Boaler J., Dieckmann J.A., LaMar T., Leshin M., Selbach-Allen M. and Pérez-Núñez G. (2021). The Transformative Impact of a Mathematical Mindset Experience Taught at Scale. Front. Educ. 6:784393.

LaMar, T., Leshin, M., & Boaler, J. (2020). The Derailing Impact of Content Standards – an Equity Focused District held back by Narrow Mathematics. International Journal of Educational Research, Open. Volume 1, 2020, 100015

Selbach-Allen, M. E., Williams C. A., & Boaler, J. (2020)  What Would the Nautilus Say? Unleashing Creativity in Mathematics! Journal of Humanistic Mathematics. 10 (2), 391-414.

Boaler, J. (2019). Prove it to Me! Mathematics Teaching in the Middle School 24 (7) 422-429.

Boaler, J. (2019). Unlocking Children’s Math Potential. The Review, 69-77

Boaler, J., Cordero, M., & Dieckmann, J. (2019). Pursuing Gender Equity in Mathematics Competitions. A Case of Mathematical Freedom.  MAA, FOCUS, Feb/March 2019.

Boaler, J. (2018). Introduction to Special Issue. Dispelling Myths About Mathematics. Education Sciences.

Boaler, J.; Anderson, R. (2018). Considering the Rights of Learners in Classrooms. Democracy and Education. 26 (2)

Boaler (2018). Developing Mathematical Mindsets: The Need to Interact with Numbers Flexibly and Conceptually. American Educator.

Boaler, J.  & Selling, S. (2017).  Psychological Imprisonment or Intellectual Freedom? A Longitudinal Study of Contrasting School Mathematics Approaches and Their Impact on Adult’s Lives. JRME. 48 (1), 78-105.

Boaler, J., Chen, L., Williams, C., & Cordero, M. (2016). Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning. Journal of Applied & Computational Mathematics 5 (5)

Boaler, J., & Sengupta-Irving, T. (2016). The many colors of algebra: The impact of equity focused teaching upon student learning and engagement. The Journal of Mathematical Behavior, 41, 179-190.

Anderson, R.K.; Boaler, J.; Dieckmann, J.A. (2018). Achieving Elusive Teacher Change through Challenging Myths about Learning: A Blended Approach. Educ. Sci. 8, no. 3: 98.

Boaler, J., Dieckmann, J., Pérez-Núñez, G., Liu Sun, K. & Williams, C. (2018). Changing Students Minds and Achievement in Mathematics: The Impact of a Free Online Student Course. Front. Educ 3:26.

Boaler, J., Chen, L., Williams, C. & Cordero, M. (2016). Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning. J Appl Computat Math 5: 325.

Boaler, J. (2013). Ability and Mathematics: the mindset revolution that is reshaping education. FORUM, 55, 1, 143-152.

Plenary Talk, ICME 12, July 2012, COEX, Seoul, Korea.

Boaler, J., Altendorff, L. & Kent, G. (2011). Mathematics and Science in the United Kingdom: Inequities in Participation and Performance. Oxford Review of Education.

Boaler, J. (2008). When Politics Took the Place of Inquiry: A Response to the National Mathematics Advisory Panel’s Review of Instructional Practices. Educational Researcher.

Boaler, J. (2008). Promoting ‘Relational Equity’ and High Mathematics Achievement Through an Innovative Mixed Ability Approach. British Educational Research Journal. 34 (2), 167-194

Boaler, J & Staples, M. (2008). Creating Mathematical Futures through an Equitable Teaching Approach: The Case of Railside School. Teachers’ College Record. 110 (3), 608-645. Cited in the “Supreme Court of the United States” in the case of parents vs the Seattle Court District, (Nos 05-908 & 05-915)

Boaler, Jo & Sengupta-Irving, Tesha (2006). Nature, Neglect and Nuance: Changing Accounts of Sex, Gender and Mathematics. in Chris Skelton & Lisa Smulyan (eds). Handbook of Gender and Education. Sage Publications (pp 207-220).

Boaler, J. (2006). “Opening Our Ideas”: How a detracked mathematics approach promoted respect, responsibility, and high achievement. Theory into Practice, Vol. 45, No. 1.

Boaler, J. (2005). The ‘Psychological Prison’ from which they never escaped: The role of ability grouping in reproducing social class inequalities. FORUM, 47, 2&3, 135-144.

Boaler, J. (2002). Exploring the Nature of Mathematical Activity: Using theory, research and ‘working hypotheses’ to broaden conceptions of mathematics knowing. Invited Paper. Educational Studies in Mathematics, 51(1-2), 3-21.

Boaler, J. (2002). Learning from Teaching: Exploring the Relationship Between Reform Curriculum and Equity. Journal for Research in Mathematics Education, 33(4), 239-258.

Boaler, J. (2002). Paying the Price for “Sugar and Spice”: Shifting the Analytical Lens in Equity Research. Mathematical Thinking and Learning. 4(2&3), 127-144.

Boaler, J. (2002). The Development of Disciplinary Relationships: Knowledge, Practice and Identity in Mathematics Classrooms. For the Learning of Mathematics, 22(1), 42-47.

Boaler, J (2001). Mathematical Modeling and New Theories of Learning. Teaching Mathematics and its Applications. 20 (3), 121-127

Boaler, J., & Greeno, J. G. (2000). Identity, Agency, and Knowing. Multiple Perspectives on Mathematics Teaching and Learning, 1, 171.

Boaler, J., Wiliam, D., & Brown, M. (2000). Students’ experiences of ability grouping – disaffection, polarization and the construction of failure. British Educational Research Journal, 26, 5, 631-648.

Boaler, J (1998). Open and Closed Mathematics: Student Experiences and Understandings. Journal for Research in Mathematics Education Vol. 29, No. 1, 41–62

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Some of the amazing evidence of brain plasticity comes from studies of London Black Cab drivers. To become a black cab driver in London you need to study for between two and four years and at the end of that time take a test called The Knowledge. To pass The Knowledge you must memorize over 25,000 streets and 20,000 landmarks in Central London. Scientists found that after this complex spatial training the hippocampus of the taxi drivers had grown significantly. The hippocampus is a part of the brain that specializes in acquiring and using complex spatial information. When drivers retire, many years later, the hippocampus shrinks back down again.

The studies conducted with Black Cab drivers, of which there have now been many, showed a degree of brain flexibility, or plasticity, that stunned scientists. They had not previously thought that such brain growth was possible. This led scientists to shift their thinking about ability and the possibility of the brain to change and grow.

Around the time that the Black Cab studies were emerging, something happened that would further rock the scientific world. A nine-year old girl, Cameron Mott, had been having seizures the doctors could not control. Her physician, Dr. George Jello, proposed something radical. He decided he should remove half of her brain, the entire left hemisphere. The operation was revolutionary—and ultimately successful. In the days following her operation, Cameron was paralyzed. Doctors expected her to be disabled for many years, as the left side of the brain controls physical movements. But as weeks and months passed, she stunned doctors by recovering function and movement that could mean only one thing—the right side of her brain was developing the connections it needed to perform the functions of the left side of the brain.

Doctors attributed this to the incredible plasticity of the brain and could only conclude that the brain had, in effect, “regrown.”

The new brain growth had happened faster than doctors imagined possible. Now Cameron runs and plays with other children, and a slight limp is the only sign of her significant brain loss. To learn more about this story visit the Today Show website.

The new findings that brains can grow, adapt, and change shocked the scientific world and spawned new studies of the brain and learning, making use of ever-developing new technologies and brain scanning equipment. In one study that is highly significant for those of us in education, researchers at the National Institute for Mental Health gave people a 10-minute exercise to work on each day for three weeks. The researchers compared the brains of those receiving the training with those who did not. The results showed that the people who worked on an exercise for a few minutes each day experienced structural brain changes. The participants’ brains were “rewired” and grew in response to a 10-minute mental task performed for just 15 days over three weeks. Such results should prompt educators to abandon the traditional fixed ideas of the brain and learning that currently fill schools—ideas that children are smart or dumb, quick or slow.

