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.

My Story of Change in Education: Student Voice and Physical Space

This story was put together as part of the Canadian Education Association's "What Standing in the Way of Change in Education" Conference in Calgary, October 21 - 22, 2013.

I only remember a few specifics from my first few weeks in the classroom. I had big ideas but the execution was definitely messy. I remember trying to keep track of things that worked and didn’t in those first few months and the second category was certainly larger than the first. I remember the first time I gave the students a math problem that asked that they apply an understanding of place value. Within ten minutes, three children were crying because they didn’t understand and were afraid to get the wrong answer. Somehow I hadn’t anticipated that my fifty students from all over the city would have such varying experiences and attitudes towards learning.

On Light, Shadows and Experience

Co-authored by Jenna Callaghan
Cross-posted on the Calgary Science School's Connect!

We began a recent investigation into Light and Shadows in Grade 4 by posing the question “What is Light?” to our students. Before beginning the conversation, we reminded students that the world is not nearly as concrete or easily-understood as over-simplified statements of "fact" might often imply. We talked about how scientists are by nature inquisitive, always open to possibility and a reinvention of old ideas. We suggested that throughout our inquiry, they too might have the potential to share a completely new perspective, contribute to making new discoveries and either support or disprove current thoughts. With two of us in the classroom, we were able to capture some of our students’ opening ideas about 'Light' and have embedded them below.

iPads for Learning

At the start of our first year with a 1:1 iPad program in Grade 4, Jon Van de Raadt made a comment that resonated and has set the tone for our year. "I think you'll find that iPads are not a junior version of the MacBook and should not be used as such. The iPad is a significantly different tool, and if you are prepared to embrace it, you might even find it more useful than the laptop has been." Two thirds of the way through our grade four pilot year with iPad 3's, I can say with certainty that I agree wholeheartedly. iPads belong in the classroom. I cannot imagine a more useful tool for representing, consolidating, expanding or creating understanding on the fly. I have no doubt that the rapidity with which app and software developers react to user feedback and update accordingly makes it one of the few technological tools out there that can effectively react to the ever-changing needs of youth in education and the evolving 21st century classroom. 

What follows is by no means a comprehensive or complete overview of what the iPad might be used for in the grade four classroom, but just a few brief exemplars from some of the artifacts we have had students create and present with the use of their devices. All apps (with the exception of the iWork apps) were free purchases. All investigations were co-developed with Amy Park and have been posted and archived in Edmodo and Google Apps. 

How to Love a Plant

Amy Park and Deirdre Bailey

Earlier this year we had a pretty cool opportunity to connect with Mount Royal University professor Dr. David Bird to co-present on Plant Growth and Changes for the Calgary Science Network. We were most eager for an opportunity to ask Dr. Bird what one one thing he wished his university students came in with that we might be able to foster in elementary school. His answer was unhesitatingly curiosity. He wished his students arrived at university with a desire, confidence and ability to ask scientific questions. Overwhelmingly, many of them arrived reluctant to explore, preferring instead to wait for instructions on what to think or how to deliver in order to "pass the course".

We were excited to be able to share the following (recently completed) video with Dr. Bird and to have a conversation about the kind of thinking we tried to cultivate in our students throughout the process. Having effectively guided them toward deeper understanding of scientific research earlier in the year with our Rotten Tomatoes Decomposition Lab, our inquiry into plant growth invited students' to use their experience with the 'Game of Science' to develop their own questions this time around. They were free to choose how to structure and represent their research autonomously, based on their previous experience.

The results were messy and in some cases, disorganized. Maintaining controlled conditions was difficult, keeping outdoor plants alive in May was a challenge. Many students discovered on their own why quantitative observations are typically separated from qualitative data, how organizing information makes analyses much simpler, and which variables should be most carefully observed when documenting plant growth. Though some conclusions were surprising or skewed by human error, the real value was that our students wrapped up these experiments with an insatiable desire to discover more about plants, undeterred by the awareness that everything might fail.

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.

Google just keeps killing it...

