Making Math Meaningful

An inquiry-based approach puts students in the position of mathematicians — where they make discoveries themselves.

At Packer, an inquiry-based math lesson is easily identified: students are clustered in small groups, talking openly as they work through various challenges together. The teacher moves constantly among them, listening to their conversations, offering feedback, and posing further questions. Around the room, each group’s progress is audible — a sigh signals a false start; a whoop signals success. This is the best of learning — authentic, lively, and occasionally loud.

From the Pre and Lower school through the Upper School, Packer’s math teachers emphasize inquiry-based learning — that is, they present their students with rich, investigative problems that can be approached through multiple strategies. Students work collaboratively to solve these problems, experimenting and building on one another’s ideas. They challenge and question one another’s thinking. Through this process — and the profound “aha moments” that it leads to — students deepen their understanding of math.

But practically speaking, how does inquiry-based math really work? Several members of the faculty gathered at the invitation of the Packer Magazine to share their experiences, insights, and convictions about the benefits of this approach. That conversation follows, with stops along the way in four math classrooms, from Third Grade through Twelfth Grade.

What are the essential features of inquiry-based learning?

Amy Hand, Math Department Chair: If we’re offering bullet points, I would say, Starting with interesting questions.

Paul McElfresh, Middle School Math Teacher: For me a really important piece is setting up a classroom community and a set of expectations about what math class looks like. That involves students taking risks and asking questions. Very early in the year, you have to disabuse kids of the notion that there’s always going to be one right answer and one right way to solve a problem.

Stella Liberman, Second Grade Associate Teacher: I would say that time for students to explore concepts is the most important part. When we talk about inquiry in the Lower School, what especially comes to mind are manipulatives: physical objects that represent non-physical ideas.

Ally Rohrbach, Middle School Math Teacher: To build on what Paul said, I think that the questions we pose need to allow students to solve problems in different ways, so they know that there’s not just one approach to the problem. Lessons need to be structured so that students have opportunities to discuss their thinking with their partner, group, or whole class, to make connections between different strategies and techniques. When students are able to make these connections and consider the relative efficiency of their strategies, their mathematical understanding and flexibility are deepened.

Sam Shah, Upper School Math Teacher: Right. Implicit in what everyone is saying is that kids are doing the heavy lifting in the classroom. The teacher is not the sole authority or “the sage on the stage.” The kids can see each other as resources. A lot of [what makes inquiry work] is that kids are working to discover things with each other.

Maria Stutt, Upper School Math Teacher: Often my biggest challenge is keeping my mouth shut — staying out of it and letting them go. There’s a tendency to want to jump in, but a few moments of silence is almost always rewarded with an outpouring of enthusiastic sharing. I am often amazed by how much I learn — what my students are thinking and why, or new ways to explain concepts and reach more students. Sometimes I even learn new approaches to solving a problem!

Chris Natale, K-6 Math Coach and Coordinator: Right — and when I do open my mouth, it’s more about asking questions than it is giving information. As Amy and Ally said, it’s a lot about asking the right questions.

Ian Rumsey, Upper School Math Teacher: There’s a moment for most students in my class when they go beyond the question I pose by posing their own question. When they begin to take that additional step, it starts to click.

Brendan Kinnell, Upper School Math Teacher: Having enough time is another key component of inquiry. You need to devote substantial time to let the ideas simmer, marinate, and develop for the kids. It’s a big investment.  Our new schedule helps. The 90-minute block [which occurs once each seven-day cycle] gives students enough time to fully understand an engaging problem, wrestle with different approaches, and implement more involved problem-solving plans.

Elisha Li, Third Grade Head Teacher: This is true at all grade levels. It’s essential to allow time for ideas to linger, grow and change. It doesn’t happen in one lesson! As the facilitator of learning in the classroom, I feel it’s also key to observe students’ thinking in order to craft the next learning experience. Even with an established curriculum, it’s ultimately up to the teacher to decide how to frame each lesson so that it fits with the class’s development of mathematical ideas. It’s very engaging for students when their ideas are front and center, and each lesson builds off of them.

