By Martha Motley, posted August 31, 2015 –

I recently had the pleasure of seeing the beauty of the progression of the Common Core’s content standards and Standards for Mathematical Practice (SMP) that students are currently being taught here in North Carolina. My second grader, like her older brother, loves math. Yes, Mom is a math coach, but I genuinely believe that her love for mathematics hasn’t been taught or caught from her mom. I think that the conceptual understanding she is growing in and the way that math makes the world make sense has given her this excitement over the subject. My child is not overly concerned with being right. When she does solve a math problem that she has created herself (a regular hobby), she always tells the listener her solution and then says, “Don’t you want to know how I know?”

This is a game for her. Sitting at the table in a Mexican restaurant the other day, she counted the twenty-five nacho chips left in the bowl. She then proceeded to tell us that we could each have six (four of us were at the table) and that there would be one left. “Do you know how I know?” came her regular question.

She explained two or three ways that she had thought about it. One involved 12 + 12 + 1 and then halving the twelves to make 6 + 6 + 6 + 6. Another strategy she shared was giving each of us five chips: “Then I gave out twenty, and there are five left; so we each get one more, but one is left.” She had done no written work. She had no answer key, A+, or sticker. She just loves to think about math, and she is really making sense of it in ways that many people beyond her age think about it.

Back to this afternoon: Inspired by partitioning, Maddy began to make a “fraction book.” She drew squares and rectangles on the pages and shaded parts of the equal partitions she had carefully sectioned off. She began with halves, but on the next page, I saw a rectangle split into two rows of five. She shaded a row of five and left the other half blank. Below she wrote, “5 10ths = half.” She didn’t know how to write five-tenths as a fraction, but she wrote “5 10ths” and saw it as half. She did the same with two-fourths and three-sixths. When she turned the page and I saw a rectangle with three of five sections shaded, I must admit I had a ping of fear that she was going to tell me that three-fifths was also one-half. I bit my lip and asked her. Her reply solidified my position on our current standards. “No, Mom; I can’t have half of five! Five isn’t an even number. I can’t make two equal groups out of those five parts. One is going to be left out!”

OK. I stood corrected. I was assured that my daughter not only grasps and understands standard 2.OA.3 but also connects it to the mathematics she is currently delving into. Again, it was not a problem her teacher had assigned or even a second-grade situation, but she “gets” it. Is she perfectly ready for third-grade equivalent-fraction assessments? Probably not. Do I think she will probably relate what she knows about mathematics and the strong foundational understanding that she has to just about any problem that is thrown her way (or thought up on her own)? Most definitely. Again, not because she has a math brain but because it makes sense; because it’s not about right and wrong all the time; and most important, because she has been taught to think, persevere, and reason.

How can you get your students to begin thinking this way? The eight SMP play an immense role in this. Here are a few things I believe students must have to engage in problem solving:

Students need context. Numbers in isolation can cause students to miss the connection between the world they live in and mathematics.

Students need time. They must be given the opportunity to think deeply. They don’t need to be asked to finish quickly or made to feel as though they are taking too long. Instead, we should encourage them to think about the mathematics another way, to draw a visual model or write an equation to represent their thinking, and to talk about it. If a few students finish quickly and have taken these these additional steps already, give them an opportunity to create some of their own related problems.

Students need to feel comfortable in making mistakes. Clearly, we want the solutions to their problems to be accurate, but often when students make mistakes and begin to explain their thinking, they catch the error. I find that students don’t mind sharing their errors with the rest of the class when they see mistakes as a part of their learning experience. When given the opportunity to prove whether their method for solving is true—and always true—they can think about why it does or does not work. This takes time. I am not suggesting that teachers should grade each problem but instead that they ask important questions, which can be as simple as “How did you think about that?”

Give it a try! Allow students to solve problems in context. Give them time to think. Allow them to make mistakes, ask them about their thinking, and give them time to find accurate solutions.

Martha Motley is a K–grade 6 instructional math coach in Kannapolis, North Carolina, City Schools. She spent almost fifteen years in the classroom, teaching exceptional children as well as regular education second and fourth grades, before beginning her coaching role. Motley has always enjoyed problem solving in mathematics. During the past few years, as her knowledge of mathematics and best math practices has grown, she has enjoyed teaching and sharing strategies with teachers to bring positive change to math instruction in her school district.