Math Book
In EMTH 310 class, we were required to read this textbook. Reading Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching by Jo Boaler. Each week, it was required for us to relect upon the designated chapter that was assigned that week. Throughout the process of reading this book, it provided valuable knowledge, ideas and different perspective when teaching mathematics. After reading this valuable book, I believe this is a valuable resource for teachers in providing them different ideas when teaching math. Below I have shared my thoughts and connection within reading this book. |
My thoughts
Chapter 2 & 3
In chapter 2; “The Power of Mistake and Struggle” it talks about the misconception about making mistakes in math. For most of us, at some point in our math education we have almost felt “defeated” when constantly getting the wrong answer. This conception of “making mistakes” in math has made lots of us have a different perception on math and teaching this to our future students. However, in chapter 2 it talks about how as teachers we need to be positive and promote making mistakes in math class. On page 12 it says; “when we make mistakes, our brains spark and grow. Mistakes are not only opportunities for learning, as students consider the mistakes, but also times when our brains grow, even if we don’t know we have made a mistake” (Boaler). This is relevant because as teachers we are also going to make mistakes within our career, so having a positive mindset about making “mistakes” we not only be good for yourself, but for your students. In the chapter it also talks about methods that teachers can change students' view on making a mistake. One of the examples consist of giving feedback to students with their “favourite mistake”. I think this is such a good idea because it still provides feedback that it is wrong but it shows that even though you got the answer wrong, I like how you did this. In conclusion of this chapter, it is important to promote the positive impact of making mistakes in math. Creating a positive impact on mistakes creates students to have a “growth mindset” rather than a “fixed mindset”.
In chapter 3; The Creativity and Beauty in Mathematics it talks about why math has become a “dead subject”. In relation to chapter 2 it discusses how math has created a negative impact on people and their perspective of doing it. This chapter argues that over many years, math has only been focused on certain aspects. On page 21-22 says; “when we ask students what math is, they will typically give descriptions that are very different from those given by experts in the field. Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what is, they will say it is the study of patterns that it is an aesthetic, creative and beautiful subject” (Boaler). As teachers we need to change this perception of math being“calculations, procedures, or rules” and make math more informative and relevant to our students' lives. This chapter builds on this concept; in the text it talks about a TED talk about mathematic has four stages “posing a question, going from the real world to a mathematical model, performing a calculation and going from the model back to the real world, to see if the original question was answered”(27). From these stages the TED talk explains that most educators only focus on the third stage “performing calculation” because it is the more important part of math. However, Boaler argues and expresses that math is ‘beautiful’ and making math more relatable to students will allow them to accept it more than just a ‘dead subject’.
After reading chapters 2 and 3 I was able to relate to them quickly and deeply. In chapter 2 it talks about how students have the fear of making mistakes. I can relate to this because when I was in grade 3, my teacher would do this activity with multiplication and flashcards. We would go out the hall and individually sit with the teacher to do the,. For me, I used to get so nervous doing this because I feared getting them wrong. Looking back on this my teacher would be so supportive when I would get one wrong, she would always say “practice makes perfect” and to be honest that is very true. I think once I got the fear of getting something wrong I was accepting of myself and my school work because in the end everyone makes mistakes. Another connection I made was in chapter 3 when it talks about students and parents questioning why they have to explain their work even when it's right? This is very relatable because I can remember being a student doing questions in my textbook and always being mad when I had to “explain my work”. For me, that never made sense to me because I never thought explaining how I got my answer relevant. After reading the chapter and having more of an understanding it makes sense to why we have to do this.
