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# The Round Table Problem

## Introduction

Students can watch this video as many times as they would like.

## Part I: Entry

You can optionally ask students what they want, and come up with a shared question. Alternatively, you can use the question I already asked in the exit ticket. To modify the question, download the Word file. I am actually going to modify the question based on the responses I get from students. I feel like a better question would be, "What is the area of the table (in cm^2)?" or "If I take out one section of this table, how much surface area will be left (in cm^2)?"

Download the exit ticket for Word (modifiable): 2016-11-27-the-round-table-exit-ticket

Download the exit ticket for PDF (printable): 2016-11-27-the-round-table-exit-ticket

## Part II: Attack

Ask students, "What resources do you need?". Click the first three links to the resources I have, or download the Word or PDF files to print them out in advance for students.

Resource 1

Resource 2

Resource 3

Resource 4

## Part III: Check, Reflect, Review, Extend

Students can check, reflect, and review in different ways. Ultimately, they need to conclude with something along the lines of, "My hypothesis was reasonably accurate, because ... However, I learned that ..." or "My hypothesis was inaccurate, because ... And I learned that ...". I don't usually deduct marks for an inaccurate hypothesis, so long as the student corrected their mistake along the way.

There are a couple possible extensions to this problem. At this point in the year, I expect students to be able to start generating these extensions themselves. However, in this case, I am going to get students to solve the following extension.

• Can I make a square table, with length and width 196 in. and with a square hole in the middle, that has the same surface area?

# The Pizza Problem

## Part I: Entry

For this problem, I brought in a Costco pizza and put it on the table at front. We started class with a number talk, and once students started to get curious about the pizza, I asked them what they wanted to know. The dimensions of the pizza and the pizza box are available in the captions of the images below, but if you do this problem in class, I would strongly recommend just bringing in a pizza.

We did this problem in the context of our "From Patterning to Algebra" unit. The grade 8 Curriculum and Connections expectation is to develop formulas for the circumference of a circle, area of a circle, volume for right prisms, and surface area of a cylinder.

## Part II: Attack

Students worked together in groups of 3 to solve this problem. They had a total of maybe three hours to work on this problem in class over the course of a week. The problem the students decided to work on was, "What are the area and circumference of the pizza?", and the extension was, "If we had 27 mini-pizzas (one for every student in the class), with the same total volume as this regular-sized pizza, what would the dimensions of each mini-pizza be?"

## Part III: Check, Reflect, Review

On Friday, we looked through student solutions together, and we found a rough estimate for the circumference, surface area and volume. The purpose of this activity was not to necessary find exact solutions for those dimensions, but rather to develop general formulas.

# From Patterning to Algebra

## From Patterning to Algebra: Unit Expectations

Here are the Curriculum and Connections expectations for this unit.

## (1) Number Talks and Chronological Photo Gallery

This is a gallery of everything we've studied during this unit, in the order we've studied it. Hopefully, this will show how the four different types of activities we do in class are tied together (number talks; strategy walls and math journals; 3-act math and inquiry problems; and exit tickets).

When I do a 3-act math problem or an exit ticket, I am usually aiming for a conceptual understanding of the problem. I often find that this needs to be supported by a series of number talks, which help teach the skills underlying the concepts.

## (2) Strategy Walls and Math Journals

[Watch this space for updates.]

## (3) 3-Act Math / Inquiry Problems

• The Coke Problem

[Watch this space for updates.]

• The Pizza Problem

This problem was complex enough for its own post.

## (4) Exit Tickets

• The Hexagonal Prism Tank Problem

The purpose of this problem is to help students understand the relationship between the area of a base of an object and its volume.

# The Watermelon Problem

## Prologue

Curriculum and Connections Expectations: Fractions

• use fraction skills in solving problems involving measurement

## Part 1: Entry

To introduce the problem, I just brought in a watermelon and sat it on the table until students had enough questions. Once they were ready, I put "What do you want to know?" on the board, and as they asked questions, I gave them markers so they could put their questions on the board.

Journal entry #1: The Watermelon Problem.

Based on these questions, students decided to study the problem, "How can you cut up the watermelon so that each of the 27 students can have an equal slice?"

What's The Watermelon Problem? Here are the questions the students asked.

## Part 2: Attack

Here are some resources for the watermelon we chose. It's honestly better to get your own watermelon though.

Here are some examples of student work and ideas. Many of them came up with some form of idea to do a third of a third of a third.

## Part 3(a,b): Review (Check, Reflect)

After finishing this problem, it's time for students to check if the solution worked the way they expected, and to reflect on what went well and what can be improved on. You can see in the video that cutting the watermelon into thirds would require a very high level of precision to do well. There are a whole bunch of hindrances. For the sequel, encourage students to find a better way to divide the watermelon. Can they add the precision necessary to cut the slices precisely equally? Can they find a better way to distribute the watermelon?

## Part 3(c): Review (Extend)

Once students are ready, they can try to find a strategy that can work easily for any number of students.

## Elements of Solutions

Here is one possible such strategy. The result is pretty easy to generalize to any number of students. However, there are still a couple of precision-related problems with this strategy.

Here is a different strategy. Still pretty easy to generalize to any number of students, but with different pros and cons.

# French Fiction: Fairy Tales (Les contes)

This is still the beginning of our unit on French fairy tales (« Les contes »). Students will analyze narrative text patterns (character, setting, plot, point of view, conflict) and use dialogue to tell their own fairy tale.

