My goal this year has not just been for me to provide opportunities for my students to enter into the Standards for Mathematical Practice – but for my students to begin to recognize when they are entering into the practices. I gave my students a list of the practices along with a short explanation of each at the beginning of the year. I have that same list posted in the front of my room. This language has become part of how we talk to each other about what we are doing in class – and what our goals are for learning in each unit.
Every quarter this year, I asked my students to reflect on a time when they participated in one of the mathematical practices. When we started, I told them I had no idea what these journals were supposed to look like. I gave them a few reflection questions for each practice, but I didn’t know if they were the right questions to ask. It has been an absolute pleasure to read their responses – especially to get to learn about times when they recognized that they were participating in a mathematical practice and I didn’t observe that happening.
I want to share some of those reflections with you.
1. Hannah writes about the mathematical practice “Make sense of problems and persevere in solving them.” She says, “As I was working in class I didn’t even notice that I was using a mathematical practice. I am going to be honest, I don’t like math in the first place, so when I don’t know how to work something I get very discouraged. I kept working, and even though I was one of the last people to finish, I still was proud of myself. I am learning not to give up and to keep going even though I can’t figure it out.
The students were given a matrix of the vertices of a triangle. You can see from Hannah’s work that they were trying to figure out by what matrix they could multiply the given matrix in order to produce the vertices of a certain transformed images.
Hannah continues – “Next time I am struggling with a problem I will think of this incident, take a deep breath, and persevere through it.”
2. Franky writes “This semester’s math class has been one of the most challenging classes I have taken. With that being said, it has also been my most beneficial, and it is one of my favorites. No matter how difficult a problem may be, I have learned to continue trying and persevering, for I should be able to figure the problem out. Franky goes on to describe a problem – and then he says to begin solving “of course, draw a triangle”. I have plenty of students who used to never think about drawing a diagram to make sense of a problem – some of them know it will help but just don’t want to take the time to do it – we are having to change the habits and practice of our students.
Franky continues – “There is no greater feeling than solving a math problem correctly, especially if it is difficult. This class has taught me time and time again to just keep on trying.”
I think it is significant for our students to know that we are going to give them problems for which they must persevere in solving.
3. Erin talks about the practice model with mathematics. While she was cycling she made a connection between linear distance (which we had studied in trigonometry) and how the sensor on her bike was able to tell her the distance traveled.
4. While he was at band practice, Sam made a connection between the band’s formation on the field and the transformations that we were studying in geometry. He writes “I am in the center of a diamond shape. We have to translate across the field.” – Note that this was early in the year and he had not yet heard my “a diamond is a gem and not a geometric shape” lecture.
5. Emilee writes about attending to precision. In her mind, she knows that negative cosecant of an angle is different from inverse cosecant of an angle, but she said negative cosecant, knowing very well she meant inverse cosecant. She writes “My ignorance of precision led to confusion among my table, but I am slowly learning to pay more attention to my words. Saying things, thingys, and whatchamacallit are not acceptable anymore.”
6. Kaci writes about “look for regularity in repeated reasoning”. We figured out that half of a square is a 45-45-90 triangle, and students were trying to determine the other two sides of the triangle given one side length of the triangle. She says “To find the length of the hypotenuse, you take the length of a side and multiply by sqrt(2). The sqrt(2) will always be in the hypotenuse even though it may not be seen like sqrt(2). In her examples, the triangle to the left has sqrt(2) shown in the hypotenuse, but the triangle to the right has sqrt(2) in the answer even though it isn’t shown, since 3sqrt(2)sqrt(2) is not in lowest form. She says, “I looked for regularity in repeated reasoning and found an interesting answer.”
7. Jordan writes about “Construct viable arguments and critique the reasoning of others”. She says, “If you can really understand something you can teach it. Every person relates to and thinks about problems in a different way, so understanding different ways to get to an answer can help to broaden your knowledge of the subject. Arguments are all about having good, logical facts. If you can be confident enough to argue for your reasoning you have learned the material well.”
8. And Franky says that construct viable arguments and critique the reasoning of others is probably our most used mathematical practice. If someone has a question about a problem, Mrs. Wilson is always looking for a student that understands the problem to explain it. And once he or she is finished, Mrs. Wilson will ask if anyone got the correct answer, but worked it a different way. By seeing multiple ways to work the problem, it is easier for me to fully understand.”
These student reflections have been very significant in my CCSS-M journey this year. They give me some hope that my students are at least beginning to understand that how we do and learn math is important to me. These student reflections give me hope as our journey continues …