Deirdre Bailey

I had a conversation with a teacher today which prompted this post. It was similar to conversations I'd had in the past about twitter, its advantages and how it has been one of the most valuable learning tools for me as I continue to inquire into teaching and learning. I find it impossible to contain my enthusiasm for how twitter has revolutionized my information consumption as I tailor it to my interests and engage in dialogue deepening my understanding. My enthusiasm for twitter might be eclipsed only by my enthusiasm for google docs. I continue to be overwhelmingly excited by the facility with which google docs allows me to  guide student collaboration, research and writing while tracking their progress, providing feedback and involving parents. It has also made assessment a total breeze.

I work at an innovative and creative institution. Perhaps it is because of the freedom we have to explore possibilities that we are often unaware of the depth to many of the simplest resources that are available. It is also without a doubt hard to be constantly adapting one's practice to ever-updating technology applications. Certainly, in this day and age, the sheer volume of resources available in education can be overwhelming. Navigating options and rating their relative value is always intimidating, particularly on the heels of a full 7 hours in front of students. But google docs is so worth it and I really am that sure. And while my presence on twitter often gives me the impression that all educators are connected, deeply familiar with technological resources and employing them in their every day practice; it occurred to me today that this might not be representative of all cases. As I continue to stumble upon features in google docs that make my job incredibly easier, I thought it might be valuable to share the simple features I have discovered and am so thankful for. This is either a biased and likely limited overview of how GDocs has been used in our classroom this year, for those who have not had the time or opportunity to explore, or it is a simple plug for the brilliance of google documents and their value in the classroom.

                                                                                                    

For our most recent science lab data collection. We used GDocs to create spreadsheets...

...which we linked to our google site 

and shared these with student groups who edited in their own colours, uploaded photos, and separated their qualitative data from their quantitative data by inserting additional sheets

Qualitative Sample

Quantitative Sample

Students used the functions feature in google spreadsheets to verify their calculations:

 They were able to graph various data using the insert chart function

We used google forms to create a survey allowing students to provide feedback on others' work...

...which transferred their feedback to a sortable spreadsheet which we were able to cut and paste into their original Science Lab spreadsheets.

The second time around, we have had students create and share their own google documents to design plant growth experiments.

We had them insert a link from this google doc to a spreadsheet in which they would track their data...

We have been able to provide feedback on particular sections of their work using the new comments feature which allows us to send them notifications and which allows them to reply and/or resolve* the comment.
*Resolving a comment removes it from the document but does not remove it from the discussion stream.

We have also been SO grateful for the revision history function which allows us to see every edit that has been made to the document since creation including accidental deletions and who has made each change and when... Furthermore, previous revisions can be restored!! Pure genius.

Google docs can be edited from any computer which makes working from home simple. It is so easy to hold kids accountable for the work they do. It takes collaboration to a whole new level and it is only getting better. I should mention that our students are only 9 and 10 years old and they have had no trouble navigating these features through their own google accounts. I can't even imagine what the future might hold for a creative mind provided with these tools. Please comment if I've missed anything particularly useful or obvious. It would make my day.

Students into Scientists

Deirdre Bailey and Amy Park
Cross-posted on Calgary Science School's Connect!


It began with a conversation on how best to develop deep understanding of biological cycles. When we considered simply slicing fruit and vegetables in half and leaving them out in the open to observe the resulting changes, we never anticipated the smells, the new life, or the learning that would ensue.


The suggestion that we use Google Docs to facilitate collaboration and data management through the scientific process provided an excellent opportunity for introducing students to the incredible value of  facilitating collaborative research and documentation through technology. Have a look at our students' work, linked directly to our classes' Science Lab Website.

Although our decomposed specimens are now carefully buried in soil, the memory of the experiment is alive and well among our grade 4 students.  From Day 1 to Day 12, these young scientists were engaged, excited, and passionate about their discoveries. Throughout the process, students developed deep understanding of decomposition, the scientific process, collaboration, problem solving, data analysis, and so much more.

The following video outlines the entire process and the amazing learning that resulted.  

Waste and Our World Movie Project

Deirdre Bailey and Amy Park
Cross-posted on Calgary Science School's Connect!

