Deliberate Practice In Mathematics Classrooms

This piece is a work of cyborg intelligence: Oldridge x Chat GpT 3.5

One of the most useful constructs for mathematics educators is Ericsson’s notion of deliberate practice, which he developed through his research, and is summarized in his great book Peak.

Before explaining this idea, let’s consider this example from my own schooling:

-each day, at least from about grade 4 to 10, I was given a worksheet. The whole class received the same worksheet, regardless of how much skill or knowledge they had about the topic on the worksheet.

-if we were “lucky”, the teacher gave us a break in older grades and said something like: “odd numbers only, from 1–50.”

-for those 25 questions, if I had already mastered the skill or concept, I likely got all or almost all of them correct.

-if I did not understand the skill or concept, I still had to try the worksheet. I likely would get 0/25 correct in both cases.

This is an example of a one-size-fits-all practice, which greatly contrasts Ericsson’s approach. Ericsson’s concept of deliberate practice emphasizes purposeful and focused efforts to improve performance through targeted activities and continuous refinement.

Note that Ericsson differentiates purposeful from deliberate practice by adding a “teacher or coach” to the latter (purposeful practice could easily be self-driven) and this coaching role for teachers, monitoring learning, giving feedback, and giving advice to move forward is the key pedagogical move to get kids to practice the right things, at the right times.

In the context of mathematics, deliberate practice involves systematically addressing weaknesses, pushing beyond one’s comfort zone, and receiving feedback to refine understanding and skills. You might consider this a “just in time” approach to practice in math classrooms, rather than a “one-size-fits-all” approach. This requires really knowing the learners in front of you, to give them what they need to practice, at any given time.

Consider briefly a sports example, from basketball:

It’s time to practice skills. I am weaker at free throws. My teammate needs to practice mid-range counters, working on his pivot foot. In these instances we are used to the idea that coaches can set us to work practicing our weaknesses, giving feedback on form, footwork, etc.

Here is part of how Deliberate Practice is explained in Ontario’s High Impact Instructional Practices in Mathematics (2020) document:

Here are some ideas about how deliberate practice can be applied to mathematics education along with examples:

Identifying Weaknesses:

Deliberate practice begins by identifying specific weaknesses in mathematical understanding or skills. This could include difficulties with specific operations (e.g., multiplication), understanding certain concepts (e.g., fractions), or struggling with problem-solving. Once you have identified weaknesses, you can advise on what and how to practice.

Focused and Challenging Exercises:

Once weaknesses are identified, deliberate practice involves engaging in focused and challenging exercises that target those specific areas. For example, a student struggling with multiplication might engage in a series of progressively complex multiplication problems, or engage in targeted activities to build an understanding of underlying concepts (groupings for example, or using an array).

Repetition with Variation:

This is likely familiar to you if you grew up working from math textbooks. Deliberate practice often includes repetition with variation. In mathematics, this might involve practicing the same type of problem but with different numbers or structures. For instance, if a student is working on solving equations, they could practice solving equations with various coefficients and constants. In this example, it’s possible that a student might need feedback on a specific aspect of solving equations, or working with negative integer terms, for example.

Immediate Feedback:

Feedback is crucial in deliberate practice. Immediate feedback helps learners correct errors before they are solidified. Experienced math teachers often home in on specific errors as they occur. Here is one common one in primary grades: if rushed to the standard adding algorithm, students often need the steps broken down by building knowledge of place value. Often they also need additional practice with adding facts to 20, which are simple to practice.

Setting Specific Goals:

Deliberate practice is goal-oriented, and depending on the age of your students, you may need to help them notice they need to set goals. Perhaps this could mean, for example, helping them practice a tricky times table like the 7s, or perhaps this could mean working on 2-digit addition without regrouping as a scaffold to get to regrouping.

Incremental Progress:

Chunking is a well-established idea that fits with how humans learn. Deliberate practice involves incremental progress over time. This often means increasing the complexity of questions over time. Perhaps we can bring in the coaching analogy again-sports coaches are very adept at breaking skills down into component parts, and building those skills into a more “game-ready” package.

When deliberately targeting skills or concepts, you will notice that students improve steadily.

Reflection:

Regular reflection on performance is a key aspect of deliberate practice, which is to say that we can help our students to think about what they are learning.

The deliberate practice framework encourages a purposeful and continuous effort toward improvement and takes us far beyond a “one size fits all” approach. Note that Ericsson also talks of “naive” practice. Novice learners are naive, in that they often need guidance on what and how to practice. The more self-motivated form is called “purposeful” practice-like my son setting out to learn Spanish with an app assist from Duolingo. The expert practitioner, a teacher in this case, gives guidance and feedback that makes the practice deliberate and purposeful.