## Overview

You can refer to the FSL Year-at-a-Glance post for a more in-depth look at our long-range plans for this year. However, here is a brief overview of what we will be doing.

• Language Talks (Études linguistiques): Understanding and using language conventions
• Reading Fiction: Analyzing Narrative Text Patterns
• Writing: Fairy Tales
• Speaking: Deliver Oral Presentations
• Listening: Using Listening Comprehension Strategies and Listening for Meaning

## Language Talks / Études linguistiques

The purpose of French Language Talks is to help students understand the basics of French grammar, punctuation, spelling, verb conjugation, as well as verb and noun agreements. I prefer to do Language Talks instead of worksheets, since the former do a better job mimicking real-life situations. Where a worksheet might give 20 questions that repeat essentially the same skill, a single Language Talk can cover multiple skills simultaneously.

We do Language Talks almost every class. The exception is if that extra twenty minutes of time is absolutely needed for another activity. Over the course of September to December, I hope to cover punctuation, spelling common words, lexical problems, and grammatical spelling of verbs. As I develop a better understanding of what this group of students already knows and what they need to work on, our focus will inevitably shift toward one or two of these concepts.

Here are some examples of the grammar talks we have done so far. One of the cool things about doing Language Talks this way is that I often learn new rules myself. For example, when writing the date, I have always copied the English convention: today's date is Sunday, September 11, 2016. Thus, French, I have always written Dimanche, le 11 septembre, 2016, with commas. As we studied the conventions as a class, we ended up looking up the rule, and it turns out that you never add commas in the date in French (with one exception, detailed in the link).

## Reading Fiction: Analyzing Narrative Text Patterns

Over the course of this unit, we read a number of fairy tales in French and tried to identify any common themes.

## Writing: Fairy Tales

These are our graphic organizer models for how to plan a fairy tale.

And here is the evaluation, with success criteria. All of the fairy tales we have read in class are also available to use as models.

## Speaking: Delivering Oral Presentations

[We haven’t started this yet. Watch this space for updates.]

## Listening: Using Listening Oral Comprehension Strategies and Listening for Meaning

[We haven’t started this yet. Watch this space for updates.]

# Fractions, Decimals and Percents

We have officially started our first number talks as a class. I will need to update this post as the unit progresses, but here is what we have so far.

## Fractions, Decimals and Percents: Unit Expectations

Here are the expectations we will cover in this unit, which is based on the Curriculum and Connections resource. There are four pillars to what we going to be covering: (1) Number Talks, (2) Strategy Walls and Math Journals, (3) 3-Act Math Problems and (4) Exit Tickets.

## (1) Number Talks

We have done our first few number talks for this unit. The idea for Number Talks is simple: I present a math question, students take a moment to solve it individually, and then we discuss strategies. In my classroom, this sort of replaces "drill work" (usually taught through worksheets or a series of problems which become slowly more difficult). Instead of giving the students a bunch of questions to work on at once, we take the time to study one good question together.

## (2) Strategy Walls and Math Journals

"In order to learn from experience, in order to have fresh possibilities come to mind when appropriate, it is necessary to sensitize yourself to notice opportunities to respond to situations rather than to react, to choose to act rather than to be caught up in and driven by old habits. Thus, by offering tasks and follow-up prompts it is possible to promote awareness of the use of people's natural powers. As they become more sensitized to and aware of their own use of these powers, those powers develop in flexibility and usefulness." - Thinking Mathematically, Mason & Burton, p. xv

I believe that student learning is what happens in a student's head. The learning process is, in a sense, the action of a student organizing her or his thinking. With that objective in mind, the objective of strategy walls (as a group) and math journals (as individuals) is to give students an opportunity to consolidate their learning and communicate their thinking. While I enjoy working on problems with students, and may write comments and questions in their math journals, I will never give a math journal any type of grade.

## (3) 3-Act Math / Inquiry Problems

We just did "The Watermelon Problem" (click link for full article), which was long enough to merit its own post.

## (4) Exit Tickets

• The Security Camera Problem

Here is a copy of our first exit ticket. The objective here was to get students to start using their problem-solving methods to find the best place to place a camera. I am a big fan of problems that are easy to understand but difficult to solve. As students went through this problem, they began to understand its nuances, which allowed them to push their understanding further.

• The Square Mural Problem

This was our second exit ticket for this unit. I based it on my first attempt at a 3-act math problem last year, The Square Mural Problem. I showed students the video from that problem, but I didn't let them ask questions as a group, since I expected them to attack the problem independently.

The purpose of this problem is ultimately to bring students to an understanding of Least Common Multiples and Greatest Common Factors, even though neither of those concepts are immediately apparent in the problem. The extension for this problem was for students to find an efficient way to create the smallest square mural that could be made from any tile. For example, how would you find the smallest square mural made of tiles with a height of 180 mm and a width of 378 mm?

As students struggled with this problem, they started to realize that they didn't have enough information based on the video. As those students came up and asked about the extra dimensions, I gave them either (or both) of these additional resources.

# Grade 8 Welcome Letter

Dear Grade 8 Parents and Guardians, Welcome back! We hope you had a safe and enjoyable summer, and we look forward to getting to know you and your children this year. We will be the grade 8 homeroom teachers for this year. Below, you will find answers to many of the questions you may have about your child’s grade 8 year.

