How Middle Schoolers Learn Math at LWS

A conversation with middle school teacher Gary McSweeney at Living Wisdom School in Palo Alto. All of the photos were taken during a single day’s one-hour class, except for the photo of volunteer teacher Richard Fouquet.


Q: Please explain how you teach math in middle school at LWS.

A: Our program is unique, in that it enables each child to go through the book at their own pace. If they have lots of talent and they’re motivated, they can go as fast as they want, without being held back. They can do problems, get practice, become thoroughly grounded in concepts, and take a test that demonstrates their proficiency.

Middle school students do recharging exercises before math class.

That’s the short of it. The process works because we’re focused on the individual and we have the resources to help each child individually. During math class, we have at least two teachers present in the classroom at all times.

Besides our math team of Eric, Leslie, Richard, and me, some of the older students also work peer-to-peer with each other. At that age, they sometimes prefer to learn from another student rather than the teacher. It can be very helpful to the student who’s being taught, because they can sometimes grasp a concept more easily when another student explains it. It also helps the helper, because making concepts clear and expressing them to another student reinforces their own learning.

We’re constantly testing to make sure the students have a firm grasp on concepts, whether they’re advanced, average, or having trouble with math. After they take a test, we correct it, score it, and enter the results in a math diary where we note any concepts they’re struggling with, whether it’s percentages, ratios, proportions, fractions, the Pythagorean Theorem, or shapes, etc.

Students at Living Wisdom School in Palo Alto, California meditate before math class.
…followed by meditation to focus and calm the mind. (Click to enlarge.)

We note their weak areas, and we hand the tests back and say, “Try to correct the ones you got wrong, without any explanation from us.” They go back and look at the problems, and at this point all they know is that they got the question wrong, but they don’t know why. They try to solve it on their own by consulting their resources, and if they can self-correct, they bring the test back and we enter the new score. Then we go over every single problem they got wrong, and we explain it until the concepts are clear and they understand the correct approach.

That’s the system in a nutshell. But a key point is that there’s a great deal of self-direction. And it can raise questions for parents, because it can seem a bit chaotic initially. But if you understand how it works, you can see that it’s very focused and directed.

It’s also super-efficient. In a traditional math class, the teacher would present one or two concepts in a lecture, and then they would give the students a few problems, and they would repeat the process every day of the school year.

In our math classes, we don’t have lectures. So it might appear that we aren’t giving them, quote unquote, “direct instruction.” But when I mentioned this to Richard Fouquet, who’s one of our most experienced mentors, he said, “But that’s all we do. It’s all direct instruction.” Because although it’s not in the form of a lecture, it’s constant, daily, one-on-one tutoring with every student.

Our system works extremely well once the students get used to the classroom atmosphere and understand that it’s their initiative that matters. And it’s our job as teachers to inspire them to show initiative. But, again, the major selling point is that it’s very efficient. The kids aren’t forced to do hundreds of problems that they already know how to do, because they’re always working at the forward edge of their learning. They’re working on the areas where they’re a little weak, as opposed to “I know percentages already. Why are we spending two days on percentages?”

Volunteer math teacher Richard Fouquet helps a student at Living Wisdom School in Palo Alto, California.
All math instruction at LWS is individual. Volunteer Richard Fouquet helps Jaydie with a difficult concept. (Click to enlarge.)

But it does require initiative and effort. And that’s the job of the teacher, to awaken initiative in each child. That’s where the art of teaching comes in. You see that such-and-such student isn’t trying, and the math diary confirms that they haven’t completed a test in a while, and so you have to find out what will motivate him.

All of the students take national tests that tell us how well they’re doing in competition with a very broad group of students worldwide. We don’t do standardized testing in our school, but we take the Math Olympiads and the AMC 8 and 10. A further test of our math program is the high school entrance exams. And, on the whole, they do very well in all of these tests.

I understand how our system can raise questions for parents, because it’s very unlike what most of us grew up with. Parents sometimes ask us, “If the system depends on the students being highly motivated, how can you guarantee that all of them will get through the material? Won’t the students who aren’t talented and engaged tend to fall behind?” But after thirteen years teaching in our system, I can say that it works extremely well, because our student-teacher ratio is about six or seven to one, and we’ve learned to motivate a very broad range of students with unique talents and personalities.