Our Grade 4 classes have spent the past three months inquiring into the topic of "Waste and Our World" with the goal of developing an appreciation and understanding of their roles and responsibilities as global citizens of this environmentally fragile and increasingly endangered planet. We had hoped that our inquiry would help students feel empowered to sustain environmentally conscious action as part of their everyday lives. 

Our first mission was to collect garbage from the school ground. Students were shocked to discover that we had collected over 4lbs of lunch-related waste in one day and that this amount was repeated by the following class the next day! This activity made it clear that action for the environment was needed and could have a very real effect in our immediate community. As a group, we concluded that one minute public service announcements highlighting a need for change in our daily practices would be the best way to inform our school peers and inspire action. 

In order to build understanding and knowledge prior to designing their PSA's, students worked through a variety of tasks which included:

• creating a decomposition timeline using human waste collected in the immediate area
• discussing the composition of various human waste items and the factors affecting their breakdown
• digitally documenting human waste samples from our school grounds
• completing a retrieval chart with hypotheses on their specimen's origin and ultimate destination

Once students had a good understanding of the impact our human waste can have on the environment, we turned our attention to natural waste and consideration of how our natural world discards, reuses and recycles. As part of our exploration we:

• explored the Weaselhead Natural Area digitally documenting natural waste samples 
• completed a retrieval chart with hypotheses on their origin and ultimate destination
• connected with two expert speakers, one from the Centre for Affordable Water and Sanitation Technology, and Pat Allen, from Worms@Work

Armed with a new understanding of how human and natural waste fit in our world, students watched some exemplar PSAs and advertisements aimed at communicating powerful environmental messages. We then worked together to build a rubric outlining what exemplary work might look like!

CATEGORIES	Lots to improve, keep working/trying.  (Kindergarten-ish)	You’re okay but...	Good job, met expectations	Excellent!  THIS IS YOUR TARGET!!!
Content Research	Boring and meaningless facts! Used only a single source. Copied!	Partially researched, lacking information,  “is that it?” incomplete	Well researched, descriptive, relevant.	Interesting and helpful facts! Engaging! Has obviously used multiple sources. Original and in own words.
Content Script	Uses “tired” words. Lacks description. Information is unclear. Off topic and confusing. Unconvincing. Audience is bored...	Unconvincing, “I’m confused”	Sufficient, concise, competent, clear message.	Uses juicy words. Descriptive. Is clear on the facts. On topic and makes sense. Persuasive. Exciting - captivates the audience’s attention.
Images	Inappropriate, fuzzy, irrelevant, unrealistic,  doesn’t go with the subject/script/topic	Unpleasant, may not be appropriate to message	Appropriate, pleasing, satisfying	Realistic! Relevant! Memorable! Detailed! Appropriate! Clear! Interesting!
Voice	Lacks expression, boring, mumbly, too quiet or too loud, “notalkinglikethis” and “no.... talking.... like.... this.....”	A little mumbly, a hard time following, muddly, needs practice	Clear, nice, good expression.	Lots of expression, enthusiastic, good volume, enunciated well, speaks at a great pace
Groupwork	Making one person do all the work, “me first”, hoarding, arguing, fighting, using only one person’s idea, nobody is cooperating, bossing, poor communication, all talking at once, “my idea is better”, not sharing, off task	Inefficient, needs help and support, argumentative,	Nice to people, shares, demonstrates positive body language	Cooperative, inclusive, everybody has a voice, communicative, compromising, taking turns, no arguing, incorporating everyone’s ideas, on task, “you first” mentality

FInally, students supplemented what they had learned with the info on their retrieval sheets and internet resources to create one-page storyboards communicating their ideas effectively and concisely. Once their storyboards had been approved, they worked in collaborative groups to add voice to their own images creating short movies highlighting differences between human and natural waste and the impact they both have on our world. 

Students had the opportunity to showcase their first products in front of their classmates, after which they were provided with constructive feedback that was related to the rubric and based on the work. Throughout this process, students learned how to give feedback as well as how to receive it, and use it to improve. They were able to learn by reflecting on the work they and their classmates had created, and were then given an opportunity to use what they'd learned to refine their final movie project.