Want to print this letter (PDF)? 2016-06-29-grade-8-welcome-letter

Want to modify it for your own students (Word)? 2016-06-29-grade-8-welcome-letter

## List of Teachers

Here is the list of your child’s teachers for this school year. The easiest way to contact us is by e-mail. The format for a teacher’s e-mail address is first.last@ddsb.ca.

• Homeroom: Mrs. Kelley or Mr. Barr
• English Language: Mrs. Kelley
• French: Mr. Barr
• Mathematics: Mr. Barr
• Science and Technology: Mr. Marsh
• History: Mlle Samaha
• Geography: Mlle Samaha
• Health: Mr. Barr
• Physical Education: Ms. Dean
• Dance: Ms. Dean
• Drama: Ms. Dean
• Music: Ms. Tremblay-Beaton
• Visual Arts: Ms. Dean

Clear communication between parents and teachers is key to a successful year. As such, we would like to bring your attention to some of the ways that we can communicate about your child’s progress this school year.

## School Supplies

In order to ensure your child’s success, the following supplies are recommended.

• Pens, pencils, erasers
• Binders (either one large binder with dividers, or several small binders)
• Looseleaf paper or exercise book
• Graph paper or exercise book
• Calculator with visible operations, geometry set (ruler, protractor, compass)

## Newsletters and Websites

• Kelley and Mr. Barr begin each unit with a newsletter that outlines all of our learning goals in each strand. We also provide a list of activities that you can do with your child to help him/her better understand the concepts taught throughout the unit. The first one of these will come home as a paper copy; all subsequent newsletters will be posted on the class website.
• The main Walter E. Harris website is ca/school/waltereharris. From here, the “Teacher Pages” link leads to the pages of each teacher
• All of our leaning goals, success criteria, exemplars, mentor texts, rubrics, and major assignments will be organized by strand and posted on our websites. If you are curious about what we’re working on in class or have any questions about assignments, please check the websites out regularly.

## Assignments and Evaluations

• All marked assignments will be sent home with a rubric and percentage grade so you are aware of your child’s progress. We kindly ask that you sign and return the rubric the next day so that we know you saw the mark. Each child will keep his/her marked assignments in a portfolio at school. It is crucial to his/her success that you return the evaluations promptly. If you would like more time to look over the assignment, please write a note in the agenda.
• Assignment due dates are not flexible unless there are extenuating circumstances. We will also send home a pink slip anytime work is not submitted on time, to ensure you are aware. The pink slip will be filled in by your child indicating which assignment was not submitted, the date, and the reason why. We ask that you kindly sign that return the pink slip the next day.

Sincerely,

Mrs. Kelley and Mr. Barr

We also occasionally send home reminders and updates by e-mail. Please provide email addresses for any parents and guardians who would like to be contacted by e-mail.

 Student’s Name: Parent or Guardian’s Name Email Address

# WEHPS Grade 8 Class Schedules 2016 - 2017

Here are the (tentative) schedules for the grade 8 classes this year. These are subject to change over the next few weeks, since there are probably still kinks to iron out.

## Grade 8 Barr Homeroom

This is the schedule for the grade 8 Barr homeroom class.

Download for Excel (modifiable): 2016-09-01 Élèves Barr - horaire hebdomadaire

Download as PDF (printable): 2016-09-01 Élèves Barr - horaire hebdomadaire

## Grade 8 Kelley Homeroom

This is the schedule for the grade 8 Kelley homeroom class.

Download for Excel (modifiable): 2016-09-01 Élèves Kelley - horaire hebdomadaire

Download as PDF (printable): 2016-09-01 Élèves Kelley - horaire hebdomadaire

## Mr. Barr's Schedule

This is Mr. Barr's personal timetable.

Download for Excel (modifiable): 2016-09-01 M Barr - horaire hebdomadaire

Download as PDF (printable): 2016-09-01 M Barr - horaire hebdomadaire

## Mrs. Kelley's Schedule

This is Mrs. Kelley's personal timetable.

Download for Excel (modifiable): 2016-09-01 Mrs Kelley - Weekly Schedule

Download as PDF (printable): 2016-09-01 Mrs Kelley - Weekly Schedule

# Four Teaching Strategies to Improve this Year

I just finished “Quiet: The Power of Introverts in a World That Can’t Stop Talking” (by Susan Cain), and the book has gotten me thinking back to what did and didn’t work last school year. Last year, I taught French Immersion and mathematics to two classes of grade 8 students. One of the big advantages to this was that I was essentially working with the same 50 students for two to three hours each day. This allowed me to thoroughly test several teaching strategies and evaluate their effectiveness. I was really happy with some of the strategies I tried out, and unimpressed by others. Here are some of my conclusions regarding these strategies, through the lens of the introvert-extrovert continuum.

## Introversion and Extroversion

There is a lot in “Quiet” that is worth reflecting on within the context of the classroom. For example, there is a huge push in education right now toward group work. While there are good intentions behind this (I believe that kids should be provided with opportunities to learn from each other), it was interesting to see that this might not be the best way for every child to learn.

Yet, as Quiet will explore, many of the most important institutions of contemporary life are designed for those who enjoy group projects and high levels of stimulation. As children, our classroom desks are increasingly arranged in pods, the better to foster group learning, and research suggests that the vast majority of teachers believe that the ideal student is an extrovert. (“Quiet”, p. 6)

On one level, this is not surprising: there is no single teaching strategy out there that will cater to every student every time. However, it is true that we tend to put an emphasis on group work over individual work, and on talking over thinking.