If brains can change in three weeks, imagine what can happen in a year of math class if students are given the right math materials and receive positive messages about their potential and ability.

This article contains excerpts from Jo Boaler’s new book, Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching

Teaching Children With These Messages

In a recent summer camp the youcubed team taught mindset and brain messages to local 6th and 7th grade students. They also – importantly – taught math in way that supported the new evidence. The camp had a huge impact on the students. After 18 days of math teaching the students improved their scores on standardized tests by an average of 50%.

You can see the camp in this music video:

Here are some quotes from the students:

“They taught us how, um, how math is for everybody, and I believed that I wasn’t a math person before but now I believe that anybody can do math, and, and that helped me a lot. And the way I thought that math was all about right answers and wrong, but it’s really about ideas and it’s very creative, and that helped me like it a lot more.” –Olivia

Four or five brains work better than one, so when we combine all of our ideas …. When we got an idea, we shared it with the group and then we compared all of them and if we got, a few people got the same one, we try convincing the other person and we see who’s right. –Antonio

Yeah, um, math is .. like now that I’ve seen a whole new different perspective of it, I really like it ‘cause it’s really, it’s a really creative subject, and there’s a lot of ways to figure out a math problem, and it’s, um, it’s really fun. –Isabel

References

Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. San Francisco, CA: Jossey-Bass.

Boaler, J (2015). What’s Math Got to Do with It? How Teachers and Parents Can Transform Mathematics Learning and Inspire Success. New York: Penguin

Karni, A., Meyer, G., Rey-Hipolito, C., Jezzard, P., Adams, M., Turner, R., & Ungerleider, L. (1998). The acquisition of skilled motor performance: Fast and slow experience-driven changes in primary motor cortex. PNAS, 95(3), 861–868.

Abiola, O., & Dhindsa, H. S. (2011). Improving classroom practices using our knowledge of how the brain works. International Journal of Environmental & Science Education, 7(1), 71–81.

Maguire, E., Woollett, K., & Spiers, H. (2006). London taxi drivers and bus drivers: A structural MRI and neuropsychological analysis. Hippocampus, 16(12), 1091–1101.

Woollett, K., & Maguire, E. A. (2011). Acquiring “The Knowledge” of London’s layout drives structural brain changes. Current Biology, 21(24), 2109–2114.Book Image

To read about studies of schools that have given students positive messages about their ability, combined with mathematics teaching that supports the brain messages, see:

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29(1), 41–62.

Boaler, J., & Greeno, J. (2000). Identity, agency and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171–200). Westport, CT: Ablex Publishing.

Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student learning (revised, expanded edition). Mahwah, NJ: Erlbaum.

Boaler, J. (2002). Learning from Teaching: Exploring the Relationship Between Reform Curriculum and Equity. Journal for Research in Mathematics Education, 33(4), 239-258.

Boaler (2005). The ‘Psychological Prison’ from which they never escaped: The role of ability grouping in reproducing social class inequalities. FORUM, 47, 2&3, 135-144.

Boaler, J. (2006). Opening Their Ideas: How a de-tracked math approach promoted respect, responsibility and high achievement. Theory into Practice. Winter 2006, Vol. 45, No. 1, 40-46.

Boaler, J. (2006). Urban Success. A Multidimensional mathematics approach with equitable outcomes. Phi Delta Kappan, 87, 5.

Boaler, J & Staples, M. (2008). Creating Mathematical Futures through an Equitable Teaching Approach: The Case of Railside School. Teachers’ College Record. 110 (3), 608-645.

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We now know that the messages we give students can change their performance dramatically, and that students need to know that the adults in their lives believe in them. Researchers are learning that students’ ideas about their ability and potential are extremely important, much more than previously understood. As well as the messages we give students about their potential, brain research is now showing that messages students pick up from their parents about math and their parents’ relationships with math can also change students’ math learning and achievement.

In an important study researchers found that when mothers told their daughters they were not good at math in school, their daughter’s achievement declined almost immediately (Eccles & Jacobs, 1986). In a new study neuroscientists Erin Maloney and colleagues found that parents’ math anxiety reduced their children’s learning of math across grades 1 and 2, but only if parents helped their children on math homework (Maloney, Ramirez, Gunderson, Levine, & Beilock, 2015) If they did not help them on homework, the parents’ math anxiety did not detract from their children’s learning.

The parents’ math knowledge did not turn out to have any impact, only their level of math anxiety.

Both studies, again, communicate the importance of the messages students receive, as it was not math knowledge that harmed the students’ performance but the parents’ anxiety. We do not know what parents with math anxiety say to their children but it is likely they communicate the negative messages we know to be harmful, such as “math is hard” or “I was never good at math in school.” It is critical that when parents interact with children about math they communicate positive messages, saying that math is exciting and it is an open subject that anyone can learn with hard work, that it is not about being “smart” or not and that math is all around us in the world. For more parental advice on ways to help students with math see the parent page.

Teachers also need to give positive messages to students at all times. Many elementary teachers feel anxious about mathematics, usually because they themselves have been given fixed and stereotyped messages about the subject and their potential. When I taught in my online teacher/parent class that mathematics is a multidimensional subject that everyone can learn, many of the elementary teachers who took it described it as life-changing and approached mathematics differently afterward. Around 85% of elementary teachers in the United States are women, and Beilock, Gunderson, Ramirez, & Levine (2009) found something very interesting and important. The researchers found that the levels of anxiety held by women elementary teachers also predicted the achievement of the girls in their classes, but not the boys (Beilock et al., 2009). Girls look up to their female teachers and identify with them at the same time as teachers are often and sadly conveying the idea that math is hard for them or they are just not a “math person.” Many teachers try to be comforting and sympathetic about math, telling girls not to worry, that they can do well in other subjects. We now know such messages are extremely damaging.

Teachers and parents need to replace sympathetic messages such as “Don’t worry, math isn’t your thing” with positive messages such as “You can do this, I believe in you, math is an open, beautiful subject that is all about effort and hard work.”

This article contains excerpts from Jo Boaler’s new book, Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching

References

Beilock, L. S., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2009). Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences, 107(5), 1860–1863.

Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. San Francisco, CA: Jossey-Bass.

Eccles, J., & Jacobs, J. (1986). Social forces shape math attitudes and performance. Signs, 11(2), 367–380.

Maloney, E. A., Ramirez, G., Gunderson, E. A., Levine, S. C., & Beilock, S. L. (2015). Intergenerational effects of parents’ math anxiety on children’s math achievement and anxiety. Psychological Science, 0956797615592630.

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It may seem obvious that students achieve at higher levels when teachers believe in them, but few would have predicted how much a simple message from teachers can change students’ whole trajectories and achievement.

A study many years ago that probably would not be allowed today, showed something remarkable about teacher beliefs. Researchers told teachers that some classes of students – across six grade levels – had been tested and were capable of greater intellectual growth than other students. In reality, those students were of the same achievement levels as other students and were randomly chosen for the classes. At the end of one year the students’ scores on IQ tests matched the teachers’ false beliefs. When teachers were told that students had higher intellectual ability their students scored at significantly higher levels on IQ tests than students whose teachers were not told anything (Rosenthal and Jacobs, 1968). This study is a powerful illustration that teachers’ expectations and beliefs about students matter.

