Monthly Archives: June 2015

What’s Math Got To Do With It? Reflections and Thoughts, Part 3

I recently finished reading Jo Boaler’s book What’s Math Got To Do With It? and I need a way to reflect on what I had learned and consider ways to implement her research in my own teaching. This is Part 3 of a (probably) 4-part series.

  • Part 1
  • Part 2
  • Part 4 (in progress ok, let’s be honest – I never wrote it)

I’m going to use quotes from the book and expand on them with my own thoughts. Chapters 3 through 6 primarily outline the findings that Jo has gathered from her research over the years, and I’m going to cover Chapters 5 and 6 in this post. The last few chapters (later post) discuss how to actually implement the things Jo has uncovered in her research with children.

Chapter 5 – Stuck in the Slow Lane: How American Grouping Systems Perpetuate Low Achievement

This chapter was depressing too. It’s actually really frustrating to read about all the things we do in this country that are extremely widespread and extremely counter-productive. Imagine that.

In the US, we group students based on their prior achievement. This is usually called ability-based grouping which, frankly, says all you need to know about the system. We assume that students who are placed in a certain group have a certain ability, ignoring everything that we have learned recently about how the brain works, and how mindset influences a student’s success. Any type of tracking or ability-based grouping leads to a fixed mindset (in all students – those at the top, bottom, and middle) which contributes to their lack of success in school. (It’s self-perpeutating.) Students at the bottom and in the middle believe they can never improve, and their teachers only give them easier work, so they wind up being right – they can’t improve. Students at the top have to maintain the image of being the best, so they never challenge themselves and become obsessed with appearing to be perfect. See? Damaging.

When students are split up in this way, they also don’t have a variety of students to work with, which has major advantages. “In mixed-ability classes the students are organized to work with each other and help each other” (p. 112). The students who need help can ask other students for it. This has two major advantages in my experience – the teacher doesn’t have to be everywhere at once, and the students tend to understand it better if they hear it from another student anyway. The student who is doing the helping also benefits because “the act of explaining work to others deepens understanding” (p. 112).

In order for mixed-ability grouping to be effective, two conditions must be in place.

  1. “The students must be given open work that can be accessed at different levels and taken to different levels” (p. 116). The students who struggle with the material need to have a place to start, and the students who get it easily need to have a way to extend the problem as much as they want.
  2. “Students are taught to work respectfully with each other” (p. 117). I’m worried about this one because I feel like I don’t know how to teach this. I need to make sure that I am hearing all the interactions among my students so I can correct any that are disrespectful.

Chapter 6 – Paying the Price for Sugar and Spice: How Girls and Women Are Kept Out of Math and Science

This chapter was fascinating. This was the chapter that I was reading on a short flight home from vacation and kept handing the book to my roommate who was sitting across the aisle from me and telling her to read passages.

In short: The biggest difference that Jo has found between the way girls learn math and the way body learn math is that boys are ok with just being given a procedure and being told to do it, while girls want to know why it works.

Quotes from students themselves demonstrate this best:

He’ll write it on the board and you end up thinking, “Well, how come this and this? How did you get that answer? Why did you do that?” (p. 123)

It’s like you have to work it out and you get the right answers but you don’t know what you did. You don’t know how you got them, you know? (p. 124)

K: I’m just not interested in, just, you give me a formula, I’m supposed to memorize the answer and apply it, and that’s it.
JB: Does math have to be like that?
B: I’ve just kind of learned it that way. I don’t know if there’s any other way.
K: At the point I am right now, that’s all I know. (p. 126)

Math is more, like, concrete, it’s so “It’s that and that’s it.” Women are more, they want to explore stuff and that’s life kind of, like, and I think that’s why I like English and science. I’m more interested in, like, phenomena and nature and animals and I’m just not interested in, just, you give me a formula, I’m supposed to memorize the answer and apply it, and that’s it. (p. 127)

Jo discusses a study that was done based off what she had uncovered about gender roles in these courses, and concludes, “…the girls wanted opportunities to inquire deeply, and they were averse to versions of the subjects that emphasized rote learning” (p. 128). On the other hand, boys “would tell me that they were happy as long as they were getting answers correct. The boys seemed to enjoy completing work at a fast pace and competing with other students, and they did not seem to need the same depth of understanding” (p. 123). In fact, Jo found that about one third of boys also valued understanding why similar to the way the girls did.