The book doesn’t overly concern itself with the precise definitions of “introvert” and “extrovert”, but rather with the fruits of the research. That being said, it is useful to start with definitions.

When I started writing this book, the first thing I wanted to find out was precisely how researchers define introversion and extroversion. I knew that in 1921 the influential psychologist Carl Jung had published a bombshell of a book, psychological Types, popularizing the terms introvert and extrovert as the central building blocks of personality. Introverts are drawn to the inner world of thought and feeling, said Jung, extroverts to the external life of people and activities. Introverts focus on the meaning they make of the events swirling around them; extroverts plunge into the events themselves. Introverts recharge their batteries by being alone; extroverts need to recharge when they don’t socialize enough.

But what do contemporary researchers have to say? I soon discovered that there is no all-purpose definition of introversion or extroversion […] There are almost as many definitions of introvert and extrovert as there are personality psychologists, who spend a great deal of time arguing over which meaning is most accurate. Some think that Jung’s ideas are outdated; others swear that he’s the only one who got it right.

Still, today’s researchers tend to agree on several important points: for example, that introverts and extroverts differ in the level of outside stimulation that they need to function well. Introverts feel “just right” with less stimulation, as when they sip wine with a close friend, solve a crossword puzzle, or read a book. Extroverts enjoy the extra bang that comes from activities like meeting new people, skiing slippery slopes, and cranking up the stereo. “Other people are very arousing,” says the personality psychologies David Winter, explaining why your typical introvert would rather spend her vacation reading on the beach than partying on a cruise ship. “They arouse threat, fear, flight, and love. A hundred people are very stimulating compared to a hundred books or a hundred grains of sand.” (“Quiet”, p. 10 - 11)

I still have no perfect solution for creating a classroom atmosphere that is stimulating enough for the most extroverted students while being calm enough for the most introverted. But I came away with some food-for-thought on the main teaching strategies I use in my classroom: (1) Number Talks (or Études grammatiques in French class), (2) Taking Notes and Using Strategy Walls, (3) Three-Act Math Problems, and (4) Group Work.

## 1. Number Talks

(1A) Last Year’s Working Hypothesis

Number Talks might be interesting. I should try them out over the course of a month and see if they are useful.

(1B) Reflection / Analysis

The principle behind Number Talks is to provide students with one or two good problems each day to work on mentally, before sharing a solution verbally with the class and discussing strategies on how to reach that conclusion. Over the course of a unit, this allows students to confront a range of problems, and to see common mistakes. Number Talks also allow students to mentally test the validity of other students’ answers.

Many psychologists would also agree that introverts and extroverts work differently. Extroverts tend to tackle assignments quickly. They make fast (sometimes rash) decisions, and are comfortable multitasking and risk-taking. They enjoy “the thrill of the chase” for rewards like money and status.

Introverts often work more slowly and deliberately. They like to focus on one task at a time and can have mighty powers of concentration. They’re relatively immune to the lures of wealth and fame.

Our personalities also shape our social styles. Extroverts are the people who will add life to your dinner party and laugh generously at your jokes. They tend to be assertive, dominant, and in great need of company. Extroverts think out loud and on their feet; they prefer talking to listening, rarely find themselves at a loss for words, and occasionally blurt out things they never meant to say. They’re comfortable with conflict, but not with solitude.

Introverts, in contrast, may have strong social skills and enjoy parties and business meetings, but after a while wish they were home in their pajamas. They prefer to devote their social energies to close friends, colleagues, and family. They listen more than they talk, think before they speak, and often feel as if they express themselves better in writing than in conversation. They tend to dislike conflict. Many have a horror of small talk, but enjoy deep conversations. (p. 11)

I feel like the Number Talk format works well for introverted and extroverted students. Giving students time to reflect on the problem individually, takes some of the pressure off of having to speak in front of their peers. At the same time, I found that the more extroverted students ended up sharing their answers most of the time. It strikes a certain balance.

(1C) Teaching practices that worked well

• Do each Number Talk on a sheet of chart paper, or find some way of recording students’ thoughts, so that they can be referred to in the future.
• Use Number Talks to build a theme. Each day, look at what students already know (what they are good at as well as what they are struggling with), look at what your final objective for the unit is, and make today’s Number Talk be a problem that will (hopefully) enable students to build their next step to reaching that objective.
• Don’t worry if a Number Talk doesn’t go well on a given day, but plan the next day’s Number Talk to help students reach the goal you envisioned.
• Staple past Number Talks together in a visible area of the classroom, so that students can refer to them, and so that you can look back and make sure you’re on the right path to teaching students what they need to learn.
• Prepare a “Part I” and “Part II” for each Number Talk. If Part I is too easy, then move straight on to part II. If Part I takes up all of the allotted time, then Part II becomes the next day’s Part I.