How do you help create an “environment of inquiry” for your students?

Chris Natale: Some problems don’t have a single right answer, but in fact plenty of problems do. So we need to help kids be increasingly comfortable with being wrong, or with not knowing. As a student, if your expectation is that you’re always going to be right, you’re never going to take those risks to stretch into new territory. Generally speaking, kids don’t want to be seen as wrong, or not smart, and may hold back on things about which they’re unsure. What we do to make kids comfortable with inquiry in Kindergarten through Sixth Grade is really the seed that grows for them in Middle and Upper Schools.

Ian Rumsey: Inquiry involves some risk and some personal vulnerability that kids are not necessarily comfortable with, initially. As a teacher, I think it’s important to sublimate your ego, to get rid of your own personal attachment to being right, to be able to say to a student “I don’t know, but I’d love to find out!” Modeling that willingness is really important. Kids pick up on it and become more comfortable saying, “I don’t know — let’s find out!”

In Elisha Li and Jennifer McConnell’s Third Grade class, students develop multiplication skills through hands-on investigation. In “the window-washer problem,” they must figure out the number of window panes that need cleaning — but the windows are obscured. Students solve the problem using a range of techniques. Ms. Li calls on students to share their methods and together the class explores each approach. Then students pair up at desks and on the floor to play a multiplication game in which they race to fill a 10x10 array with combinations of smaller arrays.

Elisha Li: Something I do is model my own curiosity as a learner. I’ll think aloud about how I’m thinking about problem, or I’ll make mistakes and try out different answers.

Another way is to create a culture of growing. I have a motto for myself as a teacher and for my students: Progress, not perfection. We have a class expectation in my Third Grade this year: “Try hard things and be okay making tweaks!” This has really helped create a culture of learning, trying, and even making mistakes. Some educators call this the “growth mindset.” It’s hard to make mistakes at first, but we try to normalize it as a part of learning. When we solve a problem like 8x5, I will often say, “Seems like many students got the answer of 40. Now, let’s get to the interesting part: How did you find the answer?”

Brendan Kinnell: As teachers, when we engage in inquiry ourselves, we are thinking mathematically “in the wild.” We don’t know which tools we’ll need to use or which representations might be best. In these moments we are acting as we hope our students will act: creating, trying, adapting, etc. It reminds us first-hand of the value in that process.

For instance, [Math Chair] Amy Hand recently walked into the Upper School math office and said, “Have you guys heard this incredible problem?”* At first we [all thought], “I don’t have time to solve that!” But in the back of our minds we knew we would become obsessed with it. Conversations about the problem started, then they started to trickle to the Computer Science department, and then one person found the solution — and another, and another.

Sam Shah: And everyone’s path to the solution was different!

Brendan Kinnell: Exactly. As professionals who are committed to trying to understand a problem deeply, we really appreciate when a student takes a wrong path but learns something.

Ally Rohrbach: Just today a few of us in the Middle School were talking about putting on our “student thinking caps” and trying to solve problems in the different ways our students might. Even though we know how to solve a simple Seventh Grade math problem, thinking about it deeply and searching for other ways to approach it allows us to connect that problem to their prior knowledge and build upon it.

Dawn Knight’s Sixth Grade students are learning how to divide a number by a fraction. Rather than introduce the algorithm that will unlock this mystery for them, she presents increasingly challenging scenarios — from splitting a half-pound bag of nuts among four people to measuring 3½ cups of flour using a ⅓ -cup measure.

In pairs, the students try different ideas and tools to solve these problems. They draw diagrams, subdivide them, and count the sections. Eventually they spot a relationship among the numbers that suggests an algorithm for dividing by a fraction. By the end of class, everyone has had the “aha moment” that helps lay the foundation for deep understanding.

