Part 1: Three Aspects of Rigor (Ambitious Instruction)

When it comes to mathematics instruction, what are we aiming for? Exactly what do we want students to know and understand and be able to do?

• Do we want students to have a big-picture understanding of the mathematics they are learning?
• Do we want students to be skillful in using procedures and algorithms?
• Do we want students to be experts at problem solving and who apply mathematics to real contexts, problems, situations, and models?

Well, the California mathematics standards requires us to aim for all three!^[1] We call this ambitious mathematics instruction.^[2] Ambitious instruction stands in sharp contrast to the traditional routine found in many classrooms. That traditional routine generally consists of some sort of warm-up problem, a short discussion of the previous night’s homework, a bit of a teacher lecture with guided practice, followed by individual practice.^[3] Instruction that uses this traditional approach is mostly focused on learning and practicing procedures that focuses on “answer getting” with little to no connection with meaning. Although students may learn the procedure, they often do not understand why it works or how to recognize when to apply the procedure in word problems.

The ambitious instruction envisioned by our mathematics framework requires students to experience mathematical rigor that focuses with equal intensity on three aspects: conceptual understanding, procedural fluency, and application.^[4] A stool with three legs is often used as a visual.

Before we talk about what rigor is, let’s start by describing what rigor ISN’T:

• It isn’t longer homework assignments with bigger numbers.
• It isn’t introducing advanced topics at earlier grades.
• It isn’t about increasing difficulty.

Now let’s focus on each of the three aspects of rigor.

Conceptual Understanding^[5]

Conceptual understanding is the comprehension of mathematical concepts, operations and relations. Students with conceptual understanding know more than isolated facts and methods. Students see the connections among concepts and procedures and can give arguments to explain why some facts are consequences of others. This goes beyond memorizing procedures or individual facts to a focus on sense-making. Teaching conceptual understanding enables students to understand a concept from multiple perspectives and look for patterns that can help them understand future problems.^[6]

Procedural Fluency^[7]

Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently. Students also need to know reasonably efficient and accurate ways to add, subtract, multiply, and divide multi-digit numbers, both mentally and with pencil and paper. Students need to see that procedures can be developed that will solve entire classes of problems, not just individual problems.

Application (Problem-solving)

The standards call for students to use math in situations that require mathematical knowledge. Correctly applying mathematical knowledge encourages students to develop a solid conceptual understanding and procedural fluency. To engage in application:

• Students need opportunities to apply mathematical knowledge and/or skills in a real-world context.
• Materials should promote activities that call for the use of mathematics flexibly in a variety of contexts in both routine and non-routine problems.
• Students are given opportunities to use math to make meaning of and access content.

Remembering that the California Mathematics Framework states that all three aspects should be pursued with “equal intensity.” How do we do that? What does it look like?

Do the three aspects occur in a sequence?

Or do they live in some sort of cycle?

Or is it more of a Venn?

In my next blog, I’ll dig into how we can pursue all three aspects with equal intensity and what it might look like.

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