Curious, Enthusiastic, Independent Mathematics Students

I envision a school culture in which curiosity and exploration are not only supported, but are cultivated. A student who arrives to school brimming over with youthful enthusiasm for new ideas and information are received as a gift and not as a challenge or a puzzle. Students who are able to take learning in a direction of their own, outside the bounds of what the teacher has prescribed in the assignment or activity, are to be encouraged and applauded. Students who are not yet able to define their own independent roadmaps are to be supported in a learning process toward the creation of their own unique paths to learning.

It is my experience that no two individuals, and similarly no two groups of students learn in the same way. If a classroom functions the same way with multiple classes it is likely that the teacher is defining the direction with little input from student thinking or creativity. Without opportunities to experience the ways in which their unique ideas influence the trajectory of the learning students feel, and will become, powerless to learn independently. They have little opportunity to experience the value of their own ideas, and instead they learn to model themselves after the teacher. Student learning will reflect and replicate the teacher’s way of thinking about the topic, and students’ ability to think creatively and independently will be steadily eroded.

My next few blog posts will be focused on my most successful strategies for cultivating curiosity and independence in mathematics learning, as well as ramifications of holding the above described philosophy.  (For example: The need for comfort with being unable to predict the direction a particular class may take.)

8 Characteristics of an Equitable Mathematics Classroom

First, I want to be absolutely clear that “math for everyone” is meaningless if it achieves equity but sacrifices excellence. Such an approach provides opportunity to no one. It underestimates the ability of the students in the class while simultaneously depriving them of the opportunity to pursue further work that depends on highly developed mathematical reasoning and problem solving skills.

So what does an equitable mathematics classroom look like? What happens in that space, among the people in that class, that leads students to develop strong conceptual understanding of mathematics, the creativity and determination to solve problems unlike any they have ever seen before, and also allows every single student to develop curiosity about and confidence in mathematics? What role does the teacher play, and how can the teacher create this reality for and along with the students?

Three of the characteristics described by National Council of Teachers of Mathematics (NCTM), in the document “Principles to Actions” under the category “Access and Equity Principle” are as follows:

  • Develop socially, emotionally and academically safe environments for mathematics teaching and learning – environments in which students feel safe to engage with one another and with teachers.
  • Model high expectations for each student’s success in problem solving, reasoning and understanding.
  • Promote the development of a growth mindset among students.

The seven characteristics I list below increase both excellence and equity and they can be achieved with the bare minimum of resources, in any classroom environment, anywhere in the country. There are certainly other considerations that come into play in order to fully prepare students to be independent and active participants in the world they are already living in (technology use, in particular), but for now I will focus on what is absolutely necessary for achieving excellent, rich mathematics learning which every student can both relate to and appreciate.

In a mathematics classroom that genuinely makes high-level mathematics learning accessible to every single student there are a few characteristics that will be present:

  1. Many voices are part of the conversation, and every student feels that they have something unique to contribute.  No student sits quietly in the back without knowing how he or she can make a valuable and important contribution to the work that gets done each and every day.  The student who is less comfortable participating in whole class discussion may shine in smaller groups or when giving feedback to another classmate.
  2. Many approaches to solving any individual problem are explored. Problems are not solved, and questions are not answered, with just a single, most efficient strategy. Multiple methods of approaching the problem from different angles are given full consideration.
  3. Thorough discussions trump quick answers. Similar to number 2, but also includes the process of making sense of the question in the first place, or verifying the accuracy of an answer.
  4. Students utilize a variety of strategies for recognizing if things make sense. It must become second nature to students to realize that they should not walk away from the solution to a problem until at least one, and preferably multiple methods have been used to confirm that the work and the answer make good sense.
  5. Feedback is rich with positive commendations. When a student gives an incorrect answer, or an inaccurate suggestion, the first response should not be “no”.  Such a negative response simply sends the student into a space from which they are unlikely to hear or absorb the advice, recommendations or corrections that will likely follow.  I firmly believe that all mistakes are based on some accurate thinking and reasoning.  The goal of the person providing feedback must be to learn more about the student’s thought process so that positive commendations can be provided first, and only after that is it appropriate and helpful (and necessary) to offer corrections and suggestions for improvement.
  6. A growth mindset permeates the atmosphere. This is a huge topic in and of itself, and Carol Dweck’s work, as it is applied to the goal of allowing all students to achieve at high levels in mathematics, is tremendously inspiring.
  7. Feedback, both commendations and recommendations, are thorough and detailed. All students need access to detailed information about what they are doing well, and what they can improve.  The traditional marking of answers correct or incorrect in math classrooms must become a thing of the past.
  8. Mistakes are embraced and are treated as rich learning opportunities. In her paper “The Mathematics of Hope” Jo Boaler states: “Research has recently shown something stunning—when students make a mistake in math, their brain grows, synapses fire, and connections are made; when they do the work correctly, there is no brain growth (Moser et al. 2011). This finding suggests that we want students to make mistakes in math class and that students should not view mistakes as learning failures but as learning achievements (Boaler 2013a).”
    The challenge of accomplishing this fully, so that the students themselves genuinely experience mistakes as learning experiences, depends greatly on the tone of the feedback.

