My first book to read this summer is Success from the Start: Your First Years Teaching Secondary Mathematics, by Rob Wieman and Fran Arbaugh. I picked this book up at the NCTM conference in Nashville in November 2015, but I have put off reading it until I was ready to begin making plans for my first year teaching. I haven't yet secured a job because it is so early in the summer, but it is my plan to read this book first to lay a foundation for the rest of the summer's reading; then, I will read it again at the end of the summer to bookend my adventures in reading and learning. Today I read the introduction and first part of this book. So far, I think it will be a great read to highlight some of the most important parts of mathematics teaching.
Introduction
The introduction talks about linking theory and practice by creating an analogy that really struck me. The authors say that just giving students algorithms to solve exercises is just like telling them to follow directions - it works out well unless they make a wrong turn, or there is an odd situation such as road construction. Instead, we need to give students a map - aka problem solving skills - so that they can figure out the best way forward when they encounter an issue. While this analogy is also attempting to connect the ideas of theory and practice when it comes to learning about teaching and actually teaching, I think this is a great way to frame the idea that we shouldn't always give students algorithms with which to solve problems. They need a map that will allow them to figure out lots of different routes to the solution! (This analogy can be found on page x).
Chapter 1: How Should Middle and High School Math Class Look and Sound?
This chapter provides a vignette of a middle school classroom run by a beginning teacher. As the scenario plays out, the reader is asked to reflect on different moves that the teacher made, and how these moves affected student learning. Many of the ideas brought forth have been discussed at length in some of my education classes, so there wasn't a whole lot of new information for me to absorb and reflect on.
Mainly, I was interested in the different phrases the teacher used in interacting with the students. At one point, the teacher asks, "Who will share?" when she wants students to provide some thoughts. I thought this was a unique way of asking for input and ideas - it is assertive without being demanding; it makes me feel as though someone will share, simply because of the word "will." Another time, the teacher asks a student to "run the conversation." I really liked this. In my graduate classes, we talked a lot about whole-class discussions, but mostly involving those run by the teacher. Realistically, there is no reason why a student shouldn't be able to run the conversation. This lets the student be more engaged and take more ownership. Of course, students would have to be taught the proper way to run and participate in conversations through modeling. I also liked that the teacher reminded students who finished the exercise/problem early to "make sure they could explain their thinking," because this gave them something to do that kept them focused on the mathematics. Sometimes I have seen teachers tell students to work on other things (read, etc) when they are finished, which can take away from their math time.
This vignette involved students solving linear equations, which is a common topic in middle school. One of the students brought up the idea that linear equations are like fact families (1+3=4, 4-3=1, 4-1=3), which allowed students to draw connections between previous skills from elementary school. I really liked this connection.
Mainly, I was interested in the different phrases the teacher used in interacting with the students. At one point, the teacher asks, "Who will share?" when she wants students to provide some thoughts. I thought this was a unique way of asking for input and ideas - it is assertive without being demanding; it makes me feel as though someone will share, simply because of the word "will." Another time, the teacher asks a student to "run the conversation." I really liked this. In my graduate classes, we talked a lot about whole-class discussions, but mostly involving those run by the teacher. Realistically, there is no reason why a student shouldn't be able to run the conversation. This lets the student be more engaged and take more ownership. Of course, students would have to be taught the proper way to run and participate in conversations through modeling. I also liked that the teacher reminded students who finished the exercise/problem early to "make sure they could explain their thinking," because this gave them something to do that kept them focused on the mathematics. Sometimes I have seen teachers tell students to work on other things (read, etc) when they are finished, which can take away from their math time.
This vignette involved students solving linear equations, which is a common topic in middle school. One of the students brought up the idea that linear equations are like fact families (1+3=4, 4-3=1, 4-1=3), which allowed students to draw connections between previous skills from elementary school. I really liked this connection.
Chapter 2: The Learning of Mathematics
The main thrust of this chapter involved the five parts of developing connected knowledge - the kind of knowledge that is necessary for success in mathematics. These five parts are explained in the concept map below.
