Mathematics and the Path to Equity

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For mathematics can, on the one hand, be though of as an incredible lens through which to view the world; an important knowledge, available to all, that promotes empowered young people ready to think quantitatively about their work and lives and that is equitably available to all students through study and hard work.  On the other hand, mathematics can be thought of as a subject that separates children into those who can and those who cannot, and that is valuable as a sorting mechanism, allowing people to label some children as smart and others as not smart.

The Myth of the Mathematically Gifted Child

Some people, including some teachers, have built their identity on the idea they could do well in maths because they were special, genetically gifted in mathematics, and the whole “gifted” movement in the United States is built upon such notions.  But we have a great deal of evidence that although people are born with brain differences, such differences are eclipsed by the experiences people have during their lives, as every second presents opportunities for incredible brain growth. (Thompson, 2014; Wo0llett & Macquire, 2011)

Equitable Strategies

  1. Offer all students high-level content
  2. Work to change ideas about who can achieve in mathematics
  3. Encourage students to think deeply about mathematics
  4. Teach students to work together
  5. Give all students encouragement to learn maths and science
  6. Eliminate (or at least change the nature of) homework

Mindset Works

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James Anderson is certified by Carol Dweck’s Mindset Works to provide training in developing Growth Mindsets in schools and organisations. This relationship allows James to provide unique and powerful resources, opportunities and training for schools and organisations.

About Mindset Works

Mindset Works’ mission is to enable a world in which people seek and are fulfilled by ongoing learning and growth. The Company translates lessons from leading universities into programs that schools can use to increase student motivation and learning.

imgres-1The Company’s Brainology® program motivates middle school and high school students to take an active role in their own academic development and gives them the tools to succeed. Children learn relevant neuroscience along with study skills, a combination that sparks their passion for learning and increases student achievement. The Mindset Works® EducatorKit is an online course for teachers to learn instructional practices that foster student ownership over their own learning.

These programs are available to students, educators, administrators, parents and anyone interested in learning how to improve their intelligence and abilities and are available through James Anderson in Australia and New Zealand.

References and Credits

The foundational references for Mindset Works are:

Blackwell, L., Trzesniewski, K., & Dweck, C. (2007). Implicit Theories of Intelligence Predict Achievement Across an Adolescent Transition: A Longitudinal Study and an Intervention. Child Development, Vol. 78, No. 1, pp. 246-263.

Dweck, C. S. (2006). Mindset: The New Psychology of Success. New York: Random House.

The Mindset Works icons and resources used throughout this site are used with permission from Mindset Works.

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Rich Mathematical Tasks

Teachers are the most important resource for students.  They are the ones who can create exciting mathematics environments, give students the positive messages they need, and take any math’s task and make it one that piques students’ curiosity and interest.

Interestingly, I found that mathematics excitement looks exactly the same for struggling 11-year-olds as it does for high-flying students in top universities – it combines curiosity, connection making, challenge, and creativity, and usually involves collaboration.  These, for me, are the 5 C’s of mathematics engagement.

Boaler, Mathematical Mindsets

5 cases of true mathematics excitement (This is just an overview of those cases.  What is valuable and probably can’t be conveyed in this blog post is the discussions that occurred during these tasks.)

Case 1. Seeing the Openness of Numbers

With a Silicon Valley start-up group that were looking to develop an online mathematics course the author when asked to what makes a good maths question, they stopped the conversation and asked the group if they could ask them all a math question.  The author enacted a mini version of a number talk.  They asked everyone to think about the answer to 18 X 5 and to show her, with a silent thumbs-up, when they had an answer.  The team then shared their methods and there were at least six different methods.  The author drew them up visually.  It was the sharing and discussion around these methods that created the engagement and excitement in understanding that there were so many ways to think about an abstract number problem.

