Математическое мышление. Книга для родителей и учителей

by Jo Boaler, Джо Боулер, Наталья Яцюк (Translator)

Многие ученики (и их родители тоже) уверены, что у них нет способностей к математике. Хотя на самом деле каждый может наслаждаться математикой и добиваться хороших результатов в ее изучении. Это доказала Джо Боулер, профессор и исследователь из Стэнфорда, изучавшая то, как учатся тысячи учеников и студентов и то, как можно раскрыть их потенциал.

Существует большой разрыв между правильными подходами и теми, что наблюдаются в школе и воспитании сейчас. Боулер адаптировала прорывную концепцию «Гибкого сознания» Кэрол Дуэк для обучения математике, показав, как можно на практике помочь учащимся почувствовать себя увереннее.

Математика часто кажется скучной, потому что для многих это набор незапоминаемых формул и счетных процедур с не всегда понятной целью и сомнительной применимостью в жизни, а также с неумолимостью оценки: неверный ответ в конце перечеркивает все старания.

Автор предлагает преподавать математику иначе — в первую очередь как логику. Она рассказывает о тех шагах, которые нужно делать школам, преподавателям, учителям, чтобы улучшить математическое образование для всех.

Genres: Education, Mathematics, Teaching, Nonfiction, Professional Development, Parenting, School

352 pages, Hardcover

First published October 12, 2015

About the author

Dr Jo Boaler is a Professor of Mathematics Education at Stanford University and co-founder of www.youcubed.org. Formerly the Marie Curie Professor of Mathematics Education for England, a mathematics teacher in London comprehensive schools and a researcher at King's College, London. She is the author of eight books including What's Math Got To Do With It? (2015) and Mathematical Mindsets (2016). She is the recipient of the NCSM award for equity, the author of the first MOOC on mathematics learning for teachers and parents, a White House presenter and an advisor to the PISA team at the OECD.

Reviews

[Mark Schlatter · Rating 3 out of 5]

First note: This is a book firmly on the side of reform in mathematics education. If you like the idea of reducing homework (a la Alfie Kohn), increasing equity in the classroom, doing away with (or ignoring) standardized testing, upping the amount of group work, and decreasing the emphasis on drill, you will probably like this book. If you're not a fan of all of the above, you probably will have a struggle to appreciate Boaler's message.

Second note: There is a lot about the concept of mindset (growth versus fixed) here, including a good introduction to what Boaler calls "low floor, high ceiling" problems: easy to understand, but with enough depth that students can challenge themselves. However, I'm not sure I would recommend this to someone just starting to learn about mindset or someone looking for a more nuts and bolts approach. You have a lot of philosophy, high level pedagogy, and research to get through. (At times, I felt the mindset approach was more of a skeleton that was fleshed out with Boaler's more expansive wishes for mathematics education reform.) A more down-to-earth introduction might be The Growth Mindset Coach.

Third note: About that research... Twice during the book, I came across Boaler's summaries of studies that seemed too good to be true. Both times, I found the original papers. Both times, I didn't see how the papers supported the claims Boaler made. I believe there is an unfortunate tendency in education to overstate the findings of neuroscience, and I was saddened to see that here.

Now, I got a lot out of this book. I agree with a lot of Boaler's philosophy, and I gained much from seeing her approach to mathematics. Among the suggestions made in the book, I plan to use participation quizzes, the "low floor, high ceiling" problems mentioned above, reflective questions on homework, and the norms that she emphasizes throughout. I'm guessing the book will have a significant impact on my teaching this next year. But, I was still somewhat irritated by the tone (we have to reform everything and then it will all work!) and the research issues.

[Marika Gillis · Rating 4 out of 5]

Mathematical Mindsets is a book about teaching math that centers around the fixed vs. growth mindset ideas in Carol Dweck's book, titled Mindset. I loved Carol Dweck's book so when a book study on Mathematical Mindsets was started at my school I jumped on board, despite the fact that I rarely teach math anymore.

Considering the content and lack of application to my current role, I found a lot of value in this book. I think my own mindset about math was challenged by reading this book. It is hard for me to think of math as the open and creative subject it is described as in this book and I love that my brain had to stretch to accommodate those ideas. I think if my math teachers had approached the subject more like the teachers used as examples in this book, my belief in my own math abilities would have been drastically different. I think it's important for teachers to know and understand their effect in this way on their students. I also really appreciated what this author shared about homework and the data she used to back up her understanding of homework as being completely useless (amen to that!).

