In my geometry class, we talk about the Segment Addition Postulate.
If we take a segment 2 in. long and put it with a segment 3 in. long, we end up with a segment 5 in. long: the part plus the part equals the whole.
But as I reflect on what CCSS have done for my students, my coworkers, and me, I lean less towards Euclid and more towards Aristotle in my beliefs: the whole is greater than the sum of its parts. We are better together than we are alone. Mississippi educators are better with educators from all over the U.S. than we are alone. Mississippi students are better when their teachers are learning alongside other teachers all over the country than they are when we only share within our own school and state.
What has happened in my classroom over the past two years of implementing CCSS regarding student ownership of the mathematics we are learning is more visible than I would have predicted. The positive changes are not only undeniable; they are important, and they have been transformative for my students and for me.
The CCSS for mathematics are anchored by the Standards for Mathematical Practice, which is how we want students to learn math. The first math practice is to make sense of problems and persevere in solving them. How many times do you have to make sense of problems and persevere in solving them in the work that you do? How did you learn to make sense of problems and persevere in solving them?
Most students that I’ve encountered think that math is about a teacher demonstrating how to work a problem and a student mimicking the teacher. They are surprised on the first day of class to encounter problems that not only can be solved in more than one way but for which they are encouraged to solve in more than one way.
Another math practice is to construct a viable argument and critique the reasoning others. Most students that I’ve encountered think that math is about getting an answer. They are surprised when I encourage them to talk about how they are getting an answer – and even more surprised when I ask another student whether they agree or disagree with the process. We learn math by talking about how we are doing math. We even learn math when someone discusses a wrong method, and we make sense of the misconception. We celebrate learning from and correcting mistakes.
Another math practice is to look for regularity in repeated reasoning. If you took a high school geometry course, you learned about 45-45-90 and 30-60-90 special right triangles. In how many of your classes were you given the opportunity to think about the triangles formed by cutting a square in half on its diagonal or cutting an equilateral triangle in half on its height?
Were you given formulas for the relationships between the legs and hypotenuse of the right triangle (as you can see that I used to do from the picture of the transparency from which I used to teach)?
Or were you given an opportunity to make sense of the relationships – and maybe even figure them out yourselves by looking for patterns and talking about what you notice with other students?
Another math practice is to look for and make use of structure. What is the formula for the area of a triangle? Why? Can you calculate the area of a trapezoid? Looking for and making use of structure is about providing students opportunities to make sense of the formulas and equivalent expressions that we have traditionally given them and asked them to memorize without providing them the opportunity to make sense of why.
I won’t spend our time here talking about how the standards are not curriculum. If you don’t know what this means, maybe it would be helpful to hear that CCSS simply tells us that the Pythagorean theorem is a topic for introduction in grade 8 and then students learn more about it in a high school geometry course, but CCSS doesn’t tell us on what day of the year to teach the Pythagorean theorem or how to teach the Pythagorean theorem. As a teacher, I have full control over what standards I teach when and how I teach them. The standards are “the floor, not the ceiling”. They are the minimum that we expect our students to do in order to graduate as college and career ready. Students will still have the opportunity to take calculus and/or dual enrollment college math courses while they are in high school.
The Math Practices have been transformative for my students and for me. And what’s better is that CCSS says that all students will learn math using the math practices. Not just those who have a good teacher. Not just those who go to a good school. CCSS allows me to learn alongside educators from all over the United States who tweet and blog about their efforts to provide opportunities for their students to make sense of mathematics. I use free sites such as Illustrative Mathematics and the Mathematics Assessment Project that make it easy to search for tasks and lessons by standard.
And so the journey to help my students make sense of mathematics using the CCSS and the Standards for Mathematical Practice continues … And you are invited to come see it in action any day. Just be forewarned, my students will expect you to fully participate in the lesson – no one is just a spectator in our math class.