We urgently need to reorient mathematics and other subjects so that they focus on understanding and deep inquiry. When such changes are made girls choose STEM subjects in equal numbers to boys, and this is a goal that we should prioritize in the United States, not only for the futures of girls and women but for the future of the STEM disciplines. (p. 136-137)

How can I do this in my classroom? I need to focus on teaching WHY things work the way they do, not just how they work. I need to have students discuss and chew on these explanations and make sure that they hold themselves and each other to the ideal that reasoning is more important than calculation.

What’s Math Got To Do With It? Reflections and Thoughts, Part 2

I recently finished reading Jo Boaler’s book What’s Math Got To Do With It? and I need a way to reflect on what I had learned and consider ways to implement her research in my own teaching. This is Part 2 of a (probably) 4-part series.

I’m going to use quotes from the book and expand on them with my own thoughts. Chapters 3 through 6 primarily outline the findings that Jo has gathered from her research over the years, and I’m going to cover Chapters 3 and 4 in this post.

Chapter 3 – A Vision for a Better Future: Effective Classroom Approaches

The Communicative Approach

Jo highlights a school that uses what she calls the communicative approach to teach math. In this approach, students “were given opportunities to work on interesting problems that required them to think (not just to reproduce methods), and they were required to discuss mathematics with one another, increasing their interest and enjoyment” (p. 66). The teachers in particular believed and taught from the point of view that there are many different ways of being “good” at math:

…being good at mathematics involved many different ways of working, as mathematicians’ accounts tell us. It involves asking questions, drawing pictures and graphs, re-phrasing problems, justifying methods, and representing ideas, in addition to calculating with procedures. Instead of just rewarding the correct use of procedures, the teachers encouraged and rewarded all of these different ways of being mathematical.

… Put simply, because there were so many more ways to be successful at [this school], many more students were successful (p. 67).

This is easily the hardest thing for me as a teacher – not to revert back to teaching the way I was taught. I fall into the trap of valuing the fastest way to get things done, I undervalue sense-making in favor of “we have to get through this before the bell rings”, and I allow students to just memorize how to do things. I want so badly to get away from these habits, but it means that I need to replace them with new habits. That’s where I’m struggling. I’m a very detail-oriented person, so when I think about changing aspects of my teaching, I want to be specific: Instead of saying _____, I’ll say _____. Unfortunately, teaching isn’t so prescriptive. I need to catch myself in the act, and think on my feet well enough to make the change.

Side note: Because I know this about myself, I’m going to ask for my students’ help when it comes to mindset this year. I’m going to teach a lesson about mindset and how your brain grows during the first week of school, and I’m already planning to follow it up with this statement to my students: “I have had a fixed mindset my whole life, especially about school. I’m learning how to have a growth mindset along with you guys. Because of this, I sometimes say things that come from my old fixed mindset. When I say those things, I want you guys to call me out on it. If I don’t realize I’m doing it, I can’t fix it, so I need your help to keep making that change.” I’m hoping that a high-school student’s natural love for telling the teacher that she’s wrong will make them very excited about this and help keep me in line.

The Project-Based Approach

I actually didn’t highlight anything in this section, either because I was reading so fast that I didn’t think to stop and highlight, or because there wasn’t a highlight-able (short) sentence available. Basically Jo did studies at two schools which were similar in demographic makeup and location. One was project-based and one was traditional. Spoiler: the students at the project-based school performed better on their national exam and a far higher percentage of them were interested in continuing to study mathematics after secondary school.

Jo states that her studies of the communicative and project-based approaches led to one big conclusion:

students need to be actively involved in their learning and they need to be engaged in a broad form of mathematics – using and applying methods, and representing and communicating ideas (p. 83).