In a much more recent study, researchers illustrated just how powerful a single message can be. Hundreds of students were involved in an experimental study of high school English classes. All of the students wrote essays and received critical diagnostic feedback from their teachers, but half the students received an extra sentence on the bottom of the feedback. The students who received the extra sentence achieved higher grades a year later, even though the teachers did not know who received the sentence and there were no other differences between the groups. It may seem incredible that one sentence could change students’ learning trajectories to the extent that they achieve at higher levels a year later, with no other change, but this was the extra sentence:

I am giving you this feedback because I believe in you.

Students who received this sentence scored at higher levels a year later. This effect was significant for students of color, who often feel less valued by their teachers (Cohen & Garcia, 2014). I share this finding with teachers frequently, and they always fully understand its significance. I do not share the result in the hope that teachers will add this same sentence to all of their students’ work. That would lead students to think the sentence was not genuine, which would be counterproductive. I share it to emphasize the power of teachers’ words and teachers’ beliefs they hold about students, and to encourage teachers to instill positive belief messages at all times. Furthermore, belief in students alone is not enough (Shouse, 1996). Teachers must couple these beliefs with an academic environment that values open, growth mathematics , mistakes , and high quality assessments.

Teachers can communicate positive expectations to students by using encouraging words, and it is easy to do this with students who appear motivated, who learn easily, or who are quick. But it is even more important to communicate positive beliefs and expectations to students who are slow, appear unmotivated, or struggle. It is also important to realize that the speed at which students appear to grasp concepts is not indicative of their mathematics potential (Supekar et al, 2013). As hard as it is, it is important to not have any preconceptions about our students.

We must be open at all times to any student working hard and achieving at high levels.

Some students give the impression that math is a constant struggle for them and they may ask a lot of questions or keep saying they are stuck, but they are just hiding their mathematics potential and are likely to be suffering from a fixed mindset; often these students are scared to take a risk or to get anything wrong. Some students have had bad math experiences and messages from a young age or have not received opportunities for brain growth and learning that other students have, so they are at lower levels than other students. This in no way means they cannot excel with good mathematics teaching, positive messages, and, perhaps most importantly, high expectations from their teachers and parents.

You can be the person who turns things around for students and liberates their learning path. It usually takes just one person—a person whom students will never forget.

This article contains excerpts from Jo Boaler’s new book, Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching

References

Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. San Francisco, CA: Jossey-Bass.

Cohen, G. L., & Garcia, J. (2014). Educational Theory, Practice, and Policy and the Wisdom of Social Psychology. Policy Insights from the Behavioral and Brain Sciences, 1(1), 13-20.

Rosenthal, R., & Jacobson, L. (1968). Pygmalion in the classroom. The Urban Review, 3(1), 16-20.

Shouse, R. C. (1996). Academic press and sense of community: Conflict, congruence, and implications for student achievement. Social Psychology of Education, 1(1), 47-68.

Supekar, K.; Swigart, A., Tenison, C., Jolles, D., Rosenberg-Lee, M., Fuchs, L., & Menon, V. (2013). Neural Predictors of Individual Differences in Response to Math Tutoring in Primary-Grade School Children. PNAS, 110, 20 (8230-8235)

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Introduction

The complex ways that children understand mathematics is fascinating to me. Students ask questions, see ideas, draw representations, connect methods, justify, and reason in all sorts of different ways. But recent years have seen all of these different nuanced complexities of student understanding reduced to single numbers and letters that are used to judge students’ worth. Teachers are encouraged to test and grade students, to a ridiculous and damaging degree; students start to define themselves – and mathematics – in terms of letters and numbers. Such crude representations of understanding vastly under-describe and in many cases misrepresent children’s knowledge.

In the United States students are over-tested to a degree that is nothing short of remarkable, particularly in mathematics. For many years students have been judged by narrow, procedural mathematics questions presented with multiple-choice answers. The knowledge needed for success on such tests is so far from the adaptable, critical and analytical thinking that students need in the modern world that leading employers such as Google have declared that they are no longer interested in students’ test performance as it in no way predicts success in the workplace (Bryant, 2013).

One critical principal of good testing is that it assesses what is important.

For many decades in the United States, tests have assessed what is easy to test instead of important and valuable mathematics. This has meant that mathematics teachers have had to focus their teaching on narrow procedural mathematics, not the broad, creative and growth mathematics that is so important. The new common core assessments promise something different, with few multiple-choice questions and assessments of problem solving, but they are being met with considerable opposition from parents.

The damage does not only end with standardized testing, for math teachers are led to believe they should use tests in classrooms that mimic low quality standardized tests, even when they know the tests assess narrow mathematics.

They do this to help prepare students for later success. Some teachers, particularly at the high school level, test weekly or even more frequently. Mathematics teachers feel the need to test regularly, more than any other subject, because they have come to believe that mathematics is about performance, and they usually don’t consider the role that tests play in shaping students’ views of mathematics and themselves. Many mathematics teachers I know introduce a class with a test, which gives a huge performance message to students on the first day of class, the time when it is so important to be giving growth messages about mathematics and learning.

Finland is one of the highest-scoring countries in the world on international mathematics tests yet students do not take any tests in school. Instead teachers use their rich understanding of their students’ knowledge gained through teaching to report to parents and make judgments about student work. In a longitudinal study I conducted in England students worked on open-ended projects for three years (ages 13 through 16) leading to national standardized examinations. They did not take tests nor was their work graded. Students encountered short questions assessing procedures in the last few weeks before the examination, as the teachers gave them examination papers to work through. Despite the students’ lack of familiarity with examination questions or working under timed conditions of any kind, they scored at significantly higher levels than a matched cohort of students who spent three years working through questions similar to the national exam questions and taking frequent tests (Boaler, 1998, 2015). The students from the problem solving school did so well in the standardized national exam because they had been taught to believe in their own capabilities; they had been given helpful, diagnostic information on their learning; and they had learned that they could solve any question, as they were mathematical problem solvers.

Students with no experience of examinations and tests can score at the highest levels because the most important preparation we can give students is a growth mindset, positive beliefs about their own ability, and problem-solving mathematical tools to equip them for any mathematical situation.

The testing regime of the last decade has had a large, negative impact on students but it does not end with testing; the communication of grades to students is similarly negative. When students are given a percentage or grade, they can do little else besides compare it to others around them, with half or more deciding that they are not as good as others. This is known as “ego feedback,” a form of feedback that has been found to damage learning. Sadly when students are given frequent test scores and grades they start to see themselves as those scores and grades. They do not regard the scores as an indicator of their learning or of what they need to do to achieve, but as indicators of who they are as people. The fact that US students commonly describe themselves saying “I’m an A student” or “I’m a D student” illustrates the ways students define themselves by grades. Ray McDermott wrote a compelling paper about the capturing of a child by a learning disability, describing the ways a student who thought and worked differently was given a label and was then defined by that label (McDermott, 1993). I could give a similar argument about the capturing of students by grades and test scores.

Students describe themselves as A or D students because they have grown up in a performance culture that valued frequent testing and grading, rather than persistence, courage, or problem solving. The traditional methods of assessing students that have been used across the United States for decades were designed in a less enlightened age (Kohn, 2011) when it was believed that grades and test scores would motivate students, and that the information they provided on students’ achievement would be useful.

Now we know that grades and test scores demotivate rather than motivate students and that they communicate fixed and damaging messages to students that result in lower achievement in classrooms.

In studies of grading and alternatives to grading researchers have produced consistent results. Study after study shows that grading reduces the achievement of students. Elawar and Corno, for example, contrasted the ways teachers responded to math homework in sixth grade, with half of the students receiving grades and the other half receiving diagnostic comments without a grade (Elawar & Corno, 1985). The students receiving comments learned twice as fast as the control group, the achievement gap between male and female students disappeared, and student attitudes improved.