Chapter 4
In chapter four; “Creating Mathematical Mindsets: The importance of Flexibility with Numbers” it talks about the limitations that teachers provide to educate their students in math. At the beginning of the chapter, Boaler discusses the misconception that students have when learning mathematics. Boaler believes that in order for students to have a ‘growth mindset’; “children need to see math as a conceptual, growth subject that they should think about and make sense of it. When students see math as a series of short questions , they cannot see the role for their own inner growth and learning” (34). This is critical because restricting students with their ability to learn can create issues with no having an open mindset to learn. To support Boalers theory, she explains that the students that have been restricted in the ‘one way’ of learning something that have been redirected in only thinking in one way. For example, in the chapter, researchers examined several students (low and high achievers) in their ability to solve a simple math question (21-6). Examining their process in getting the answer, researchers noticed that all the students got the answer, but it varied in the length of time and process of doing it. The researcher noticed that high achievers would break down the question to be (20-5) and found that the lower achievers would understand the same question. Boaler expresses that this example is a common issue in most classrooms and has created an issue with students not having that ‘growth mindset’. To change this Boaler believes that it is important for teachers to adapt their teaching in mathematics to a wide variety of examples, lessons and related concepts that will help students understand the true meaning of what they are learning.
In the middle of the chapter Boaler states; “many students do not describe mathematics as a “real joy” [...] therefore students do not engage in conceptual thinking and instead approach mathematics as a list of rules to remember are not engaging in the critical process of compression, so their brain is unable to organize and file away ideas” (37). This is important because in the chapter it talks about the importance of being flexible in teaching your students. This is also important because every student learns in a different way. There are four types of learners are; visual, reading/writing, kinaesthetic and auditory. In math especially, teachers focus on only one type of learner which restricts students in their learning.
For most people, they have the perception that in order to consider being good at math the speed and simplicity are the key factors. However, Boaler argues that this is not the case, due to the fact that it creates ‘math anxiety’ for some students. In this section of the chapter it claims that being ‘quickest’ and solving math does not make you the smartest. In the chapter it states; “math facts by themselves are a small part of mathematics, and they are best learned through use of numbers in different ways and situations. Unfortunately, many classrooms focus on math facts in isolation, giving students the impression that math facts are the essence of mathematics, and, even worse, that mastering the fast recall of math facts is what it means to be a strong mathematics students” (38). With that being said, math anxiety is a big factor to students and their implication in learning math. In the chapter it talks about the simplicity in lessons and questions that have caused an issue when it comes to testing for students. The chapter talks about how the course work has been designed to teach math concepts in the simplest way. Boaler argues that this does not work for all students. Lessons and tests need to be changed because students learn in different ways and providing alternative approaches in teaching concepts will help more students understand. The chapter also talks about how researchers have discovered that the right and left side of the brain works together when solving problems, so being a teacher it is important to be mindful of that.
In my own personal experience I have felt the pressure of math anxiety. When I was in grade 3 I began feeling the pressure in math, especially when it came to multiplication. My teacher would constantly time us, and it was important to be under a minute. At a young age I had the perception that I needed to be quick when solving simple math questions. From reading this chapter I have realized that being the fastest in answering math questions does not make you the smartest. However, now being a future educator, I need to be mindful about my future students and their math abilities. Seeing your students quality in being able to solve the question is more important than seeing how fast they can do it. Creating this mentality with my future students will help them have a better mathematical mindset.
Chapter 5
In chapter 5; Rich Mathematical Tasks it talks about critical methods that are important to help students succeed in mathematics. At the beginning of the chapter it talks about six case studies that Boaler has done over the years. These case studies have created important concepts to help teachers with the ability to teach math to their students. The first case talks about the importance “seeing the openness of numbers”. In this case study Boaler talks about how teachers need to be open minded and elaborate when teaching students math. Many of the teachers participating in this case study, expressed their fear and frustration of students failing their math classes. Boaler simply states; “students were failing algebra not because algebra is so difficult, but because students don’t have number sense, which is the foundation for algebra” (58). With this, Boaler expressed that in order students to gain ‘number sense’ they need to see different math concepts in different ways. For example, in this case study Boaler asked the teachers to solve 18x5 visually. By teachers doing this it showed that there are multiple concepts in explaining this concept. The second case “Growing Shapes: the Power of Visualization” talks about the importance of visualization for students in math. In this case, Boaler expresses the importance of creating visuals for students to learn. Providing this to students gives them a better understanding, when grasping new concepts. Boaler examines summer school students in this case study to examine their development of concepts when using manipulative and visuals. In the end result of this method, students were able to grasp concepts and were able to succeed. The third case study “A Time to Tell?” shows the significance of allowing students more time to explore and develop concepts. Boaler states; “researcher found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problem, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn” (66). In the four cases; “Seeing a Mathematical Connection for the First time” Boaler talks about the importance of being able to create personal connections to help students understand math in different perspectives. Boaler believes that students understand concepts better when they are able to relate to what they are learning. The fifth case; “The Wonder of Negative Space” talks about the importance of working, forwards or backwards in math. The case talks about how some students grasp different concepts quicker than others. Boaler expresses that it is important to challenge students even when students understand concepts. Creating this challenge for students allows them to develop a deeper understanding of what they are learning and creating more of a ‘math sense’. The final case discussed in the reading, case 6; “From Math Facts to Math excitement”talks about how students were able to learn better when math lessons were ‘fun’. Boaler examines that students love learning math in a game sense. Teachers that provide ‘game like’ lessons will engage students and their willingness to learn.
