether two non-identical conjectures could mean the same thing and both be completely correct. We modified conjectures to make them true. 

I thought we'd breeze through multiples of one. We'd planned on coloring in multiples of one, then multiples of two, then moving on to defining odds and evens before the end of class. We didn't even get past multiples of one. There is so much material that is so often neglected in "simple" math. What a waste to present kids with a definition and miss the magic of abstract mathematical discussion and debate among nine year olds. 

How will we know what they understand, how will they know that they do, unless we allow them to debate, to defend and to support their ideas?

We can't. They won't.

I have three things to think about for next time:

• How can I guide a discussion without directing it?
• How do I end a conversation without eliminating further possibilities for exploration within the topic?
• How do I know whether everyone "gets" the conversation? How do I continue to provide opportunity for everyone to contribute?