Our pedagogical mission

Why is mathematics so hard for many children? And how can we improve teaching methods to make math easier?

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 are important. The former, conceptual understanding, tells you when you should use an equation and why they work the way they do. The latter, procedural proficiency, lets you solve the equation easily and automatically. And it’s the same idea not only for equations but also for other types of mathematical content. An important point for education is that learning conceptual understanding or procedural proficiency is not a ‘zero sum game’ but precisely the opposite: the two contribute to each other. The problem is that while teachers have good pedagogical programs and tools to teach conceptual understanding, there are much fewer 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 example, in equations, the teacher would need to explain the meaning of ‘x’ as an unknown value, use good examples that can be solved via an equation, and so on. For the basic topics in math, existing pedagogical programs do this pretty well.

Teaching procedural proficiency is a different story. Procedural proficiency is essentially a cognitive skill, and like several other skills, it requires repetitive training until proficiency is obtained (think about learning to ride a bicycle). For this training to be as effective as possible, we can optimize the learning process to the learner’s cognitive abilities and limitations. For example, we can use various techniques to reduce the cognitive load, as this improves learning; we can present the learning materials in a format that is easier for the brain to store in long-term memory; and so on. The problem is that existing pedagogical programs don’t incorporate such cognitive optimization. Consequently, while many teachers understand the importance of practice, they do not have the tools and methods to allow the children practice efficiently in a way that leads to procedural proficiency.

Our lab creates 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 to meany topics learned in higher grades: the decimal number system (multi-digit numbers); arithmetic facts (the multiplication table, trigonometric facts, etc.); and arithmetic and 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.