After Boaler explains these particular cases in the reading, she talks about how you can incorporate these concepts into a lesson. She provides 5 easy suggestions; 1) open up the task so that there are multiple methods, pathway, and representation 2) Include inquiry opportunities 3) Ask the problem before teaching the method 4) Add a visual component and ask students how they see the mathematics 5) Extend the task to make it lower floor and higher ceiling 6) Ask students to convince and reason; be skeptical (90). The suggestion has been taken from Boaler’s case study that she has been a part of over the past years. Her involvement and case studies have provided us the best knowledge in helping us future teachers in making out students have a better mathematical mindset.
A few things that caught my attention when reading chapter 5; was when Boaler said; “Teachers are the most important resource for students” (57). I believe that this is a powerful statement because it expresses the impact that teachers make on their students. In my own experience, I can remember some of my teachers having a significant impact on me and their powerful presence in the classroom. I believe being a teacher we have a role to be a role model to our students. Although we have a job to teach our students, we also have the job to impact our students in a holistic sense.
Another concept that I agreed with was on page 69 when they talk about teachers having a ‘narrow mind’. This semester in school I have learned that there is endless potential in creating lessons and units for students. The curriculum is the foundation for teachers in designing lessons. I believe that teachers also have been narrow minded and playing their lesson ‘safe’. As a future teacher, I believe that I need to have an open mind when teaching my students because it allows students to learn better. School should be an exciting place for students, not something that they hate.
Chapter 6
In chapter six, the author addresses the importance of equity in mathematics. At the beginning of the chapter the author states his philosophy on how mathematics should be demonstrated in the classroom; “I want to live in a world where everyone can learn and enjoy math, and where everyone receives encouragement regardless of the color of their skin, their gender, their income, their sexuality, or any other characteristic” (93). Boaler expresses that in today’s classrooms students feel that math is a sorting mechanism to those who are “gifted” and who are not. Making students have the mentality of a typical stereotype of some students being naturally good at math. However, Boaler argues; “rather than recognizing and celebrating the nature of exceptional work and persistence, the U.S. education system focuses on “gifted” students who are given different opportunities, not because they show great tenacity and persistence but often because they are fast with math facts” (94). With this, Boaler also supported his argument, talking about how intelligence is not the factor that some teachers seek in their students, they also look upon their race and sexuality. From some of Baoler’s observations in some schools he has noticed that teachers will provide easier tests to the students that have a lower grade level and that male students that are not white will be acknowledged more. When Boaler observed the teacher’s at these schools, the teachers were confident in their teaching being effective to their students. However, Boaler stated that what they were actually doing with their students was not fair at all. The author used another observation to support his ideas to another school. At this Oakland school he explained that although this school is highly gang related, the teachers provide the best education to all of their students equally and it reflects on their students' performances. When Boaler was interviewing one of the teachers, she explained that it is important to provide the same knowledge and treat each student fairly, by doing this, it enhances their learning. Overall, Boaler’s observation in schools he noticed that there were many issues when it comes to students learning math.
Later in chapter, Boaler claims “equitable strategies” when teaching math to students. Boaler claim six strategies; offer all students high level content, work to change ideas about who can achieve in mathematics, encourage students to think deeply about mathematics, teach students to work together, give girls and students of color additional encouragement to learn math and science, and eliminate (or at least change the nature of) homework. Amongst all these strategies they eliminate the stereotyping and assumption that develop within students' mindset about math.
