A Colossal Failure of Society, and a New Ladder of Opportunity

I’ve been away from this site in recent weeks (too many weeks) while we have planned our way into a move to a smaller town in the mountains of Colorado.  Having moved just last fall from a small town in rural NH we hope this most recent move will help us to begin to feel at home here.

Buying and selling houses, and all the associated planning and packing, does little to slow my thinking about equity and math education however.  The problem is that the writing occurs in my head, in a jumbled mess of sentences, paragraphs and posts which never get sorted out through the process of actually writing them down.  It’s a lot like talking to myself, and begins to feel rather psychotic after a while.  So here I am, finally hoping to sort out some of the jumble.

What follows might not be the post I ought to start back with.  The request for more detail about the dialogue that occurs in Precalculus classes when I use “Essential Questions to Personalize Mathematics Learning” has been on my mind also, and I promise to write that soon.  However, this has been so stuck in my head that I have to start here.

I recently began reading the book The New Jim Crow by Michelle Alexander.  For those who have not read or heard about this book it addresses the development of a new racial caste system in America in the form of mass incarceration of black men.

In the book Alexander refers to, “some of the differences between slavery, Jim Crow, and mass incarceration, most significantly the fact that mass incarceration is designed to warehouse a population deemed disposable – unnecessary to the functioning of the new global economy – while earlier systems of control were designed to exploit and control black labor.”  The tragedy of this description has really stuck with me.  In this time of employers in all types of STEM fields recognizing the need to diversify their workforce, and when such jobs are regularly being filled by highly trained people from abroad because there aren’t enough US citizens with the necessary knowledge and skills, the fact that millions of people, a disproportionate number of them men of color, are in prison, locked away from participating in society in any meaningful way, is a colossal failure of our society.

I am becoming more and more convinced that a new ladder of opportunity can and must be constructed in this country by developing pathways leading from poverty to jobs in the fields of Science, Technology, Engineering and Mathematics.

Essential Questions to Personalize Mathematics Learning

The biggest breakthrough I’ve ever had in teaching high school mathematics has involved doing drastically less lesson planning.  That probably bears repeating, in a slightly different way:

When I started spending less time planning lessons I was dramatically happier with what was happening in my classroom.

One key factor, and perhaps the backbone of the entire concept, was allowing time, a lot of time, for any example or question I put before the students.  By this I mean something along the lines of spending an entire class period on a single prompt.  It’s an approach which would have seemed preposterous to me 10 years ago, and yet it was absolutely the best idea I have ever tried in teaching HS mathematics.  I have now taught classes in which we spent a full 50 minutes on a single quadratic function as the prompt.  One function.  50 minutes.  And the students understood more from those 50 minutes, and retained more of that learning over the coming weeks and months, than any class I ever taught using multiple examples.

50 minutes on one prompt led to more and better learning

Perhaps now you can see where the reduced lesson planning time comes in.  If I only need one prompt to start class I don’t need a whole lot of time to come up with it.  There are days when I would spend more time devising the right prompt, in the hope of eliciting the types of questions and observations I was most hoping to move toward, but even then it is critical to remember that this whole approach relies heavily on having a high level of comfort with letting the class go in the direction it needs to go in, and even letting individual students go in the directions they need to go in.  I’ve never taught two classes which turned out the same way, even when they started with the same prompt, using this approach.

The students in the room are different, the learning will evolve to meet their needs, and each class will move forward as a group along a unique path.

Another key factor is an approach which consistently allowed me to start class by meeting students exactly where they were.  By this I absolutely mean that each student was able to start the learning process where they were, individually, when they walked in the door, and move forward from that point.

So, how exactly, does one structure 50 minutes of time spent on a single question, example or prompt? If someone had asked me this years ago I would have imagined that it depended on asking a rich enough question.  Any question that is sufficiently challenging will take the students a whole class period, or more, to complete, and will give them plenty to talk about.  But often it will often cause some students to drop out part way through.  Other students may stick with it by following along, but won’t feel that they have contributed their own voice in any meaningful way.

Each and every student must have a meaningful voice in the conversation, and must leave feeling that he or she contributed to the learning process.