Chapter 4 – Taming the Monster: New Forms of Testing that Encourage Learning

I got very excited when I read the title of this chapter, because I’ve been doing a lot of thinking over the past two years about the purpose of assessment and how to write effective assessments.

What Is Wrong with What We Have Now?

Jo first focuses on standardized tests in the US. The format of the test (multiple-choice problems) and the reporting of scores (ranking students as above-average, average, below-average, etc.) can cause so much damage.

First, multiple-choice tests are not used in the highest-performing countries for 4 main reasons (all from p. 86):

  1. The goal is to assess understanding. In order to do so, you have to look at a student’s work, not at which letter they chose (where it’s possible that none of them mean anything to the student).
  2. Multiple-choice tests are known to be biased. There is some evidence that multiple-choice tests are biased against girls, and there is LOTS of evidence that they are biased against ethnic minority students.
  3. Timed tests create anxiety and contribute to the high levels of stress that our students currently report. (Story of my elementary-school math experience, right here. My mom is nodding her head so hard right now.)
  4. “The best thing that multiple-choice tests show is a student’s ability to complete multiple-choice tests” (p. 86). The SAT is a classic example of a timed multiple-choice test, and do you know what SAT prep courses actually teach? Strategies to choose the best answer in a multiple-choice setting. They really don’t teach much content. (I used to teach them.)

The standardized tests we use in this country don’t test thinking or reasoning, just procedures, so they don’t give us useful information. Furthermore, when we report results, we assign labels to the students. The previous test we used in Arizona reported student scores using the labels Exceeds Standards, Meets Standards, Approaches Standards, and Falls Far Below Standards.

(I could go off on a whole rant here about how these labels make it sound like we’re evaluating students’ performance against a set of standards and un-ambiguously saying that they demonstrated understanding of the standards set for their grade level, but they don’t – they actually compare students to each other. These scores are based on where a student fell compared to other students, not whether they demonstrated understanding of the standards.)

These labels that we assign are damaging to students across the spectrum (the students who did well and the students who did poorly), but Jo points out the effect that this reporting of standardized test scores had on a student that she interviewed, Simon. She came away with this conclusion:

Testing and reporting measures such as those experienced by Simon can create low-achieving students, crushing students’ confidence and giving them an identity as a low achiever.

… If you tell students they are low achievers, they achieve at a lower level than if you do not (p. 91).

If I’d been reading this book with a pen and not a highlighter, I think I might have actually scrawled a very sarcastic “NO SHIT” in the margin.

Another study “found that students who were not given scores but instead given positive constructive feedback were more successful in their future work. Unfortunately, [this study] also found that teachers gave less and less constructive feedback as students got older” (p. 93-94). This isn’t surprising. Feedback isn’t real high on my priority list when I’m grading 90 of the same test. The way we have set up high schools in this country leads to this.

Assessment for Learning

Here’s where I got excited again (which is good, because reading the first half of this chapter is really depressing) – Jo is going to tell us how to assess so that learning continues, not ends, after the assessment.

3 requirements – “Students are made aware of what they are, should be, and could be learning through a process of self- and peer assessment” (p. 96).

What does this actually look like? Clear objectives. Students should understand them, and they should show the relationships between ideas. Those objectives should be used by the students to see what they should be learning from a piece of work. They should evaluate their (and their classmates’ ) work against the objective and see whether they understood what the objective says. In doing so, they come to understand the big picture and they also take ownership of their learning.

Students need to move from being passive learnings to being active learners, taking responsibility for their own progress, and teachers need to be willing to lose some of the control over what is happening, which some teachers have described as scary but also liberating (p. 98-99).

The most important part of the assessment-for-learning approach is in the feedback from teachers. Giving a score is counter-productive because students just focus on the score and not on next steps. Giving feedback (particularly without a score) is the most beneficial in getting students to focus on what they need to do next in order to continue learning.

This is valuable information for me as I’m trying to decide how to assess in my Algebra 1 Support (intervention-type) class this year. Sounds like I need to be careful NOT to write any kind of score on my students’ quizzes, but only feedback. This feedback should consist of commenting on errors with specific suggestions for improvement and include at least one positive remark.