Sam Shah: When we’re designing lessons, especially collaboratively, one of the most interesting things is what happens when we go to the top-most view and ask, “What’s important? What’s the macro-level question we want kids to understand?” Planning our units, we start at that top level. Ian and I started to see all the ideas in one combinatorics unit emerging from a single organizing principle. Even after a math degree and a decade in the classroom, my own perspective on combinatorics completely changed! I was in awe of the framework we uncovered when we looked for “the big picture.” And because of that feeling, I was totally psyched to see how kids would interact with this material in the ways we were thinking about. I couldn’t wait to see them make the same discoveries we did!

Paul McElfresh: Something we talk about in Middle School curriculum design is breaking problems down into parts: not just the solution, but the approach, the process, the reflection. We work at giving our students the vocabulary of, “When I went into this problem, here’s what I was considering.” Or, “Here are some possible strategies.”

Another important part of inquiry, as Ian mentioned, is what lies beyond the solution: “How can I take what I did here and apply it so that I can further deepen my understanding?”

Stella Liberman:  The whole time we allot for sharing in math, the kids talk about their approaches: “What strategy did you use?” “Was that efficient?” “Oh, that wasn’t really super efficient, but I liked that you tried that.” And that’s Second Graders talking to each other!

So we don’t teach solutions as much as we teach the process of thinking about the numbers. We focus on how you got there. It seems as if that’s our main focus in our math curriculum: the strategy to an answer, not solely the answer itself.

In an inquiry-based approach to math, why is student collaboration so essential?

Chris Natale: I think collaboration is basically the essence of learning. Whether it’s math or anything else, learning is social. The way we learn and process things is through communicating with them. Whether we write words, or draw symbols, or create diagrams, we’re refining our thinking when we do that.

Amy Hand: There are different stages of inquiry, and not all of them must be collaborative to work well. We certainly allow kids to think deeply on their own when it’s appropriate — for instance, before they start collaborating with each other.

But I would second what Chris just said. Many of us have has some version of this exchange with students: “Do you understand this?” “Yes!” “Can you explain it?” “I understand it, but I can’t explain it.” Well, if they can’t explain it to somebody, they probably don’t really understand it. And so requiring students to work together in small groups is a way, as Chris said, for them not just to share their understanding, but to further cement and develop their understanding.
A lot us read Jo Boaler’s Mathematical Mindsets last summer; she writes about the importance of figuring out how to convince a skeptic. [What we regularly ask] kids is, “How would you put together a tight, convincing mathematical argument for what you know to be true?”

Elisha Li: Student collaboration also trains students to listen to each other. Listening is actually very hard. When students talk, listen, and compare ideas, they are matching up their own idea to another’s and then making a cognitive choice to either incorporate it into their thinking or not. They are allowed to disagree with another’s idea, but it’s actually not possible to disagree unless you listen first!

At the same time, I try to allow for variations in our partner work to suit students’ work styles. I know I wouldn’t want to talk the entire time in a math class. I benefit from thinking in my mind first, while other people benefit from talking out their process and figuring out what to do together.

Ian Rumsey: I think of it as one’s radius of influence as you’re working on a problem: How many people are watching what I’m doing and paying attention to what I am doing? As teachers it’s very easy for us to project our radius of influence. Maria touched on this. What’s hard is letting that go and allowing the students to bring people into their orbit.

Let’s say one student has an insight and he or she start to share it with another student. Then as they’re working on it, somebody else comes and says, “I don’t see how you got that.” The more the first student has to defend an idea to the rest of his or her group and refine it through really concise exposition, the more that collaboration leads to deep understanding. Without people around you to draw into your radius, you don’t have that opportunity [to refine through explanation] and your understanding isn’t as strong.

Sam Shah: That happened recently in my Eleventh Grade Pre-Calculus classes. I asked kids to write an equation to represent a sequence of numbers that follow a particular rule. Their approach was much better than the way I had been thinking of it, so much more efficient and so much more elegant! The idea propagated independently in both of my classes. And I always try to credit the kid who has the idea: That’s Max’s idea or That’s Destin’s idea. No one is handing them the discovery: they make the discovery themselves. They have ownership of it and are proud that other students end up using and citing it.

