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**The Mindset Revolution: Teaching mathematics for a growth mindset**

In 2006 a trade book appeared on bookshelves that would ultimately have one of the biggest impacts on education of any research volume ever published. Jo Boaler Professor of Mathematics Education Stanford University

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**The New Psychology of Success**

Carol Dweck, 2006, Mindset: The New Psychology of Success In “Mindset: The New Psychology of Success” (2006a) Carol Dweck summarized key findings from her research on the nature and impact of different mindsets. Kids with a “growth mindset” believe that ability grows with experience, kids with a fixed mindset believe that you are smart or you are not, some people can be good at math some not. The implications of this mindset are profound. While Dweck’s work on mindsets has become a high priority for many schools, I have found that teachers don’t know what it means. principals have told me in mtgs that their teachers are all on board with the ideas but what does it mean for their mathematics teaching? In this presentation I will describe some of the recent work I have been doing with math teachers, and highlight some of the pedagogical structures that support the development of growth mindsets.

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**The myth of mathematics**

Being good at math is a “gift” – some people are naturally good at math, some are not In mathematics education, in our role as teachers, teacher leaders, principals, - any part of math education we deal with a very damaging myth. The myth is this: Math is a gift, some people are naturally good at math and some are not – this is one of the most damaging ideas in math education –

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Shake It Up Chicago The idea is everywhere and we are going to look at it in math teaching for the rest of the session. Before that lets see how strong the idea is in students’ perhaps most important influences – TV and film. I cant seem to watch any kids show at the moment without seeing math come into it. First clip is a tweenie Disney show that I have the very questionable pleasure of watching with my daughters. Certainly get new insights into the sources of kids sexist. Racist, homophobic views when spending any time with disney shows, but anyway … lets consider the messages in here What is she communicating to the millions of young vulnerable viewers? that math is something you can do or not. Not to mention the pain, the idea that math is all about performance …

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Hollywood Math features dominantly– but the messages are not always that math is bad ….

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Hollywood another

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Hollywood Also about the fixed nature of math, Cecie in shake it up a good example, this is another… along with some other damaging , sexist messages thrown in

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Brain Plasticity So instead of learning from disney and hollywood lets consider the science of math learning. The brain isn't actually made of plastic but the idea of neuroplasticity refers to the life-long capacity of the brain to change and rewire itself. In fact there have been stunning examples of the brain’s ability to create new neurons and connections in recent years.

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**When learning happens …**

When learning happens a synapse fires – electric currents –and create new neurons and connections between neurons. A synapse fires

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Synapses are like footprints in the sand – the brain follows the footprints and makes them deeper the more they are followed The paths of synapse are like footprints in the sand, they can become firm paths that are substantial or without use they wash away.

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**Plasticity Learning creates and strengthens synapses**

The plasticity of the brain means that these connections grow into adult-hood If Pathways aren’t followed they may be discarded “use it or lose it” The rain is so flexible it is changing by the minute losing, gaining, your brains can be changing in here tonight by connections you might make

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Brain growth So lets look at some examples of brain plasticity. The first is very sad but important to know about – these 2 images are of the brains of 2 equivalent 3-year old children, one from a loving home and one neglected.

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**London ‘Black Cab’ Drivers**

Now let me introduce you to the life of a London cab driver All-London drivers' have to learn 320 routes (or runs). This will help them learn the 25,000 streets and 20,000 landmarks and places of interest in the six mile radius of Charing Cross.

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**– 25,000 streets and 20,000 landmarks**

“The Knowledge” – 25,000 streets and 20,000 landmarks They have to pass a test called “the Knowledge” It takes between two and four years to pass the All-London Knowledge. Once you are licensed you can work anywhere in the Greater London area.– There is no grid structure in London

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Brain Growth London taxi drivers have a larger hippocampus than London bus drivers bus drivers as you probably know, follow a set route. It is because this region of the hippocampus is specialized in acquiring and using complex spatial information in order to navigate efficiently.

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**A 6-year old girl Had half of her brain removed**

Amazed doctors and scientists - within months she had recovered functions from the “missing” side of the brain A 3rd example was the 9-year old girl who had half of her brain removed – one hemisphere. her left side was paralyzed . Within weeks she had grown the connections back and developed all of her functions back – that was because of the growth of her brain.

