Please forgive the indulgence, but this is a different kind of blog post. Over the winter, I read What’s Math Got to Do With It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject by Dr. Jo Boaler. Where was SO MUCH inspiring that I just collected all of the quotes that spoke to me and I share them here without commentary from me. Dr. Boaler’s words do not need my elaboration!

“Traditional educators believe that some students do not have the brains to be able to work on complex mathematics that enables brain connections to develop. Students can graph high-level ideas, but they will not develop the brain connections that allow them to do so if they are given low-level work and negative messages about their own potential.” (pg. xvii)

‘Most students when asked what they think their role is in math classrooms say it is to answer questions correctly.” (pg. xviii)

“What research tells us is that when a mistake is made there are two potential brain sparks: the first one comes when we make a mistake but are not aware of the mistake; the second comes when we realize we have made a mistake. How can this be? How can our brains grow when we do not even know we have made a mistake? The best knowledge we have on this question tells us that our brains grow when we make mistakes because those are times of struggle, and our brains grow the most when we are challenged and engaging with difficult, conceptual questions.” (pg. xiv

“Unusually for a math class it was the students, not the teacher, who had solved the problem.” pg. 2)

“This class worked so well because students were given problems that interested and challenged them and, also, they were allowed to spend part of each lesson working alone and part of each lesson talking with one another and sharing ideas about math.” (pg. 3)

“There are two versions of math in the lives of many Americans: the strange and boring subject that they encountered in classrooms, and an interesting set of ideas that is the math of the world and is curiously different and surprisingly engaging.” (pg. 5)

“The mathematicians interviewed gave many reasons for collaboration, including the advantage of learning from one another’s work, increasing the quality of ideas, and sharing the ‘euphoria’ of problem solving.” (pg. 26)

“Some students think their role in math classrooms is to memorize all the steps and methods. Other students think their role is to connect ideas.” (pg. 41)

“Students want to know how different mathematical methods fit together and why they work.” (pg. 43)

“…students have their problem-solving abilities drained out of them. They think that they need to remember the hundreds of rules they have practiced and they abandon their common sense in order to follow the rules.” (pg. 43)

“One problem is that students often need to talk through methods to know whether they really understand them. Methods can *seem* to make sense when people hear them, but explaining them to someone else is the best way to show whether they are really understood.” (pg. 46)

“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.” (pg. 49)

“Just like stepping through the wardrobe door and entering Narnia, in math classrooms trains travel towards each other on the same tracks and people paint houses at identical speeds all day long. Water fills tubs at the same rate each minute, and people run around tracks at the same distance from the edge. To do well in math class, children know that they have to suspend reality and accept the ridiculous problems they are given. They know that if they think about the problems and use what they understand from life, they will fail. Over time, schoolchildren realize that when you enter Mathland you leave your common sense at the door.” (pg. 51)

“In California in 2001, there was a staggering correlation of 0.932 between students’ scores on the mathematics and language arts sections of the tests used. Correlations this high between two tests tell us that the tests are assessing virtually the same thing….the mathematics tests are really language tests.” (pg. 88)

“Teachers set out mathematical goals for students, not a list of chapter titles or tables of contents, but details of the important ideas and the ways they are linked. For example, students might be given a range of statements that describe the understanding that they should have developed during a piece of work…The statements are clear for students to understand, and they communicate to them what they should be learning from a piece of work.” (pg. 96)

“Students need to move from being passive learners 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 ultimately liberating.” (pp. 98-9)

“If students are not given opportunities to learn challenging and high level work, then they do not achieve at high levels.” (pg. 109)

“In a mixed-ability group, the teacher has to *open* the work, making it suitable for students working at different levels and different speeds. Instead of pre-judging the achievement of students and delivering work at a *particular* level, the teacher has to provide work that is multileveled and that enables students to work at the highest levels they can reach.” (pg. 111)

“Instead of one person serving as the resource to thirty or more students, there are many. The students who do not understand as readily have access to many helpers. The students who do understand serve as helpers to classmates. This may seem like it is wasting the time of high achievers, but the reason these students end up achieving at higher levels is because the act of explaining work to others deepens understanding. As students explain to others, they uncover their own areas of weakness and are able to remedy them and they strengthen what they know.” (pp. 112-3)

“For mixed-ability classes to work well, two critical conditions need to be met. First, the students must be given more open work that can be accessed at different levels and taken to different levels. Teachers have to provide problems that people will find challenging in different ways, not small problems targeting a small, specific piece of content.” (pg. 116)

“In addition to open, multilevel problems, the second critical condition for mixed-ability classes to work is that students are taught to work respectfully with each other.” (pg. 117)

“The low-achieving students came to believe that in order to be successful, they needed to count very precisely.” (pg. 142)

“…one of the important things people do as they learn mathematics is compress ideas. What this means is, when we are learning a new area of math, such as multiplication, we may initially struggle with the methods and the ideas and have to practice and use it in different ways, but at some point things become clearer, at which time we compress what we know and move on to harder ideas. At a later stage when we need to use multiplication, we can use it fairly automatically, without thinking about the process in depth. (pg. 142)

“One way of thinking about the learning of mathematics, visually, is to think of a triangle…The larger space at the top of the triangle is new mathematics you learn, that you need to think about and connect to other areas, and it takes up a big space in your brain. The smaller area at the bottom of the triangle represents mathematics that you know well and has been compressed. (pg. 143)

“It is this compression that makes it easy for people to use concepts they learned many years ago, such as addition or multiplication, without having to think about how they work every time they use them. Gray and Tall found that the low-achieving students were not compressing ideas. Instead, they were so focused on remembering different methods, stacking one new method on top of the next. Our brains can only compress concepts, not rules or methods, and the low-achieving students were not thinking conceptually, probably because they had been led to believe that mathematics is all about rules.” (pp. 143-4)

“…keep telling students that math is very exciting, and it is important to work hard because it is hard work that leads to high achievement.” (pg. 187)

“Try not to lower the cognitive demand of a problem when helping. Try not to do the hard thinking for the child, leaving her with a calculation.” (pg. 190)

Boaler, Jo. What’s Math Got To Do With It?. Penguin Books, 2015.