The 4 star rating comes only from the repetitive nature of this book. At times Jo Boaler seemed to drive her point home again... and again... and again. But since her points resonated with me, I was mostly okay with it.

[Emily Thomas · Rating 4 out of 5]

Mathematical Mindsets (MM) is a great book for pre-service teachers to read about effectively teaching mathematics using the growth mindset. This book is relatable and easy to read, as it incorporates real-world examples and concepts, and provides tips on how to invite students to engage and thrive in mathematical learning. As a pre-service teacher, one can relate to the struggles that you will find in the classroom, including how to deal with a student with a fixed mindset and students who view themselves as a “low” math student with low self-confidence. MM breaks the mathematical classroom stereotypes and teaching patterns that have overtaken today’s typical classrooms. While school mathematics focuses on numbers and calculations, MM criticizes this surface level learning and promotes real mathematics by focusing on patterns and developing a deeper number sense. There are many memorable concepts in MM that we will incorporate into our future classrooms. We will promote a growth mindset within our classrooms in order to value students’ mistakes and teach that they are learning opportunities that help their brains grow. We will dispel the myth that there is only one correct strategy when solving problems and teach multiple strategies and representations for each concept in mathematics. Overall, MM is a valuable resource for both pre-service teachers and current teachers to use as a reference to better teach mathematics to students.

[Ardyth · Rating 1 out of 5]

... le sigh...

TL;DR on my opinion on where this goes wrong: inquiry based, productive struggle type learning environments are a slow moving, demoralizing waste of time for *most* students. Direct instruction, moving from concrete to abstract, from conceptual in tandem procedural with working memory to fluent recall from long term memory -- this is the better way to go.

But direct is not a synonym for BORING. This is where Boaler, and so many other "productive struggle" proponents go most wrong, in my opinion... they conflate these terms.

Noodling problems is good -- AWESOME, even -- and some of these look like fun. But noodling is awesome only in the sweet spot. Not too easy, not too hard... that magic, constantly-in-motion place where the puzzler doesn't instantly *know* the answer, yet is confident she can *reach* it, and then (before that confidence completely expires) does so.

^^ This, and only this, gives the little rush of "I did it!" which helps develop persistence. Mess up this sweet spot, and those "inquiry" lessons serve only make the student feel inadequate (or bored).

How exhausting to have to figure out that sweet spot every minute of every daily lesson for my single student. For 30? Breathtaking in its complexity.

(And is math class really where we should be worried about developing grit? Do we focus on grit in phonics lessons? I definitely did not, and would not really recommend it. I focused early reading lessons on on those delightful feelings of achievement. Grit is developed while writing essays or researching a project... multi-day tasks that *should* be hard and *should* have a grand feeling once finished)

Anyway. That's the tip of the iceberg with this book. These all seem like super fun investigations. They also seem extremely time consuming and not terribly clear in the end. Good for a weekly or biweekly math game allotment, perhaps? But for core approaches.. ehhhh.

I'll stop now, because I'm only some rando on the internet so who cares? Links below to knowledgeable sources who have addressed these questions again and again AND AGAIN.

PS - if you're a CM family following a CM Arithmetic-first arc, congratulations! The science backs you up.

* * * * *
See Sal Khan here, discussing why Algebra I is and always has been the most popular course on Khan Academy (it has very little to do with Algebra I and everything to do with storage in long term memory):

https://www.the74million.org/article/...

"Myths That Undermine Maths Teaching" is a meta-analysis of 100+ studies done during the past twenty years:

https://www.cis.org.au/publication/my...

And don't miss Daniel Willingham's Why Don't Students Like School?: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom for very clear and thorough discussion about all the ways our vague "go and do" without the "likewise" (direct instruction) part waste the time and energy of our children.

[Jeri Vick · Rating 5 out of 5]

Very thought-provoking and left me with at least as many questions as it did answers. I do wish that she addressed the challenges encountered in implementation and lessons learned along the way. Her contention that we do learn best from mistakes makes sense. I just wish she had shared some of those experiences too to round out the picture.

[Declan · Rating 4 out of 5]

Why we need to change math instruction to focus on depth and understanding not speed.

[Isaac Solberg · Rating 5 out of 5]

Wow! My mind is blown on what it means to engage with math. This is such an encouraging book for myself, as a teacher as well as for parents, and students who hate the intimidating subject.