In the chapter, Boaler interviews about 30 people asking one basic question “can you tell me how you feel about math?” (99). Reflecting on the question to myself, I would believe that I would be confident in some areas of math. However, reading this chapter I was very surprised to see all of the stuff that I could relate to. For example, in the chapter it discusses that many students do not have an open mindset when learning math because they feel like they are not good at it. For me, looking at my own experiences I was never the student that was excited to learn math, but I never hated it. For me, it was always frustrating because I always had someone in my class that would try to be the smartest. Now after reading this chapter it has made me realize that sometimes the smartest student sometimes does not always understand the full concept of what is being taught. On page 101, on the bottom of the page shows a diagram. I was surprised by this diagram because I have never looked at interrupting math in this sense before. After understanding this diagram, that is something to consider when teaching in the future.
Another thing that I learned from this chapter was incorporating those particular strategies into my classroom. I believe that working in groups is so important in developing relationships with other students. On the other hand, another thing that I found important from this chapter that I can bring to my future classroom is to be excited in what I am teaching. Reflecting back to my own experience, with teachers teaching math, I can only remember a few teachers actually being excited and making math fun. So for me, in my future classroom I think it is important that I make lessons fun, so students can be excited about what they are learning.
Chapter 7
In chapter 7: From Tracking to Growth Mindset Grouping Boaler talks about the importance that educators need to be more equitable in providing a growth mindset for students. This chapter, Boaler talks about how teachers and our education system has provided a narrow opportunity for all students to develop all students equally. She argues that although all students learn at different rates, we should not be cutting off students' success based on their assessments and grading at a young age. At the beginning of the chapter Boaler talks about the topics 'detracking’ and ‘growth mindset grouping’ and fixating this in schools. The purpose of detracking in schools is to provide an equal opportunity in mathematics. Having tracking throughout schools over the years has created division between students that are good and not good at math. Creating issues for students and their futures. Boaler supports this statement by talking about a recent study done in New York, where a school eliminated all advanced classes in schools. As a result, the study showed that students who had never taken advanced classes were succeeding better in classes due to them being able to enjoy what was being taught. The things that were discussed in the chapter was changing students' growth minds grouping. Boaler talks about the importance of this because she believes that when students are divided at a young age whether they are good at math or not developed a close mindset throughout the years. Boaler supports this by using an example of a teacher expressing her frustration of teaching math to students that are ‘not the math people’. The teacher talks about how she knows that her students are not dumb when it comes to math, they just have developed a closed mindset due to be divided at a young age of being in the lower class of mathematics.
Later in the chapter Boaler begins discussing effective ways of teaching math classes and how it can eliminate de-tracking. She begins by talking about “Teaching Heterogeneous Groups Effectively: The Mathematics Task”. In this method it is broken down into 3 levels; 1) Providing open ended tasks, 2) Offering a choice of Tasks, 3) Individualized Pathways. In each task it talks about how students can effectively develop an open mindset by creating options, choices and freedom when it comes to their reading. Next, Boaler talks about “Teaching Heterogeneous Groups Effectively; Complex Instruction. She believes it is a nessectitie in the classroom because it provides students to be informed about what is being taught and allows students to debrief with classmates and group work. She also mentions the importance of ‘Multidimensionality’ in mathematics. The purpose of this is to provide a variety of options for students to learn. It has been argued in many of these chapters that students learn at different rates, so providing the option will help students learn better. Boaler also talks about the importance of incorporating group work and students working together in understanding math. It is not only beneficial in providing students to learn from one another, but it also creates an impact on students' communication skills.
In my connection to this chapter I agree with Boaler’s argument that advanced classes should be eliminated in schools. Last week, for chapter six, I was the group leader. One of my questions was do you believe advanced classes are beneficial to students why or why not? With most of my group believing not, and now reading what Boaler had to say about it I can completely agree that it is not beneficial at all. In my own experience I never took advanced classes but I always had the mentality that the students that took those classes were smarter than me. Which now thinking back on that, I think that is why I developed a closed mindset when I got to high school. Overall this chapter was good in explaining the importance that teachers need to adjust in their classroom. Although sometimes the easiest route, by providing easy work to all students seems better, we need to acknowledge and work harder in providing better opportunities to all of our students.