And in terms of the sense of risk we were talking about before, I think student collaboration actually lowers the risk of being wrong. The people [in their small groups] are their peers. They’re not speaking publicly in a big, scary space. It’s comfortable. It’s more social. It’s a lot more fun to be talking about things with people. It’s true that ideas build off each other. Someone might say: ”Wait — yes! Then we can do this!” It’s cool when that happens.

Another thing that’s really great about inquiry is that each group gets to move at its own pace. It’s not, “Now let’s all do problem 7. Now everyone move to on problem 8.” They get to take more time on the things they need more time with. As a teacher, you facilitate that on a class-wide scale.

For students who struggle in math, does an inquiry-based approach make learning harder?

Amy Hand: I would argue any day that inquiry-based learning is even more critical for students who struggle. Stronger students have the ability to make connections themselves, no matter the manner in which they’re taught. With other students, it’s especially critical that they’re able to attach the mathematical ideas being introduced to their intuition, and then exploit their mathematical intuition — to attach meaning to a context that they’re familiar with — and have the math emerge from there.

You [have to] start with a question that’s accessible. It connects to something they’ve already learned or an experience they’ve had, from which they can build on their own understanding. I think the best inquiry-based learning does that really well.

Brendan Kinnell presents a challenge to his Ninth Grade Geometry Advanced students: find the center of rotation between two points, using translucent slips of paper and a compass. For several minutes, no one succeeds, but the playful process of tinkering with these tools sits comfortably with them. Mr. Kinnell circulates among the groups, asking if they have a theory. “We did, and we were collectively incorrect,” chuckles one student, undiscouraged.

Then one group finds one center of rotation. Another thinks they have found three. Then the breakthrough: “There are infinite centers of rotation, and they’re all on the perpendicular bisector!”

Ian Rumsey: I’ve found in some cases students who have struggled historically actually do better with an inquiry-based approach than students who are used to success. There’s a struggle and frustration inherent in inquiry — running up against the wall repeatedly — and some students are more accustomed to having to persevere through a lack of understanding. They are often quicker to adjust to inquiry-based learning, whereas students who are super capable and rarely experience that struggle find that the adjustment to not knowing is unfamiliar and initially uncomfortable.

What are some of your greatest moments teaching math with an inquiry-based approach?

Amy Hand: Last year in Sixth Grade, we implemented a fantastic new curriculum called Connected Math. And one student, as he was leaving my classroom one day, spontaneously said, “I like this class because we don’t learn about math, we do math.” That was the best thing I had ever heard about a class.

Ian Rumsey: My personal favorite moment was being totally wrong about something. I had asked my class to find the line tangent to a conic section, and one student presented a vector-based solution that — well, I don’t know where it came from! He played with it and he tinkered with it until he realized, “Oh, if I make these two things vectors, the whole thing falls into place.”
I looked at the solution and said, “That can’t possibly be right.” And the whole class rallied around him, saying, “But it works!” And then together they investigated why it worked and proceeded to prove this theorem that I had never seen before. I don’t think I have ever seen it again. It was brilliant! It was so creative — and it all began with my being wrong. They were practicing that skepticism Amy was talking about — towards me. It was amazing.

Paul McElfresh: Early this year, we had a project where students had to come up with all the possible rectangular prisms that would meet a set of criteria. They approached the problem in all these different ways and made posters to share their solutions. At first, they were so focused on their own solutions and which one was “the best.” But through the process of sharing their strategies, the students realized that by borrowing from one another’s ideas, they could begin to piece them together and actually create the solution, “Transformer”-style [Laughter.]
So they collectively realized, “There is some benefit to listening to one another and not just saying our own ideas.” That was great.

