By: Marc L., Math Tutor, Dentist, Executive MBA from the University of Ottawa
In the previous blogs, we discussed the first category of developing math skills which was: Understanding concepts where the student becomes aware of individual facts and information presented and where there is also a fundamental understanding of the mathematical idea presented.
We also discussed the second category: Understanding Procedures which involves 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.
In my last blog, we discussed the third category: Understanding Strategy which involves a competency developed by the student to formulate a mathematical problem, know how to represent it and solve it.
As a recap, the development of math skills can be broken down into 5 categories which 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 this blog, we will discuss the fourth category: Understanding Logic which is all about developing the capability of thinking logically about the relationships between concepts, processes, strategy and the situation at hand.
How Do We Understand Logic
Logic involves using reason and common sense to make things happen correctly. The actual definition of logic is: reasoning conducted or assessed according to strict principles of validity.
In order to better explain this skill, I will keep on using the analogy of baking a chocolate cake. You started with the concept of having to mix ingredients together, the procedures necessary to combine the ingredients and the strategy on how to mix the ingredients in a timely manner using the technologies and techniques you have available to you.
Using a Real-Life Example to Explain Logic
How are your reasonable assumptions on baking a cake validated? In cooking, the answer lies in the look, texture and taste of the final cake product. If it tastes good, you must have done it right. If the cake tasted bad, is undercooked, is burnt or is flat as a pancake, then logic would dictate that you did something incorrectly.
When you visit some kitchens, there are pots and pans all over the place, countless mixing bowls, as well as ingredient containers, can be seen everywhere. There is no rhyme or reason for a cooking strategy that can be found anywhere. If something goes wrong, it will be almost impossible to trace the source (or sources) of the mistakes.
Conversely, some kitchens are very well organized. You only see the pots and pans required and the ingredients are already pre-measured and ready to go. If something goes wrong, it is much easier to find out where the mistake took place. Such a kitchen will most likely produce consistent results. This kitchen will be more capable of experimenting with techniques and ingredients to achieve even better results.
Not long ago, my daughter-in-law made baked macaroni and cheese. She carefully cooked the noodles “al dente”, combined 3 types of cheeses onion, spices and baked it for the requisite amount of time. It was served hot on our plates and tasted wonderful except that the consistency was not as creamy as I remembered it. No one complained about the lack of creaminess as it tasted so good.
Later that evening, someone asked why there was a cup of warm milk in the microwave. The milk had been forgotten and not added to the noodle and cheese mix prior to baking. I am sure this mistake will never happen again.
Logic and Reasoning in Math
In mathematics, you use logic and reasoning to determine if the strategies you have developed make sense. As you use your strategies to solve the problem, reasoning will let you know if you are on the right track. If you are not on track, then go back, re-assess and develop news strategies based on accumulated knowledge from past experiences.
Understanding logic is the glue that bonds the previous categories together. It definitely gives you a sense of how interactive concepts, procedures and strategy are in the quest to solve mathematical problems in a logical manner. Developing a strategy toward a solution usually requires a certain fluency in your procedures and calculations while understanding logic involves determining if the procedure is the correct one and adequate for the task at hand.
In my next blog, I will present understanding the real world which involves developing skills to think of mathematics as a tool to solve problems. We will discuss how your attitude toward the use of mathematics can make a huge difference in your life and career.