Q: When I was in the classroom, the two teachers who were present spent all their time helping the students. They didn’t need to spend time on discipline issues. Does that suggest that the students are, in fact, motivated?

A: The freedom to progress at their own pace serves two purposes. As I mentioned earlier, some kids at this age prefer to learn from a friend. Their friends may have a unique way of explaining things because they share ways of talking and thinking, and because they’re friends.

There’s more than one way to get the right answer. We try to find the way the child learns most readily and that they’re comfortable with. If they want to learn from another student, the danger is that the concept might be explained incorrectly, or the student who’s playing the “teacher” won’t have a depth of understanding. But the defects of the system are minor and they tend to get ironed-out when we correct the tests and go over every problem with them. We explain the concepts and we ask them to demonstrate their awareness of the concepts.

Math at Living Wisdom School in Palo Alto, California.
Students can come and ask Gary or the other teachers for help. (Click to enlarge.)

Our curriculum is approximately sixty-percent review, so while they’re constantly tackling new material, they’re also reviewing what they’ve learned. If there’s a hole in their learning, the math diary tells me exactly which issues we need to address.

A second major benefit of the system is that it develops a wonderful atmosphere in the classroom. It’s a group energy that is competitive without being cutthroat. The kids are keenly aware of where everyone is in the book, so when they see a younger student going fast, it motivates them. And because they’re helping each other, it develops an enthusiasm for the subject, because it’s tied in with their natural urge to socialize and help their friends.

In any class, you’ll always have some solid mathematicians, and you’ll have students who say “Yeah, I’ll do it.” Then you’ll have students who are on the edge. “I don’t see the point.” “Math isn’t my thing.” Our system creates a magnetism that tends to draw in all of the kids, even if they aren’t comfortable with math initially.

At the start of summer break, I said, “Who wants to take home a math packet?” I didn’t put any pressure on them, but virtually all of them grabbed tests to work on over the summer. I’ll know in the fall how they did, but a lot of them will work on math on their own, because they enjoy it, they share the enthusiasm, and they like the challenge.

It’s a unique system, but it’s probably not as unique as it once was. Public schools in the area are starting to use the Kahn Academy system, which is a little bit like what we’re doing, except that we were doing years before Kahn.

Our system evolved with the help of a previous math and science teacher, Dharamaraj Iyer, who had an advanced degree from MIT and was a brilliant teacher. It was born of the fact that we have sixth, seventh, and eighth-graders in the same classroom. But the truth is that even if I had all seventh or eighth graders, we would always have a range of aptitudes for math.

Frey studies the mathematics of finance.
Advanced student Freya studies the mathematics of finance. (Click to enlarge.)

Some are passionate, some will enjoy it without being brilliant mathematicians, and some will struggle. And I’ve seen how our system serves all of them. It certainly has served the highly motivated, gifted math students. But I’ve seen that it serves all three populations beautifully.

For the gifted students, we’ll get whatever textbook they need. We aren’t wedded to a single text or approach. If you want to serve the individual child, you have to individualize the instruction.

Q: As I walked about the classroom, I looked over the shoulder of a sixth-grader who had her book open to a chapter on the mathematics of finance.

A: Freya is a bit of a prodigy. She aced the AMC 8, the Olympiad E, and the Olympiad M. She got 100 percent on all three of these very difficult tests. Her AMC 8 score was the top score among sixth-graders in California. There are over 153,000 students who take the AMC 8 globally, and she was one of just 32 girls with a perfect score among the 67,544 girls who took the test worldwide. And only the top math students in a school take the AMC. It’s not as though we can take credit for her results, because it’s obviously the fruit of her talent and her hard work. But we are able to nurture the exceptionally gifted child.

Our kids, by and large, do very well across the range of abilities. We have lots of kids who scored in the 50th to 98th percentile. We offer the tests to all the grades, and we have some third graders who scored above the 50th percentile. These are exceptional results, because the average score for all students is 10.67 of 25 possible correct answers, and primarily middle schoolers take the test.