(1D) Teaching practices that didn’t work so well

• I tried taking pictures of Number Talks and posting them in a class OneNote. My hope was that this would allow students to access the full history of Number Talks at home and at school, whenever they needed. What ended up happening was that, since it takes too long to boot up a computer and go look at the OneNote, students stopped referring to past Number Talks entirely. Solution: go back to writing Number Talks on chart paper and stapling them together on a wall.
• Picking specific strategies for students to learn: I usually have a way that I envision solving a problem. And I usually try to pick problems that will encourage students to use specific strategies. But it can often take a class a while to develop effective and efficient strategies. When I’ve tried to show students what my favourite strategies are, it usually goes better when they have had a good amount of experience with that type of problem. If I try to explain my own strategies too soon, (1) often, many of the students don’t understand what I’m trying to do, and (2) it takes the fun away from the students’ developing their own strategies. Solution: toward the end of a section of Number Talks, it helps to have a conversation around what the students think their best strategies are. It helps to encourage them to develop two strategies that they like, since one strategy can help you find the answer and the other can help you check your work.
• Waiting until the students are “ready”. I was trained to teach first, then ask questions after. With Number Talks, I might warn students that they are going to see something new, and that a problem might be a bit more difficult than usual, but I have found it inefficient to try to backfill all of students’ possible missing knowledge before starting a unit. Now, I just give problems, and the skills come. Starting the Pythagorean Theorem? Put the question on the board, see what students come up with. Will the Number Talk sometimes be too difficult? Yes. Will it ever be a failure? No, because if students get stuck on (for example) what a square root is, they will ask you to explain it, and will hopefully be more inclined to listen to the answer.

(1E) Next Steps

• This year, I hope to extend the idea of Number Talks in math to “Études grammatiques / Études linguistiques” in French. My hope is that I can apply the same strategy to language (I know that I love the problem-solving aspect of grammar and conjugation).

## 2. Taking Notes and Using Strategy Walls

(2A) Last Year’s Working Hypothesis (Initial Question)

I hate writing notes, students hate copying notes, why don’t we just skip taking notes and do awesome math problems during the time we save?

(2B) Analysis

Well, we avoided wasting time writing down notes that many students didn’t want to copy down. But this was a fine example of throwing out the baby with the bathwater. There is a lot of value in taking notes, specifically notes that students create and structure themselves. We were able to compensate by sort of having “group notes” (if you will) on our strategy walls, but this isn’t as effective.

(2C) Conclusion

Having students copy down notes exactly as I’ve written them is inefficient. Compiling what students have learned into a strategy wall is useful. But not taking any notes at all is somewhat ineffective. There has to be an optimal solution somewhere in between.

(2D) New Hypothesis

I’m going to keep creating strategy walls. For note taking, I would like to try a type of “math journal”, where students can take their own notes, including summaries of Number Talks, definitions, and work they’ve done on inquiry problems and three-act math problems. I’m hoping that this will allow students to construct their own overall understanding of mathematics. The bottom line is that students have to do some form of written reflection at the end of the unit, where they can figure out what the most important concepts in the unit were, and write down some of the implications.

## 3. Three-Act Math Problems

(3A) Last Year’s Working Hypothesis (Initial Question)

What would happen if, instead of starting a new unit with a series of definitions, theorems and examples, I instead start with an awesome problem? I think it will destabilize the students (which is probably a bad thing), but also make them ask better questions and be more engaged (which is a good thing). Is the trade-off worth it?

(3B) Analysis

I started doing a lot of three-act math problems this year, using many from Dan Meyers’ website, as well as creating some myself. It turns out that they are hands-down awesome, and that this is how I want to teach math from now on. That being said, there are certain disadvantages to teaching math this way which need to be minimized.

Three-Act math problems are most easily managed from a teaching perspective by having students work in small groups of 2-3 people per group. I am currently unsure of whether this is best for the students (I will discuss this in the next section), but it definitely is easier for me. By having, say, 8 groups that I need to circulate between in a given class, I can give feedback to each group two or three times in an hour, and help them flesh out their ideas.

(3C) Things that worked well

• Letting students work in groups of 2 – 3. Students seemed to learn well from each other, and I was able to circulate effectively between groups.
• Using the “Continuum and Connections” framework. I needed a way to structure my year that would allow me to be flexible, and this resource did the job.
• Writing students “next steps” directly on their paper. I found that by adding questions directly to students’ papers, I was able to track their progress better and was better able to evaluate their final product. I would literally write things like, “This is inaccurate. What do you think went wrong? Fix it,” right on their paper. We would discuss possibilities, and then I would tell them to go write down (and hopefully prove) what they thought.
• Having some groups present their work to the class, and having all groups give their conclusion before watching the third act of the 3-act math lesson. That way, students had something at stake when they saw the actual solution.
• Reviewing as a class some of the things we learned, and some of the things that went wrong.

(3D) Things that didn’t work so well

• Letting student work in groups of 4 or more. There was always at least one student not working in these groups.
• Trying to give grades based on group work. I’ve spent a bunch of time thinking about this, and nothing I’ve come up with makes sense. Group activities like this can be good to help develop strategies, skills and knowledge, but they offer little value for giving a grade. If anyone comes up with a good solution, please let me know.
• Sometimes students who could talk with more authority were taken more seriously than students who had difficulty expressing themselves.

(3E) Conclusion

Three-Act Math problems are absolutely fantastic, and I can see these being one of the pillars of my math classroom for years to come. However, room needs to be left for students to work on problems individually, whether that be through an exit ticket, a math journal, assignments or tests.

## 4. Group Work

(4A) Last Year’s Working Hypothesis

If I provide students with opportunities to work on cool problems in small groups of 3 or 4 people per group, and I structure the learning environment, then students will invest more into their work and will learn more from what they do.