In chapter 2; “The Power of Mistake and Struggle” it talks about the misconception about making mistakes in math. For most of us, at some point in our math education we have almost felt “defeated” when constantly getting the wrong answer. This conception of “making mistakes” in math has made lots of us have a different perception on math and teaching this to our future students. However, in chapter 2 it talks about how as teachers we need to be positive and promote making mistakes in math class. On page 12 it says; “when we make mistakes, our brains spark and grow. Mistakes are not only opportunities for learning, as students consider the mistakes, but also times when our brains grow, even if we don’t know we have made a mistake” (Boaler). This is relevant because as teachers we are also going to make mistakes within our career, so having a positive mindset about making “mistakes” we not only be good for yourself, but for your students. In the chapter it also talks about methods that teachers can change students' view on making a mistake. One of the examples consist of giving feedback to students with their “favourite mistake”. I think this is such a good idea because it still provides feedback that it is wrong but it shows that even though you got the answer wrong, I like how you did this. In conclusion of this chapter, it is important to promote the positive impact of making mistakes in math. Creating a positive impact on mistakes creates students to have a “growth mindset” rather than a “fixed mindset”.
In chapter 3; The Creativity and Beauty in Mathematics it talks about why math has become a “dead subject”. In relation to chapter 2 it discusses how math has created a negative impact on people and their perspective of doing it. This chapter argues that over many years, math has only been focused on certain aspects. On page 21-22 says; “when we ask students what math is, they will typically give descriptions that are very different from those given by experts in the field. Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what is, they will say it is the study of patterns that it is an aesthetic, creative and beautiful subject” (Boaler). As teachers we need to change this perception of math being“calculations, procedures, or rules” and make math more informative and relevant to our students' lives. This chapter builds on this concept; in the text it talks about a TED talk about mathematic has four stages “posing a question, going from the real world to a mathematical model, performing a calculation and going from the model back to the real world, to see if the original question was answered”(27). From these stages the TED talk explains that most educators only focus on the third stage “performing calculation” because it is the more important part of math. However, Boaler argues and expresses that math is ‘beautiful’ and making math more relatable to students will allow them to accept it more than just a ‘dead subject’.
After reading chapters 2 and 3 I was able to relate to them quickly and deeply. In chapter 2 it talks about how students have the fear of making mistakes. I can relate to this because when I was in grade 3, my teacher would do this activity with multiplication and flashcards. We would go out the hall and individually sit with the teacher to do the,. For me, I used to get so nervous doing this because I feared getting them wrong. Looking back on this my teacher would be so supportive when I would get one wrong, she would always say “practice makes perfect” and to be honest that is very true. I think once I got the fear of getting something wrong I was accepting of myself and my school work because in the end everyone makes mistakes. Another connection I made was in chapter 3 when it talks about students and parents questioning why they have to explain their work even when it's right? This is very relatable because I can remember being a student doing questions in my textbook and always being mad when I had to “explain my work”. For me, that never made sense to me because I never thought explaining how I got my answer relevant. After reading the chapter and having more of an understanding it makes sense to why we have to do this.
Chapter 4
In chapter four; “Creating Mathematical Mindsets: The importance of Flexibility with Numbers” it talks about the limitations that teachers provide to educate their students in math. At the beginning of the chapter, Boaler discusses the misconception that students have when learning mathematics. Boaler believes that in order for students to have a ‘growth mindset’; “children need to see math as a conceptual, growth subject that they should think about and make sense of it. When students see math as a series of short questions , they cannot see the role for their own inner growth and learning” (34). This is critical because restricting students with their ability to learn can create issues with no having an open mindset to learn. To support Boalers theory, she explains that the students that have been restricted in the ‘one way’ of learning something that have been redirected in only thinking in one way. For example, in the chapter, researchers examined several students (low and high achievers) in their ability to solve a simple math question (21-6). Examining their process in getting the answer, researchers noticed that all the students got the answer, but it varied in the length of time and process of doing it. The researcher noticed that high achievers would break down the question to be (20-5) and found that the lower achievers would understand the same question. Boaler expresses that this example is a common issue in most classrooms and has created an issue with students not having that ‘growth mindset’. To change this Boaler believes that it is important for teachers to adapt their teaching in mathematics to a wide variety of examples, lessons and related concepts that will help students understand the true meaning of what they are learning.
