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Developing Math Skills Series 1/5 – Understanding Concepts

By: tutor Marc L profile picture Marc L., Math Tutor, Dentist, Executive MBA from the University of Ottawa

First of all, let us debunk the most common idea people have about math: “You are born with the math gene or you are not.” This is the typical scenario that gives a person a reason to quit trying. Math skills, just like most other skills such as music, writing and athletics, are developed over time through experience and learning from successes and failures. There is no doubt that some people have more talent and aptitudes than others otherwise we would all be a Mozart or an Albert Einstein. These talents and aptitudes need to be developed and nurtured. Would Mozart have become a great composer had he never been exposed to music? What if Albert Einstein had never gone to school? They were products of their environment and life lessons.

Math skills also need to be developed but the key to success is creating the correct environment to nurture your abilities and watch them grow and flourish. Attitude towards learning will always trump aptitude.

In a previous blog, I mentioned that I do not teach math, I teach the understanding of math. The development of math skills can be broken down into 5 categories that have a high degree of inter-dependency. These 5 categories are necessary to develop great math skills. They are:

  1. Understanding concepts
  2. Understanding procedures
  3. Understanding strategy
  4. Understanding logic
  5. Understanding the real world

In this blog, I will discuss the first category: understanding concepts, which involve the comprehension of the concept, the operations required within the concept and the relations found within the concept itself. In future blogs, the other concepts will be discussed as well as how they interact with each other.

First Concept: Basic Math

The first math concepts we learned about at a young age were counting, addition, subtraction, multiplication and division. Once mastered, the concepts could be used in more intricate math concepts such as fractions, equations, set theory, BEDMAS, prime numbers, Pi, real and imaginary numbers, x and y graphs, algebra, trigonometry and functions.

Most of these concepts do require abstract thinking in order to fully understand the concept. Abstract thinking is performed by all of us on a daily basis. Let me give you a few examples:

Here is a list of items for you to think about:

4 dogs

4 cats

4 goldfish

I have not given you a question related to this list but you brain will start performing in an abstract thinking manner. This is where you let your imagination run wild.

  • These are all animals
  • There are 4 of each animal
  • There are 12 animals
  • These could all be pets
  • The cats could eat the goldfish
  • The dogs might chase the cats
  • They will need a lot of food
  • What kind of dogs are they?
  • What kind of cats are they?
  • Who will clean up their mess?

Without having a clear direction on what the problem is, your mind will make up many more scenarios to the list. Abstract thinking will lead you to concrete answers once you know the question.

If we ask specific questions, then your brain can focus on figuring out what is required.

How many are mammals?  Answer: 8

How many have scales?  Answer: 4

How many are felines? Answer: 4

How many evolved from wolves? Answer: 4

How many can fly? Answer: 0

How to Overcome Math Anxiety

In a previous blog, I spoke about math anxiety. This anxiety can and will prevent you from abstract thinking and prevent you from understanding the concept.

A few years ago, I was tutoring a middle-aged man who had feared and avoided math for most of his life. He had performed miserably in grade and high school when it came to math. He ultimately quit math in grade 10 swearing to himself that he would never need or have to learn math again.

Unfortunately, his job now required him to take on new roles that required him to have to take a remedial course on how to use math in business. I could feel his fear and he had developed a mental block when it came to learning anything about math.

I started with a simple problem which would involve basic math to obtain the correct answer:

You are a car salesperson who needs to sell 100 cars at $15,000 each. If you sell all of them, what will be the total amount of money generated from these sales? We sat down and figured out that this was a simple multiplication problem and we had to multiply $15,000 by 100 to get the final answer which of course was $1,500,000.

The next step was to present him with a similar problem and see if he could solve it on his own:

You are a truck salesperson and you have 50 trucks to sell at $25,000 each. If you sell all of them, what will be the total amount of money generated from these sales? I told him that the mathematical concept was the same as the first problem we had solved together.

He said: “This is not the same type of problem”

I asked: “Why not?”

He said: “The first problem was all about cars, this problem is all about trucks. They are not the same at all!!”

His fear of math was so strong that he ignored the numbers presented and the fact that this was a multiplication problem and concentrated on one thing: cars and trucks are not the same thing therefore the problem cannot be the same. He ignored what he did not understand and only looked at what he did understand. It took a lot of work to get him to look at numbers as tools to solve any problem.

To Sum It Up

Understanding concepts means that students are aware of not just individual facts and methods, but they grasp the mathematical idea presented and become acutely aware of the importance of the combined information as well as the contexts where it becomes useful. Such students have learned how to organize their thoughts as a whole and to relate them to things they already know. Their retention of the knowledge increases and enables them to better connect the information in a manner that is easier to remember and how to use it in future problem-solving.

In my next blog, we will present understanding procedures which involves developing skills to carry out procedures in an efficient manner.

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