Ask Not What Your Task Can Do for You...

…But What You Can Do for Your Task!

Many math instructional programs include special tasks designed to promote the rich thinking and student discourse described by the Standards for Mathematical Practice. And those tasks are valuable—but what about the rest of the curriculum? Are there other opportunities to engage students in these practices? The Math Practices aren’t intended to be something students “do” once in a while; they describe the ways students engage in and make sense of mathematics every day. Imagine a classroom where rich mathematical thinking is woven into every stage of learning and every type of task, from conceptual exploration and strategy building to procedural fluency and application.

Throughout this series, we will show how a teacher can leverage any task as a springboard for rich mathematical thinking. Join us as we examine one problem through multiple lenses and consider how attending to, interpreting, and shaping student thinking can engage learners in various aspects of the Mathematical Practices.

What Are the Mathematical Practices?

The Standards for Mathematical Practice describe eight habits of mind evidenced in mathematically proficient students as they engage deeply with mathematics. Designed to transcend grade level or mathematical topic, the standards capture various nuances of thinking and reasoning that students should develop along their math learning journeys.

Introduced in 2010 as part of the Common Core State Standards for Mathematics¹, the Standards for Mathematical Practice (SMP) represent the processes and proficiencies described in two earlier foundational frameworks in K-12 mathematics education: the National Council of Teachers of Mathematics’ 2000 Process Standards² and the National Research Council’s 2001 Five Strands of Mathematical Proficiency³. More recently, many states have developed their own similar collections of practices and habits of mind. Although the wording and structures may vary across these collections, the goal is always the same: to identify the thinking and behaviors of mathematically proficient students. And as educators, we have the power to use any mathematical problem or task as an opportunity to engage students in this type of thinking; but so much of that thinking is unlocked by asking the right questions.

Let’s Consider an Example

To begin to see the power of advancing questions in helping students to engage in the mathematical practices while they work, consider the following vignette from a seventh-grade math classroom.

Vignette: Grade 7, Ratios and Rates on a Biking Trail

Mr. Bell, a seventh-grade teacher, has just finished engaging students in a data collection activity to compute and compare unit rates, followed by a lesson on the concepts of rates and unit rates. Students are now working together in small groups on Problem #12 shown below.

Mr. Bell wants to use this as an opportunity to engage students in making sense of problems and persevering in solving them (SMP.1) while also being sure they are not just computing blindly but considering the units and contexts, reasoning both abstractly and quantitatively (SMP.2). As Mr. Bell circulates throughout the room, he is attending, interpreting, and responding to shape student thinking as he notices each group’s progress.

Group A: Finding an entry point
SMP.1 — Make sense of the problem

Mr. Bell approaches Group A and notices that students have read the problem several times but haven't written anything down. One student says, “I don't even know what we're supposed to do.” Another begins pointing to numbers in the problem, suggesting they should multiply or divide them.

Rather than telling them which operation to use, Mr. Bell attends to the fact that the students haven't yet established what the situation is describing. He asks: “What is the situation? Explain it to your partner in your own words.” As they explain, Mr. Bell listens for whether they understand that two riders are traveling toward one another from opposite ends of the same trail. He then asks: “What do you know about each rider? What don't you know?”

The students begin identifying the known rates and recognizing that the distance each rider travels is unknown. Mr. Bell follows with: “What is true about the distances when the riders meet? How will you represent this relationship mathematically?”

Now the students have an entry point. Rather than prescribing a starting point or a procedure, Mr. Bell has given them space to understand the meaning of the problem and find their own solution pathway.

Group B: Computing without thinking about the quantities
SMP.2 — Reason abstractly and quantitatively

In Group B, students are already working. Mr. Bell notices that they have immediately begun calculating with the numbers in the problem.

Their work contains several computations, but when he asks what one of the numbers represents, the group is unable to answer. This tells Mr. Bell that they're making procedural efforts but may not be connecting their computations to the quantities in the situation.

He asks: “Is there enough information to compare the rates?” The students initially say yes and point to the numbers they've calculated. Mr. Bell continues: “As you compute your answers, what are the units?” The students stop and examine their work. They realize they have inadvertently calculated the distance that one rider travels in 1/16 of an hour, rather than 1 hour.