My favorite quote from this chapter is, "If the person doing the task already knows how to arrive at an answer, then nothing is problematic about the task" (pg. 11), which implies that the students are not actually problem solving when they are just working out exercises. I think this is something that all teachers need to keep in mind - are students really doing real problem solving?
Another discussion in this chapter revolves around the idea that teachers need to give students the opportunity to engage with these five parts of developing connected knowledge, because that is the only way they will learn. In essence, "the key to their learning is in how they were thinking and what they were thinking about" (pg. 15). We as teachers have to give students opportunities to engage with the mathematics, and we have to monitor students to make sure that these opportunities are being used to their full potential.
Lastly, this chapter talks about redefining what it means to be a successful math student. This is oh-so-important for those students who may not be able to achieve good grades but have a good, logical, mathematical mind - if they can see that there are other ways to succeed in math, they will be more likely to invest in their practice and learning. The book provides a list of what successful mathematics students should be able to do, which aligns closely with the skills in the concept map above. I really like this list - it could be used as a checklist and provided to the students at the beginning of the year as a set of expectations. Then, throughout the year, students could be evaluated as to how they work towards these skills, so that they see there are multiple ways for them to be successful math students beyond just mastering number skills and algorithms.
Another discussion in this chapter revolves around the idea that teachers need to give students the opportunity to engage with these five parts of developing connected knowledge, because that is the only way they will learn. In essence, "the key to their learning is in how they were thinking and what they were thinking about" (pg. 15). We as teachers have to give students opportunities to engage with the mathematics, and we have to monitor students to make sure that these opportunities are being used to their full potential.
Lastly, this chapter talks about redefining what it means to be a successful math student. This is oh-so-important for those students who may not be able to achieve good grades but have a good, logical, mathematical mind - if they can see that there are other ways to succeed in math, they will be more likely to invest in their practice and learning. The book provides a list of what successful mathematics students should be able to do, which aligns closely with the skills in the concept map above. I really like this list - it could be used as a checklist and provided to the students at the beginning of the year as a set of expectations. Then, throughout the year, students could be evaluated as to how they work towards these skills, so that they see there are multiple ways for them to be successful math students beyond just mastering number skills and algorithms.
Chapter 3: The Teaching of Mathematics
It was hinted in Chapter 2 that teachers need to ensure that the students, not the teachers, are spending time thinking about the math, and this point is further explained and illustrated in Chapter 3. Essentially, the authors explain a "triangle of instruction" with the teacher, students, and mathematics occupying each vertex, and the sides representing the interactions between these (teacher-math, teacher-students, students-math). Even though the teacher-math and teacher-students interactions are important, it is stressed that the students-math interaction is the most important, and that this is where teacher focus should lie during instruction and class time. The teacher should always be working to stress the importance of the students doing and learning the mathematics themselves by redirecting questions to others and asking students to evaluate and explain.
This chapter briefly mentions that teachers too often try to break down topics into small pieces so that they are easier for students, but the issue with this tactic is that students do not have the capacity to put the pieces back together to see the whole picture. For this reason, it is important to make sure that the teacher works to help students make the connections to construct the bigger picture, so that the math topics they learn are not so disjointed.
The chapter ends with a discussion of assumptions - that assumptions can be dangerous to student success. While I wish there were some examples of the types of assumptions the authors mean, I can see their point in challenging teachers to identify and challenge their assumptions. Hopefully this topic is explained in more detail later.
This chapter briefly mentions that teachers too often try to break down topics into small pieces so that they are easier for students, but the issue with this tactic is that students do not have the capacity to put the pieces back together to see the whole picture. For this reason, it is important to make sure that the teacher works to help students make the connections to construct the bigger picture, so that the math topics they learn are not so disjointed.
The chapter ends with a discussion of assumptions - that assumptions can be dangerous to student success. While I wish there were some examples of the types of assumptions the authors mean, I can see their point in challenging teachers to identify and challenge their assumptions. Hopefully this topic is explained in more detail later.