Tricks Aren't for Kids: Mathematical Thinking IS!

We are simply not giving students enough credit for the knowledge they bring to the classroom. If you have ever seen a toddler share pieces of food or pieces from a game or toy, they are dividing! Granted, they usually benefit from leftovers and decide that the remainder should be theirs. My three year old daughter knows when something has been subtracted, taken away, or is less than what she wants. When she wants five miniature pancakes and I accidentally give her four, she immediately informs me that she is one short.

My first few years of teaching, I taught math the traditional way: show kids the algorithm for how to solve the problem and let them practice, practice, practice. While this was effective for some students, it wasn't enough for every student. If a kid didn't understand, I would slow down, show them again, and give them more practice. Some students never got it and were deemed “bad at math.” Once I started learning to pose word problems and use students’ strategies to teach and pull out the math concepts I needed to teach, I saw successes in math that had never been attained through traditional teaching methods. Now, if a kid gets an incorrect answer, I can analyze their thought process through their strategy and find exactly where their misconceptions and mistakes are. It shows me that they do have some understanding of the problem, and I can guide them through the part that is a challenge for them. It becomes more than just a correct answer. For example: My fifth grade students were multiplying 6 ½  x 8 ¼.  One girl’s work is in the following photo.

Instead of teaching a trick that many learn, FOIL (First, Outer, Inner, Last), to teach the steps in solving this problem, she derived it on her own! Since we have done so much work on the meaning of multiplication, she knew she needed 6 ½ groups of 8 ¼ and when she split up the whole numbers and fractions she still had to account for the 6 groups of 8 and 6 groups of ¼ as well as the ½ of a group of 8 and the ½ of a group of ¼. Another important note is that she has purposefully never been told to multiply denominators as in ½ x ¼, and because of this she has reasoned on her own that when you take a half of a fourth, it would then be an eighth. And then, when she needed to add the ⅛ to ½ she figured out on her own that ½ would be equal to 4/8 and that she would then have ⅝.  While having a conversation with her about her strategy and looking at her work, she showed me far more mathematical understanding than I would see if she were simply using a trick.

Now, not all students are at her level of thinking, but I can use her work to teach others! There is so much power and ownership taken when students’ strategies are used to guide math lessons and instruction. Kids have the innate ability to think through math problems, so the idea that we have to show them how to “do math” is false. By using their strategies to explain the reasoning of a problem and as a starting point for understanding math, kids see it as “doable” and understand that if their classmate can do it, so can he/she. They no longer see the teacher as the source of the right answer or the right way, but instead become independent, confident problem solvers. The role of the teacher should be a facilitator of learning, or someone who guides students and challenges students’ levels of thinking. We can model how to correctly write students’ thinking into accurate equations/number sentences.

The teacher must carefully plan the context, problem type, and numbers used in problems posed to meet grade-level standards. For example, in the student’s work above, I carefully chose whole numbers and fractions in an area problem context that would bring out multiple fifth grade standards:

Common Core State Standards

5.NF. 1-add and subtract fractions with unlike denominators

5.NF.2-solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators

5.NF.4a- find the area of a rectangle with fractional side lengths...and show that the area is the same as..multiplying the side lengths.
5.NF.6- solve real-world problems involving multiplication of fractions and mixed numbers by using visual fraction models or equations to represent the problem.

Once a problem is created, teachers should solve it in multiple ways from the point of view of children to anticipate how students will solve it. (A good resource for creating problems and seeing possible student strategies is Extending Children’s Mathematics: Fractions and Decimals by Susan Empson and Linda Levi and Children’s Mathematics: Cognitively Guided Instruction by Thomas P Carpenter, et. al.)This will help the teacher plan ahead which strategies to share aloud with the class for discussion. By selecting a few strategies, students can compare/contrast and critique the reasoning of others (Common Core Standards for Mathematical Practice # 3), get ideas on how to solve the problem (connects to Math Practice #4), look for and make use of structure (Math Practice # 7), look for and express regularity in repeated reasoning (Math Practice # 8), and the benefits go on.

There is much debate about the new Common Core State Standards and the idea that we are implementing “new math.” Yes, it’s true that math needs to be taught differently, and as I mentioned above, we are not giving kids enough credit for the knowledge and abilities they bring to the classroom. With careful planning on the teacher’s end, they can discover math concepts without being told “how to do/solve” a problem.