Problem-based learning can put students on the path to math success. In this post, we’ll dig a little deeper into what it is, what it’s not, and teachers’ role in putting it into action. 

You can read the first post in this series here.

Tackling real-world questions

In our previous post, we established that a problem-based math curriculum sets math students up for long-term success. We showed that lessons in a problem-based learning model introduce students to interesting and often real-world problems or tasks that require them to draw on background knowledge, previously learned content, and/or new information. 

Problem-based learning vs. teaching as presenting

With traditional show-and-tell pedagogy, the teacher describes the procedures and formulas to answer problems and then gives students an opportunity to practice what they’ve been shown. This model is very common—most middle school and high school math teachers report using it as their primary mode of instruction.

With this approach, instruction is focused on getting answers through isolated skills and processes, so many students fail to develop the conceptual foundations required for the math to make sense. This often means students don’t know when a piece of knowledge is useful to a new, novel problem.

While teachers may be able to make a given lesson fun (for example, by turning it into a game), the math in the lesson is often uninspiring. Students may remember the game, but forget the math.

The limits of telling students how to do things

The occasional use of direct instruction is not always a bad option for teachers. Not all concepts and skills require substantial inquiry in order to stick or make sense.

But there are limits to deploying direct instruction as the primary mode of teaching. They include the following:

1. For routine algorithmic problems such as calculating the sum of two multi-digit numbers, teaching has to involve a certain amount of telling—but just telling students how do something doesn’t set them up for success. Students remember algorithms better over the long term when those procedures are grounded in conceptual understanding. If students forget a procedure, conceptual understanding can help them recover it.
2. Not every problem is routine or has an algorithm. Word problems are a big part of math, and word problems aren’t routine. No algorithm can make solving them a mechanical process. Instead, students have to comprehend the situation and create equations or models that reflect the relationships presented in the problem.
3. In middle school math, algorithms become even less prevalent. As soon as rational numbers enter the scene, even a numerical calculation like -1(-1 – 1) has elements of strategy. 

Algebra often presents a student with choices, as when solving an equation like 3(x + 1) = 6. Will they begin by using the distributive property to rewrite the equation as 3x + 3 = 6? Or should they begin by dividing both sides of the original equation by 3 to obtain x + 1 = 2?

With problem-based pedagogy, choices about how to solve a word problem or which calculation strategy to pick can become learning moments. If some students do it one way while other students do it another way, both groups can learn by discussing how the two methods relate. 

How Amplify Math can help 

Amplify Math lessons help teachers cultivate and structure these student conversations. The program includes easy-to-follow instructional supports that make implementing a problem-based program more effective and enjoyable for both teachers and students. The lessons are designed to elicit creative thinking and get students collaborating. 

By working on problems that are intriguing, engaging, and relevant, students see how the math they are learning in class connects to their everyday lives. Students are placed into situations where they need to reason, collaborate, revise their thinking, and apply what they’ve learned.

Webinar series recap, part 1 of 3

Problem-based math learning helps teachers set the stage for memorable learning experiences and transfer the responsibility for the learning to students, which has been shown to help develop students’ problem-solving and math reasoning skills.

Our webinar series explores how this type of instruction engages all students in grade-level math every day, and how instructors can go about implementing problem-based learning in the classroom. In part 1 of the webinar series, award-winning teacher Kristin Gray asks—and answers—the question: What does problem-based learning unlock for students?

Experience and explanation form a learning cycle

Imagine you’ve just gotten a new piece of technology: a phone, a TV, a computer. How do you learn to use it? Do you read the entire user guide first? Jump in and never touch the guide? Or turn it on and try some things, referencing the guide as needed? 

If the last option sounds like you, that’s very common—and it’s an example of learning through problem-solving. 

“It’s something we naturally do,” says Gray.  “We’ve had a phone before so we would pick up this new phone and try doing things that we know worked on our last phone, and then we would experiment: Does it work the same on this phone? This bouncing between experience and explanation is really the foundation of how we learn through problem-solving.”

What learning through problem-solving looks like in the math classroom

If we think of instructional methods in the math classroom along a spectrum, on one end we might have a classroom where students are left to solve a problem and discover the relevant math on their own. On the other end, the instructional method might be to show students how to get the answer and then practice doing similar problems. 

The methods at both extremes are challenging, and it’s hard for instructors to go from one to the other, says Gray. “We need to install a soft landing space in the middle of these extremes—and we can think of that space as learning through problem-solving, or problem-based learning.” 

What does that look like in the math classroom? 

