By: Marc L., Math Tutor, Dentist, Executive MBA from the University of Ottawa
I always mention 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:
- Understanding concepts
- Understanding procedures
- Understanding strategy
- Understanding logic
- Understanding the real world
In the previous blog, we discussed the first category of developing math skills which were: Understanding concepts where the student becomes aware of individual facts and information presented but where there is also a fundamental understanding of the mathematical idea presented.
In this blog, I will discuss the second category: Understanding Procedures which involve having knowledge of procedures, when and where to use them and having a sense of how to use these procedures with efficiency and efficacy while maintaining an element of flexibility.
What are the Differences between Concepts and Procedures?
The best way to compare and contrast the concepts and the procedures is by comparing it to a real life situation:
You are asked to bake a cake. Let’s make it a chocolate cake which is one of my favourites. You have never baked a cake before. Having watched people in your household bake a cake before, or seen it on reality TV or YouTube, you have a fundamental understanding of the concept that you will need to mix certain food ingredients together, place it a baking pan and put it in the oven and, voilà!, you have a cake. Not so fast, you now need to know certain procedure in order to have a chance at any success with your first cake.
In order to understand the procedures involved, you will need to look it up online or in a cookbook or watch it on YouTube. You will be given the ingredients list, their measurements, temperature settings, baking times and the order in which things are added and mixed together. If you just toss it all together and bake it, you may end up with a very flat, hard failure that could be used as a hockey puck or a door-stop.
Maybe your first attempt at baking a cake will be by using a store-bought cake mix. At a later time, you will want to try a home-made cake. This is where the procedure becomes more complicated and you will need to have an understanding of the strategy and logic of putting it all together for a great chocolate cake. These concepts of strategy and logic will be discussed in future blogs.
What if I ask you to bake an apple pie? The concept is fundamentally the same but the procedure will be somewhat different even though there may be many similar ingredients. Some of the skills you developed in baking a cake from scratch will become useful in baking a pie. Math concepts and procedures often have overlapping ideas.
Understanding Procedures Takes Time to Develop
Some mathematical skills need to be developed over time as they should become second nature without the need for tables or calculators. Mental arithmetic capabilities are essential to developing an understanding of procedures. Would you use a calculator, fingers or diagrams to calculate 2+2? Over time, simple arithmetic calculations become second nature and help you quickly estimate the answer to problems. Students need to become proficient at simple calculations such as 4+5, 10-4, 8X4 and 27/3. The student needs to efficiently know how to add, subtract, multiply and divide multi-digit numbers either in their head, pencil and paper or a calculator.
A student who understands procedures will be able to compare and contrast different methods of calculations. These methods include mental calculations of sums, differences, multiplication products and division quotients. You can also develop methods of utilising computers, calculators or other visual aids such as blocks, sticks and balls.
Without a thorough understanding of procedures, a student will have difficulty in understanding mathematical ideas and solving mathematical problems. Some may argue that understanding concepts and understanding procedures are in constant battle as one is about abstract thinking and the other is on memorization; they are actually meant to work in tandem. Students who simply memorize formulas perform at a lesser level than students who understand the formulas.
When I was in high school (many years ago), I had a classmate who did very well on all our tests and exams. She would memorize all of the math problems learned in class and would write them out in tests as the teacher would always give us exams with material we had learned. She would spend 4 hours every night going over these problems without really understanding them. We both entered the science program at the same university and she quit by the end of the first term. The amount of material covered in university compared to high school was 10 times greater. No one could possibly memorize that much material as well as all the material we covered in physics, chemistry and biology. Before she left the program she admitted that there were not enough hours in a day to memorize all of the courses.
When you understand, you then develop the ability to modify and apply these procedures to other situations. The retention of knowledge increases and enables one 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, I will present understanding strategy which involves developing skills to carry out the how and why of using procedures in an efficient manner.