In the middle of the chapter Boaler states; “many students do not describe mathematics as a “real joy” [...] therefore students do not engage in conceptual thinking and instead approach mathematics as a list of rules to remember are not engaging in the critical process of compression, so their brain is unable to organize and file away ideas” (37). This is important because in the chapter it talks about the importance of being flexible in teaching your students. This is also important because every student learns in a different way. There are four types of learners are; visual, reading/writing, kinaesthetic and auditory. In math especially, teachers focus on only one type of learner which restricts students in their learning.
For most people, they have the perception that in order to consider being good at math the speed and simplicity are the key factors. However, Boaler argues that this is not the case, due to the fact that it creates ‘math anxiety’ for some students. In this section of the chapter it claims that being ‘quickest’ and solving math does not make you the smartest. In the chapter it states; “math facts by themselves are a small part of mathematics, and they are best learned through use of numbers in different ways and situations. Unfortunately, many classrooms focus on math facts in isolation, giving students the impression that math facts are the essence of mathematics, and, even worse, that mastering the fast recall of math facts is what it means to be a strong mathematics students” (38). With that being said, math anxiety is a big factor to students and their implication in learning math. In the chapter it talks about the simplicity in lessons and questions that have caused an issue when it comes to testing for students. The chapter talks about how the course work has been designed to teach math concepts in the simplest way. Boaler argues that this does not work for all students. Lessons and tests need to be changed because students learn in different ways and providing alternative approaches in teaching concepts will help more students understand. The chapter also talks about how researchers have discovered that the right and left side of the brain works together when solving problems, so being a teacher it is important to be mindful of that.
In my own personal experience I have felt the pressure of math anxiety. When I was in grade 3 I began feeling the pressure in math, especially when it came to multiplication. My teacher would constantly time us, and it was important to be under a minute. At a young age I had the perception that I needed to be quick when solving simple math questions. From reading this chapter I have realized that being the fastest in answering math questions does not make you the smartest. However, now being a future educator, I need to be mindful about my future students and their math abilities. Seeing your students quality in being able to solve the question is more important than seeing how fast they can do it. Creating this mentality with my future students will help them have a better mathematical mindset.
Chapter 5
In chapter 5; Rich Mathematical Tasks it talks about critical methods that are important to help students succeed in mathematics. At the beginning of the chapter it talks about six case studies that Boaler has done over the years. These case studies have created important concepts to help teachers with the ability to teach math to their students. The first case talks about the importance “seeing the openness of numbers”. In this case study Boaler talks about how teachers need to be open minded and elaborate when teaching students math. Many of the teachers participating in this case study, expressed their fear and frustration of students failing their math classes. Boaler simply states; “students were failing algebra not because algebra is so difficult, but because students don’t have number sense, which is the foundation for algebra” (58). With this, Boaler expressed that in order students to gain ‘number sense’ they need to see different math concepts in different ways. For example, in this case study Boaler asked the teachers to solve 18x5 visually. By teachers doing this it showed that there are multiple concepts in explaining this concept. The second case “Growing Shapes: the Power of Visualization” talks about the importance of visualization for students in math. In this case, Boaler expresses the importance of creating visuals for students to learn. Providing this to students gives them a better understanding, when grasping new concepts. Boaler examines summer school students in this case study to examine their development of concepts when using manipulative and visuals. In the end result of this method, students were able to grasp concepts and were able to succeed. The third case study “A Time to Tell?” shows the significance of allowing students more time to explore and develop concepts. Boaler states; “researcher found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problem, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn” (66). In the four cases; “Seeing a Mathematical Connection for the First time” Boaler talks about the importance of being able to create personal connections to help students understand math in different perspectives. Boaler believes that students understand concepts better when they are able to relate to what they are learning. The fifth case; “The Wonder of Negative Space” talks about the importance of working, forwards or backwards in math. The case talks about how some students grasp different concepts quicker than others. Boaler expresses that it is important to challenge students even when students understand concepts. Creating this challenge for students allows them to develop a deeper understanding of what they are learning and creating more of a ‘math sense’. The final case discussed in the reading, case 6; “From Math Facts to Math excitement”talks about how students were able to learn better when math lessons were ‘fun’. Boaler examines that students love learning math in a game sense. Teachers that provide ‘game like’ lessons will engage students and their willingness to learn.