These tests, the Olympiad in particular, challenge the kids with creative word problems, where it can be a challenge to understand what they’re asking for. They’re out-of-the-box problems. And we don’t do any special prep for those tests. We teach our regular math curriculum and allow the kids to apply what they’re learning in class. Many of the kids they’re competing against spend time mastering the Olympiad and AMC approach. Freya, the student who aced all three tests, definitely worked at preparing on her own, because it’s a timed test, and you’ve got to have a strategy if you want to finish the 25 questions in 40 minutes.

Q: Is Freya enthusiastic about math?

A: She loves math. Over the years, we’ve had many students who just seemed to be born to do math. We had a boy who was doing AP (advanced-placement) calculus in eighth grade. It’s unusual, but there’s an amazing group of mathematicians in this area.

Q: Due to the Silicon Valley gene pool?

Teacher Eric helps Luke understand a math concept at Living Wisdom School in Palo Alto, California.
Math team teacher Eric helps Luke with a difficult problem. (Click to enlarge.)

A: Yes, and there are lots of challenges for them. There’s the California MathCounts competition, the Math Olympiads, the AMC 8 and 10, and AIME (American Invitational Mathematics Examination). There are also many instruction options, including the EPGY (Education Program for Gifted Youth) at Stanford for advanced math students. The kids meet once a week to talk about math with Stanford professors. The kids are invited, and Freya is part of that program. There’s very strong interest in math in this area, as you can imagine, and some of the parents are gifted in math.

Then there’s the other end of the spectrum. One of our girls didn’t get a single question right on last year’s Olympiad, but then a very interesting thing happened. This year she got 14 questions right out of 25. Which was an absolute miracle, and a very respectable score, well above average, in the top 25% of students, on this difficult test. Timed tests were definitely not her strength, and she had a lot of trouble with confusing word problems. But she began to build a momentum, to the point where she got 14 out of 25 right. It was remarkable, and it happened because of the classroom environment where she could move at her own pace and begin to get a spark of inspiration and realize, “I can do math, too.”

She’s not anti-math, but the truth is that whereas you have these kids who just live for math, there are many who don’t understand the relevance, or they don’t have any interest at all. We have kids who come here hating math, possibly because they’ve been traumatized at other schools with big classes where the system rigidly expected them to keep up or be flunked if they couldn’t.

I’ve seen kids start the year in our math program with their shoulders hunched and tense, and then they start to relax, let go of their hatred, and begin to think, “Well, it’s not that bad, and I can at least finish the book and stay at grade level.” Sometimes they’ll even start to feel, “This is interesting. This is fun. This has relevance.”

A student raises her hand for help in math class at Living Wisdom School in Palo Alto.
Elizabeth asks for help. (Click to enlarge.)

Q: For the girl who improved so much, did she have hundreds of hours of extra tutoring? Or did her progress come as part of the regular classroom curriculum?

A: She had zero tutoring. It was purely from the classroom. I bring up her case because it’s often only the prodigies who get mentioned with glowing praise. In this area, it’s really quite amazing what the kids are doing. Calculus in eighth grade is not unheard-of. But this girl was someone for whom math was completely foreign. She had no motivation. Zero. And being in a classroom where the atmosphere is about encouraging the individual and sharing with peers is what truly helped her.

When you’re in a classroom where math is seen as fun, and there’s a ton of individual attention, it creates a soil where all of the kids can be challenged and thrive at their own level. There’s a structure, with the math diary, that tells us exactly where each student is – because we aren’t daft. But there’s a supportive atmosphere, as opposed to an intimidating one, where we’re always saying, “I’ll help you. Or would you rather have your friend help?” “Oh, I’d like to have my buddy help me.” So another student will come over and explain a concept. She’s quite smart, it’s just that the way she processes numbers is really interesting.

Eric helps Elizabeth in math class at Living Wisdom School in Palo Alto.
…and Eric responds. (Click to enlarge.)

It’s all about individualizing instruction. I have another student who’s very smart, but numbers are a big challenge, and I’m developing a strategy for him. I’ve worked with him for a year, because it can take a while, honestly. But after about a year we started to see some progress, and I’m hoping that next year we’ll pick up where we left off.

