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.