After Boaler explains these particular cases in the reading, she talks about how you can incorporate these concepts into a lesson. She provides 5 easy suggestions; 1) open up the task so that there are multiple methods, pathway, and representation 2) Include inquiry opportunities 3) Ask the problem before teaching the method 4) Add a visual component and ask students how they see the mathematics 5) Extend the task to make it lower floor and higher ceiling 6) Ask students to convince and reason; be skeptical (90). The suggestion has been taken from Boaler’s case study that she has been a part of over the past years. Her involvement and case studies have provided us the best knowledge in helping us future teachers in making out students have a better mathematical mindset.
A few things that caught my attention when reading chapter 5; was when Boaler said; “Teachers are the most important resource for students” (57). I believe that this is a powerful statement because it expresses the impact that teachers make on their students. In my own experience, I can remember some of my teachers having a significant impact on me and their powerful presence in the classroom. I believe being a teacher we have a role to be a role model to our students. Although we have a job to teach our students, we also have the job to impact our students in a holistic sense.
Another concept that I agreed with was on page 69 when they talk about teachers having a ‘narrow mind’. This semester in school I have learned that there is endless potential in creating lessons and units for students. The curriculum is the foundation for teachers in designing lessons. I believe that teachers also have been narrow minded and playing their lesson ‘safe’. As a future teacher, I believe that I need to have an open mind when teaching my students because it allows students to learn better. School should be an exciting place for students, not something that they hate.
Chapter 6
In chapter six, the author addresses the importance of equity in mathematics. At the beginning of the chapter the author states his philosophy on how mathematics should be demonstrated in the classroom; “I want to live in a world where everyone can learn and enjoy math, and where everyone receives encouragement regardless of the color of their skin, their gender, their income, their sexuality, or any other characteristic” (93). Boaler expresses that in today’s classrooms students feel that math is a sorting mechanism to those who are “gifted” and who are not. Making students have the mentality of a typical stereotype of some students being naturally good at math. However, Boaler argues; “rather than recognizing and celebrating the nature of exceptional work and persistence, the U.S. education system focuses on “gifted” students who are given different opportunities, not because they show great tenacity and persistence but often because they are fast with math facts” (94). With this, Boaler also supported his argument, talking about how intelligence is not the factor that some teachers seek in their students, they also look upon their race and sexuality. From some of Baoler’s observations in some schools he has noticed that teachers will provide easier tests to the students that have a lower grade level and that male students that are not white will be acknowledged more. When Boaler observed the teacher’s at these schools, the teachers were confident in their teaching being effective to their students. However, Boaler stated that what they were actually doing with their students was not fair at all. The author used another observation to support his ideas to another school. At this Oakland school he explained that although this school is highly gang related, the teachers provide the best education to all of their students equally and it reflects on their students' performances. When Boaler was interviewing one of the teachers, she explained that it is important to provide the same knowledge and treat each student fairly, by doing this, it enhances their learning. Overall, Boaler’s observation in schools he noticed that there were many issues when it comes to students learning math.
Later in chapter, Boaler claims “equitable strategies” when teaching math to students. Boaler claim six strategies; offer all students high level content, work to change ideas about who can achieve in mathematics, encourage students to think deeply about mathematics, teach students to work together, give girls and students of color additional encouragement to learn math and science, and eliminate (or at least change the nature of) homework. Amongst all these strategies they eliminate the stereotyping and assumption that develop within students' mindset about math.