For some people, the logic of numbers is a foreign language. One of the textbooks we use is from the CPM organization (College Preparatory Mathematics). It approaches each concept in a variety of ways so that a broad range of students can find the approach that works best for them. We also make use of some excellent algebra tiles that can help some of the kids. You try to come up with a way of explaining it that they can grasp, so they’ll catch the spark and really start rolling.

Sometimes it’s deeply entrenched. “I don’t like math, and I’m not doing it, period!” It can take days, weeks, months, even years to get them to open up and say, “Well, I’ll try.” It might happen because of a peer’s influence, or some special technique. But we’re open to all approaches. We do whatever works for the individual child, whether it’s Kahn Academy, a special textbook, or something else.

Q: I observed you talking to a boy who’d gotten lazy in math so that his grades were slipping. You told him he had great potential, and that he could excel if he would work as hard as he’d done formerly. What was that about?

Gary tests a student’s grasp of concepts at Living Wisdom School in Palo Alto, CA.
Gary tests a student’s grasp of concepts. (Click to enlarge.)

A: Those are interesting cases. Education for Life, the book that describes our philosophy, asks, “Why do we teach algebra?” Because there are always kids who want to know. The author explains that it’s about abstract thinking. It’s about logic. So it’s not just about math as a skill for playing with numbers. It’s about learning skills that will help them in many areas.

Most kids nowadays have endless distractions – computer games, Minecraft, and so on. Math is a long way down the list. So it’s about finding ways to motivate them and get them to do the work. Because if you aren’t working, you’re never going to develop an interest.

I studied French for fourteen years, and I can barely speak a word, because I resisted it with all my might. Motivating the students is a big part of any good math program. The toughest cases are always the kids who have no initiative. And I feel it’s my job to find out which program will work for them.

Because we don’t give direct instruction with lectures, and because we don’t give homework that’s based on the lecture of the day, it’s true that some kids can start to fall through the cracks. That’s why we adopted the daily math diary, because it tells me exactly how much work the student is doing. And I can immediately take steps to get them back on track.

Sometimes I’ll encourage the parents to get involved. For some kids, that’s been the missing ingredient. I’ll tell the mom or dad, “He isn’t doing enough math.” Sometimes they’ll have to get an outside tutor. Sometimes all it takes is getting the parent involved enough to say, “You have math homework tonight. You have twenty minutes to do it. Let’s get it done. Let me see it when you’re finished.”

Q: Last time we talked, you mentioned a program where the students can do their homework during school hours.

Students in math class at Living Wisdom School in Palo Alto.
Student share a laugh while continuing to work on math. (Click to enlarge.)

A: In middle school it’s a big issue for kids to get their homework done. The kids can stay in the classroom two days a week from 3 to 5 and finish their homework. It’s included in the tuition, and I supervise. The only rule is that you must be doing homework. It can be science, history, language arts, math, or art, but it has to be a homework assignment. The students like it and feel it’s successful.

The different styles of learning are very apparent. Some are quiet and focused, patiently going from assignment to assignment. Many of them move, talk, socialize, and have several things going on, but they feel they’re getting their work done. It works, but there are definitely a variety of approaches to learning.

Q: Helen told me it’s popular with the parents.

A: In the middle-school years, all of the brain development is in the back part of the brain, and the development of the frontal areas, where the discriminative faculties are localized, goes dormant. So they aren’t always weighing their actions and the consequences – what if I come to school without my homework?

At this age, organization is a big issue for many of them – they’ll get their homework done and forget to put it in their folder. Study hall tries to address some of that. It helps them develop good study skills, and if they do their homework at school they can leave it here. And it’s a blessing for the parents to be able to come home and have dinner with the child without nagging: “Did you do your homework?”

Q: When I visited math class, there were always two teachers. Is that usual?

A: On Monday through Friday there are two of us, Eric Munro and I. Eric is a parent of two boys who graduated from LWS. He has a degree in electrical engineering from MIT, and he’s retired from the tech industry. Richard Fouquet comes in on Monday, Tuesday, and Wednesday. Richard went to Stanford and Harvard and has advanced degrees. In the early days of Silicon Valley he worked with Robert Noyes, who co-founded Fairchild Semiconductor and Intel Corporation. Richard has been coming in for ten or twelve years as a volunteer. Then Leslie Peters usually comes in once a week, and she has many years of experience as a math teacher. So that’s three of us three days a week, two of us two days a week, and one day a week with a different second person. So it works out to a very good student-teacher ratio.

