Chapter 5 discusses whether or not students should be grouped together based on prior attainment, which is sorting students into different groups based on their ability level. In the United States we group students later on in their years into different level classes, while in Japan every type of student is placed together because they believe it allows students, "to help each other, to learn from each other, to get along and grow together-mentally, physically and intellectually" (108). I personally agree with Japan's way of thinking. 

I believe that placing the same type of students together is unnecessary. Placing all different kinds of students together makes a classroom more diverse and this can help the different types of students learn from each other. I believe that lower achieving students can learn from higher achieving students. Clearly, mixed-ability classrooms are working out very well for Japanese students. 
Chapter 4 is entitled, "Taming the Monster". The monster being, assessments. This chapter discussed what is wrong with the American testing system. America uses multiple choice exams, which does not really assess understanding. Also, standardized assessments give questions involving unfamiliar scenarios and does not focus on the mathematics part of the question. Giving students an unfamiliar situation confuses students and makes it hard for them to visualize the situation.  

Assessments should be able to show a students understanding and assess them on what they have learned. I personally remember struggling with multiple choice exams when I was younger, and would do a lot better if an exam was a short answer assessment. I think it is important to make assessments with questions that are familiar to the students, yet challenging. This way, students will be able to show what they have learned about the subject area. 
Chapter 3 discussed three different types of mathematical teaching approaches in three different schools. The first school Boaler visited was Railside High School which used a communicative approach in teaching. Students would need to explain their work to other students. Students were also placed in different groups to discuss mathematical problems. The students of this school described math as being language instead of being a "set of rules". This way of teaching appeared to be very effective. 

I have experienced both a traditional way of being taught math and I have experienced the "communicative approach" Personally, I did not enjoy either of these methods. Though, the communicative approach seems to work at Railside. In high school, we were sat in different groups and we would be forced to work together. The problem was, it was high school, and a lot of people (at least at my high school) did not get along. I found myself a lot of the time not being able to discuss math because someone in my group thought they were "too cool" to talk to me. Another issue was the students that did not care about their education whatsoever. Those students would not do any work and I found myself doing their work for them, since they would just copy me. Using groups like this in high school is difficult because of these situations. For this method to work, I believe that it really depends on the class and their motivation to be successful in school. 

The other two approaches Boaler explored were the "Project-Based Approach" and the "Typical Traditional Approach". With a project-based approach there is a lot less organization, but this approach seemed to be very effective. This approach allows students to explore and choose what they want about mathematics. It allows students to be creative with math and makes it enjoyable. With the traditional approach the classroom was more controlled, but was a lot less creative and was very traditional. The Project Based students ended up doing better on examinations than the traditional mathematics students. This shows that creativity in the classroom, especially a math classroom, helps students be more successful.  
Chapter 2 is entitled, "What's Going Wrong in Classrooms?". This chapter introduced the phenomenon of "math wars" which is described as, "a series of unproductive and heated exchanges between advocates of different mathematics approaches" (Boaler 31). The war in a California classroom was between the "traditional curriculum" and a "reformed curriculum". Groups of people are trying to save the traditional math curriculum which consists of students sitting in isolation, doing math problems from their books. In a reform curriculum students would discuss math and teachers would try to make math more meaningful to the students. 

I believe that discussion in a mathematics classroom is essential. Going through traditional math classes when I was younger was very unhelpful for me. I remember sitting at my desk, in silence, and not doing anything because I had no idea how to do the problems. Getting all of the students engaged and discussing math I feel would definitely benefit all of the students. I believe that more students would understand the math better and it would be more meaningful to them. Educators need to realize math is not just memorization and just repeating problems and skills. There is so much more to math. I believe that getting students more engaged in math classrooms is essential for success of all students.