A study by Ruth Butler added a third condition, which gave students grades and comments –as this could be thought of as the best of both worlds (Butler, 1987, 1988). In this study, students who received grades only and those who received grades and comments scored equally badly, and the group that achieved at significantly higher levels was the comment only group. This showed that when students received a grade and a comment, they focused on only the grade. Butler found that both high-achieving (the top 25% GPA) and low-achieving (the bottom 25% GPA) fifth and sixth graders suffered deficits in performance and motivation in both graded conditions, compared with the students who received only diagnostic comments. Further research showed that that students only needed to think they were working for a grade to lose motivation, resulting in lower levels of achievement (Pulfrey, Butera, and Buchs, 2011).

The move from grades to diagnostic comments is a crucial one, and is a move that allows teachers to give students an amazing gift – the gift of their knowledge and insights about ways to improve.

Teachers, quite rightly, worry about the extra time this can take, as good teachers already work well beyond the hours they are paid for. My recommended solution is to assess less; if teachers replaced grading weekly with diagnostic comments given occasionally, they could spend the same amount of time, remove the fixed mindset messages of a grade, and provide students with insights that would propel them onto paths of higher achievement. Teachers who have made these changes see increases or no changes in test performance and significant increases in motivation and confidence.

When we give assessments to students we create an important opportunity. Well-crafted tasks and questions accompanied by clear feedback offer students a growth mindset pathway that helps them to know that they can learn to high levels, and, critically, how they can get there. Unfortunately most systems of assessment in U.S. classrooms do the opposite of this, communicating information to students that causes many of them to think they are a failure and they can never learn math. I have worked with teachers in recent years who have shifted their methods of assessment from standard tests with grades and scores to assessments that are focused upon giving students the information they need in order to learn well accompanied with growth mindset messages. This resulted in dramatic changes in their classroom environments.

Math anxiety, formerly commonplace among students, disappeared and was replaced by student self-confidence, which led to higher levels of motivation, engagement and achievement.

I am a strong supporter of teachers and know that the No Child Left Behind era stripped the professionalism and enthusiasm of many teachers as they were forced (and I choose that word carefully) to use teaching methods that they knew to be unhelpful. An important part of my work with teachers now is to help them regain their sense of professionalism. My aim in working with teachers is to help them see themselves as creators again, people who can design teaching environments infused with their own ideas for creative, engaging math. I have watched teachers come alive when they are encouraged in these ways.

In a new film by Vicki Abeles, director of Race to Nowhere (watch the trailer to learn more! ), her team interviewed the middle school students in a district I was working in, helping the teachers shift their teaching and assessment. In the film one girl, Delia, talks about getting an F for her homework in the previous year, and how it had caused her to stop trying in math and – shockingly – all of her classes across the school. In the interview she poignantly said:

“When I saw the F on my paper I felt like a nothing. I was failing in that class so I thought I may as well fail in all my other classes too. I didn’t even try.” Later in the film she talks about the change in her math class and how she now feels encouraged to do well. “I hated math,” she says “I absolutely hated it, but now I have a connection with math, I’m open, I feel like I’m alive, I’m more energetic.”

Beyond Measure Source: Image courtesy of Reel Link Films

Delia’s use of the word “open” in describing how she felt about math is a sentiment I hear frequently from students when they are taught mathematics without the impending fear of low test scores and grades. But it goes further than assessment – when we teach creative, inquiry math students feel an intellectual freedom that is powerful. In interviews with 3rd graders who experienced number talks in class, I ask the students how they feel about number talks. The first thing young Dylan said in the interview was “I feel free.” He went on to describe how the valuing of different mathematics strategies allowed him to feel he could work with mathematics in any way he wanted, to explore ideas and learn about numbers. The students’ use of words such as ‘free’, and ‘open’ demonstrate the difference that is made when students work on growth mindset mathematics; this goes well beyond math achievement to an intellectual empowerment that will affect students throughout their lives (Boaler, 2015).

The perceptions students develop about their own potential affect their learning, their achievement, and, of equal importance, their motivation and effort—as Delia describes in the film. When she got an F in math, she gave up not only in math but also in all of her other classes; she felt like a failure. This is not an unusual response to grading.

When students are given scores that tell them they rank below other students, they often give up on school, deciding that they will never be able to learn and they take on the identity of an underperforming student.

The grades and scores given to students who are high achieving are just as damaging. Students develop the idea that they are an “A student” and begin a precarious fixed mindset learning path that makes them avoid harder work or challenges for fear that they will lose their A label. Such students often are devastated if they get a B or lower, for any of their work.

In another research study on grading Deevers found that students who were not given scores but instead given positive constructive feedback were more successful in their future work. He also, sadly, found that as students got older teachers gave less constructive feedback and more fixed grading. He found a clear and unsurprising relationship between teachers’ assessment practices and student attitudes as students’ beliefs about their own potential and the possibility of improving their learning declined steadily from 5th to 12th grade (Deevers, 2006).

We want students to be excited about and interested in their learning. When students develop interest in the ideas they are learning, they increase their motivation and their achievement. There is a large body of research that has studied two types of motivation. Intrinsic motivation comes from interest in the subject and ideas you are learning; extrinsic motivation is the motivation provided by the thought of getting better scores and grades. Because mathematics has been taught for decades as a performance subject, the students who are most motivated in math classrooms are usually those who are extrinsically motivated. One result of this is that students who feel positive about math class are usually only those students who are getting high scores and grades. Most of the teachers who believe in grades, use them because they think they motivate students to achieve. They do motivate some students – those who would probably achieve at high levels anyway, but they de-motivate the rest. Unfortunately the extrinsic motivation that the high achieving students develop is not helpful in the long term. Study after study shows that students who develop intrinsic motivation achieve at higher levels than those who develop extrinsic motivation (Pulfrey, Buchs, & Butera 2011; Lemos & Verissimo, 2014), and that intrinsic motivation to learn ideas motivates students to pursue subjects to higher levels and to stay in subjects rather than drop out (Stipek, 1993).

Assessment for Learning

A few years ago two professors from England – Paul Black and Dylan Wiliam –conducted a meta-analysis of hundreds of research studies on assessment. They found something amazing: a form of assessment so powerful, that if teachers used it the impact would be so great that it would raise the achievement of a country in international studies from the middle of the pack to a place in the top five. (Sir Paul Black and Professor Dylan Wiliam were both good colleagues of mine at London University; Paul Black was also my dissertation advisor and mentor.) Black and Wiliam found that if teachers were to use what is now called “assessment for learning” the positive impact would be far greater than other educational initiatives such as reductions in class size (Black, Harrison, Lee, Marshall, & Wiliam, 2002; Black & Wiliam, 1998a, 1998b). They published their findings in a small booklet that sold over 20,000 copies in the first few weeks in England. Assessment for Learning is now a national initiative in many countries; it has a huge research evidence base and it communicates growth mindset messages to students.

A little background will be helpful. There are two types of assessment—formative and summative. Formative assessment informs learning and is the essence of assessment for learning or A4L. Formative assessments are used to find out where students are in their learning so that teachers and students can determine what they need to know next. The purpose of summative assessment, in contrast, is to summarize a student’s learning—to give a final account of how far a student has gotten, as an end point. One problem in the United States is that many teachers use summative assessment formatively; that is, they give students an end score or grade when they are still learning the materials. In mathematics classrooms teachers often use summative tests weekly and then move on to the next subject without waiting to see what the tests reveal. In A4L, students become knowledgeable about what they know, what they need to know, and ways to close the gap between the two. Students are given information about their flexible and growing learning pathways that contributes to their development of a growth mathematics mindset. In the weeks and months that students are learning in a course, it is very important to assess formatively, not summatively. Further, the A4L approach, which can also be thought of as assessing for a growth mindset, offers a range of strategies and methods.

One important principle of A4L is that it teaches students responsibility for their own learning.