Students will tackle interesting problems, raise questions about the math required, receive an explanation, and apply it back to the problem—as with the example of learning new technology. 

“When we show students how to get the answer, we send the message that math is solely about answer-getting and learning processes. Answers are important, but we want to use problems to teach the math, not just teach students to get the answer,” says Gray. 

Practice is also key, she adds: “This place in the middle pulls the best from both extremes and puts them into a structure that supports teachers in teaching and students in learning.”

Why students should learn through problem-solving

Learning through problem-solving has the potential to engage all learners in math, says Gray. It influences the way teachers and students think of themselves as mathematicians and what it means to know and do math. 

In the 2000 NAEP survey, 70 percent of fourth and eighth graders reported that they enjoy activities that challenge their thinking, and enjoy thinking about problems in new ways. 

“Students are already naturally curious and like solving challenges and trying things in new ways, so that’s a great start,” says Gray. 

“No matter how kindly, clearly, patiently, or slowly teachers explain, they cannot make students understand,” says Gray. “Understanding takes place in the student’s mind as they connect new information with previously developed ideas. Teachers can help, but understanding is a by-product of solving problems.” 

Add understanding is motivating. It inspires perseverance and confidence. It supports making connections, not learning concepts in isolation. 

When students are given a new problem and are able to use prior knowledge to help solve it, that “promotes the development of autonomous learners,” says Gray. 

How Amplify Math supports problem-based learning

Amplify Math supports teachers in the planning and delivery of problem-based lessons. It also enables teachers to monitor student progress and differentiate instruction based on real-time data. 

Lessons start with warm-ups that tap into prior knowledge and move into problems that require collaboration to solve. Teachers monitor, engage, and ultimately synthesize student work into the main idea. There are also ample opportunities for practice and reflection. 

Learn more about Amplify Desmos Math.

Register to watch the rest of the series here. 

Visit Gray’s site, Math Minds, here.

Webinar series recap, part 2 of 3

Our webinar series explores how problem-based learning engages all students in grade-level math every day, and how instructors can bring problem-based learning into their classrooms.

We reviewed part 1 of the series in this blog post. Now, in part 2, we dig deeper into this key aspect of problem-based learning: transferring responsibility for learning to the students.

So…now what? “If you watched Kristin Gray’s webinar,” says educator Kathleen Sheehy, “You may be thinking, ‘I learned so much about the power of problem-based learning. Where do I get started?’”

In this webinar, Sheehy joins fellow educator Ben Simon to explore how teachers can truly make that key shift toward student-centered instruction. “It is a journey. So we are going to talk about the small shifts that teachers and others can make that add up to something big,” Sheehy says.

The role of the teacher in student-centered learning

Most adults were not taught to do math this way as kids—and many teachers were not taught to teach math this way. When teachers have a lot of content to get across in limited time, it can feel risky to shift to a style that requires a bit of letting go.

“Student-centered instruction helps us embrace the idea that people can come at math ideas from different directions,” says Sheehy. “It’s collaborative and social. It focuses on problem-solving with an emphasis on multiple strategies and flexible thinking.”

Problem-based math learning may not be the sage-on-a-stage model, where the teacher stands up front and acts as the only math expert in the room—but it doesn’t mean the teacher relinquishes control, either. You can have both student-focused instruction and solid classroom management.

“It’s not a free-for-all. It’s very structured,” says Sheehy. “The teacher also plays a role in providing instruction and then guiding their students to the key takeaways they want for them.”

Building stakeholder investment

To be most effective, problem-based learning needs to be not only focused on the student but supported by the community as well. This means you aren’t the only one who needs to adjust to the new approach.

What actions can you take to build stakeholder investment? How can you get the principal, other teachers, parents, and kids (who are also accustomed to another style of learning) involved and excited?

Be able to articulate a really compelling reason why student-centered instruction is right for your students. The following are just a few research-backed examples:

• It helps students develop deeper and longer-lasting mathematical understanding.
• It helps students grow as problem-solvers, engaging them in productive struggle and collaboration and learning core life skills.
• It helps students develop a growth mindset, which reduces math anxiety, boosts math confidence, and helps them relinquish the idea that someone either is or is not a math person.

When the teacher is the supporter of knowledge, not the gatekeeper, students lead the learning process and feel more confidence with and connection to math, says Sheehy.

How and where do you communicate these ideas? Sheehy and Dixon have found that providing a short hands-on math experience with problem-based learning examples can be very effective. This enables stakeholders to experience the difference themselves, especially when conducted in a low-stakes scenario like a parent math night or PD training.