What’s Math Got To Do With It? Reflections and Thoughts, Part 1

I recently finished reading Jo Boaler’s book What’s Math Got To Do With It? and I need a way to reflect on what I had learned and consider ways to implement her research in my own teaching. This book was fantastic. In every chapter Jo made at least one point that just blew my mind. I was highlighting things in the PREFACE, people. I don’t usually even read the preface.

I’m going to use quotes from the book and expand on them with my own thoughts. I’m going to cover the introduction and first two chapters in this post, and I’m planning follow-up posts to cover the rest of the book.

Update:

Introduction: Understanding the Urgency

…we do not and cannot know what mathematics students will need in the future, but the best preparation we can give them is to teach them to be quantitatively literate, think flexibly and creatively, problem solve, and use intuition as they develop mathematical ideas. (p. 8)

Huh. That list kinda sounds like skills that student actually need to be successful in life, not just math. This is always the problem with asking the question, “When will I use this in real life?” The answer is usually, “You’re not going to.” Most people don’t need to know how to solve a quadratic equation, graph a sine curve, or integrate a function in “real life”. They do need to know how to think flexibly and creatively, problem solve, and use intuition as they develop ideas. If this was the focus of math class instead of computation, maybe we could finally stop asking that question.

Chapter 1 – What is Math? And Why Do We All Need It?

What is math? “…ask mathematicians what math is and they are more likely to describe it as the study of patterns.” (p. 19) Notice this doesn’t say computation. It doesn’t say memorizing rules or steps. It doesn’t say any of the things that we typically associate with math class. Mathematicians describe true mathematics as looking for and analyzing patterns in the world around them (not always related to numbers).

Imre Lakatos, mathematician and philosopher, describes mathematical work as “a process of ‘conscious guessing’ about relationships among quantities and shapes.” Those who have sat in traditional math classrooms are probably surprised to read that mathematicians highlight the role of guessing. … [W]hen children who have experienced traditional math classes are asked to estimate, they are often completely flummoxed…. This is because they have not developed a good feel for numbers, which would allow them to estimate instead of calculate, and also because they have learned, wrongly, that mathematics is all about precision, not about making estimates or guesses. Yet both are at the heart of mathematical problem solving. [emphasis mine] (p. 25)

This is the hardest part of any conversation about how math class should be taught. It’s hard to explain to people that their beliefs about what math truly is are wrong, usually because they never got far enough in math (usually graduate-level work) to see how creative and exploratory math really is.

True mathematics is: exploratory and creative, collaborative, and all about asking questions.

People commonly think of mathematicians as solving problems, but as Peter Hilton, an algebraic topologist, has said, “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.” … Bringing mathematics back to life for schoolchildren involves giving them a sense of living mathematics. When students are given opportunities to ask their own questions and to extend problems into new directions, they know mathematics is still alive, not something that has already been decided and just needs to be memorized. [emphasis mine] (p. 27)

This is what I need to be focusing on in my Algebra 1 Support classes this year (where I have the freedom to explore without the constraints of a curriculum calendar) – “pos[ing] and extend[ing] problems of interest to students” (p. 27-28). I need to give them the chance to explore, not just memorize and regurgitate.

Now, this analogy is my favorite, right here. I’ve seen it before, I think in A Mathematician’s Lament by Paul Lockhart. (Incidentally, that book is the one that first began to teach me that math could be something more than memorizing and regurgitating.) I love the comparison of the way we teach math to a hypothetical way we could teach music. It really helps to underscore the way we suck all the interestingness out of math.

Imagine music lessons in which students worked through hundreds of hours of sheet music, adjusting the notes on the page, receiving checks and crosses from the teachers, but never playing the music. Students would not continue with the subject because they would never experience what music is. Yet this is the situation that continues, seemingly unabated, in mathematics classes.