The truth is that the teacher builds a relationship with the individual child – so they know when to motivate them and how, when to give them some slack, what their weak areas are, and they are patient enough to work them over the long term.

Charlotte works with quiet focus on a math problem at Living Wisdom School.
Charlotte works with quiet focus on a math problem at Living Wisdom School. (Click to enlarge.)

Richard is one of the most patient people I’ve met. He has what I would call the old-school approach. There’s lots of blarney going around about math instruction and academics these days. But there are really no shortcuts except hard work. Malcolm Gladwell’s book Outliers looks at what makes high-achieving people successful, and it confirms the importance of sheer hard work. In our class, Freya is truly gifted at math, but she also works very hard to prep for those tests. Last year, we had another very gifted student, Sahanna. Her dad told me, “It’s true, she is smart, but she also works very hard to prepare for these competitive tests.”

So the kids, lo and behold, who apply themselves and work hard, do well. What a shock! And sometimes the most painful thing for a teacher is to see kids who have potential and slough it off. Math comes easy, so they’re content to enjoy a measure of success without doing more. But the kids who work hard do very well.

Q: As I observed in the classroom, it seemed that any student who wanted help could walk over and sit at your table, or they could approach Eric or Richard and get help, and they felt comfortable doing so. Several boys were working at the round table where you were sitting, and they were comfortable sitting with you and being themselves.

A: That’s another key thing, to build an openness. And sometimes you have to be proactive about it. The math diary tells us that some kids will sit in class and not ask for help. I’ll check the math diary and realize that I haven’t heard from so-and-so in a while. So I’ll talk to them. “You’ve been working on chapter seven for quite a while. Please ask for help. Richard’s here, Eric’s here, Leslie’s here, I’m here. Tell me who you want to work with.”

On the surface, it can look a little chaotic, but that’s just because it’s all individualized, direct instruction. And what’s really happening is that a lot of math is getting done. On the days when you were here, you probably noticed that some of the girls were talking during math. They were sitting at a table by themselves and chatting. There’s the old model where you’re supposed to sit down, be absolutely silent, and get your work done. And it can lead to a lot of resistance and sneakiness. Dharmaraj Iyer, our former math and science teacher, who trained me, told me that he’d been told, and it seemed true to him, that if you let them do a little socializing they’ll get plenty of work done.

So you have to be a bit intuitive. Is it all chatter? Or is there chatter, and they’re learning math? Simple observation can tell us what’s going on, and the math diary tells us how they are progressing.

Eric watches as a student demonstrates her grasp of a problem-solving approach, at Living Wisdom School in Palo Alto.
Eric watches as a student demonstrates her grasp of a problem-solving approach. (Click to enlarge.)

One of the boys this year had lots of parental support, and he just caught fire. He did five chapters the first semester and twelve the second semester, with very high marks. We aren’t just concerned with passing kids along to the next grade by having them take tests and crank out work. They have to genuinely understand the material. At any rate, this boy really caught fire, and it was no secret how it happened. The parents hired a math tutor, and it helped, but they were also very supportive. When they realized he was staying up all night playing computer games, they got rid of that problem, and his scores just took off. He became highly motivated, because when you’re successful at something, lo and behold, you like to do it. And when you’re not successful, you don’t like to do it.

So we try to have every child experience success. It’s very important. Otherwise it’s just a slog for them.

Sometimes it will look as if a student isn’t doing much, even though they’re highly motivated and engaged. I have a student in art class who, whenever I say “Okay, let’s start,” will sit there for a long time without moving. And it took me a while to realize what was going on. It was a concern, because middle schoolers can fritter away lots of time. But I realized that when he starts an art project he has to get his inspiration. And there’s something similar that can happen in math. They’re working with heady stuff, and it can take a bit of “thinking time” to come to grips with a problem.

One summer, I went through all the textbooks, and I would get stuck from time to time, in part because the questions were sometimes poorly phrased, or they were complex and hard to understand. Sure enough, I would go for a walk, go for a hike, go meditate, and the next morning I’d see the answer.