Sheehy also suggests asking them what they think the impact of student-centered learning would have been for them when they were students. “We’ve heard people say things like, ‘I would have been way less anxious about math if I’d learned it this way,’” she says.

Making a plan to start the shift

“We’re not expecting to create a masterpiece overnight. It takes time to develop the teacher and student skills and to establish everything that needs to be in place,” Sheehy says, “You can’t get better at all the things all at once.”

Where to start? “Size up the shift,” she says, and make a plan.

“Using very clear look-fors can enable educators to decide where to focus,” says Sheehy. “‘What would I look for if I walked into a classroom that is beginning to engage in student-centered instruction?’”

Here are a few key elements to look for:

1. Management of materials, routines, and classroom setup in a way that facilitates collaboration.
2. Establishment of a classroom community (using norms charts, etc.) around the core idea that everybody belongs there and is a mathematician.
3. A teachable structure that models the thinking process and creates predictability, allowing students to focus.

Sheehy and Dixon have found that a focus on these three areas helps teachers name what they are trying to improve in a systematic way.

“Once I tackle this first area and feel successful with that, I know what I’m going to tackle next, and after that,” says Sheehy. “These look-fors can help you make informed decisions that, little step by little step, can help you eventually get to where you want to be.”

How Amplify Math supports problem-based learning

Amplify Math is designed to support problem-based learning, so you’re making that shift every time you teach. The program specifically supports teachers in the planning and delivery of problem-based lessons, and enables them to monitor student progress and differentiate instruction based on real-time data.

Lessons start with warm-ups that tap into prior knowledge, then move into problems that require collaboration to solve. Teachers monitor, engage, and ultimately synthesize student work into the main idea. There are also ample opportunities for practice and reflection. 

Learn more about Amplify Desmos Math.

Register to watch the recording.

Subscribe to Math Teacher Lounge.

Webinar series recap, part 3 of 3

We hope you’ve enjoyed reading about—and watching—parts one and two of our three-part webinar series on student-centered learning. The earlier segments explored the thinking and framework behind student-centered instruction.

In this section—a sneak peek at a new lesson from Desmos Math 6–A1—we explore what it actually looks like in practice (and in a fish tank).

Read on for a look at how problem-based math instruction creates memorable learning experiences, and how you can find inspiration to do the same in your classrooms. (Impatient to find out? You can also just go straight to the full recording!)

Carlos’s fish: A different type of real-life problem

The idea for this lesson arose from the real-life experience of Desmos Classroom engineer Carlos Diaz, who found himself in possession of a “magic” toy aquarium. (For more of the entertaining backstory, watch the demo!)

The aquarium contained small fish that grow when you add water—by up to 400%, according to the package.

Takeaway 1: We are always surrounded with inspiration for student-driven math lessons, we just have to keep our eyes open.

Takeaway 2: Green did keep his eyes open, and they were drawn immediately to that 400%. He was skeptical—”At 400% larger, will they even fit?”—and then inspired. “We need to test this thing out,” he thought.

A stream of other questions came forth: Does the scale factor apply to lengths, volumes, something else? Would the growth be linear, or exponential? (Would Carlos ever have to clean the tank?)

The power of open-ended questions

We can’t tell you how large the fish grew (spoiler!) but we can tell you that they did (metaphorically) bust out of their tank and into a lively math lesson.

In the lesson, students look at the toy and are asked: What do you see? What do you notice? What do you wonder?

This type of question helps form the basis of student-centered learning. Here, students are not presented with a fixed set of variables and parameters and asked to solve strictly within them. Rather, they’re presented with a relevant or real-world problem and invited to reference background knowledge, previously learned content, new information, and even imagination.

Potential for exponential growth

From there, a teacher can guide students to make connections between a situation in context and the type of solution or equation that might be relevant. Students can explore collaboratively why one strategy might work better than another.

In this case, a teacher can help students determine that they’ll need to calculate exponential growth (mass), and support them in deciding the best way to do so. Then, having arrived thoughtfully at an approach, they can actually solve the problem and find an answer.

In other words, teachers leading student-driven learning transfer responsibility to those students. Teachers set up the lessons and activities and then provide just enough information and scaffolding to allow students to learn and reinforce math concepts, apply knowledge, and discover new approaches.

Let’s put it this way. Science has found that—contrary to popular belief—goldfish can remember things for not just weeks or months, but years. With student-focused learning, your students will, too.

Learn more.

Register for a free trial for access to this and other lessons. 

Learn more about Amplify Desmos Math. 

Watch the webinar.

Subscribe to Math Teacher Lounge.