Creating mathematical mindsets - annotations#

Annotations Boaler 2015#

Keith Devlin has written a range of books showing strong evidence that we are all natural mathematics users, and thinkers (see, for example, Devlin, 2006) (Boaler, 2015, p. 33)

The best and most important start we can give our students is to encourage them to play with numbers and shapes, thinking about what patterns and ideas they can see. (Boaler, 2015, p. 34)

Successful math users have an approach to math, as well as mathematical understanding, that sets them apart from less successful users. They approach math with the desire to understand it and to think about it, and with the confidence that they can make sense of it. Successful math users search for patterns and relationships and think about connections. (Boaler, 2015, p. 34)

When students see math as a series of short questions, they cannot see the role for their own inner growth and learning. They think that math is a fixed set of methods that either they get or they don't. (Boaler, 2015, p. 34)

When students see math as a broad landscape of unexplored puzzles in which they can wander around, asking questions and thinking about relationships, they understand that their role is thinking, sense making, andgrowing.Whenstudentsseemathematicsasasetofideasandrelationshipsandtheirroleas one of thinking about the ideas, and making sense of them, they have a mathematical mindset. (Boaler, 2015, p. 34)

So how do we develop mathematical mindsets in students so that they are willing to approach math with sense making and intuition? (Boaler, 2015, p. 35)

Afterextensivestudyofthedifferentstrategies that the students used, the researchers concluded that the difference between high- and low-achieving students was not that the low-achieving students knew less mathematics, but that they were interacting with mathematics differently. (Boaler, 2015, p. 35)

Instead of approaching numbers with flexibility and number sense, they seemed to cling to formal procedures they had learned, using them very precisely, not abandoning them even when it made sense to do so. The low achievers did not know less, they just did not use numbers flexibly---probably because they had been set on the wrong pathway, from an early age, of trying to memorize methods and number facts instead of interacting with numbers flexibly ( (Boaler, 2015, p. 35)

Number sense reflects a deep understanding of mathematics, but it comes about through a mathematical mindset that is focused on making sense of numbers and quantities. It is useful to think about the ways number sense is developed in students, not only because number sense is the foundation for all higher level mathematics (Feikes & Schwingendorf, 2008) but also because number sense and mathematical mindsets develop together, and learning about ways to develop one helps the development of the other. (Boaler, 2015, p. 36)

Notably, the brain can only compress concepts; it cannot compress rules and methods. Therefore students who do not engage in conceptual thinking and instead approach mathematics as a list of rules to remember are not engaging in the critical process of compression, so their brain is unable to organize and file away ideas; instead, it struggles to hold onto long lists of methods and rules. (Boaler, 2015, p. 37)

Math facts by 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 isolation, giving students the impression that math facts are the essence of mathematics, and, even worse, that mastering the fast recall of math facts is what it means to be a strong mathematicsstudent. (Boaler, 2015, p. 38)

For about one-third of students, the onset of timed testing is the beginning of math anxiety (Boaler, 2014c). 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). (Boaler, 2015, p. 38)

We know that when learning happens a synapse fires, and in order for structural brain change to happen we need to revisit ideas and learn them deeply. But what does that mean? It is important to revisit mathematical ideas, but the "practice" of methods over and over again is unhelpful. (Boaler, 2015, p. 42)

Most textbook authors in the United States base their whole approach on the idea of isolating methods, reducing them to their simplest form and practicing them. This is problematic for many reasons. First, practicing isolating methods induces boredom in students; many students simply turn off when they think their role is to passively accept a method (Boaler & Greeno, 2000) and repeat it over and over again. Second, most practice examples give the most simplified and disconnected version of the method to be practiced, giving students no sense of when or how they might use the method. (Boaler, 2015, p. 42)

When learning a definition, it is helpful to offer different examples---some of which barely meet the definition and some of which do not meet it at all---instead of perfect examples each time. Mathematics teachers should also think about the width and breadth of the definition they are showing, and sometimes this is best highlighted by non-examples. When learning a definition, it is often very helpful to see both examples that fit the definition and others that do not fit, rather than just presenting a series of perfect examples. For example, when learning about birds it can be helpful to think about bats and why they are not birds, rather than to see more and more examples of sparrows and crows. (Boaler, 2015, p. 45)

The students who were taught to practice methods over and over in a disciplined school in which there were high levels of "time on task" scored at significantly lower levels on the national mathematics examination than students who practiced much less but were encouraged to think conceptually. One significant problem the students from the traditional school faced in the national examination---a set of procedural questions---was that they did not know which method to choose to answer questions. They had practiced methods over and over but had never been asked to consider a situation and choose a method. (Boaler, 2015, p. 45)

The oversimplification of mathematics and the practice of methods through isolated simplified procedures is part of the reason we have widespread failure in the United States and the United Kingdom. It is also part of the reason that students do not develop mathematical mindsets; they do not see their role as thinking and sense making; rather, they see it as taking methods and repeating them. Students are led to think there is no place for thinking in math class. (Boaler, 2015, p. 46)

There is a lot of evidence that homework, of any form, is unnecessary or damaging; (Boaler, 2015, p. 46)

But there is hope: schools that decide to end homework see no reductions in students' achievement and significant increases in the quality of home life (Kohn, 2008). (Boaler, 2015, p. 46)

Research also shows that the only time homework is effective is when students are given a worthwhile learning experience, not worksheets of practice problems, and when homework is seen not as a norm but as an occasional opportunity to offer a meaningful task. (Boaler, 2015, p. 46)

Two innovative teachers I work with in Vista Unified School District, Yekaterina Milvidskaia and Tiana Tebelman, developed a set of homework reflection questions that they choose from each day to help their students process and understand the mathematics they have met that day at a deeper level. Th (Boaler, 2015, p. 46)