(4B) Analysis

I’ve been struggling this past year, trying to find the best way to implement group work. I believe that kids should construct their own mathematical structure as much as possible. But that takes a mixture of individual work, large group work and small group work. And every student has their own way of learning, and their own preferences for how much individual work, small group work and large group work they want to participate in. The question in, how can we accommodate everyone as best we can?

What does this mean about group work? Some students learn better on their own (but you already knew that). Other students learn better in groups. And no matter our own temperament, we need to learn to work with people who are all across that spectrum. The danger is that there could be misunderstandings between people on opposite ends of the spectrum: introverted students might let extroverted students run the conversation; extroverted students might feel like they are doing all the work because introverted students aren’t expressing themselves.

If we assume that quiet and loud people have roughly the same number of good (and bad) ideas, then we should worry if the louder and more forceful people always carry the day. This would mean that an awful lot of bad ideas prevail while good ones get squashed. Yet studies in group dynamics suggest that this is exactly what happens. We perceive talkers as smarter than quiet types—even though grade-point averages and SAT and intelligence test scores reveal this perception to be inaccurate. In one experiment in which two strangers met over the phone, those who spoke more were considered more intelligent, better looking, and more likable. We also see talkers as leaders. The more a person talks, the more other group members direct their attention to him, which means that he becomes increasingly powerful as a meeting goes on. It also helps to speak fast; we rate quick talkers as more capable and appealing than slow talkers.

All of this would be fine if more talking were correlated with greater insight, but research suggests that there’s no such link. In one study, groups of college students were asked to solve math problems together and then to rate one another’s intelligence and judgment. The students who spoke first and most often were consistently given the highest ratings, even though their suggestions (and math SAT scores) were no better than those of the less talkative students. These same students were given similarly high ratings for their creativity and analytical powers during a separate exercise to develop a business strategy for a start-up company.

A well-known study out of UC Berkeley by organizational behaviour professor Philip Tetlock found that television pundits—that is, people who earn their livings by holding forth confidently on the basis of limited information—make worse predictions about political and economic trends than they would by random chance. And the very worst prognosticators tend to be the most famous and the most confident—the very ones who would be considered natural leaders in an HBS classroom. (p. 51 - 52)

This next anecdote provides some insight into how we could train students to avoid the trap of thinking the most talkative students are necessarily right.

(4C) Things that worked well

• Randomly generating student groups: This way, students knew that they would always be working with different people, so if they didn’t get along with someone, at least they knew it would be temporary
• Structuring the introduction to the group project: Things worked better when everyone started more-or-less on the same page
• Not formally evaluating student work: When students knew that the objective of group projects was to learn and to get prepared for a future evaluation, they found it motivating and it allowed them to take more risks

(4D) Things that didn’t work so well

• Creating groups larger than 3 people: There was always someone without a job
• Not giving enough time to think before asking for questions: Next year, it might be good to start a math journal entry and allow students to generate their own questions before starting the group discussion

(4E) Conclusion

Keep group work, but try to make the work more efficient. Set up student desks in the classroom in groups of 3 (maximum), to facilitate transitions.

(4F) New Hypothesis

If I have students start all three-act math problems in their own math journals first, before randomly assigning them to groups of 2 – 3 students, then students will be able to bring their own initial impressions to their group, and make better problem-solving decisions.

# Planning a Flexible Framework for Intermediate Mathematics

This is an overview of what we will be discussing during our curriculum framework professional development session on August 25th. We will be at the DDSB Education Centre, room 2015 from 8:30 AM to about 2:00 PM. I hope you will all be finished your initial planning before 2:00 PM, but I put some extra time as padding, to avoid cutting short any great discussions.

## Why I Enjoy Math

There are two main experiences that I can point to that really drove me to want to teach mathematics. The first was my grade 12 calculus course. What I enjoyed the most about that class was when we were given time to work on tough problems. I was lucky enough to be in a class where most (if not all) of the students really wanted to be there, and really wanted to learn. The teacher didn't pull any punches, and gave us really tough problems to work on. I remember working on those problems both individually and with other students for hours. The sheer difficulty of the problems made them intensely satisfying to solve.

The second really positive experience I had was one of my university math courses about the process of problem-solving. The course was entirely based on the book, "Thinking Mathematically", by Mason, Burton and Stacey. In that book, I found that the authors had summed up the essence of what is satisfying about mathematics: solving intricate and difficult puzzles by finding the simple logic that underlies them. The book showed where the art and the science of mathematics meet. (It sometimes feels like many of the new ideas of how to engage students in math are pulled from this book. The problems are simple enough that they can be understood and attempted by grade 7 students, but are complex enough that they can lead to university-level solutions.)

So, I do math because I enjoy solving intricate puzzles, and I find it extremely satisfying when seemingly impossible problems can be solved simply by looking at them in the right way. And the reason why I teach math is that I enjoy seeing students get the same satisfaction out of these puzzles.

## The Perfect Lesson Plan: a Hypothesis

Unfortunately, there is a disconnect between the way I enjoy learning mathematics and the way that I felt I was expected to teach mathematics. I get excited about learning math when I'm faced with a challenging problem that I can generate good hypotheses for, but that takes a while to figure out the underlying structure. I enjoy teaching math when I see my students going through the same process: they have a tough problem that is easy to understand, but which is tough to solve. Take, for example, the first problem in "Thinking Mathematically".