Key concept: Everyone has the ability the grow their math understanding to the deeper ways no matter their history.

I also loved Boaler’s writing ability to bring concepts/theory to practical classroom use. So excited to implement creative lessons filled with mistakes, concepts, and growth mindset in my own math classroom.

[farmwifetwo · Rating 3 out of 5]

I am at pg 97 and I want to get out my pom pom's and go "math is great, math is easy" over and over again. I do appreciate that there are many you need to get that thought into but... I also appreciate the math 'attitude'. Whether it be parents or "professionals". I have been told again and again it must be nice how smart my son is but they have already forgotten how far behind he was all through public school. When I tell them he's been taught and I have redone years of curriculum (2 to 4 btwn the 2 of them and fractions) they don't believe me it is that simple. Not just one or 2... ALL of them. Nobody, to date, believes me or has replicated what I did for their kid. Instead I get "well it's ok, math is hard and I don't understand it."

Is it a book for parents... at this stage (not quite half way).... not certain... IMO, it's directed towards other professionals in the education field. It does mention her online class, some you tube videos and other materials that she bases her points on. If I spent the time and resources to look into those materials, then I could give you a better answer.

Will I finish it... no. I am going to return it today to the library. Why?? This book was recommended by the local librarian, it does not apply to me and mine and I am bored with lots of other books to read. I am the over educated, under employed, SAHM with the STEM degree... math is easy... with a complete dislike for funky math without the basics. She (the librarian), on the other hand, thought it was excellent.

So, don't take my opinion.

[Ricardo Garcia · Rating 3 out of 5]

Good information, but a difficult read.

[☘Misericordia☘ ⚡ϟ⚡⛈⚡☁ ❇️❤❣ · Rating 5 out of 5]

Can't say enough good things about this book!

[Denis Vasilev · Rating 3 out of 5]

Размышления о том, как должна преподаваться математика. Пространные, все уже видел в других книгах на тему

[Corrie · Rating 5 out of 5]

I read this for a story I'm writing, and wow do I wish this had been around when I was in high school bored to tears in my math classes.

[Stuart Macalpine · Rating 5 out of 5]

A phenomenally pragmatic, moral and optimistic book specifying thinking and approaches to teaching Maths that will help children learn, based on her vast experience working at Stanford, with schools and with the OECD Pisa (ISA) tests. Intellectually honest and refreshing.

Interesting that she sees a desire for equity (and inclusion) as at the heart of such excellence, just like some learning focused schools with NFI.

[Ellen Little · Rating 5 out of 5]

This will change the way I teach math!!!

[Melissa Riggs · Rating 4 out of 5]

Not an easy read by any means, but so very powerful if you are a teacher of mathematics OR if you ar a parent of a child in school. This book will make you rethink the phrase "I'm not a math person". I'd highly recommend it.

"Mathematical Mindsets provides practical strategies and activities to help teachers and parents show all children, even those who are convinced that they are bad at math, that they can enjoy and succeed in math. Jo Boaler--Stanford researcher, professor of math education, and expert on math learning--has studied why students don't like math and often fail in math classes. She's followed thousands of students through middle and high schools to study how they learn and to find the most effective ways to unleash the math potential in all students.

There is a clear gap between what research has shown to work in teaching math and what happens in schools and at home. This book bridges that gap by turning research findings into practical activities and advice. Boaler translates Carol Dweck's concept of 'mindset' into math teaching and parenting strategies, showing how students can go from self-doubt to strong self-confidence, which is so important to math learning. Boaler reveals the steps that must be taken by schools and parents to improve math education for all. Mathematical Mindsets:
Explains how the brain processes mathematics learning
Reveals how to turn mistakes and struggles into valuable learning experiences
Provides examples of rich mathematical activities to replace rote learning
Explains ways to give students a positive math mindset
Gives examples of how assessment and grading policies need to change to support real understanding
Scores of students hate and fear math, so they end up leaving school without an understanding of basic mathematical concepts. Their evasion and departure hinders math-related pathways and STEM career opportunities. Research has shown very clear methods to change this phenomena, but the information has been confined to research journals--until now."

