September means back to school, and we’re diving into the meat of the Practice Standards! Reasoning skills have always been part of the curriculum, but they have become a prominent focus of the Common Core. These skills have also become an easy target for critics of the Standards. But why are they so important?
Reason abstractly and quantitatively
The second Practice Standard focuses on students’ reasoning skills. It encourages us to develop the ability to reason through situations and not just solve problems. Rote memorization has its place, but an ability to reason and break down situations will give opportunities for broader success.
Without a focus on reasoning skills, each problem is seen as a separate and unique algorithm. But when we utilize reasoning skills, many problems become related. For example, three problems from our curriculum seem to be completely different on the surface:
“How many groups of 4 can you make in 16 balloons?”
“Two thirds of a pound of candy costs 60 cents. How much does a pound and a half cost?”
“â…— of ___ = 48”
But all are solved in the same manner, when reasoning is used!
- We need 4 groups of equal amounts to make 16 balloons. We can split the whole into 4 groups by dividing, so 16/4 = 4. So there are 4 in each group.
- If two thirds of a pound is 60 cents, then one third is 30 cents. A whole pound is three thirds, so it costs 30*3 = 90 cents. Half of a pound would be 90/2 = 45. So a pound and a half is 90+45 = 135. So $1.35
- We have 48, and that is three groups of one-fifth of the whole. So each group is 48/3 = 16. This means that one-fifth is 16. Since they are equal parts, the whole has five of these groups. So 16*5 = 80.
We used reasoning in all three to identify that we have groups of equal parts that make a whole. While there are standard algorithms to use in solving each of these problems, when we apply reasoning skills it all becomes related.
But beyond using reasoning skills to solve a problem, students must be able to make sense of the answer in terms of the quantities. If given different units, being able to convert to a common unit. Reasoning quantitatively also means to be able to check for reasonable answers - if they find that three baskets of 12 apples is 200 apples, they should know that they need to check their work. They should also be able to check the reasonableness of their answers - if they solve a similar figures problem and find that a man is 24 ft tall, students should be able to recognize that it’s not a realistic answer.
Part of this standard that has caused uproar of late is the increased focus on Estimation. Using estimation, students can use their reasoning skills to see if their answer would make sense. Many parents and teachers dislike this emphasis, preferring that the student goes straight to the precise answer. But estimation and checking for reasonable results is a key 21st Century skill! Everyone carries calculators, and they are being used more and more frequently in classrooms as well as real life. Being able to estimate an answer and check for reasonable results is our best check on mistakes. I know many times I’ve typed numbers wrong (112 instead of 1.12, for example), and knowing how to catch those mistakes is very valuable.
Reasoning in mathematics is seeing the basis and connections between steps of an algorithm, and interpreting the solutions in terms of the quantities. Part of what sets the Mathnasium Method apart is rooted in the ability to lead students to connections and help them understand not just what to do, but why. Those connections form a cornerstone of Mathematical success!