This problem is simple enough that students can understand the question easily, but complex enough that it takes some time to figure out. That's precisely what makes it so enjoyable to work on. But when you go in to the Ontario Math Curriculum (Grade 8 Number Sense is on page 111), it's not immediately clear whether this question really teaches kids to "solve problems involving percents expressed to one decimal place (e.g., 12.5%) and whole-number percents greater than 100 (e.g., 115%)". It becomes easy to start asking myself if I've really taught that students what I was supposed to teach. Did I go over the concept too superficially? Should I have gone into more detail? What about the kids who struggle with the concept of decimals, fractions or percents--how do I help them succeed? And if I try to focus on the students who are struggling and fill the gaps in their understanding, what should the kids who find this easy be doing?

(You may have already heard an easy answer to this, such as, "You just work with the struggling students in small groups while the independent workers work independently"; unfortunately, the reality of working with students in the classroom often precludes easy answers. I have a couple elements of solutions and hypotheses to propose. I also have a couple good questions. But I can confess right now that I don't have any easy answers or solutions that will 100% work in your class for 100% of your students 100% of the time.)

## Where I Got Stuck

Last September, I taught Chapter 5: Measurement of Circles in September. And I did Chapter 1: Number Relationships in October. And after that, I realized that I had no interest in continuing. I couldn't figure out why. It didn't make any sense to me--I love doing math. I love teaching math. I love the rigour involved in preparing lessons that are pedagogically sound, and that explore the nuance of the mathematics curriculum. Why wasn't I looking forward to the next unit?

So I did what I usually do: I started hypothesizing. Maybe I didn't change the program enough from last year (I don't like teaching the same thing the same way twice). Maybe I wasn't precise enough in my examples or definitions (perhaps students weren't fully understanding). Maybe I didn't want to write notes on the board (in retrospect, that was certainly part of the answer). Maybe I should ditch the textbook entirely and plan my year based only on the curriculum. No matter how much I personalize the program and add in my own activities, by starting with a textbook, I'm automatically following someone else's plan, and that limits my creativity. (I looked at my long-range plans, and they looked dreadfully boring.) Maybe I'm experienced enough now that I don't need to curriculum as a guide (I have a pretty good feel for what a grade 8-level problem should be). Maybe if I taught using just the curriculum, I could cover the entire thing in one year. Well, anyway, as much of the curriculum as anyone can cover, since there are so many curriculum expectations, how can anyone cover them all anyway?

When I prepare my units, I try to start with the end in mind. I look around for any great resources (after all, why create something from scratch when I can take something good and make it great with the same amount of effort?), and I try to match them up against the curriculum and organize them in a way that should make sense to students. You can sort of see my process when I compiled the resources, "Nelson Math Grade 8 Long-Range Plans", "Jump Math Grade 8 Long-Range Plans", and "Ontario Curriculum Long-Range Plans". I tried to take the best parts of all three of those documents, and structure them in a way that would make everything as clear as possible to the students. As far as I was concerned at the time, a perfect lesson should cover all the curriculum expectations in an organized and structured manner, and should define all terms, hypotheses and theorems along the way. A perfect lesson should be rigorous, precise, and provide excellent examples for students to follow. You can see the result of that hypothesis in "Chapter 1: Number Relationships".

## A Possible Solution

So here is the working hypothesis I tested last year: if I can find a way to teach mathematics so that the kids learn the same way that I like to learn, then they will have more fun and I will have more fun. (Oh, and also, we will all learn more and other stuff too.)

With that in mind, I started to re-design my curriculum so that, instead of teaching a bunch of theory, taking notes, and then trying out some problems, we tackled the big problems first.

The advantage to doing it this way is that the students started asking better questions, generating better hypotheses, and becoming more independent. The short-term downside was that many of the students felt destabilized. The structure I was using to teach had changed, and they justifiably felt unsure of what to expect. To be fair, I felt a bit destabilized myself. I like having structure.

And that brings us to the reason why we are all here today. How can we teach the way we want to teach while still retaining structure? How can we avoid the extremes of "too much flexibility" (which makes it difficult to plan, and to make sure that we are going to teach what needs to be taught) and "not enough flexibility" (where we can't adapt what we are teaching to the needs of the students)?

Well, there's a resource that helps tremendously with that problem.

## The "Curriculum and Connections" Resource

This collection of documents, created by the Ontario Ministry of Education, is a fantastic way to organize a flexible framework for mathematics. The documents cover grade 7 through grade 10, and distinguish between applied and academic at the high school level. The best part is that each of the documents is broken up into the strands for each grade.

For example, grade 7 and grade 8 mathematics have a section devoted to each of the following: (1) Fractions; (2) Integers; (3) Patterning and Algebraic Modelling; (4) Perimeter, Area and Volume; (5) Proportional Reasoning; and (6) Solving Equations. Furthermore, each of those sections are broken down into the 5 strands we are supposed to evaluate in those grades: Number Sense and Numeration; Measurement; Geometry and Spatial Sense; Patterning and Algebra; and Data Management and Probability.

The way the "Curriculum and Connections" resource is put together on the EduGAINS website, is unfortunately difficult to work with on a daily basis. I needed something that I could just put on the wall, and check off as I went, something that I could refer to quickly and easily. So, last year, I took some time to compile all the units (by grade level) into a series of 1-page documents containing all of the important information. Click here for the full "Curriculum and Connections" resource.