[Mallory · Rating 5 out of 5]

This was way more interesting than it ought to be! I love how the author debunks that there are math people and not math people--like there is some special gene for being good at math. That fast doesn't equal smart. Making mistakes grows your brain. Homework and tests do not motivate or improve learning. And how we need to teach math differently so that it's an active, creative discovery, not a sit and watch and regurgitate. I thought it was super interesting how tracking in schools (putting kids in the 'smart' class or 'slow' class) is so detrimental to both the students told they aren't smart enough AND the ones that are told they are. Also that kids benefit a lot from working in groups to arrive at an answer from different ways of thinking and actually be able to explain why their answer makes sense. It's a bit mind boggling how US culture is so engrained with how math "ought to be taught, even though I hated math when I was taught that way" that it's such a hard path for teachers or schools to change to this more effective method--of helping kids actually understand math, not just memorize and forget procedures.

[Michael Sheppard · Rating 4 out of 5]

"We need to free our young people from the crippling idea that they must not fail, that they cannot mess up, and that only some students can be good at math. We need to introduce them to creative, beautiful mathematics that allows them to ask questions that have not been asked, and to think of idea that go beyond traditional and imaginary boundaries."

[Fred · Rating 4 out of 5]

There is a lot of strong research about effective mathematics teaching and learning. I kept making notes about ideas I wanted to share. Boaler’s writing style can be off putting - which is the only reason I haven’t given this five stars.

[Stacy Stoker · Rating 5 out of 5]

This book has given me a lot to ponder regarding my own view of math and my teaching practices. I think I will need to re-read it- my brain is on overload!

[Menolly Lysne]

I enjoyed this book and there are a tonne of fantastic suggestions. It just took me a while to finish. I ended up restarting as I could not remember what I had already read.

[Katharina · Rating 5 out of 5]

If math ed interests you, check out my blog (http://metamathed.wordpress.com) for more.

Here is the first of hopefully many book reviews I will write for this blog. I’ve just finished Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching, by Jo Boaler. Despite its lack of an Oxford Comma in its subtitle, I think it is an excellent read full of great ideas. Every few years, I read a pedagogical text that really shakes my core as a teacher, and Boaler’s book has truly started to push my pedagogy in a way I did not anticipate. (The last book that really transformed my teaching practices was Myron Dueck’s “Grading Smarter, Not Harder,” which I will likely review in an upcoming post, even though I read it in early 2015).

[Before you read this, please note that I’m also reading Visible Learning for Mathematics, by Hattie, Fisher, & Frey and Making Number Talks Matter, by Humphreys and Ruth, so these other texts will likely influence my writing more than if I had only been reading one book whilst writing this review.]

Summary

Boaler begins her text with a description of growth and fixed mindset practices and discusses the danger of making fixed mindset comments to students, even if those comments seem positive. She discusses the power of mistakes, and the various ways students can grow by taking mathematical risks. Boaler also discussed the issues that arise when we grade students for mathematical performance, and suggested instead “Assessment for Learning,” which seems to be verbal feedback covering the following three areas: “where students are now, where students need to be, and ways to bridge the gap.”

Waxing Poetic (Prosaic?) on Boaler’s Ideas

One tidbit that I need to read more about before I really feel comfortable accepting is that making mistakes help your brain to grow, even if you don’t know you made a mistake. From what I understand, this is because of a disequilibrium created when your brain expects one thing but thinks another. I’m not sure if I believe this is true of all mistakes; maybe just larger mistakes. I fail to believe that the student who told me yesterday that 7 x 2 = 13 developed intellectually as a result. Or maybe she did, and I’m making a mistake by saying that, and now my brain is growing.

Tracking

The biggest effect that Boaler’s book had on me is a complete change in my opinions on tracking. I was in the G/T (“Gifted and Talented”) track as a student myself, and my students are “leveled” into accelerated or regular classes. Prior to reading this book, I was silently in favor of this system. I’ve always felt that I succeeded in life in part due to the advanced classes I took at an early age. However, after considerable reading on critical thinking and mindset, I realize that I was not necessarily at an advantage as a result of these placements.

Looking back on my time as a student, I realize that Boaler’s assessment is spot-on: as an “advanced” student, I did not have time to process the material deeply. I did not think about the tasks being asked of me, but rather was really good at determining the correct algorithm to use for a given situation. This was brought to my attention as a mathematics major in college, when I was hired to work in the math tutoring room and discovered that, though I was doing well in the classes I was taking, I was often stymied when helping students who were taking entry-level college math classes. (To those of you reading this and saying, ‘hmm, haven’t you been teaching entry-level college math classes for the past few years...why did they hire you?’ know that I ask myself that regularly, but also that I remedied that issue long before they hired me.) Additionally, I had a professor once who told me and my classmates that we weren’t math majors because we were good at math, we were math majors because we had learned to be good students. At the time, I was offended! I was good at math! That was a part of my identity. But I suppose that’s Boaler’s point. That professor’s comment has stuck with me for well over a decade and for a few years now, I’ve known that he was right.

