The Mathematical Practices
Lesson 4 of 6
Objective: SWBAT determine the mathematical practices used in a class.
What is a mathematician?
Are you a mathematician?
I have these 2 questions on the board. After students have thought about these we will discuss these.
I want students to realize that they are mathematicians when they ask questions, look for patterns, solve problems. They need to also understand that part of being a mathematician is to make mistakes.
I will ask what is a mathematical practice? I do not expect students to list the 8 from the standards but I want the students to think in general terms about how to do mathematics.
After giving the students a few minutes to express their initial ideas, I hand out a sheet that has the 8 mathematical practices from the Common Core State Standards (CCSS). I ask the students what they notice about each practice. Then, I add commentary similar to my notes below.
MP 1- Make sense of problems and persevere in solving them. I ask the students to read the first practice and think about what it means. We discuss what it means. I ask is it alright to be given problems you have never seen? How should you approach problems you have never seen? What is perseverance?
MP 2- Reasoning abstractly and quantitatively is hard for students to understand. We discuss what the words mean. I explain when we do examples to see the pattern we are reasoning quantitatively, when we write an equation for the pattern we are then reasoning abstractly.
MP 3-Construct viable arguments and critique the reasoning of others. Students immediately think about Geometric proofs. I explain that we will look at other proofs but it is important to validate what you find. For critiquing arguments, I ask students how to solve 3(x-8)=24. I let a student put their process on the board. I then put another method on the board and say who is right? We discuss when each method is good to use. The two methods we look at are distributing first compared to dividing by 3 first. This is usually a good discuss. Most students will distribute first. We discuss when dividing first is more efficient. I explain that we have just critiqued the work of others.
MP 4- Modeling in mathematics. The students will say make pictures. I agree and then ask if we write an equation for a situation are we modeling?
MP 5-Use appropriate tools strategically. I explain that you do not use a hammer to screw in a screw or a screwdriver to hammer in a nail. It might work but it is not the best method. All the skills you have already learned you need to use to solve problems. Deciding which is best is what this standard is about.
MP 6-Attend to precision. The student will saying making sure you get the right answer. I will explain it is also about labeling you answer using the appropriate units. Defining your variables when you write and equation and labeling your axes when you graph are also part of the practice.
MP 7-Look for and make use of structure. I explain that this is where we can see how multiplying by 1/2 is the same as dividing by 2.
MP 8-Look for and express regularity in repeated reasoning. I discuss how I like to look for patterns. I look at previous work to see if the problem I am working on is similar to something I have already done.
I want students to see that each practice starts with a mathematical proficient student. According to this who is doing these practices? I explain that all of us will do these practices. I then tell the students that I will need help with attending to precision. I am bad about not labeling and I need to get better at that. The students need to correct me whenever I forget.
Can we learn from mistakes?
After discussing the mathematical practices I ask the class:
Are famous mathematicians always correct?
Here is a story from the recent past about one of the most famous theorems in mathematics: Fermat's Last Theorem states has no integer solutions for n>2. This theorem was originally found in a side note of a mathematics book in 1637. In the note Pierre de Fermat stated that the proof was to large to put in the space. For many years mathematicians tried and failed to come up with a proof. No one was successful until Andrew Wiles published a "proof" in 1993. Several months later, Wiles' proof was shown to have a major flaw (MP3). Fortunately, Mr. Wiles worked with his colleagues to correct the mistake and create a correct proof of Fermat's Last Theorem (MP1).
I now share a couple of video clips of famous people to see how even if you fail at first you can become very successful. The first video is a Nike commercial with Michael Jordan. The second is a presentation from "Motivating Success." Many famous people are shown and discuss how they at first had failure before becoming very successful. We many times only see success and don't realize how many times someone has failed before the person succeeds.
Once we have viewed the videos and talked about Fermat I focus on how this class will work:
- Problem solving processes often result in many unsuccessful attempts before a correct solution is found.
- Sharing your own work and critiquing the work of others is an important part of mathematical thinking and learning.
- We can all learn from others mistakes
I want students to understand that it is alright not to be correct all the time. Some students will not share their work unless I have checked it for being accurate. I want students to understand that everyone learns from errors. I also need students to be respectful and help each other learn.
I ask students what mathematical practices are you doing when you correct another person's error? This let's them see that they have been doing some of these practices for a number of years.
I let students know that we will be discussing the mathematical practices throughout the school year. I will be grading students on the math practices. I share the scoring guide for MP1. Students read through the scoring guide and discuss any questions.
I review what we have done so far this year. I say that we have been working at getting ready for learning mathematics in my class. Learning math requires study skills and mathematical practices.
I make sure students know that I will always help them. I make sure they understand that I really want them to grow as a student and as a mathematician through their experiences in my class.
This is the last lesson for this unit. We do not have an assessment. I will informally assess the unit throughout the year by informally analyzing students' use of the the ideas discussed. I will look to see if students are using the study skills, and the math practices. I will note if a student mentions using an Internet site when we are discussing new ideas
Students are asked to get out a piece of paper and reflect on the following questions. Names are not necessary. I explain that I am getting some data about the unit so I can improve the unit for next year.
Why did I feel these lessons were important?
What did you learn?
What was the most useful information?
What would make this unit better?
I give students about 5 minutes to write. I feel that student feedback helps me in my instruction and gives the students a way to be a part of the class.
I end the class by previewing the next unit. Our course starts with exponential and logarithmic functions. We begin by reviewing some concepts with functions and then extend our list of functions to logarithmic and exponential. We end the unit by seeing how these functions are used to solve problems.
Students prefer to know what they are going to learn. Some have some prior knowledge and this gives them a chance to remember the material before we begin.