Stella Liberman: We do this one unit called Parcels and Presents that even kids who aren’t very “math happy” get so excited about. They create their own number stories based on images they’ve made. And they do them during choice time, which in the Lower School is their unstructured playtime. It’s great seeing how much they want to do it, and it has nothing to do with me! I just want to make sure their math is correct, and I get them to ask another student. It’s perfect!

Sam Shah: The year before last, there was one unit in Geometry — the cross-chord theorem for circles — that was so boring. In general we try not to give the kids too much, but in this case we sort of led the kids to the conclusion. And we weren’t happy with it because it didn’t feel like natural discovery.
Last year we completely revised the lesson so that the students would discover this theorem themselves. We didn’t want anyone to look it up, so we created a mathematical creature called a Blermion. It was just a circle with a quadrilateral inscribed in it, and from there you can actually come up with the theorem.
At the end of the year, we wanted to test their habits of mind so we gave them Merblions, where you have a circle and you have to put a quadrilateral outside of it, touching all four sides. I just let them loose for two class periods to discover everything they could about the Merblion. They made little Google presentations that they were really excited about. One group would present a problem and another group would come up with a solution. It was just really cool.

In Sam Shah’s Multivariable Calculus class — the equivalent of a second-year college course — eight seniors make the challenging leap to working with figures in three and four dimensions. Mr. Shah asks them to relate a topographical map to the level curves of  $$z=\sqrt{1−x^{2}−y^{2}}$$. They debate: Does the equation describe “the top half of a grapefruit” or merely “the skin of the top half — if the skin were infinitesimally thin”? [Answer: the latter, a dome.] In pairs, the students then calculate the figure’s level curves based on different values of z.

Several times a month, their inquiries into advanced calculus extend beyond pure math: the students also lead discussions about thought-provoking books such as The Calculus of Friendship and A Mathematician’s Apology.

Does inquiry-based math make better thinkers across disciplines?

Chris Natale: I do feel that our teaching in math somehow transcends all the content; it goes back to those habits of mind again. So being persistent, or coming up with a cogent, clear argument about why you believe what you believe — those are things you do in math and they transcend to other subjects. So the idea of geometry and number work — that stuff is all important but it’s not the only thing.

Stella Liberman: I just think that math helps students see patterns and also helps them explain their thinking. In Second Grade, those things often are relatively easy to do in math, but not in other disciplines. But the skills translate: if the kids are not understanding another concept, or are confused about something in reading or writing, they can ask questions, which is something they do naturally in inquiry-based math.

Paul McElfresh: By engaging in this practice of tinkering and thinking critically, young people start to become creators of things, not just consumers. As they get older, they’re going to have to identify problems and come up with solutions.
Inquiry prepares them for this. In the inquiry process, they learn to explore a situation and all of its components. They have to communicate and collaborate with one another. They have to come up with possible ideas, and then tweak and evaluate, and then arrive at a solution and test it. They’ll need these skills more and more in terms of addressing whatever personal, professional, academic, or social-emotional dilemmas they’re going to come across. I think that we’re giving them the tools to be patient and resilient in that process and to participate actively in it.

Elisha Li: The ultimate goal is to help students understand their learning process and motivate them to learn more. I see so many parallels in learning, no matter the subject. Ultimately, when students are curious and motivated, they tend to ask questions, or wonder about the world around them, or research to learn more, or try out different ideas and solutions. This is true in math as much as it is in reading and writing.

Amy Hand: I see inquiry-based learning generally manifest in the confidence that kids have approaching unfamiliar problems. It’s impossible for me to imagine that their confidence in approaching new challenges doesn’t translate across disciplines.

I can throw any kind of math problem at them now — even ones that I know they don’t yet have the tools to solve. And they’re trained. They look at it. They figure out what they can figure out. They start playing around. They draw a picture if they can. They use numerical examples if they can. They look for a pattern if it’s appropriate.

They have such confidence approaching new tough work, and I think that comes from the approach we’ve used the last few years.

The Dennis Years: A Visual Timeline

Click image above to view key dates during Dr. Bruce L. Dennis's 15-year tenure as Head of School at Packer.