In the chapter, Boaler interviews about 30 people asking one basic question “can you tell me how you feel about math?” (99). Reflecting on the question to myself, I would believe that I would be confident in some areas of math. However, reading this chapter I was very surprised to see all of the stuff that I could relate to. For example, in the chapter it discusses that many students do not have an open mindset when learning math because they feel like they are not good at it. For me, looking at my own experiences I was never the student that was excited to learn math, but I never hated it. For me, it was always frustrating because I always had someone in my class that would try to be the smartest. Now after reading this chapter it has made me realize that sometimes the smartest student sometimes does not always understand the full concept of what is being taught. On page 101, on the bottom of the page shows a diagram. I was surprised by this diagram because I have never looked at interrupting math in this sense before. After understanding this diagram, that is something to consider when teaching in the future.
Another thing that I learned from this chapter was incorporating those particular strategies into my classroom. I believe that working in groups is so important in developing relationships with other students. On the other hand, another thing that I found important from this chapter that I can bring to my future classroom is to be excited in what I am teaching. Reflecting back to my own experience, with teachers teaching math, I can only remember a few teachers actually being excited and making math fun. So for me, in my future classroom I think it is important that I make lessons fun, so students can be excited about what they are learning.
Chapter 7
In chapter 7: From Tracking to Growth Mindset Grouping Boaler talks about the importance that educators need to be more equitable in providing a growth mindset for students. This chapter, Boaler talks about how teachers and our education system has provided a narrow opportunity for all students to develop all students equally. She argues that although all students learn at different rates, we should not be cutting off students' success based on their assessments and grading at a young age. At the beginning of the chapter Boaler talks about the topics 'detracking’ and ‘growth mindset grouping’ and fixating this in schools. The purpose of detracking in schools is to provide an equal opportunity in mathematics. Having tracking throughout schools over the years has created division between students that are good and not good at math. Creating issues for students and their futures. Boaler supports this statement by talking about a recent study done in New York, where a school eliminated all advanced classes in schools. As a result, the study showed that students who had never taken advanced classes were succeeding better in classes due to them being able to enjoy what was being taught. The things that were discussed in the chapter was changing students' growth minds grouping. Boaler talks about the importance of this because she believes that when students are divided at a young age whether they are good at math or not developed a close mindset throughout the years. Boaler supports this by using an example of a teacher expressing her frustration of teaching math to students that are ‘not the math people’. The teacher talks about how she knows that her students are not dumb when it comes to math, they just have developed a closed mindset due to be divided at a young age of being in the lower class of mathematics.
Later in the chapter Boaler begins discussing effective ways of teaching math classes and how it can eliminate de-tracking. She begins by talking about “Teaching Heterogeneous Groups Effectively: The Mathematics Task”. In this method it is broken down into 3 levels; 1) Providing open ended tasks, 2) Offering a choice of Tasks, 3) Individualized Pathways. In each task it talks about how students can effectively develop an open mindset by creating options, choices and freedom when it comes to their reading. Next, Boaler talks about “Teaching Heterogeneous Groups Effectively; Complex Instruction. She believes it is a nessectitie in the classroom because it provides students to be informed about what is being taught and allows students to debrief with classmates and group work. She also mentions the importance of ‘Multidimensionality’ in mathematics. The purpose of this is to provide a variety of options for students to learn. It has been argued in many of these chapters that students learn at different rates, so providing the option will help students learn better. Boaler also talks about the importance of incorporating group work and students working together in understanding math. It is not only beneficial in providing students to learn from one another, but it also creates an impact on students' communication skills.
In my connection to this chapter I agree with Boaler’s argument that advanced classes should be eliminated in schools. Last week, for chapter six, I was the group leader. One of my questions was do you believe advanced classes are beneficial to students why or why not? With most of my group believing not, and now reading what Boaler had to say about it I can completely agree that it is not beneficial at all. In my own experience I never took advanced classes but I always had the mentality that the students that took those classes were smarter than me. Which now thinking back on that, I think that is why I developed a closed mindset when I got to high school. Overall this chapter was good in explaining the importance that teachers need to adjust in their classroom. Although sometimes the easiest route, by providing easy work to all students seems better, we need to acknowledge and work harder in providing better opportunities to all of our students.