I love teaching my accelerated classes. I always have. Boaler ruined that for me! Seriously. But I’m not sure that’s a bad thing. I used to get through my remedial and grade-level math classes so I could have fun with the advanced students. I now feel negligent for feeling that way, as I was doing no one any favors. The students in my accelerated class have to go through the material too quickly and truly struggle when asked to think critically. My “regular” students are essentially tracked into a lower group and are thus held to lower standards than these other students. There’s plenty of research out there to indicate that expectations have a huge impact on learning. I am angry with myself for my years of vastly different expectations for different students, but honestly there is no point beating myself up over mistakes I’ve made in the past when there is so much I can focus on improving now.

Okay, trying to find my way back from this tangent…I need to find a way to adjust how my school tracks students. Boaler cites multiple studies which all imply that the students who benefit most from heterogeneous grouping are the more advanced students, because it helps them learn to explain themselves. For now, I will try to push all of my students to master difficult material and think critically, but in the long run, I am on a mission to eliminate tracking in my school.

Grading

It has been a while since my love affair with Grading Smarter, Not Harder, so there’s very little I remember in great detail. That said, one of my big take-aways was that a student’s grade should reflect their mastery of content and not their effort. That idea made me reassess my homework grading methods, which I will talk about in a different post at some point. Boaler goes even further and says that we should not give students a number grade for math at all. She points out that feedback about how students are processing ideas and expressing themselves is much more valuable. I fully agree, but I’m not sure how effective I will be at convincing my school that I should no longer give number grades to my students.

Concerning my lab: last year, my school decided that we should go from a “Satisfactory/Unsatisfactory” rating for labs to a numerical grade on a 0-100 scale. I was hesitant about it at that point due to sheer laziness. Now I’m hesitant about it because I don’t really think kids should receive a grade for it when they already receive a narrative and constant verbal feedback. This is something I’m not going to even attempt to change at the district level, but I can certainly change in my classroom.

Homework

Well. I’ve adjusted my homework policies over the years, shifting from daily homework, to homework except on the weekends, to my current plan of two homework assignments per week. I’ve found that as I’ve decreased the number of assignments, more students are completing them with greater effort and accuracy. A teacher I work with is firmly against homework and I’ve thought about this position a great deal. I’ve always found homework to be an equity issue: some students go home to both their parents and sit at a desk in a quiet room, given time and space to work on their assignments, with mom and/or dad in earshot to answer questions and provide support as needed. Other students go home to be caretakers for their siblings while mom and/or dad goes to work. These students have a lot more on their proverbial plates, having to make dinner and perform other parental duties without their actual parents present to help, let alone to answer homework questions. Those reasons alone always give me pause about assigning homework.

[Let me also say that our country claims that we should work for 8 hours, rest for 8 hours, and have 8 hours to do with what we wish. If that’s true for adults, should we not respect that same philosophy for children? Isn’t being a child one of the most joyous times of life? Why should kids be weighed down with excessive homework?]

Boaler goes so far as to claim that homework has no impact on student learning. I’m not sure if I’m just romanticizing her ideas or she’s just a good writer, but her claims have made me want to stop giving homework altogether. I won’t be able to do that this year: we’re too far in, and I often get complaints that I don’t give enough homework (yes, really; from students and parents). If I’m going to eliminate homework from my class norms, I’ll need to back that up by citing more research, so you can expect that much of my upcoming reading will be about homework, equity, and whether or not it impacts student learning.

Number Talks

Boaler’s discussion of number talks was enough to get me to pick up Making Number Talks Matter again (I put it down not because I was uninterested, but because I was very pregnant) and start this blog, so I don’t feel like I need to write too much here for now. More to come in many ways, shapes, and forms.