## Covering All the Curriculum Expectations

One of the big advantages to teaching this way is that I no longer worry about "covering the curriculum". There are only six units in the entire year, and I can break them up however I want. In addition, I know I'm going to cover each strand once in every unit, so I'm never going to be in a position where, "Oh, no! I have to teach fractions sometime in the next three weeks so that I can get a grade for the report card!"

Using the Curriculum and Connections framework, I'm going to cover every strand multiple times each semester, so I really just need to make a judicious choice of what I'm evaluating.

Here are two examples of units that I've taught this past year, using the "Curriculum and Connections" framework.

### 1. Understanding the Relationship Between Percents, Fractions and Decimal Numbers

This unit was based on the Curriculum and Connections grade 8 "Fractions" section. I co-constructed the learning goal and success criteria with the students, but I already had a good idea going in what I wanted to do. The big idea was to understand that "fractions, decimals and percents are just different ways of representing numbers" (you can see that on the top-right, in yellow), and that the same number can always be represented (at least) three different ways.

The student work you see on the board is for two problems, "Nana's Lemonade" and "Dueling Discounts", created by Dan Meyer. For an overview of how I structured this unit, see the post in its original context. At

### 2. From Patterning to Algebra

This unit was based on the grade 8 Curriculum and Connections section, "Patterning and Algebraic Modelling". For the students, I named it "From Patterning to Algebra" because that was more meaningful to them. The big idea was to "Understand that pattern rules, tables of values and graphs are simply different ways of representing patterns" (in yellow at the top left). We created our success criteria together, and we added definitions as the unit progressed, to help students learn the terminology that goes with the ideas they were expressing. You can see how I structured that unit in this post. I also included the following evaluation in that unit.

## Distinguishing Between Skills (Computing) and Reason (Problem-Solving)

Now that we have a framework upon which to base our teaching, the next problem is, "What do we put in that framework?". This is where we get more into the art of teaching, rather than the science. I'll do my best to show you what I do, and hopefully some of that will be useful for you, but most probably, at least some of this won't work with your style of teaching, so please look at this with a grain of salt.

The Ontario mathematics curriculum identifies 7 important mathematical processes: (1) problem solving; (2) reasoning and proving; (3) reflecting; (4) selecting tools and computational strategies; (5) connecting; (6) representing; and (7) communicating. I find it difficult to address all of these processes using just one teaching strategy. I suspect that it would be difficult to reach every student in the classroom using just one teaching strategy as well.

There are two main types of activities that I use to structure student learning. The first is called "Number Talks", and is based on the book "Making Number Talks Matter" by Humphreys and Parker. The book is available on Amazon, but if you bug your school's numeracy coach enough, they might be able to find a copy for you. The second type of activity I like to use are "Three-Act Math" problems, which is (sort of) Dan Meyer's take on three-part math lessons.

Number Talks are 20-minute lessons that cover the skills or computing aspect of solving relatively straightforward problems. Essentially, I chose a problem that I hope will get students to come up with good strategies. Each successive Number Talk builds on the one before it, until students have developed a variety of effective and efficient ways to solve "skill" problems. I will do one or two examples with you during our workshop, but in the meantime, this is how I introduced Number Talks to my class this past year.

Number Talks have replaced the "exercises" portion of my math class. I used to spend a lot of time worrying about the exact best way to structure exercises so that students develop a variety of strategies, and are challenged but not overwhelmed (here is an example of a series exercises for exploring square roots and squares). Now, with Number Talks, I instead use what students have learned so far, and where I want them to go, to choose a problem that I think will help them develop the type of strategies that will help them with a particular skill.

Three-act math lessons are where students get to work in groups to solve complicated problems and develop reasoning, proving and reflecting skills. With good three-act math problems, students form a hypothesis, test the hypothesis, maybe solve the problem, and then reflect on their solution. Often, at the end, they will realize that their solution was inaccurate, partially accurate or reasonably precise, at which point they can reflect on the usefulness of their mathematical models. Over time, they often realize for themselves the statistics aphorism:

All mathematical models are wrong. Some are useful.

I'm going to do an example of a three-act math problem with you as well, and we'll talk about how to make them as effective as possible for your students.

## Trials and Errors

In no particular order, here are a couple of the questions that I am struggling with right now. I have elements of answers, but I don't have a coherent global theory that answers all of the problems.

• How to emphasize group work vs. individual work? (Should one count more than the other? How much time should students spend working individually or in teams?)
• How do you make sure that every student has learned from their work? (What kinds of activities help you determine what a student has, in fact, learned?)
• How do you evaluate students when you are constantly giving them feedback? (If you give them questions or next steps, how does that affect your grading of their work?)

## Overview and Comments

Here is what I hope to cover during this in-service:

• Introduction to "Curriculum and Connections" (large group)
• Choose an order in which to do the 6 units proposed in Curriculum and Connections (randomly generated groups of 2-3)
• Go through the planning process for one of those units:
• Do a Number Talk to teach the skills section of the problem (large group)
• Do a 3-act math problem to teach the problem-solving side of it (randomly generated groups of 2-3)
• Do an evaluation (exit ticket or math journal entry) to evaluate summatively (plan in groups of 2-3, do the problem individually)

I hope to add pictures of the work we did together to this post, so that we can refer back to it later. If you have questions or comments, please add them below.

## UPDATE: Pictures from the Workshop

Thanks for coming out, everyone! This was a lot of fun, and I learned a lot.