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:**

**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.**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.**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.**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.**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.**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.**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.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).”**Mistakes are embraced and are treated as rich learning opportunities.**

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