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.
As a recap, 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 this blog, we will discuss 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.
The best way to compare and contrast the concepts and the procedures is by comparing it to a real life situation so we will refer to the example provided in the previous blog on procedures and we will add a third dimension to the story, namely, strategy.
Using a Real-Life Example to Understanding Strategy
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 in a baking pan and put it in the oven and, voilà!, you have a cake. Not so fast, you now need to know certain procedures 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.
At this point in time, you will need to develop a strategy on how all of these ingredients will be combined to bake the best chocolate cake ever. At this point, you have all of the ingredients and the instructions. The instructions may not contain all of the information you need:
- When do I pre-heat the oven?
- Should I use plastic, ceramic or metal mixing bowls?
- Should the eggs be at room temperature or straight out of the refrigerator?
- I need to use milk. Should it be skim, 1%, 2% or whole milk?
- Do I need a conventional, convection or natural gas oven?
- At what height do I adjust the baking racks?
- They want a greased baking pan. Do I use butter, margarine, shortening or spray cooking oil?
- How much flour is needed to coat the greased pan?
- How fast should I beat the eggs?
- Do I mix by hand or use a hand mixer, Kitchen Aid or a food processor?
- What does fold together mean?
- What is sifting flour?
- Is there a difference between baking powder and baking soda?
- How do I separate egg yolks?
- How can I tell when the cake is done?
- How long do I cool the cake for?
- What do you mean by: “Now you need to make your own icing”?
Now that you are aware of the ingredients, instructions and variables, you will start to test out different ways to make this cake. Over time, you will become more proficient at it and develop strategies that make sense to you. Once you are challenged to cook something else, you will be able to draw on these experiences to achieve success in another recipe. You will also learn that there are many ways to bake a cake.
How Do You Develop Strategies in Mathematics?
When you are dealing with routine problems, your strategy is fairly simple as you can usually rely on easy past experiences to solve the problem. You may from time-to-time dabble in solving these routine problems a different way.
However, when faced with non-routine problems, you may have to map out the problem and possibly look at several strategies to solve it. You understand the concept(s) presented, you have all the knowledge (ingredients) needed to solve the problem but you need to strategize in order to put it all together. Here is an example of different strategies to solve the same problem:
You are told that there is a display of flowers at the local outdoor market. There are a total of 53 pots where the white pots have 3 flowers and the red pots have 5 flowers. The total number of flowers in the display is 180. You want to go there and buy 15 red pots with the 5 flowers in each. Is this purchase possible? If it is not possible, how many red pots can you buy?
If you understand the problem being presented, you need to somehow calculate how many pots have 5 flowers. The tools you have at hand are as follows:
- Simple mathematical deduction using addition and subtraction
- Trial and error estimations
- Simple algebraic equation solving
This solution involves reasoning in that we know that all pots have at least 3 flowers. This means that if all pots have 3 flowers, then the total number of flowers should be (3×52) which are 156. The actual number is 180 so that we are short by (180-156) 24 flowers. How many red pots are required to make up the 24 flower difference? The answer is (24/(5-3)) which is 12. Therefore only 12 red pots are available for purchase.
A less sophisticated method would guess through estimations. Your first guess could be 30 white pots and 22 red pots. This means (30×3) + (22×5) which equals 200 flowers total. Because 200 is greater than 180, we need to reduce the number of red pots and increase the number of white pots. Let us try 35 white pots and 17 red pots. This means we now calculate (35×3) + (17×5) which is 190 total flowers. Still too high, but we are getting closer. We can now try 40 white pots and 12 red pots. We will now calculate (40×3) + (12×5) which now equals 180 flowers. This means that the display has 40 white pots with 3 flowers and 12 red pots with 5 flowers. Under this scenario, you still cannot buy 15 red pots.
This is a more sophisticated approach that requires some basic algebra to solve it. In algebra, we like to assign letters to variables so let the white pots be w and the red pots be known as r. From the information given by the original problem, we know that w + r=52 is the total number of pots.
We also know that 3w + 5r=180 is the total number of flowers in the display. With 2 equations, you can isolate w in the first equation and substitute it in the second equation. This will generate a value of r=12. I will let you calculate it on your own to verify. If you cannot, you may need tutoring on algebra.
Students who understand strategy for non-routine math problems will develop a competency and will come up with several approaches to solve the problem. They can use deductive reasoning, guess and check techniques or rely on standard algebraic equations to solve the problem presented. Without an understanding of concepts and procedures, one could never develop a strategy for mathematical problem-solving. You need to be flexible.
In my next blog, I will present understanding logic which involves developing skills to think logically around concepts, processes, strategies and the situations presented.