Low Floor, High Ceiling

The idea of creating a quality question that essentially self-differentiates is one that I’ve tried since reading Boaler’s book and has already transformed my teaching. This allows all students to approach a problem, but some students to take it to different levels than other students. I tried this recently in my classroom, revamping a lesson that I’ve been using for years that revolved around a quality question, but was poorly handled on my part. In it, students have to determine how much given items cost a family when they went on vacation. The activity was designed to help students write algebraic equations to represent real-world situations. I used to teach this lesson by having students work in groups, but forcing them to start by writing and then solving an algebraic equation (like a fool). After reading Boaler’s suggestions about Low Floor, High Ceiling Activities, I took my supports away and encouraged students to solve these same problems in any way they wanted to, challenging them to solve the problem in at least two ways and make sure they got the same answer both ways. For some students, that task took the whole class period. For others, though, this opened the door to me challenging them to first draw a visual representation of their situation, and then write an equation to represent the visual. This, in conjunction with some questioning techniques that were either from this book or Visible Learning (I’m too exhausted and lazy right now to check), led me to take a thoroughly mediocre lesson and turn it into one of the best moments of my teaching career.

Conclusion

If nothing else, Boaler has revitalized my passion for teaching mathematics. As is likely obvious from my lengthy post, I have a lot of opinions on Boaler’s writing and many ideas for how it will impact my teaching. Believe it or not, I have a lot more to say on the topic, but I am sick and tired and this post is already too long. If you teach any level of math, you should read this book. I read a lot of educational books, but this one has affected me in the most profound way in at least 2 years, and potentially ever.

What I’m hoping to read next

Boaler cited many different works that I made note of, and I’ll list some below for later reference.
What’s Math Got to Do with It? By Jo Boaler
A couple articles by Alfie Kohn (weirdly enough, I had already picked out my next educational read, and it is a book about homework by Kohn)

[Alex · Rating 4 out of 5]

one of the more useful books on teaching math i've read so far, albeit focused on k-12. i hope to find something equivalent focused on higher ed

[Greg Chandler · Rating 4 out of 5]

Carol Dweck proposed the idea of Mindsets in her book by that name. A fixed mindset assumes that potential is predetermined - you're smart or not, athletic or not... and people with a fixed mindset allow this perception to limit them. People with a negative fixed mindset are quick to give up because they don't believe they can succeed. People with a positive fixed mindset are also quick to give up when their mindset is challenged by difficult tasks. They avoid taking risks in fear that they will fail, and disprove their positive self-image.

People with a growth mindset believe that they can improve their abilities, through understanding the mistakes they've made and by working to understand the concepts they struggle with.

Boaler looks at these ideas under the lens of teaching and studying mathematics. She provides some clear direction for anyone who wants to encourage a growth mindset in their students. Most of the strategies she purports would be just as applicable to other subjects. There are, though, some concrete examples that relate directly to teaching math.

Fostering a growth mindset involves providing students with challenging work and specific feedback that helps direct their continued progress. It eschews praise for students' intelligence in favor of recognition for their effort and learning strategies. It requires students to use meta-cognition - evaluating their own thinking processes. Teachers encourage students to look at how they think about a problem, praise their strategies and encourage them to celebrate and learn from mistakes.

As with Dweck's book, there are times when the writing seems more like a sales pitch. It's hard to picture most students "celebrating" mistakes. But using mistakes as "teachable moments" is a powerful teaching strategy. I often hope students will exhibit common mistakes during whole class work, so that I can address those mistakes (and examine why students commonly make them). I let students know that they are not the only ones making those mistakes, and that the whole class can learn from it. One goal of the growth mindset is that students are willing to make mistakes. While a mistake is an attack on a fixed-mindset student's self-image, it is seen as an opportunity for improvement and learning to students with a growth mindset.

Focusing on grades (as well as the implied intelligence or talent they represent) is a key to maintaining a fixed mindset. Students who see their worth defined by grades are quickly labeled as either smart or dumb, as successes or failures. Teachers too often buy into this thinking as well. When I was studying towards teacher certification, I observed classes where the teacher on the first day warned me that the students in one particular low-level class were "not the sharpest tools in the box". His attitude played out as he went over the previous day's homework without involving the students, modeled new material for a few minutes, left students on their own to start a worksheet (instead of providing individual help or direction) and told them, to finish the sheet as homework. His low expectations were clear. He doubted they could be successful, so didn't waste his time trying to help them.

While most teachers, hopefully, are less jaded than this, they are still probably often guilty of judging students by the grades they receive or have received in the past. Students are probably much harder on themselves. When I taught middle school 8th grade math, I had one section of lower-level students. When most of them failed an assessment, they asked "what do you expect? You know we're the dumb class." I denied this and pointed out that the assessment they failed was the same one I gave all my classes. I then asked them to try the quiz again, but instead of focusing on the math to choose the answer they thought made the most sense. They aced it! This showed them (and me) that, though they lacked some of the procedural skill of the other students, their critical thinking was just as good.