Mr. Bell doesn't correct the calculation for them. Instead, he prompts them to attend to the meaning and units of the quantities. He then asks: “What do you know about the distances that might help? Talk about the distance that each rider goes per unit of time.”

Now the students begin considering not simply “What numbers can I calculate?” but “What do these numbers represent, and how are the quantities related?”

Group C: Part of the way there
SMP.1 — Persevering in problem solving
SMP.2 — Reasoning about quantities and their units

Group C is well into the solution strategy. Mr. Bell notices that students have converted both rates to unit rates in fractions with the same denominator. They have circled 26/3 mph. When Mr. Bell asks them, “Which rider has traveled farther?” they say, “You traveled farther because your speed is faster.”

Mr. Bell notices that the students have made a meaningful comparison, but their response answers only part of the question. “What else do you know that can help you find how much farther you ride than your friend?” he asks.

The students are silent.

Mr. Bell gives them a moment, then returns to the quantities they have already calculated. "What are the units? What does that tell you about each rider?” he asks. The students explain the meaning of the rates. Mr. Bell then asks, “What else do you know about the two riders besides their rates?” “They've been riding for the same amount of time,” a student says. The students begin discussing the relationship between speed, time, and distance. Mr. Bell listens as the students now look beyond the comparison of rates to determine the difference in distances.

Mr. Bell has not prescribed a solution pathway but has encouraged productive thinking about the scenario that can lead them to a solution pathway.

Group D: A correct solution, but an opportunity hiding inside it
SMP.2 — Creating a coherent representation of the problem

Group D appears to have solved the problem. Mr. Bell notices that one student has used an equation while another has created a table.

Both approaches appear to lead to the same answer. Rather than simply confirming that they're correct, Mr. Bell asks: “How did you decide to compare the two riders’ progress?” Then: “What kinds of mathematical work did you decide to do? Why?”

The students explain their method in each approach. Mr. Bell then encourages them to examine the relationship between the two strategies. He asks: “How does your equation represent what your partner's table shows, and vice versa?”

Mr. Bell has turned the focus away from just getting a correct answer; students begin to flexibly make connections among their representations and explain why their mathematics works.

Group E: Finished, but not yet making sense of the answer
SMP.1 — Checking if an answer makes sense
SMP.2 — Representing a situation symbolically, then contextualizing

Finally, Mr. Bell reaches Group E. They have an answer circled and seem ready to move on. He asks: “What helped you? What confused you?” Then: “I see you used a variable. What does your variable represent in the context of the situation?”

The students realize that they have used the variable t but haven't actually articulated what t represents. Mr. Bell asks them to explain their answer in terms of the original situation.

After this, Mr. Bell asks a final question: “So, what does your numeric answer tell you about the two riders when they meet?” The students must return from the symbolic and numeric computations back to the real-life context.

Teacher Noticing to Advance SMP Thinking

As he walks around the classroom, Mr. Bell makes several observations about what students are doing and saying in each group and considers what those observations might mean: What do students understand? Where is their reasoning developing? What might they be struggling to understand? Why are they using a particular strategy? How can I connect where students are with the next step in their mathematical thinking?

This is an important shift in how we can think about mathematical tasks. A task doesn't have to be labeled “rich” to become a springboard for rich mathematical thinking. The potential is often waiting to be uncovered. If a teacher attends carefully, interprets what that reveals, and responds with advancing questions, even a seemingly ordinary problem can be the vehicle to a powerful mathematical experience. Teacher noticing is "the secret weapon that separates good teaching from great teaching... it’s about deeply understanding [students'] academic and emotional assets and needs, and responding with intentional actions that drive learning forward."⁴

While it is not necessarily true that any task is automatically rich, we can say that a well-designed task has more mathematical potential than may be immediately visible, and teacher noticing is what helps surface that potential. The task opens the door. Teacher noticing—and the questions that follow—invites students to walk through it.

In the next article in this series, we will explore SMP.3 and SMP.4 and see how a teacher leverages a third grade task for constructing viable arguments and attending to precision.