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Neuroplasticity refers to the lifelong capacity of the brain to change and rewire itself in response to learning and experience. So here is the irony - Children can develop half of a brain yet we think kids can’t develop a few neurons to learn algebra. I am now going to share with you a letter written by a high school math dept that I won’t name, to illustrate for you the task that is before us in educating teachers about the potential of students. We can see the racist, classist, elitist issues as well as fixed mindset

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**From a local, public high school math dept in 2012**

We know taking advanced math classes is the best predictor for success in college. Nothing would make us happier than being able to produce only graduates that have calculus on their transcripts! However, brain theory supports the reality that confounding student situations interfere with their ability to focus and succeed as they move through advanced mathematics in high school. We live in an affluent community. Most of our students are fortunate to come from families where education matters and parents have the means to support and guide their children in tandem with us their teachers. Not all of them. We are concerned about the others who for reasons that are often objective (poor math background, lack of support at home, low retention rate, lack of maturity etc) cannot pass our Algebra II regular lane course. Many of them are VTP students or under-represented minorities. Others are serious, committed special ed students (etc) Racist overtones are huge – shows us clearly where we are in our need to educate teachers about students’ potential

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**Brain research tells us:**

Every child can excel in mathematics in school, from elementary to high school The potential of the brain to grow no matter where they start from is huge

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**Ability? Each new learning experiences changes your “ability”.**

We use fixed ability language all the time – high and low kids etc So scientists know that kids can develop the wiring to learn any math in school, but many teachers believe in natural ability and schools are set up on a model of fixed ability thinking. Of course there are Kids who seem naturally good and fast but they are good and fast because those synapses have been firing since toddlers. First I would like to knock down another myth – that working on math quickly equates to ability – my evidence for that is that mathematicians are the slowest people I have ever worked with. Lauren Schwartz a fields medal mathematician reflecting on his school learning Yes some kids are faster and some kids are born with more brain power but the difference they may have is nothing compared to what they can get from experience as these examples tell us. Here the thing – people think of kids as having “ability” – a word I hate, but that ability can shift in every experience and every lesson. That is the beauty of learning and of the brain. But most aspects of US schools run on a model of brain stability.

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**Laurent Schwartz ‘A Mathematician Grappling with his Century’**

..I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Even when I was the first to answer the teacher's questions, I knew it was because they happened to be questions to which I already knew the answer. But if a new question arose, usually students who weren't as good as I was answered before me. Towards the end of the eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time. I never talked about this to anyone, but I always felt convinced that my imposture would someday be revealed: the whole world and myself would finally see that what looked like intelligence was really just an illusion. If this ever happened, apparently no one noticed it, and I’m still just as slow. (...)At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn't have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant. Naturally, it's helpful to be quick, like it is to have a good memory. But it's neither necessary nor sufficient for intellectual success.

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**How important are the ideas that students hold about ability?**

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**Carol Dweck: Mindset Mindset: The New Psychology of Success. (2007)**

Fixed - math ability is a “gift” Growth – math ability or “smartness” grows with experience Growth mindset behaviors – persistence, learn from mistakes, determination to keep going, encouraged by other’s success Affects students from across the achievement spectrum Role of parents in encouraging fixed mindset

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**7th grade students with a growth mindset outperform those with a fixed mindset in math**

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**The impact of a mindset intervention (same math teacher, same curriculum)**

Eight 25 min periods one per week in the spring of 7th grade

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**Research on Mindset and equity**

African American students show sharpest increase in grades and valuing school A growth mindset eliminates any gender gaps eg in highest SAT levels

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Mindset and gender High achieving 5th grade girls did not cope well with challenge The higher their IQ the more difficulty they had, in boys the reverse was true At the end of 8th grade there was a gender gap but only among fixed mindset students

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**Mindset and gender Calculus at Columbia Stereotyping is alive and well**

Stereotyping only affected those with a fixed mindset, their confidence eroded over the semester and they abandoned plans to pursue STEM subjects Women here who have been through university math depts can probably identify with that

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Implications Seeing math as a gift not only makes students vulnerable to lack of confidence but vulnerable to stereotypes too Having a growth mindset is what we want all teachers and all students to have

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**The big message Intelligence is malleable, but …**