There is a wealth of information about how to accomplish these goals, and I plan to describe some simple and specific strategies that have worked for me in upcoming posts.

References:

Boaler, J. 2013a. “Ability and Mathematics: The Mindset Revolution That Is Reshaping Education.” FORUM 55(1): 143–52.

Darling-Hammond, Linda (2009-01-01). The Flat World and Education: How America’s Commitment to Equity Will Determine Our Future (Multicultural Education Series) (p. 3). Teachers College Press. Kindle Edition.

Ferguson, Ron. Toward Excellence With Equity: An Emerging Vision for Closing the Achievement Gap. Harvard Education Press, 2008.

Moser, J., H. S. Schroder, C. Heeter, T. P. Moran & Y. H. Lee. 2011. “Mind Your Errors: Evidence for a Neural Mechanism Linking Growth Mindset to Adaptive Post Error Adjustments.” Psychological Science 22: 1484–9.

National Council of Teachers of Mathematics.  Principles to Actions: Ensuring Mathematical Success for All. Reston VA. 2014.

Walsh, Barry. “Getting to Excellence with Equity.” https://www.gse.harvard.edu/news/uk/15/01/getting-excellence-equity , January 26, 2015

Walsh, Barry. “Getting to Excellence with Equity.” https://www.gse.harvard.edu/news/uk/15/01/getting-excellence-equity , January 26, 2015

Why Math for Everyone?

In Smartblog on Education, February 18th, Joshua Thomases states the following: “There is a new majority in our nation’s public schools. Recent data from the Southern Education Foundation reveal 51% of all students are eligible for free or reduced lunch — schools’ basic benchmark of low-income status.”

In Math Matters: The Links Between High School Curriculum, College Graduation, and Earnings, Heather Rose and Julian R. Betts find a strong relationship between taking advanced math courses in high school and earnings 10 years after graduation.

In The Flat World and Education Linda Darling-Hammond states, “… by 2012, America will have 7 million jobs
in science and technology fields, “green” industries, and other fields that cannot be filled by U.S. workers who have been adequately educated for them.”

When I put these facts together I wonder what it will take in order to position math, science, engineering and technology education (STEM) as a 6-lane highway out of poverty for millions of American children. I imagine a highway without traffic jams, and with systems in place to get tired travelers back on the road with both efficiency and care. I struggle with the notion that it could be moving at 70 mph, because those who get there first will likely not have had the richest experience. Perhaps we should imagine millions of bike riders, efficient, yet thoughtful, riding down millions of country roads.

In a generation we could flood the workforce with creative problem solvers and increase racial diversity in STEM fields exponentially. All it would take is transformational change in mathematics education.

How do we accomplish this, you wonder? Why do I imagine that everyone could successfully navigate this mathematical path out of poverty? By what means do I believe we could achieve social justice and equity through transformational change in math education? Isn’t this the very subject in which, according to Jo Boaler, two thirds of students fall below grade level by the time they reach middle school?