Boaler sees grades as so destructive that they should be eliminated. While i don't see this as a reasonable alternative at the high school level, where students will be judged by prospective colleges largely on their GPAs and subject grades, I think it might be more practicable in middle school. Students in these middle grades are usually promoted whether they master material or not, whether they earn A's or F's. Grades are often overly influenced by homework and classwork completion, and don't always reflect students' understanding of material. In this setting, the negative influence of grades may outweigh the usefulness. I actually attended an ungraded school for 5th through 8th grade. There were very few curricular requirements. report cards provided only a short essay about the students' progress based on informal assessment. I think there may be some merit to a gradeless system, if it was coupled with regular formative assessment and targeted instruction individualized to student progress. Students who master the middle school curriculum could still take Algebra and prepare for the honors-level high school courses.

Boaler also rails against regular testing - advocated testing as little as possible. While many see the state-mandated standardized (and largely multiple choice) tests as having a negative impact on student achievement, particular when they drive a "teaching to the test" drill-and-kill strategy, eliminating all testing would be counter-productive. Regular assessment provides the information needed to tailor instruction to student needs. This suggests not an elimination of assessment, but an emphasis on formative testing.

Boaler is on the "no homework" band wagon, seeing it as unproductive busy work that dampens students love of learning and a discriminatory practice as some students have less resources to help them complete it. I think homework is an important part of math instruction: from personal experience I know that students who turn in homework tend to do better on quizzes and tests. It certainly is possible, however, to overdo it - and for homework to become an undue burden. My own kids often came home with large packets of math problems that might take more than an hour to do. A smaller set of problems will allow students to practice retrieving the information they learned in class without overwhelming them.

The best strategy Boaler offers for encouraging a growth mindset in students is the use of low-floor, high-ceiling problems. These are problems that allow students to succeed at different levels, starting with in depth understanding of a simpler problem and progressing into higher-level thinking (abstraction) and extending to creative work. She provides a few examples of this sort of problem and links to sites where additional problems can be found. Generally, she proposes working on students for more in-depth conceptual understanding. Inquiry and hands-on experiences can help with this. It is, however, important to help students tie these conceptual understanding to the procedural exercises that they are more commonly expected to perform on. Boaler does not address this, assuming that conceptual understanding will lead to procedural capability (which it sometimes fails to do) or that this procedural fluency is not needed.

The notion of mindsets may be relatively new, but explores ground previously covered. Piaget's work fell rather out of favor as educators viewed his learning stages as too rigid: there is no magical age (12 years) at which every student can think abstractly. But his notion that students construct new knowledge by tying it to previous schema seems directed related to the idea of a growth mindset. The notions of intrinsic vs. extrinsic motivation also are reflected. Students who are intrinsically motivated by the desire to learn are much more successful than those who are extrinsically motivated by grades.

I would recommend this book to anyone who is hoping to inculcate a growth mindset in their students. It offers more practical advice than Dweck's book and is focused more specifically on teaching and learning. Dweck's work, re-examined through the lens of teaching and learning math, provides persuasive argument for developing a growth mindset, encouraging students to learn from instead of hide from mistakes, and promoting the intrinsic motivation of learning from challenging problems.

[Margaret Coleman · Rating 4 out of 5]

This book did an excellent job educating us as learners and as preservice teachers. As learners, we learned the importance of building and maintaining a growth mindset, as well as, the importance of recognizing the value of failure and resilience through difficult tasks. Another key fact we learned from this text it is important for any learner to monitor their progress and growth so that they are connected and responsible for their own learning.
This book helped us as pre-service teachers by reminding us that it is important to create an environment that encourages students to adopt a growth mindset. As pre-service teachers, we want students to encounter challenging tasks that have a positive influence on their learning. We also learned the importance of summative and formative assessments within a lesson, this helps the teacher and students with monitoring learning with each lesson taught.
Throughout our teaching careers, we will remember that the environment created by the teacher is important. Students must have a supportive and nurturing atmosphere in order to feel successful in school. Lastly, we will maintain the idea that we, as teachers, need to challenge our students daily and give them positive feedback about mistakes. We need our students to feel comfortable with working on harder problems, taking chances, and making mistakes to learn at their fullest potential.

