Our pedagogical mission

Why is mathematics so hard for many children? And more importantly, must it be this way?

The answer is no: mathematics is hard, but it doesn’t have to be that hard. We can make it easier by teaching it in better ways.

Mathematical proficiency is multi-faceted: it requires understanding many concepts, but also mastering several ‘technical’ skills. For example, when you learn how to solve a simple equation such as 3x=9, you need to understand conceptually what it means—often via examples such as ‘3 dogs eat 9 bowls of food, how much does each dog eat?’—but you also need to be proficient in executing the technical procedure of solving the equation (in this simple example: divide both sides by 3, leading to x=3).

Both kinds of abilities—conceptual understanding and ‘procedural proficiency’, i.e., skilled execution of the mathematical procedure—are super important. Moreover, learning the two are not two contradicting pedagogical goals, but precisely the opposite – they contribute to each other. The problem is that while teachers have good pedagogical programs and tools to teach conceptual understanding (at least up to junior-high school level), they do not have good methods and tools to teach procedural proficiency.

Our mission is to create such methods and tools.

Mastering conceptual understanding and mastering procedural proficiency require two completely different pedagogical approaches.

Teaching conceptual understanding requires the teacher to organize the conceptual knowledge clearly and systematically, to use proper examples, and so on. For basic math contents, existing pedagogical programs do this pretty well.

In contrast, effective teaching of procedural proficiency, which is essentially a cognitive skill, requires a different approach. We must define the learning goals in terms of cognitive proficiency level, and we must specify how to optimize the learning process to the learner’s cognitive abilities and limitations, so that learning becomes as easy and time-efficient as possible. For example, the program should specify the learning goals in terms of accuracy, speed, and the cognitive strategy being used; it should indicate how exercises should be organized to minimize the memory load (e.g., the amount of new content per practice session); it should determine how to present exercises to facilitate optimal encoding of the knowledge in memory (e.g., verbal encoding? spatial encoding? kinematic encoding? combined encoding?); and so on.

Existing pedagogical programs do not use such cognitive optimization sufficiently. Most programs do not incorporate cognitive optimization at all, and in the few programs that do, the optimization is often partial and coarse. This is presumably not because the authors of these programs don’t want to expand the cognitive optimization techniques that they use, but because the research needed to support such expansion does not yet exist.

The bottom line is that while many teachers understand the importance of practice, they do not have enough tools and methods to allow the children practice efficiently in a way that leads to procedural proficiency.

Our lab’s goal is create evidence-based pedagogical tools and methods that lead to procedural proficiency in mathematics.

At present, we focus on 3 topics that cover most of the grade 1-4 curriculum and pertain also to higher grades:

• The decimal number system (multi-digit numbers)
• Arithmetic facts (the multiplication table, trigonometric facts, etc.)
• Arithmetic / algebraic procedures (multi-digit calculation, finding common denominators, solving equations, etc.)

We tackle each of these topics in three steps:

1. Define what ‘procedural proficiency’ precisely means in a particular topic. This is done via research that discovers the cognitive operations and skills involved in relevant mathematical tasks.
2. Create tools to assess specific cognitive skills and proficiencies in individual children, and to detect individual differences and even learning disorders.
3. Create evidence-based pedagogical methods that enhance these cognitive skills up to proficiency. These methods specify the type, amount, and context of practice, and how to adapt it to the cognitive abilities and limitations of learners in general as well as to those of individual learners.

It works: this process leads to highly efficient methods. For example, we have recently developed two techniques for learning the multiplication table – one technique reduces proactive interference in memory, and the other employs multi-sensory learning. These techniques, which require no additional learning effort, have resulted in respective decreases of 30% and 75% in the learners’ errors.