Methods of teaching mathematics that are both appealing and accessible to a broad population of students are known. Researchers across the country have projects and strategies along with the data to verify their effectiveness in classrooms. There are teaching methods that have been demonstrated to eliminate achievement gaps between genders, socioeconomic backgrounds and racial groups, and between native English speakers and English Language Learners (described by Edd Taylor and Valerie Otero, in their presentation “How Children Learn Math”) while achieving high levels of mathematics comprehension, effectively achieving what Ron Ferguson describes as “excellence with equity.”

To some extent the information is getting out into schools and to individual teachers, and the methods are being used effectively in classrooms. Some students are developing creativity, curiosity, perseverance, and insight around mathematics problem solving. Teachers are teaching, and students are learning mathematics with a growth instead of a fixed mindset, and there are classrooms where students realize that success and confidence in mathematics is more than just a privilege for a select group; that mathematics is accessible to everyone. Classes have experienced the richness of knowing that there are many valuable ways of looking at math concepts and math problems, students do not have to climb a ladder one rung at a time to achieve success in mathematics, and the fastest way to the answer may not represent the most complete level of understanding.

Unfortunately, this does not yet mean that the most cutting-edge, most up to date research about how to effectively teach high-level mathematics to all learners is available on a widespread basis. This work remains challenging and time consuming for the teachers who take it on wholeheartedly. Some of the most resource rich school districts have not yet embraced teaching strategies that make mathematics genuinely accessible to everyone. Schools and teachers in the most resource poor districts, where the students come from the poorest socioeconomic backgrounds, are the least likely to have the time, resources or skills needed to make this kind of mathematics learning fully available to all students.

We have reached a very high hurdle, and we have the ability to both meet and exceed this challenge. The knowledge exists, and we must get the most critical information into the hands of the schools and teachers who need it most. We must accomplish this in ways that do not depend on great expenditures of time or money that are unavailable in the districts with the poorest students. The internet, and the professional learning opportunities it affords, provide an unprecedented opportunity. More years of incremental change will allow the income gap between the wealthy and the poorest people in our country to widen still further.

Ron Ferguson of Harvard Graduate School of Education states, “…we are already in a social movement that is defining for the 21st century how we prepare young people for life. Several contemporary trends are converging and will compel us to make changes — from birth to career — in how the country prepares its young.” In The Flat World and Education: How America’s Commitment to Equity Will Determine Our Future Linda Darling-Hammond states, “We cannot just bail ourselves out of this crisis. We must teach our way out.”

Imagine millions of children, guided by their teachers, pedaling down millions of paths of mathematical learning toward a better future. Of course math education must exist in the context of a high quality overall education, but achieving equity in math education has the potential to be a powerful lever for increasing opportunity. We can choose. I believe this is a question that has only one right answer, and the benefits of achieving high-quality, equitable math education for everyone could quite possibly exceed our wildest dreams.

References:

Boaler, Jo. The American Math Crisis, forthcoming documentary. http://youcubed.stanford.edu/the-american-math-crisis-forthcoming-documentary/

Darling-Hammond, Linda (2009-01-01). The Flat World and Education: How America’s Commitment to Equity Will Determine Our Future (Multicultural Education Series) (p. 3). Teachers College Press. Kindle Edition.

Ferguson, Ron. Toward Excellence With Equity: An Emerging Vision for Closing the Achievement Gap. Harvard Education Press, 2008.

Taylor, Edd and Valerie Otero. How Children Learn Math and Science, presentation, February 18, 2015.

Thomases, Joshua. http://smartblogs.com/education/2015/02/18/making-sure-poverty-≠-destiny , February 18, 2015

Rose, Heather and Julian R. Betts, Math Matters: The Links Between High School Curriculum, College Graduation and Earnings. http://www.ppic.org/content/pubs/report/R_701JBR.pdf , 2001

Walsh, Barry. “Getting to Excellence with Equity.” https://www.gse.harvard.edu/news/uk/15/01/getting-excellence-equity , January 26, 2015