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

Case 2. Growing Shapes: The Power of Visualisation

 Middle School classroom in San Francisco Bay Area.  The focus was algebra, but not algebra as an end point, with students mindlessly solving for x.  Instead, algebra was taught as a problem-solving tool tha could be used to solve rich, engaging problems.  The students had just finished sixth and seventh grades, and most of them hated maths.  Approximately half the students had received a D or an F in their previous school year.

The task that created the case of mathematics excitement came from Ruth Parker; it asked students to extend the growing pattern shown below:

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The success of the task was in the discussions particularly with a number of difficult boys and remained on task for 70 minutes.  Some of their work below:

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Here are some important observations that reveal opportunities to improve the engagement of all students:

  1. The task is challenging but accessible
  2. The boys saw the task as a puzzle
  3. The visual thinking about the growth of the task gave the boys understanding of the way the pattern grew
  4. They had all developed their own way of seeing the pattern growth
  5. The classroom had been set up to encourage students to propose ideas without being afraid of making mistakes
  6. Taught the students to respect each other’s thinking
  7. Using their own ideas
  8. Working together
  9. Boys were working heterogeneously

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Another way of solving the problem that led to discussions about why the pattern was growing as a square, why it was (n + 1) squared.

 Case 3. A Time to Tell?

When I share open, inquiry-based mathematics tasks with teachers, such as the growing shapes or “raindrop” task just discussed, they often ask questions such as, “I get that these tasks are engaging and create good mathematical discussions, but how do students learn new knowledge, such as trig functions? Or how to factorize? They can’t discover it.”

The traditional approach has been Teachers show the method and then students use them.

Researchers have found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, became curious, and their brains were primed to learn new methods, sot that when Teachers taught the methods, students paid greater attention to them and were more motivated to learn them.

So the question is not about whether we should tell or explain methods but when is the most appropriate time.  The student showed clearly that the best time was after students had explored the problems.

Case 6. From Math Facts to Math Excitement

Game: How close to 100

2 dice and a 100 grid.  Multiply the two numbers and colour in the squares.  Working towards filling the grid.

The following are some of the students’ important words as they reflected on the game:

  • “It challenged me to make my brain think.”
  • “It was a fun way to explore math and learn.”
  • “It gave me a lot of practice with multiplication.”
  • “It’s a fun way to learn multiplication facts.”
  • “I learned that multiplication and area are related.”
  • “I know now how division, multiplication, and area are related because I can see it!”

Conclusion

When mathematics tasks are opened for different ways of seeing, different methods and pathways, and different representations, everything changes.  To summarise there are 5 suggestions that can work to open mathematics tasks and increase their potential for learning:

  1. Open up the task so that there are multiple methods, pathways, and representations.
  2. Include inquiry opportunities.
  3. Ask the problem before teaching the method.
  4. Ask a visual component and ask students how they see the mathematics.
  5. Extend the task to make it lower floor and higher ceiling.
  6. Ask students to convince and reason; be sceptical.

Resources:

Creating Mathematical Mindsets: The Importance of Flexibility with Numbers

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http://www.brainquake.com/


Babies and infants love mathematics…

Give babies a set of blocks, and they will build and order them, fascinated by the ways edges line up.  Children will look up at the sky and be delighted by the V formations in which birds fly.  Count a set of objects with a young child, move the objects and count them again and they will be enchanted by the fact they still have the same number.  Ask children to make patterns in coloured blocks and they will work happily making repeating patterns – one of the most mathematical of all acts.

For many students their first experience of school mathematics is one of confusion, as the methods to do not make sense to them.  The inquisitiveness of our children’s early years fades away and is replaced by a strong belief that mathematics is all about following instructions and rules.

Successful mathematics users search for patterns and relationships and think about connections.  They approach mathematics with a mathematical mindset, knowing that mathematics is a subject of growth and their role is to learn and think about new ideas.