We meditate before we go into math, and we do a series of energization exercises, and I honestly believe it has something to do with our math program’s success. But with a student like Freya, who’s highly motivated, I would give her a lot of slack. If I see her apparently not doing anything, I know she’s probably just finding her inspiration, or mulling over a problem.

Some others kids would do anything else but math, given a choice. So there has to be enough structure to bring them along until they can have some success and find their inspiration. But there has to be a focus. If they’re really starting to goof off, I’ll ring the bell, and if I ring it a third time, they lose their snack time. They rarely do, but it’s a reminder – we’re in math class, let’s talk about math.

Q: I stopped at a table and saw that there was chatting going on, but there was also math getting done. The kids would be deep in the book, or they would be writing or working a calculator, and maybe two students would talk a bit while the others worked, but I could see that the focus was on math, and that the chatting was secondary, a social break.

A: It’s definitely not my style. When I learn, I like total silence. But I’ve come to appreciate that not everyone learns the same way. For some kids, the social side helps their mind be engaged. There’s also a cooperative atmosphere that builds, where they help each other, and they can laugh and get their work done.

The peer-to-peer engagement is an invaluable aspect of our math program. We had a young class this year, where a lot of the students were new to the middle school. And it wasn’t until late into the year that they began to form their natural work groups. But it requires adult supervision – an adult who’ll come around and say “Are you guys getting any math done?”

Sometimes, if I don’t see any math being done, I’ll separate them. We have a girl who loves to socialize, and she actually thanks me when I help her get it under control. She’ll talk and laugh with the other kids and not get much done. But when she’s by herself you could almost say that she hyper-focuses. So that’s why I’m not always going to let her be with her friends.

Q: There’s a website called Study Hacks run by Cal Newport, an MIT professor who’s interested in how students can be successful in high school and college and have successful careers. He believes it’s less important to “follow your passion” than it is to work hard in a narrowly defined field until you master it, with the result that you end up enjoying it, and you can have a wonderful career.

A: The truth is that math is an essential life skill, and it’s an important part of the academic picture. So we don’t give the kids a choice – they all have to work hard at math. And it’s amazing the transformations that happen to the children who apply themselves. It’s not an option to ask an eleven-year-old “Do you want to do math?” Because they might say no. But we make it fun, and we encourage them and individualize it.

I think the Study Hacks guy is on to something. That was the premise of Outliers, that if you want to be highly successful and love what you’re doing, you’ve got to work hard for a long time.

Sierra accepts the challenge of a difficult math assignment. Enjoyment comes with hard work and mastery.
Sierra accepts the challenge of a difficult math assignment. Enjoyment comes with hard work and mastery. (Click to enlarge.)

Keith Devlin is a Stanford professor who’s best known as “The Math Guy” on NPR Radio. He’s written 32 books, and he’s very interested in math teaching. He spoke at our school, and he said that he didn’t like math when he was in elementary school and high school. But he was interested in biology, and a college professor told him he’d have to take math because it was a strong requirement in the field. So he reluctantly took a math class, and that’s when he discovered how much he loved math.

Someone says “I don’t really like music.” “Have you played it?” “No.” “I don’t like Indian food.” “Have you tried it?” “No.” “I don’t like to meditate.”

If you asked the average eleven-year-old if he wants to meditate at 8:30 in the morning, probably not. But we do it, and they see “Oh, this helps me, because I’m focused and I’m happier.”

As adults, we can pick and choose what we do with our lives, but these kids aren’t at that stage. So we don’t beat them up about math, but we try to awaken their appreciation for it, and a love of it.

Teacher Gary helps a student at Living Wisdom School.
Gary tutors a student in middle school math at LWS. Getting to know the individual student is the first step in helping them discover the joy of math. (Click to enlarge.)

When you’re successful, you’re more likely to enjoy what you’re doing. So we look for the areas where they can be successful and we can genuinely praise them. “Wow, you really get percentages.” “So you’ve got that going for you. Now can you help so-and-so?” As they help each other, they’re reviewing it for themselves. And when you can articulate it to someone, you’ve learned it for sure.