At its core A4L is about empowering students to become autonomous learners who can self-regulate and determine what they most need to learn and who know ways to improve their learning. Assessment for learning can be thought of as having three parts: (1) clearly communicating to students what they have learned, (2) helping students become aware of where they are in their learning journey and where they need to reach, and (3) giving students information on ways to close the gap between where they are now and where they need to be.

Developing Student Self-Awareness and Responsibility

The most powerful learners are those who are reflective, who engage in metacognition – thinking about what they know – and who take control of their own learning (White & Frederiksen, 1998). A major failing of traditional mathematics classes is that students rarely have much idea of what they are learning or where they are in the broader learning landscape. They focus upon methods to remember but often do not even know what area of mathematics they are working on. I have visited math classes many times and stopped at students’ desks to ask them what they are working on. In many cases students answer with the question they are working on. Many of my interactions have gone something like this:

JB: What are you working on?

Student: Exercise 2

JB: So what are you actually doing, what math are you working on?

Student: Oh, I’m sorry – question 4

Students are often not thinking about the area of mathematics they are learning, they do not have an idea of the mathematical goals for their learning, and they expect to be passively led through work with teachers telling them whether they are “getting it” or not. Alice White, an assessment expert, likens this situation to workers on a ship who are given jobs to do each day but don’t have any idea where the ship is travelling to.

One research study, conducted by Barbara White and John Frederikson (1998), powerfully illustrated the importance of reflection. The researchers studied twelve classes of seventh grade students learning physics. The researchers divided the students into experimental and control groups. All groups were taught a unit on force and motion. The control groups then spent some of each lesson discussing the work whereas the experimental group spent some of each lesson engaging in self- and peer assessment, considering criteria for the science they were learning. The results of the study were dramatic. The experimental groups outperformed the control groups on three different assessments. The previously low-achieving students made the greatest gains. After they spent time considering the science criteria and assessing themselves against them, they began to achieve at the same levels as the highest achievers. The middle school students even scored at higher levels than AP physics students on tests of high school physics. The researchers concluded that a large part of the students’ previous low achievement came not from the fact that they lacked ability but that they had not previously known what they should really be focusing upon.

This is true for many students and this is why it is so important to communicate to students what they should be learning. This both helps the students know what success is, and starts a self-reflection process that is an invaluable tool for learning.

There are many strategies for encouraging students to become more aware of the mathematics they are learning and their place in the learning process, several of which are in Jo’s upcoming book, Mathematical Mindsets.

Conclusion

Tests and grading can lead students to disengage from mathematics and even school itself. Assessment for Learning, by contrast, represent an incredible opportunity for teachers to provide students with information on their learning that accelerates pathways to success and gives students powerful growth mindset messages about mathematics and learning. Research shows that a change from grading and testing to Assessment for Learning has a powerful impact on students’ achievement, self-beliefs, motivation and future learning pathways.

By using assessments to empower students to learn and grow, we can help our students develop positive attitudes towards mathematics and themselves.

This article contains excerpts from Jo Boaler’s new book, Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching

Why are we doing this??

“Many students have felt a sense of anxiety/stress when it comes to learning math. Many fear that math is something that you are either good at or not. There is also this sense of pressure that students feel because they view math as something that you are either right or wrong. What many students struggle to understand is that the beauty of math is not so much on the solution but the thinking and creativity that goes into trying to solve the problem. In addition to this, students constantly feel pressure to perform due to grades. The anxiety students feel with regards to grades is like a dark cloud that hangs over them. No matter what is going on in class, or how much they are enjoying the topic they are learning, there is still this looming pressure to perform in order to receive the highest grade possible. Whether it’s higher achieving students or lower achieving students, many students’ focus is on performing in order to achieve a certain grade. What this has caused is a focus on performing (memorizing) rather than a focus on learning. We want students to enjoy learning. Isn’t that the purpose of going to school? With this being, last year we tried to make a shift by encouraging students to take risks and not be afraid to make mistakes. However, when it came time to grading tests, students were still penalized for having wrong answers. How can we encourage students to not be afraid to make mistakes and then penalize them for doing it on a test? This sent mixed messages and has been on our minds over the course of the summer. After careful thought, we’re curious to try this. What would happen if everyone got an A at the beginning of the year?! Would students no longer feel pressure or worry about their performance? Would this free students up to be more creative and to take more risks? Would this lead to students being more curious and wanting to go deeper and to understand why things work the way they do? Would this lead to more of an intrinsic motivation of students wanting to do well and to learn? Let’s Find Out…” -An extract from letter to parents, from High Tech High Math Department, Chula Vista, San Diego.

References

Black, P., Harrison, C., Lee, C., Marshall, B., & Wiliam, D. (2002). Working inside the black box: Assessment for learning in the classroom. London: Department of Education & Professional Studies, King’s College.

Black, P. J., & Wiliam, D. (1998a, October). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan, 139–148.

Black, P. J., & Wiliam, D. (1998b). Assessment and classroom learning. Assessment in Education, 5(1), 7–74.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for research in mathematics education, 41-62.

Boaler, J. (2015). Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching. San Francisco, CA: Jossey-Bass.

Butler, R. (1987). Task-involving and ego-involving properties of evaluation: Effects of different feedback conditions on motivational perceptions, interest and performance. Journal of Educational Psychology, 79, 474–482.

Butler, R. (1988). Enhancing and undermining intrinsic motivation: The effects of task-involving and ego-involving evaluation on interest and performance. British Journal of Educational Psychology, 58, 1–14.

Deevers, M. (2006). Linking classroom assessment practices with student motivation in mathematics. Paper presented at the American Educational Research Association, San Francisco.

Elawar, M. C., & Corno, L. (1985). A factorial experiment in teachers’ written feedback on student homework: Changing teacher behavior a little rather than a lot. Journal of Educational Psychology, 77(2), 162–173.

Kohn, A. (2011, November). The case against grades. Retrieved from https://www.alfiekohn.org/ article/case-grades/

Lemos, M. S., & Veríssimo, L. (2014). The relationships between intrinsic motivation, extrinsic motivation, and achievement, along elementary school. Procedia – Social and Behavioral Sciences, 112, 930–938.

Pulfrey, C., Buchs, C., & Butera, F. (2011). Why grades engender performance-avoidance goals: The mediating role of autonomous motivation. Journal of Educational Psychology, 103(3), 683–700. Retrieved from https://www.researchgate.net/profile/Fabrizio_Butera/publication/232450947_Why_grades_engender_performanceavoidance_goals_The_mediating_role_of_autonomous_ motivation/links/02bfe50ed4ebfd0670000000.pdf

Stipek, D. J. (1993). Motivation to learn: Integrating theory and practice. New York: Pearson.

White, B. Y., & Frederiksen, J. R. (1998). Inquiry, modeling, and metacognition: Making science accessible to all students. Cognition and Instruction, 16(1), 3–118.

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Mathematics educators have long known that engaging students in visual representations of mathematics is extremely helpful for their learning. When youcubed offered “How Close to 100” as an activity for learning math facts with visual representations, teachers across the world were thrilled and responded with thousands of tweets showing students learning by playing the game.

Some of the world’s top mathematicians engage almost entirely with visual mathematics.

For example, Maryam Mirzakhani, arguably the most important mathematician of our time, works almost entirely visually. In news articles she is shown sketching ideas on large pieces of paper on her kitchen table, Maryam joked to reporters that her 3 year old daughter probably thinks she is an artist. Despite the importance of visual mathematics at high levels of mathematics (and all other levels) there is a common perception that visual mathematics is only needed as a crutch for more abstract mathematics. Curriculum guides suggest that students work with physical manipulatives and drawings to help them learn abstract mathematics. Older students often develop the idea that manipulatives are for babies and can’t possibly be useful in higher levels of math. The widespread lack of appreciation of the visual nature of mathematics relates to the misconceptions that exist about the nature of mathematics.