Students, teachers, schools, parents treat math learners as though it is relatively fixed Brainstorm with others around you - which aspects of schools / math teaching encourage a FIXED mindset? Choose your top 3 What are the big culprits

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**What has mindset got to do with my math teaching?**

Messages Mindset + Math mistakes grading & feedback grouping tasks questions asked norm setting

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**Today - mindset Grouping Classroom Math Tasks Assessment & Grading**

Mistakes Messages

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**Ability Grouping Burris, C., Heubert, J., & Levin, H. (2006).**

Accelerating Mathematics Achievement Using Heterogeneous Grouping. American Educational Research Journal, 43(1), No bigger fixed mindset message than putting children into ability groups – telling them that their prior achievement will determine what they work on in their future for ever more. We have decades of research on the harmful impact of ability grouping and the countries that are most successful are those that don’t have it For the first three years of the study students were taught in tracked classes with only high track students being taught an advanced curriculum. In the next three years all students in grades 7-9 were taught the advanced curriculum in mixed ability classes and all of the 9th graders were taught an accelerated algebra course. The researchers looked at the impact of these different middle school experiences upon the students’ achievement and their completion of high school courses, using 4 achievement measures, including scores on the advanced placement calculus examinations. They found that the students from de-tracked classes took more advanced classes, pass rates were significantly higher and students passed exams a year earlier than the average in New York State. The increased success from de-tracking applied to students across the achievement range – from the lowest to the highest achievers.

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Ability Grouping In England researchers followed children through years 4 and 6 comparing those taught in sets with those grouped heterogeneously over the period of a year. Nunes, Bryant, Sylva & Barros, 2009 Need to empower schools to take brave decisions.

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Classroom Math Tasks How do you maintain a growth mindset when math class is a series of closed questions that you get right or wrong? Yeachers in our workshop If getting wrong hard to maintain growth – if getting right too

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**Most math classrooms offer math as a performance subject not a learning subject.**

Rachel Lambert’s 6 year old son: “Math is too much answer time and not enough learning time” Tasks need to give students the space to learn. childrenthink the point of math is whether they get it or not How can a 6 year old see that math is too much about performance and not enough about learning, yet we as educators have built up a whole performance model of math?

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**Growth Mindset Task Framework**

Task focuses on learning: opportunities to learn something rather than demonstrate what you know Openness Ways of seeing Multiple entry points Multiple paths / strategies Clear learning goals and opportunities for feedback. We developed a frame work for growth mindset tasks

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An example Comes from a 5 week algebra class I taught with graduate students in summer school Our goal: to teach algebra as a problem solving tool Underachieving 7th, 8th grade students Tasks – Ruth Parker, Mark Driscoll, SMILE, Points of Departure

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**How many blocks are in case 100?**

\ Think about what you see. On your own, how is the pattern growing? How would you describe it? When you have thought about it for a while try and describe it to the person sitting next to you.

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Sarah Kate Selling

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Luke Jorge Carlos To focus on the practice of representation, I’d like to introduce you three students from one of the project math classes. In this photograph, we see Luke, Jorge, and Carlos. Today I’d like to share how they engaged in representational practice by looking at both their written mathematical work and their small group interactions.

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**Recursive pattern: +5, +7…**

Case n has (n+1)2 blocks Recursive pattern: +5, +7… Case Case Case 3 Luke Jorge As Jo described earlier, at the beginning of the summer, the boys showed little interest or engagement in mathematics. They were primarily interested in socializing. But by the end of summer school, the boys were deeply engaged in the mathematics in class. And they were participating in representational practice in a wide variety of ways without the support of a teacher. On one of the final instructional days, the three boys worked together for 70 minutes on a growth pattern problem. Here are the first three cases of the growth pattern that were presented to the boys. They were tasked with finding out the number of blocks in case n. Take about 15 seconds to think about this pattern. How is it growing? Where do you see the new blocks? Notice that the number of blocks in case n is given by (n+1) squared. The recursive pattern between cases given by consecutive odd integers. On the previous day, they had worked with linear growth pattern problems. This problem was the first quadratic pattern that they had worked with in this class. During the 70 minutes that the boys spent together on this problem, they engaged in representation in a wide variety of ways. I’d like to share a brief excerpt with you to illustrate some of these ways. Before we examine the boys’ interactions, it is important to know that the boys visualized the pattern differently. CLICK Luke and Jorge both visualized the growth pattern by seeing the new blocks as the bottom row in each case.CLICK In contrast, Carlos visualized the new blocks as the top of each previous stack plus two new blocks starting new stacks. Notice that the boys represented the pattern differently. Jorge drew horizontal bars labeled with the number of blocks in that row. Luke used the graph paper to connect the different stacks. And Carlos used the original separated stack representation that was given in the original problem. Carlos