Here's an overview of the process:

Of the 22 projects, our group selected the best of the best to represent our learning and the message we hope to convey. Please help us get the word out by commenting below, sharing your feedback and your ideas on how we might be able to share these with a broader audience. Thanks for listening!

Student Exemplar 1: Nature Killers!

Student Exemplar 2: Think About the Future

See more examples of student PSA's here.

These kids get it

Deirdre Bailey

Today was amazing for a number of reasons. We finished our decomposition lab this morning. It had been an enlightening few weeks of learning what becomes of a perishable food, halved and left in the open for three weeks. Some awesome things happen to rotting fruit after it starts to smell, not least of which are fluid loss and bacteria growth. This morning, the tomato had given birth to a new family of fruit flies, all of whom remained happily trapped under the wrapping as our young scientists whooped victoriously while pressing their faces to the creeping mass of molding vegetable.

As today was the last day of our lab, we were burying our specimens in dirt with plans to test soil content and plant growth in the New Year. Most students were happy to put an end to an experiment that was becoming increasingly hard on the stomach, however one group in particular stood out as they proceeded solemnly to the dirt pile with their specimen, chanting the name "Alfred" in melancholic tones. "Aaaaalfreeed, life won't be the same without your stink... Aaaaalfreeeed, you taught us a lot about mould.... Aaaaalreed, we are going to bury you so you can grow a plant..." Alfred was their cucumber.

When I think back to the things I remember from Grade 4, I struggle to recall anything memorable outside of field experiences, Christmas holidays and family trips to Ireland. I imagine however, that if I had spent two weeks watching Alfred shrink, stink, and give birth to fruit flies, that I would remember Alfred. As adults, we seem to spend a tremendous amount of time discussing, warning and lecturing without experiencing and we forget so easily that our most powerful inspirations to date will have almost exclusively resulted from actual emotional experiences.

After lunch we had a presenter come and talk about endangered species in North America, specifically wolves. She asked the kids a number of questions which they were eager to answer, one of which was  about why maintaining the wolf population was of benefit to trees. Thirty plus hands went up.

"Because the nutrients in their poop help build rich soil for plant growth..." 

"Because they eat animals which would destroy the trees in larger quantities..."

"Because they sometimes eat plants and poop the seeds which helps distribute the growth..." 

Three months exploring the Weaselhead and connecting online in order to develop meaningful videos highlighting local natural waste and its place in our ecosystem had paid off. Natural organisms have a place in the wild, these kids had seen it.  "Wow, you guys really get it" was our presenter's response. "I have this conversation all the time with adults and it is hard for them to understand how wolves are connected to the trees."

"EVERYthing is connected to the trees!" was one indignant reply. Followed by: 

"..those adults, you have to explain a lot of things to them, there's so much they just don't understand..."

We learned a lot about native species in Alberta in 45 minutes. We learned a worrying amount about what our government says they're doing to protect them, as even to 9 year olds, many of their policies appear outdated and nonsensical. I gave the students sticky notes at the end of the presentation to jot ideas and questions about our current Endangered Species Act, what's working and what we should change. One student kept coming back for more notes... "Three's the limit right Mrs. Bailey? I can only have three ideas?" Who told this kid and at what point in his life that he could only have three ideas? He wants to write to the government on behalf of every one of the 13 male sage grouse left in our province. Then he wants to tell them that shooting wolves won't save cariboo, that Little Red Riding Hood was a liar, that wolves aren't the bad guys. He also wants to tell them to forget about the oil in the ground on the Blood Tribe land and just re-introduce the Swift Fox population the way they'd planned in 1973 and that they should be ashamed they've waited so long. He's got classmates on board.

Adults should remember that it's that simple. Kids are already curious and creative. They want to explore things and find out what's happening. Nothing about who they are is average, obedient or passive. All that is left to us, is to foster an awareness of the possibilities that surround them, to allow them to explore and then to get out of the way. Let them ask questions, make decisions, be different. There's more to their futures than resigned consumption and obedience.

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?