Visual mathematics is an important part of mathematics for its own sake and new brain research tells us that visual mathematics even helps students learn numerical mathematics.

In a ground breaking new study Joonkoo Park & Elizabeth Brannon (2013), found that the most powerful learning occurs when we use different areas of the brain. When students work with symbols, such as numbers, they are using a different area of the brain than when they work with visual and spatial information, such as an array of dots. The researchers found that mathematics learning and performance was optimized when the two areas of the brain were communicating (Park & Brannon, 2013). (for math questions that encourage this use of visual and symbolic representations see https://www.youcubed.org/tasks/). Additionally, they found that training students through visual representations improved students’ math performance significantly, even on numerical math, and that the visual training helped students more than numerical training.

What is Visual Mathematics?

At youcubed we provide many different mathematics tasks that engage students in visual mathematics. Through decades of work with students, teachers, high-tech companies, politicians and others we have learned that people are excited and inspired when they see mathematics as pictures, not just symbols. For example, consider how you might solve 18 x 5, and ask others how they would solve 18 x 5. Here are some different visual solutions of this problem.

Each of these visuals highlights the mathematics inside the problem and helps students develop understanding of multiplication. Pictures help students see mathematical ideas, which aids understanding. Visual mathematics also facilitates higher-level thinking, enables communication and helps people see the creativity in mathematics.

Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, the mathematics comes alive.
Teachers can create such mathematical excitement in classrooms with any mathematics question by asking students for the different ways they see and can solve the problems and by encouraging discussion of different ways of seeing problems.

For an example of visualizing algebra see here.

When we don’t ask students to think visually, we miss an incredible opportunity to increase students’ understanding and to enable important brain crossing.

This article contains excerpts from Jo Boaler’s new book, Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching

Park, J., & Brannon, E. (2013). Training the approximate number system improves math proficiency. Association for Psychological Science, 1–7.

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Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts

By Jo Boaler, Professor of Mathematics Education, co-founder youcubed

With the help of Cathy Williams, co-founder youcubed & Amanda Confer, Stanford University

Updated January 28th, 2015

Introduction

A few years ago a British politician, Stephen Byers, made a harmless error in an interview. The right honorable minister was asked to give the answer to 7 x 8 and he gave the answer of 54, instead of the correct 56. His error prompted widespread ridicule in the national media, accompanied by calls for a stronger emphasis on ‘times table’ memorization in schools. This past September the Conservative education minister for England, a man with no education experience, insisted that all students in England memorize all their times tables up to 12 x 12 by the age of 9. This requirement has now been placed into the UK’s mathematics curriculum and will result, I predict, in rising levels of math anxiety and students turning away from mathematics in record numbers. The US is moving in the opposite direction, as the new Common Core State Standards (CCSS) de-emphasize the rote memorization of math facts. Unfortunately misinterpretations of the meaning of the word ‘fluency’ in the CCSS are commonplace and publishers continue to emphasize rote memorization, encouraging the persistence of damaging classroom practices across the United States.

Mathematics facts are important but the memorization of math facts through times table repetition, practice and timed testing is unnecessary and damaging. The English minister’s mistake when he was asked 7 x 8 prompted calls for more memorization. This was ironic as his mistake revealed the limitations of memorization without ‘number sense’. People with number sense are those who can use numbers flexibly. When asked to solve 7 x 8 someone with number sense may have memorized 56 but they would also be able to work out that 7 x 7 is 49 and then add 7 to make 56, or they may work out ten 7’s and subtract two 7’s (70-14). They would not have to rely on a distant memory. Math facts, themselves, are a small part of mathematics and they are best learned through the use of numbers in different ways and situations.  Unfortunately many classrooms focus on math facts in unproductive ways, giving students the impression that math facts are the essence of mathematics, and, even worse that the fast recall of math facts is what it means to be a strong mathematics student. Both of these ideas are wrong and it is critical that we remove them from classrooms, as they play a large role in the production of math anxious and disaffected students.

It is useful to hold some math facts in memory. I don’t stop and think about the answer to 8 plus 4, because I know that math fact. But I learned math facts through using them in different mathematical situations, not by practicing them and being tested on them. I grew up in the progressive era of England, when primary schools focused on the ‘whole child’ and I was not presented with tables of addition, subtraction or multiplication facts to memorize in school. This has never held me back at any time or place in my life, even though I am a mathematics education professor. That is because I have number sense, something that is much more important for students to learn, and that includes learning of math facts along with deep understanding of numbers and the ways they relate to each other.

Number Sense

In a critical research project researchers studied students as they solved number problems (Gray & Tall, 1994). The students, aged 7 to 13, had been nominated by their teachers as being low, middle or high achieving. The researchers found an important difference between the low and high achieving students – the high achieving students used number sense, the low achieving students did not. The high achievers approached problems such as 19 + 7 by changing the problem into, for example, 20 + 6. No students who had been nominated as low achieving used number sense. When the low achieving students were given subtraction problems such as 21-16 they counted backwards, starting at 21 and counting down, which is extremely difficult to do. The high achieving students used strategies such as changing the numbers into 20 -15 which is much easier to do. The researchers concluded that low achievers are often low achievers not because they know less but because they don’t use numbers flexibly – they have been set on the wrong path, often from an early age, of trying to memorize methods instead of interacting with numbers flexibly (Boaler, 2009). This incorrect pathway means that they are often learning a harder mathematics and sadly, they often face a lifetime of mathematics problems.

Number sense is the foundation for all higher-level mathematics (Feikes & Schwingendorf, 2008). When students fail algebra it is often because they don’t have number sense. When students work on rich mathematics problems – such as those we provide at the end of this paper – they develop number sense and they also learn and can remember math facts. When students focus on memorizing times tables they often memorize facts without number sense, which means they are very limited in what they can do and are prone to making errors –such as the one that led to nationwide ridicule for the British politician. Lack of number sense has led to more catastrophic errors, such as the Hubble Telescope missing the stars it was intended to photograph in space. The telescope was looking for stars in a certain cluster but failed due to someone making an arithmetic error in the programming of the telescope (LA Times, 1990). Number sense, critically important to students’ mathematical development, is inhibited by over-emphasis on the memorization of math facts in classrooms and homes. The more we emphasize memorization to students the less willing they become to think about numbers and their relations and to use and develop number sense (Boaler, 2009).

The Brain and Number Sense

Some students are not as good at memorizing math facts as others. That is something to be celebrated, it is part of the wonderful diversity of life and people. Imagine how dull and unispiring it would be if teachers gave tests of math facts and everyone answered them in the same way and at the same speed as though they were all robots. In a recent brain study scientists examined students’ brains as they were taught to memorize math facts. They saw that some students memorized them much more easily than others. This will be no surprise to readers and many of us would probably assume that those who memorized better were higher achieving or “more intelligent” students. But the researchers found that the students who memorized more easily were not higher achieving, they did not have what the researchers described as more “math ability”, nor did they have higher IQ scores (Supekar et al, 2013). The only differences the researchers found were in a brain region called the hippocampus, which is the area of the brain that is responsible for memorized facts (Supekar et al, 2013). Some students will be slower when memorizing but they still have exceptional mathematics potential. Math facts are a very small part of mathematics but unfortunately students who don’t memorize math facts well often come to believe that they can never be successful with math and turn away from the subject.

Teachers across the US and the UK ask students to memorize multiplication facts, and sometimes addition and subtraction facts too, usually because curriculum standards have specified that students need to be “fluent with numbers”. Parish, drawing from Fosnot and Dolk (2001) defines fluency as ‘knowing how a number can be composed and decomposed and using that information to be flexible and efficient with solving problems.’ (Parish 2014, p 159). Whether or not we believe that fluency requires more than the recall of math facts, research evidence points in one direction: The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense.

