Math Education Theory and Principles

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There has been a lot of research done in education over the years. You have, I expect, heard about Piaget's theories of how children learn--progressing from concrete to abstract as they get older. There are also researchers whose theories apply quite directly to mathematics. I'm going to introduce you to three researchers here who have influenced how I think about teaching and learning mathematics

Zoltan P Diens (especially the perceptual variability principle):

If you read a lot of Diens' work and theories about learning mathematics, you'll see that he describes a set of stages that children go through as they learn new things in math, and that those stages are roughly similar to those that Piaget describes. There are a few new and interesting things in his work, though, that I've found useful:

• Diens along with many other researchers in math education believe that not only children, but all math learners need to progress from concrete to abstract in their understanding. Further, their progress from concrete to abstract is not just something that comes with age, but also requires appropriate mathematical learning experiences. So, in order to in order to move from a concrete to an abstract understanding of a concept, one needs appropriate experiences (lessons) with that concept.
• Perceptual Variability Principle: Diens proposed this principle to describe how different experiences help children learn. It explains why manipulatives and diagrams are important to learning. In order to build a full understanding of a principle, it is helpful to see it in many different contexts. If the same idea can be taught with manipulatives (perhaps different kinds of manipulatives) and with pictures and with numbers, then learning the same concept in all of those different ways strengthens and deepens understanding of the concept or process. This principle describes why learning with (physical) manipulatives along with written representations is helpful for developing mathematical understanding.You'll see this principle throughout this course, where I ask you to explain the same concept using two or more different representations (numbers, diagrams, virtual manipulatives). Knowing several different ways of representing these concepts will deepen your understanding and will give you examples of ways you can teach children different useful representations of mathematical concepts.
• Mathematical Variability Principle: This principle says that when you give examples and create learning experiences, you should vary the problems as much as you can without changing the underlying math concept you are trying to teach. Having five different examples of what a triangle can look like is better than having just one example. Counting different kinds of counters is good, but it's even better to practice counting other things too, like steps or claps or inches.

Lev Vygotsky's Zone of Proximal Development:

Vygotsky described the most effective learning experiences as being those within the learner's zone of proximal development (ZPD). Lessons where children are doing things that they already know how to do unaided are primarily practice, not new learning. Lessons that are beyond the learner's ability to understand and do at all are ineffective (too difficult). Tasks that the learner can do with direction or help are within their zone of proximal development, and are at the ideal level.

The zone of proximal development is important for creating appropriate problem solving experiences. If the problem is too easy, then the child will be successful, but won't learn anything new. If the problem is too hard, the child will be unsuccessful, and will not learn. If the problem just just right--just beyond what the child has done before and can do unaided, then the child can be successful, and will increase their understanding. A single task could be an appropriate problem solving task for one child, because it is in their zone of proximal development, not be an appropriate problem solving task for another child (because it is either too easy or too hard). This principle describes what is most challenging about creating appropriate problem solving lessons for children.

Other researchers (including D. Wood) have built on this theory to describe the sorts of supports that teachers or other adults provide to children to solve problems that are just outside the range of what they can solve by themselves. These supports are called scaffolding--a problem can be scaffolded, and after problems with support, children can progress to solving similar problems by themselves.

(Note from me: It's probably not effective in most situations for all lessons to lie within the students' zone of proximal development (need scaffolding). Lessons that stretch students to learn new things need to alternate with students where children practice and consolidate things that they already know and can do.)

Daniel Willingham (flexible and inflexible knowledge):

Dan Willingham isn't particularly an math education theorist, he is a cognitive psychologist (study of learning) and he writes essays for teachers summarizing things in the research that are relevant for teaching. I've been particularly influenced by two things he's written:

Inflexible Knowledge: The First Step to Expertise. In this essay he describes how learners naturally progress from inflexible (rote) knowledge to flexible (deeper) knowledge about whatever they are learning. This is relevant for math education because there's a commonly described dichotomy between "procedural" and "conceptual" knowledge. Procedural knowledge is described as the ability to "do" mathematical processes, but without showing and understanding of why they work (reasoning) or how to apply them to new situations (problem solving). Conceptual knowledge encompasses reasoning and problem solving along with computational ability. The relevance of understanding inflexible knowledge means that we should regard inflexible (procedural, rote) knowledge not as a failure to learn, but as a step in the learning process. If our students have a rote understanding, it means that there is more learning that is needed, not that the learning so far has failed. Diens' principles of perceptual and mathematical variability are helpful in creating learning experiences that help children move to more flexible (conceptual) knowledge, by creating experiences where learners encounter the same concepts and skills in different contexts.

Learning Styles Don't Exist is a video Dan Willingham made describing what research has shown (and not shown) about learning styles. The big take-away is that different ways of presenting knowledge are more or less useful depending on how well they fit the knowledge they are representing, and not so useful depending on the learner they are presented to. This doesn't mean that you shouldn't use different ways of understanding concepts (perceptual variability), but it does mean that the usefulness of a representation lies in how well it represents the concept to be learned, not in how well the representation matches the learning style of the learner. Look for lots of ways of representing concepts, but make sure they all represent the concept you're trying to teach (and don't be distracted by tasks that don't fit the concept well). Also, having different representations is valuable for many/most students (back to Deins): representations in lots of different styles are good for the whole class, not just one child.