… Students should not just be memorizing past methods; they need to engage, do, act, perform, and problem solve, for if they don’t use mathematics as they learn it, they will find it very difficult to do so in other situations…. (p. 29)

NO WONDER kids hate math! In its truest sense, math is supposed to be creative and fascinating and wonderful and we make it so boring. “We cannot keep pursuing an educational model that leaves the best and the only real taste of the subject to the end, for the rare few who make it through the grueling years that precede it.” (p. 30)

Chapter 2 – What’s Going Wrong in Classrooms? Identifying the Problems

This chapter was frustrating to read, and I’m going to skip over the section about the math wars because it just made me angry. I just want to say that I truly cannot comprehend the motivations behind a movement that wants to keep math as dry and boring as we currently teach it. In our country, we already have a negative national attitude about school in general and math in particular. Why would anyone fight to keep that? Why would anyone think it’s a good thing to be raising yet another generation who thinks that “I can’t do math, haha” is an acceptable thing to say?

I did have a couple of notable things I pulled out of this chapter, in spite of the feelings of frustration and futility that her discussion of the math wars brought about. First, “It is ironic that math – a subject that should be all about inquiring, thinking, and reasoning – is one that students have come to believe requires no thought.” (p. 42) This statement follows up a discussion about student perceptions of math, in which one student said, “In math you have to remember; in other subjects you can think about it.” (p. 40). This breaks my heart. Students who don’t understand the interconnectedness and logic of mathematics sincerely believe that they have to memorize EVERYTHING in math class. The idea of memorizing every single thing taught in a class for twelve years is enough to give me anxiety. I’ll say it again – NO WONDER kids hate math!

The worst thing about this, though, is that kids don’t start off as number-crunching robots. They learn to be this way because of years of math instruction that teaches them they are supposed to be number-cruching robots. “The fact that many students learn to suppress their thoughts, ideas, and problem-solving abilities in math class is one of the most serious problems in American math education.” (p. 46)

Two other problems with current math instruction that Jo identifies in this chapter are the lack of communication and the lack of reality in math classes. True (graduate-level and beyond) mathematics is very collaborative. One reason for this is the emphasis on reasoning:

Whenever students offer a solution to a math problem, they should know why the solution is appropriate, and they should draw from mathematical rules and principles when they justify the solution rather than just saying that a textbook or a teacher told them it was right. Reasoning and justifying are both critical acts, and it is very difficult to engage in them without talking.

… When students are asked to give their ideas on mathematical problems, they feel that they are using their intellect and that they have responsibility for the direction of their work, which is extremely important for young people. (p. 49)

I already encourage a lot of talking in my classroom, but this year I need to focus on ensuring that those conversations are going in productive directions. I need to make sure that students are reasoning and justifying, and analyzing each other’s justifications.

I really enjoyed Jo’s discussion of ridiculous contexts (think word problems) that have nothing to do with actual real life. My favorite example was this problem, on page 52: A pizza is divided into fifths for 5 friends at a party. Three of the friends eat their slices, but then 4 more friends arrive. What fractions should the remaining 2 slices be divided into? Jo’s point here is that by providing so-called real world applications, students should be able to use their knowledge of the real world to solve the problems. Unfortunately, they actually have to set aside this knowledge in what Jo refers to as Mathland. As she points out, “Everybody knows that … if extra people turn up at a party more pizza is ordered or people go without slices.” Personally, I have even more issues with this problem. Who cuts a pizza into 5 slices? Do you have any idea how difficult it is to cut a circle into an odd number of pieces? How pissed are those two people going to be when they realize they have to share their two slices with 4 more people, so they don’t get as much pizza? And when the 4 new people decided to come to the party, didn’t they text someone and find out if there was any pizza left? They could have picked one up for themselves on the way over! (Then it could have been cut into 4 slices, which is much easier to cut than 5 slices.)

The elimination of ridiculous contexts would be good for many reasons. Most important, students would realize that they are learning an important subject that helps make sense of the world, rather than a subject that is all about mystification and non-sense. (p. 56)

These first two chapters gave me a lot to chew on with regard to how we are currently teaching math and why it’s not a good approach. The next 4 chapters deal with more specifics of how we can improve our math teaching, and will be where the bulk of my reflection needs to happen.