One way to give students opportunities to developing a thinking, conceptual approach to mathematics is to engage them in maths apps and games that approach mathematics conceptually:

New research on the brain tells us that the difference between successful and unsuccessful students is less about the content they learn and more about their mindsets.  Even mathematics facts, one of the driest parts of maths, can be taught conceptually and with sense making and understanding.

https://www.youcubed.org/wp-content/uploads/2015/03/FluencyWithoutFear-2015.pdf

 

The Creativity and Beauty in Mathematics

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Chapter 3 of Mathematical Mindsets

What is mathematics, really? And why do so many students either hate it or fear it – or both?  Mathematics is different from other subjects, not because it is right or wrong, as many people would say, but because it is taught in ways that are not used by other subject teachers, and people hold beliefs about mathematics that they do not hold about other subjects.

Keith Devlin, a top mathematician, has dedicated a book to this idea (mathematics is about patterns). In his book Mathematics: The Science of Patterns he writes:

As the science of abstract patterns, there is scarcely any aspect of our lives that is not affected, to a greater or lesser extent, by mathematics; for abstract patterns are the very essence of thought, of communication, of society, and of life itself. (Devlin, 1997)

An example in life:

The spider web is an amazing feat of engineering that could be constructed using calculations, but the spider intuitively uses mathematics in creating and using its own algorithm.

Mathematics exists throughout nature, art, and the world, yet most school students have not heard of the golden ratio:

https://www.mathsisfun.com/numbers/golden-ratio.html

…and do not see mathematics as the study of patterns.

the questions that drive mathematics. Solve problems and making up new ones is the essence of mathematical life.  If mathematics is conceived apart from mathematical life, of course it seems – dead

Reuben Hersh (1999) What is Mathematics, Really?

Over the years, school mathematics has become more and more disconnected from the mathematics that mathematicians use and the mathematics of life.

Four stages of mathematics:

  1. Posing a question
  2. Going from the real world to a mathematical model
  3. Performing a calculation
  4. Going from the model back to the real world, to see if the original question was answered

 

 

 

The Power of Mistakes and Struggle

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I recently (well earlier in the year) bought this book –  Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching by Jo Boaler.  I was interested in the mindset work and also the mathematics element.  I know as a student I loved mathematics (sad I know) but as a Teacher it was often the most difficult to instil that same love of learning for mathematics.  I see it in my own children who come home and say “maths is boring”…anyway I thought that I would share this book (chapter by chapter…and only the key points)…Liz borrowed the book off me first and I know she put together some ideas she was going to try in class (maybe she will share…sorry Liz did I just dob you in 🙂 )

Anyway, starting with Chapter 2 on  the power of mistakes…

When I have told teachers that mistakes cause your brain to spark and grow, they have said, “Surely this happens only if students correct their mistake and go on to solve the problem.” But this is no the case.  In fact, Moser’s study shows us that we don’t even have to be aware we have made a mistake for brain sparks to occur…because it is a time of struggle; the brain is challenged, and this is the time when the brain grows the most.

How Can We Change the Ways Students View Mistakes?

When we teach students that mistakes are positive, it has an incredibly liberating effect on them.

The habits of successful people in general:

  • Feel comfortable being wrong
  • Try seemingly wild ideas
  • Are open to different experiences
  • Play with ideas without judging them
  • Are willing to go against traditional ideas

One idea for class

Asking students who have been working on a problem and have knowingly made a mistake to share that mistake on the board.  Allows for discussions around common misconceptions and to solve the problem together.

Other resources:

 

Dealing with Negative Co-Workers

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Negative Nelly
Negative Nelly teaches at my school
She thinks that being negative is cool
She warns of bad students, bad adults, and bad things
Each day, it’s a song of doom that she sings
She’s managed to acquire a very small following
Of other naysayers intent on wallowing
In gossip and pettiness and bitterness and complaints
All who will listen, Nelly gladly taints
So do not listen, and do not follow
In anything negative, do not wallow
Just be the most optimistic teacher you know
And down Nelly’s path, you’ll never go.

Whitaker & Breaux The Ten-Minute Inservice

You may have never yet encountered this type of teacher, but some day you might.