When teachers emphasize the memorization of facts, and give tests to measure number facts students suffer in two important ways. For about one third of students the onset of timed testing is the beginning of math anxiety (Boaler, 2014). Sian Beilock and her colleagues have studied people’s brains through MRI imaging and found that math facts are held in the working memory section of the brain. But when students are stressed, such as when they are taking math questions under time pressure, the working memory becomes blocked and students cannot access math facts they know (Beilock, 2011; Ramirez, et al, 2013). As students realize they cannot perform well on timed tests they start to develop anxiety and their mathematical confidence erodes. The blocking of the working memory and associated anxiety particularly occurs among higher achieving students and girls. Conservative estimates suggest that at least a third of students experience extreme stress around timed tests, and these are not the students who are of a particular achievement group, or economic background. When we put students through this anxiety provoking experience we lose students from mathematics.

Math anxiety has now been recorded in students as young as 5 years old (Ramirez, et al, 2013) and timed tests are a major cause of this debilitating, often life-long condition. But there is a second equally important reason that timed tests should not be used – they prompt many students to turn away from mathematics.  In my classes at Stanford University, I experience many math traumatized undergraduates, even though they are among the highest achieving students in the country. When I ask them what has happened to lead to their math aversion many of the students talk about timed tests in second or third grade as a major turning point for them when they decided that math was not for them. Some of the students, especially women, talk about the need to understand deeply, which is a very worthwhile goal, and being made to feel that deep understanding was not valued or offered when timed tests became a part of math class. They may have been doing other more valuable work in their mathematics classes, focusing on sense making and understanding, but timed tests evoke such strong emotions that students can come to believe that being fast with math facts is the essence of mathematics. This is extremely unfortunate. We see the outcome of the misguided school emphasis on memorization and testing in the numbers dropping out of mathematics and the math crisis we currently face (see www.youcubed.org). When my own daughter started times table memorization and testing at age 5 in England she started to come home and cry about maths. This is not the emotion we want students to associate with mathematics and as long as we keep putting students under pressure to recall facts at speed we will not erase the widespread anxiety and dislike of mathematics that pervades the US and UK (Silva & White, 2013; National Numeracy, 2014).

In recent years brain researchers have found that the students who are most successful with number problems are those who are using different brain pathways – one that is numerical and symbolic and the other that involves more intuitive and spatial reasoning (Park & Brannon, 2013). At the end of this paper we give many activities that encourage visual understanding of number facts, to enable important brain connections. Additionally brain researchers have studied students learning math facts in two ways – through strategies or memorization. They found that the two approaches (strategies or memorization) involve two distinct pathways in the brain and that both pathways are perfectly good for life long use. Importantly the study also found that those who learned through strategies achieved ‘superior performance’ over those who memorized, they solved problems at the same speed, and showed better transfer to new problems. The brain researchers concluded that automaticity should be reached through understanding of numerical relations, achieved through thinking about number strategies (Delazer et al, 2005).

Why is Mathematics Treated Differently?

In order to learn to be a good English student, to read and understand novels, or poetry, students need to have memorized the meanings of many words. But no English student would say or think that learning about English is about the fast memorization and fast recall of words. This is because we learn words by using them in many different situations – talking, reading, and writing. English teachers do not give students hundreds of words to memorize and then test them under timed conditions. All subjects require the memorization of some facts, but mathematics is the only subject in which teachers believe they should be tested under timed conditions. Why do we treat mathematics in this way?

Mathematics already has a huge image problem. Students rarely cry about other subjects, nor do they believe that other subjects are all about memorization or speed. The use of teaching and parenting practices that emphasize the memorization of math facts is a large part of the reason that students disconnect from math. Many people will argue that math is different from other subjects and it just has to be that way – that math is all about getting correct answers, not interpretation or meaning. This is another misconception. The core of mathematics is reasoning – thinking through why methods make sense and talking about reasons for the use of different methods (Boaler, 2013). Math facts are a small part of mathematics and probably the least interesting part at that. Conrad Wolfram, of Wolfram-Alpha, one of the world’s leading mathematics companies, speaks publicly about the breadth of mathematics and the need to stop seeing mathematics as calculating. Neither Wolfram nor I are arguing that schools should not teach calculating, but the balance needs to change, and students need to learn calculating through number sense, as well as spend more time on the under-developed but critical parts of mathematics such as problem solving and reasoning.

It is important when teaching students number sense and number facts never to emphasize speed. In fact this is true for all mathematics. There is a common and damaging misconception in mathematics – the idea that strong math students are fast math students. I work with a lot of mathematicians and one thing I notice about them is that they are not particularly fast with numbers, in fact some of them are rather slow. This is not a bad thing, they are slow because they think deeply and carefully about mathematics. Laurent Schwartz, a top mathematician, wrote an autobiography about his school days and how he was made to feel “stupid” because he was one of the slowest math thinkers in his class (Schwartz, 2001). It took him many years of feeling inadequate to come 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.’ (Schwartz, 2001) Sadly speed and test driven math classrooms lead many students who are slow and deep thinkers, like Schwartz, to believe that they cannot be good at math.

Math ‘Fluency’ and the Curriculum

In the US the new Common Core curriculum includes ‘fluency’ as a goal. Fluency comes about when stu- dents develop number sense, when they are mathematically confident because they understand numbers. Unfortunately the word fluency is often misinterpreted. ‘Engage New York’ is a curriculum that is becom- ing increasingly popular in the US that has incorrectly interpreted fluency in the following ways:

Fluency: Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions such as multiplication tables so that they are more able to understand and manipulate more complex functions. (Engage New York)

There are many problems with this directive. Speed and memorization are two directions that we urgently need to move away from, not towards. Just as problematically ‘Engage New York’ links the memorization of number facts to students’ understanding of more complex functions, which is not supported by research evidence. What research tells us is that students understand more complex functions when they have num- ber sense and deep understanding of numerical principles, not blind memorization or fast recall (Boaler, 2009). I am currently working with PISA analysts at the OECD. The PISA team not only issues interna- tional mathematics tests every 4 years they collect data on students’ mathematical strategies. Their data from 13 million 15-year olds across the world show that the lowest achieving students are those who focus on memorization and who believe that memorizing is important when studying for mathematics (Boaler & Zoido, in press). This idea starts early in classrooms and is one we need to eradicate. The highest achievers in the world are those who focus on big ideas in mathematics, and connections between ideas. Students develop a connected view of mathematics when they work on mathematics conceptually and blind memorization is replaced by sense making.

In the UK directives have similar potential for harm. The new national curriculum states that all students should have ‘memorised their multiplication tables up to and including the 12 multiplication table’ by the age of 9 and whilst students can memorize multiplication facts to 12 x 12 through rich engaging activities this directive is leading teachers to give multiplication tables to students to memorize and then be tested on. A leading group in the UK, led by children’s author and poet Michael Rosen, has formed to highlight the damage of current policies in schools and the numbers of primary age children who now walk to school crying from the stress they are under, caused by over-testing (Garner, The Independent, 2014). Mathemat- ics is the leading cause of students’ anxiety and fear and the unnecessary focus on memorized math facts in the early years is one of the main reasons for this.

Activities to Develop Number Facts and Number Sense

Teachers should help students develop math facts, not by emphasizing facts for the sake of facts or using ‘timed tests’ but by encouraging students to use, work with and explore numbers. As students work on meaningful number activities they will commit math facts to heart at the same time as understanding numbers and math. They will enjoy and learn important mathematics rather than memorize, dread and fear mathematics.