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**Recognizing different ways of seeing**

In the following slides, to best communicate the boys’ interactions, I’ve included the boys’ discourse, annotated photographs showing their body positioning and gesture, and a representation of the mathematics being discussed. The green corresponds to Carlos’s way of seeing. The blue corresponds to Jorge and Luke’s way of seeing. In the first part of the exchange, we see the boys recognize that different ways of seeing exist within the group. Carlos begins, “Watch… When Carlos responds with “What?”, this keys the boys into the fact that they may not be talking about the same thing. Notice that the boys are discussing the pattern using the third person.

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**Explaining different ways of seeing**

The boys continue explaining with Carlos taking the lead. Read slide

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**Resolving through connecting**

In the last part of the exchange, the boys work to resolve the difference in perspective. Read the slide. Notice that they finally resolve the difference by connecting each visualization with the same numerical recursive pattern.

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**A case: mathematical practices & heterogeneity**

Gauss What engages the students so strongly and for so long? And what does it have to do with growth mindset teaching? They started summer school turned off – didn’t know each other, disrupted class, now working for 90 minutes on one problem

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3 boys video You will see them – hard to make sense of the actual maths – but meandering around as problem solvers do – following one path then trying another, considering each others representations etc. Showing this to get a sense of what problem solving and engaging in practices looks like

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**When math tasks are opened for**

Different ways of seeing Different methods / pathways Different representations The opportunities for learning and developing a growth mindset are increased

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**1 ÷ 2/3 Cathy Humphreys, 7th graders**

Connecting Mathematical Ideas – video cases Mathematics as sense-making which encourages: Different ways of seeing Different methods / pathways Different representations

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**Assessment &Grading Grades Diagnostic Feedback**

Diagnostic Feedback significantly higher Grades Diagnostic Feedback & Grades Diagnostic Feedback significantly higher (Butler)

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Timed Tests?

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**A Timed Test of 50 questions to finish in 3 minutes**

From 1st to 5th grade

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From neuroscience .. Math should never be associated with speed

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From Sian Beilock:

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**no relation between stress and prior achievement**

Math stress cuts across the achievement spectrum –particularly affecting girls

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From Sian Beilock:

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4th grade:

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2nd grade:

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Mistakes “Every time a student makes a mistake in math they grow a new synapse” (Carol Dweck) I was struck by that and so have teachers been that I work with, what does it mean?

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**Every mistake grows a synapse**

Mistakes are good They are the time your brain is growing Students should be making mistakes Students hate making mistakes – because they have been brought up in a performance and not a learning culture Students & most teachers view them negatively

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Reposition mistakes In classroom norms In 1:1 interactions Eg Jaime

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**Messages www.brainology.com (Geoff Cohen)**

“I am giving you this feedback because I believe in you” Resulted in significant achievement gains, again especially for minority students. My freshmen All women and people of color no white men!

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**What is the big messages or message(s) that the students took away?**

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Stanford freshmen They were nervous and the class was at the same time as when I reveealed the academic bullying of 2 mathematicians – milgra and bishop so sensitive to “bad mathematicians” ..

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To conclude It is critically important that teachers and students are encouraged to develop a ‘growth mindset’ The ideas I have shared for teaching and assessing are not new but the reason for implementing them – to develop growth mindsets - is an important impetus for change, and an idea that many teachers and schools understand The ways teachers teach math has huge implications for students’ mindsets I know from discussions with other presenters that the next few days will be full of opportunities for learning about teaching that supports a growth mindset – engaging with the CC practices is a big part of that.

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**Udacity & Stanford ‘MOOC’**

Teaching Mathematics for the Common Core https://www.udacity.com Talk no longer than 90 seconds before engaging the learner Just been hearing that Susan Jo Russel and Virginia Barnstape and Megan Franeke recently taught on online course

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