How Students Learn to Understand Mathematics

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Mathematics is the science of patterns, and can be understood in the same way that other kinds of patterns are understood. This lesson will discuss the way that the students' minds grasp mathematics, and how to effectively teach them.

Relating Math to Art

'Mrs. Wright, I can't get this trinomial factoring. I just have no idea where you're getting these values.'

'Well, Robin, you're an artist, right? You love to paint? Think of it as mixing colors. That trinomial has an important secondary color at each end, and a mixed-up palette in the middle. So how do you find out which colors make up the secondary color in front of the first term? How would you do it on a palette?'

'I'd try mixing some primary colors, and see which mix produces that secondary color.'

'Exactly. Do that with these numbers. All secondary numbers are produced by mixing (multiplying) primary numbers, just like secondary colors are made by mixing primary colors. Break them up, then we can find out the mix that's on the palette in the middle.'

'That makes more sense. Would you show me how to do it again?'

Math understanding is the ability to explain mathematical concepts and facts, make and recognize logical connections, and identify principles that are at work in a given situation. Because math is a science of patterns, math understanding is created by drawing parallels to patterns that are already understood, such as language, art, or physical object relationships. It is a matter of finding effective grounding (connection to familiar, known information, concepts, and principles) for the new material.

Grounding: Finding Similarities in Relationships

The human mind understands concepts and principles through relationships with recognized patterns already in memory. For example, it is fairly easy to visualize a strange vehicle like the Mars Rover traveling across the Martian ground because we're familiar with cars and trucks doing the same thing on Earth.

Every math step that you're trying to teach is similar to a more familiar relationship that is already in the student's mind. They key is to connect the dots between what they already know and what they're trying to learn.

This is not exactly the same thing as what is called 'building on previous learning', because often a student only has a vague knowledge of 'previous learning'. Unless the previous learning was also well-grounded, you can't base new instruction on that ground, any more than it makes sense to build a second story on a house where the foundation was never established. Today's learning must be grounded in the familiar and understood.

Principles of Teaching the Understanding of Mathematics

The effective teaching the understanding of mathematics (like the effective teaching of anything else) requires that your instruction be specific and vivid, achievable, grounded, and carry an emotional basis for interest.

Specific and Vivid Instruction

The human mind works in pictures, made up of interlocking sensory experiences. Students remember what can be vividly imagined. In math, this means that generalities like factoring or orders of operations will be difficult to hold in the mind, unless they are somehow grounded in powerful human experience.

If the students have a bright, brilliant, dynamic picture to attach to a concept, then they are more likely to remember and understand the concept. Use as many types of sensory experiences and intelligence types as possible. Introduce sounds, smells, music, art, dance, anything that will broaden the sensory impact of the instruction.

Achievable Exercises

Mathematics understanding is tied to actually doing something. Students are called to multiply, divide, factor, differentiate, prove, and so on. In every part of this adventure, there has to be the constant feeling that 'I can do this!' By constantly proving achievability, you can connect virtually any student to any type of math functionality. Very often, there is only one small victory between a frustrated student and a complex concept.

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