Number Talks

One of the best methods for teaching number sense and math facts at the same time is a teaching strategy called ‘number talks’, developed by Ruth Parker and Kathy Richardson. This is an ideal short teaching activity that teachers can start lessons with or parents can do at home. It involves posing an abstract math problem such as 18 x 5 and asking students to solve the problem mentally. The teacher then collects the different methods and looks at why they work. For example a teacher may pose 18 x 5 and find that students solve the problem in these different ways:

Students love to give their different strategies and are usually completely engaged and fascinated by the different methods that emerge. Students learn mental math, they have opportunities to memorize math facts and they also develop conceptual understanding of numbers and of the arithmetic properties that are critical to success in algebra and beyond. Parents can use a similar strategy by asking for their children’s methods and discussing the different methods that can be used. Two books, one by Cathy Humphreys and Ruth Parker (in press) and another by Sherry Parish (2014) illustrate many different number talks to work on with secondary and elementary students, respectively.

Research tells us that the best mathematics classrooms are those in which students learn number facts and number sense through engaging activities that focus on mathematical understanding rather than rote memorization. The following five activities have been chosen to illustrate this principle; the appendix to this document provides a greater range of activities and links to other useful resources that will help stu- dents develop number sense.

Addition Fact Activities

Snap It: This is an activity that children can work on in groups. Each child makes a train of connecting cubes of a specified number. On the signal “Snap,” children break their trains into two parts and hold one hand behind their back. Children take turns going around the circle showing their re- maining cubes. The other children work out the full number combination. For example, if I have 8 cubes in my number train I could snap it and put 3 behind my back. I would show my group the remaining 5 cubes and they should be able to say that three are missing and that 5 and 3 make 8.

How Many Are Hiding? In this activity each child has the same number of cubes and a cup. They take turns hiding some of their cubes in the cup and showing the leftovers. Other children work out the answer to the question “How many are hiding,” and say the full number combination.

Example: I have 10 cubes and I decide to hide 4 in my cup. My group can see that I only have 6 cubes. Stu- dents should be able to say that I’m hiding 4 cubes and that 6 and 4 make 10.

Multiplication Fact Activities

How Close to 100? This game is played in partners. Two children share a blank 100 grid. The first partner rolls two number dice. The numbers that come up are the numbers the child uses to make an array on the 100 grid. They can put the array anywhere on the grid, but the goal is to fill up the grid to get it as full as possible. After the player draws the array on the grid, she writes in the number sentence that describes the grid. The game ends when both players have rolled the dice and cannot put any more arrays on the grid. How close to 100 can you get?

Pepperoni Pizza: In this game, children roll a dice twice. The first roll tells them how many pizzas to draw. The second roll tells them how many pepperonis to put on EACH pizza. Then they write the number sentence that will help them answer the question, “How many pepper- onis in all?”

For example, I roll a dice and get 4 so I draw 4 big pizzas. I roll again and I get 3 so I put three pepperonis on each pizza. Then I write 4 x 3 = 12 and that tells me that there are 12 pepperonis in all.

Math Cards

Many parents use ‘flash cards’ as a way of encouraging the learning of math facts. These usually include 2 unhelpful practices – memorization without understanding and time pressure. In our Math Cards activity we have used the structure of cards, which children like, but we have moved the emphasis to number sense and the understanding of multiplication. The aim of the activity is to match cards with the same numerical answer, shown through different representations. Lay all the cards down on a table and ask children to take turns picking them; pick as many as they find with the same answer (shown through any representation). For example 9 and 4 can be shown with an area model, sets of objects such as dominoes, and the number sentence. When student match the cards they should explain how they know that the different cards are equivalent. This activity encourages an understanding of multiplication as well as rehearsal of math facts. A full set of cards is given in Appendix A.

Conclusion: Knowledge is Power

The activities given above are illustrations of games and tasks in which students learn math facts at the same time as working on something they enjoy, rather than something they fear. The different activities also focus on the understanding of addition and multiplication, rather than blind memorization and this is critically important. Appendix A presents other suggested activities and references.

As educators we all share the goal of encouraging powerful mathematics learners who think carefully about mathematics as well as use numbers with fluency. But teachers and curriculum writers are often unable to access important research and this has meant that unproductive and counter-productive classroom practices continue. This short paper illustrates both the damage that is caused by the practices that often accompany the teaching of math facts – speed pressure, timed testing and blind memorization – as well as summarizes the research evidence of something very different – number sense. High achieving students use number sense and it is critical that lower achieving students, instead of working on drill and memori- zation, also learn to use numbers flexibly and conceptually. Memorization and timed testing stand in the way of number sense, giving students the impression that sense making is not important. We need to ur- gently reorient our teaching of early number and number sense in our mathematics teaching in the UK and the US. If we do not, then failure and drop out rates – already at record highs in both countries (National Numeracy, 2014; Silva & White, 2013) – will escalate. When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics. We have the research knowledge we need to change this and to enable all children to be powerful mathematics learners. Now is the time to use it.

References

Beilock, S. (2011). Choke: What the Secrets of the Brain Reveal About Getting It Right When You Have To. New York: Free Press.

Boaler, J. (2015). What’s Math Got To Do With It? How Teachers and Parents Can Help Transform Mathematics Learning and Inspire Success. New York: Penguin.

Boaler, J. (2014). Research Suggests Timed Tests Cause Math Anxiety. Teaching Children Mathematics, 20 (8).

Boaler, J. (2013, Nov 12 2013). The Stereotypes That Distort How Americans Teach and Learn Math. The Atlantic.

Boaler, J. & Zoido, P. (in press). The Impact of Mathematics Learning Strategies upon Achievement: A Close Analysis of Pisa Data.

Delazer, M., Ischebeck, A., Domahs, F., Zamarian, L., Koppelstaetter, F., Siedentopf, C.M. Kaufmann; Benke, T., & Felber, S. (2005). Learning by Strategies and Learning by Drill – evidence from an fMRI study. NeuroImage. 839-849

Engage New York. https://schools.nyc.gov/NR/rdonlyres/9375E046-3913-4AF5-9FE3-D21BAE8FEE8D/0/CommonCoreIn- structionalShifts_Mathematics.pdf

Feikes, D. & Schwingendorf, K. (2008). The Importance of Compression in Children’s Learning of Mathematics and Teacher’s Learning to Teach Mathematics. Mediterranean Journal for Research in Mathematics Education 7 (2).

Fosnot, C, T & Dolk, M (2001). Young Mathematicians at Work: Constructing Multiplication and Division. Heinemann:

Garner, R. (October 3, 2014). The Independent. ( Link to Article )

Gray, E., & Tall, D. (1994). Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic. Journal for Research in Mathematics Education, 25(2), 116-140.

Humphreys, Cathy & Parker, Ruth (in press). Making Number Talks Matter: Developing Mathematical Practices and Deepen- ing Understanding, Grades 4-10. Portland, ME: Stenhouse.

LA Times (1990) https://articles.latimes.com/1990-05-10/news/mn-1461_1_math-error

Parish, S. (2014). Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5, Updated with Common Core Connections. Math Solutions.

Park, J. & Brannon, E. (2013). Training the Approximate Number System Improves Math Proficiency. Association for Psychological Science, 1-7

Ramirez, G., Gunderson, E., Levine, S., and Beilock, S. (2013). Math Anxiety, Working Memory and Math Achievement in Early Elementary School. Journal of Cognition and Development. 14 (2): 187–202.

Supekar, K.; Swigart, A., Tenison, C., Jolles, D., Rosenberg-Lee, M., Fuchs, L., & Menon, V. (2013). Neural Predictors of Indi- vidual Differences in Response to Math Tutoring in Primary-Grade School Children. PNAS, 110, 20 (8230-8235)

Schwartz, L. (2001). A Mathematician Grappling with His Century. Birkhäuser

Silva, E., & White, T. (2013). Pathways to Improvement: Using Psychological Strategies to help College Students Master Devel- opmental Math: Carnegie Foundation for the Advancement of Teaching.

National Numeracy (2014). https://www.nationalnumeracy.org.uk/what-the-research-says/index.html

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