Some questions

  • Have you ever known a negative person? Just get that person in mind.
  • How do you feel around that person?
  • How do others feel around that person?
  • Have you ever known a negative teacher? Maybe you were taught by a negative teacher when you were in school?
  • How was that teacher viewed by you and others?
  • Did you or others try to avoid that teacher?
  • Was that teacher a little intimidating?
  • What kind of influence do you think a Negative Nelly (or Nelson) would have on a school or on students?

Most Teachers tend to avoid these negative people.  Worse yet, some even engage in negative conversations with that person.  It’s not necessarily that they enjoy it, but many don’t know what else to do.  Below are some tips on “removing their fuse”…they may know what you are doing when you try these techniques…but that’s a good thing, right?

  • The next time Nelly or Nelson speaks unkindly about a student, simply say, “I love that student.” Even if you’ve never met that student, say, “I love that student.” They will have no comeback for that, and you will have made a very powerful, positive statement about how you feel about students.
  • For teachers new to the school, Nelly is a magnet.  They love warning of the bad students they will be teaching.  When they do, there’s a trick to defusing them.  Simply say, “Thank you so much for telling me about these students.  These kinds of students are the reason I became a teacher.  You obviously must care a lot about them to have taken time out of your busy schedule to come and speak with me about them.  I’m so glad they’ll be in my classroom.  They obviously need caring teachers such as you and me.  I’ll keep you posted on their progress.”
  • If Nelson ever approaches you offering any type of gossip, you can always say, “I’d love to chat, but I’m in a rush.  See you later.”  And then leave.  Gossip is hot.  They’ll quickly feel the need to find another listening ear.  But if you’re all using this technique, their gossip will never travel!
  • When Nelly speaks unkindly about a parent, say, “Whenever I think about the fact that not all parents rear their children as we wish they did, I quickly remind myself of how lucky some students are to spend so many hours with positive people like us at school each day.”

 Implementation

The bottom line is that Nelson or Nelly continue their negativism because someone is enabling them.  There is no place for Negative Nelly or Nelson in a school setting.  Our work is too important!

Happy for your feedback though no names should be mentioned.  The work above comes from the book mentioned under the poem.  I don’t believe we have this person at our school but it is always important to remember and in your career you may be unfortunate to come across this person.  They can be personally damaging and I know from experience that they can make your day unbearable…you can lose your way and why you chose this brilliant profession.  I would not want this to happen to any of you as your job is hard enough.  I would encourage you to be a positive force in your school community and for your students.  Some of them come here just to see your smile and hear your words of encouragement.

Why Understanding These Four Types of Mistakes Can Help Us Learn

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We can deepen our own and our students’ understanding of mistakes, which are not all created equal, and are not always desirable. After all, our ability to manage and learn from mistakes is not fixed. We can improve it.

Here are two quotes about mistakes that I like and use, but that can also lead to confusion if we don’t further clarify what we mean:

A life spent making mistakes is not only most honorable but more useful than a life spent doing nothing – George Bernard Shaw

It is well to cultivate a friendly feeling towards error, to treat it as a companion inseparable from our lives, as something having a purpose which it truly has. – Maria Montessori

 

http://ww2.kqed.org/mindshift/2015/11/23/why-understanding-these-four-types-of-mistakes-can-help-us-learn/

 

Growth Mindsets for kids

5 ways to teach children the growth mindset

  • Reward effort, not attainment.
  • Encourage them to take risks with their learning.
  • It’s ok to make mistakes. Use it as an opportunity to learn what could improve the outcome next time.
  • When praising your child,  focus on the strategies and skills they employed to learn about a specific subject, rather than their innate talent or skills.
  • Be mindful of the messages conveyed at home. For example, referring to yourself as ‘bad at maths’ or ‘no good at spelling’ can reinforce the idea that intelligence is fixed.

‘I don’t divide the world into the weak and the strong, or the successes and the failures…I divide the world into the learners and non-learners.’  Benjamin Barber