[Tracey · Rating 5 out of 5]

"Mathematical Mindsets" is a great book for math educators, elementary through high school, to read and ponder. The book begins with explaining how mathematics learning takes place in the brain. This is when the discussion of fixed mindset and growth mindset is addressed. The author also discusses the power of mistakes and struggle. Helping students see mistakes as opportunities for growth helps students use a growth mindset toward learning more challenging math. Samples of activities, questioning techniques, web addresses to incredible learning sites, and assessment tools are all provided to help any math educator to move toward a growth mindset in mathematical teaching and learning.

One very interesting chapter was on the inequities in math education, and how inequities in course placement can become illegal. Many schools are still holding back children of color from advanced math courses. Studies show that more African American and Latino students are required to repeat courses than whites and Asians, thus reducing the number of students of color to enter AP Calc during their high school years. Educators must focus on math education for all, not just those who are "gifted" in mathematics. Math is not elitist, but essential for all.

[Arun Mahendrakar · Rating 5 out of 5]

If you are someone who is dealing with teaching or learning Math, this book is for you. Traditional ways of teaching mathematics is challenged at every level. Proven techniques mentioned throughout the book are a testament that Mathematics not only requires, but also improves imagination by leaps and bounds.

I really liked the concept of 'If you want to understand it, you draw it' as one teacher taught her students. The book provides many illustrations of how this can help students imbibe the concepts of mathematics.

There are numerous web links in the book that are well-worth following up after reading the book. As a parent, I'd like to know how best to teach a creative subject like Math to my kid.

I have a question for the author though. In page 29, the author mentions "The high incidence of fixed mindset thinking among girls is one reason that girls opt out of STEM subjects". But it is not clear as to why girls have fixed mindset in the first place. Women opt out of STEM subjects and seem to join one of the many 'pink-collar' jobs - Nursing, Teaching, Counselors, HR Managers - does this mean these jobs don't need people with growth mindset?

[Luan Nguyen · Rating 5 out of 5]

This book wraps up the concept of growth mindset beautifully within the context of mathematics education.
Boaler constantly reiterates Dweck's principles of growth mindset while introduces practical strategies that teachers, no matter what grades, can easily implement. She also challenges the reader about the conventional and orthodox methods of teaching that do not work; and of course, backs them up with solid evidence from research. Her take away message is to foster an inquiry-based learning within the classroom is plausible. However, meta-analysis evidence conducted by John Hattie suggests that this approach to teaching yields a low effect-size, mainly due to the bad 'timing' that students were exposed to inquiry-based learning before they had the chance to build on surface knowledge.

[George Woodbury · Rating 5 out of 5]

Although aimed at K-12 students, I found many ideas that can be incorporated in math courses in colleges as well.
The importance of discovery and reflection are ideas that are underutilized in many classes (and nonexistent in some) and have had a huge impact on my students and their mathematical development.
Giving students an opportunity to work collaboratively is a great way to encourage greater conceptual understanding and helps students learn to communicate mathematically.
Allow students to make mistakes - that is how they learn.
Attitude is important. Help students develop a growth mindset instead of a fixed mindset.

I am looking forward to checking out "What's Math Got To Do With It?"

[April Storm · Rating 4 out of 5]

As math learners, we have discovered through this book that there is no such thing as ‘math people’. Everyone can be successful in math if they understand that it requires effort, time, and growth. As learners, we now understand that we should be confident in our abilities, no matter the speed we do it or what our starting point is. From the perspective of pre-service teachers, we know that we need to take this message about growth mindsets and apply it in our own classroom. Specifically, we can focus on praising effort, talking about mistakes in a positive way, and holding all students to high standards. We want our future students to view math as a creative process rather than a performance based subject. Overall, this book would be a great read for parents, students, and teachers.

[Abigail Lutz]

“Mathematical Mindsets” is an informative text that provides concrete ideas of how to support the development of growth mindsets with evidence of the importance of this mindset development. Jo Boaler teaches readers how to structure math instruction and tasks to help build a growth mindset in your students. She places emphasis on praising student’s effort rather than their intelligence, creating an environment where mistakes are seen as opportunities for growth rather than as failures, and where low floor, high ceiling tasks make problems accessible to all students at varying levels. This is a wonderful book for pre-service and in-service teachers because it gives practical advice and specific activities for how to implement these ideas into a classroom.