So, a richer, more complex question isn’t always the best approach.  (I don’t mean to imply that it isn’t valuable, and even very important, as an approach some of the time!  More on that another time.)  Sometimes the best prompt is the simplest.  Just a single equation or function.  A description of a relatively simple scenario which can be represented with a linear equation. A sequence.  Nothing more.  And in particular no specific question.  Nothing that says, “Find the vertex of the graph of this quadratic function.”  Nothing that asks, “What is the slope of the line.” No specific question to answer about the scenario.  Perhaps just, “I went for a run yesterday, and it took me 45 minutes to run 5 miles.”  That’s it.

Then you let the students talk.  Once they get the hang of it they can start individually, or in small groups, but I found the first few times I sprung this on them it had to be done as a whole class.  I would start by asking,

“What do you notice?”

Ask the students to make some observations.  State what they see.  If the prompt was a quadratic function I would get responses like, “There’s a little 2 above the x,” or “It’s quadratic,” or “There are 3 terms.”  This is where the students get to start where they are, not where I think they ought to be.  And there are no wrong answers.  If it’s what they see then it’s correct.  The vocabulary they choose tells you a great deal about what’s going on for them when they see a quadratic function.  You can build on their observations, by first providing validation and commendation on what they have offered, and then restating it in a way which helps them move forward, or builds upon what they know.

When the observations start to die down, I ask, “What do you want to know?” or “What questions do you have?” And then, “What can you do with this?” or “What can you figure out or tell me about this?” This question, with a quadratic function as a prompt, will gradually lead to covering an entire board with information related to that one function.  And it can easily take 50 minutes.  Ideally the process will continue on to, “Are there other ways that you could think about it,” or “Are there other approaches that you could use find that information,” and, “How do you know that your work is accurate?”  Last, “What else would you like to know or do at this point?” or “Where could you take this from here?”

In summary, this highly personalized approach depends on a simple prompt and 6 essential questions:

  • What do you notice?
  • What questions do you have?
  • What can you do or figure out?
  • How have you verified that your work or answer is accurate?
  • Is there another way to approach this?
  • What else can you do or say, or what more would you like to know?


Boaler, Jo. “Depth Not Speed.” http://www.youcubed.org/category/teaching-ideas/depth-not-speed/

Kress, Nancy. “Essential Questions for Mathematics: Developing Confident, Knowledgable, Creative Students.” Presentation, NHTM Spring Conference. 2014.

McTighe, Jay and Grant Wiggins. “Essential Questions: Opening Doors to Student Understanding.” Association for Supervision and Curriculum Development. 2013.

Richardson, Will. “Still Here… Sort of” blog post.  Comments include the following quote, “I think our whole concept of teaching is going to have to change in order for us to truly help kids flourish in this world. I think it’s all focused on how people learn, and in the teacher’s case, how he or she can create the conditions required for learning more deeply in the classroom. Anybody still doing lesson plans hasn’t gotten there yet.” This quote influenced my writing.  http://willrichardson.com March 18, 2015.

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.


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

Elementary and Secondary Education Act: Time to Get it Right

The following is the text of my letter to Congress, urging revision of the Elementary and Secondary Education Act:

No Child Left Behind ushered in an era of urgency in American schools.  That urgency focused on obtaining, reporting and responding to vast quantities of testing data.  As Congress works to rewrite the Elementary and Secondary Education Act (ESEA) it is time to “get it right”.

There is broad consensus that schools need to look unrecognizably different in order to effectively prepare students for lives and work in a world we cannot even anticipate.  The workplace is changing rapidly, and the main thing we know is that today’s children will have to be much better than the current adult population at being innovative and at solving problems that they cannot anticipate far in advance.  Students will need to be able to solve problems unlike any they have ever seen before, and they will need to do it frequently and repeatedly.

The Carnegie Corporation of NY and The Institute on Advanced Study, in their report titled “The Opportunity Equation,” state the following: “The United States must mobilize for excellence in mathematics and science education so that all students — not just a select few, or those fortunate enough to attend certain schools — achieve much higher levels of math and science learning. Over the coming decades, today’s young people will depend on the skills and knowledge developed from learning math and science to analyze problems, imagine solutions, and bring productive new ideas into being. The nation’s capacity to innovate for economic growth and the ability of American workers to thrive in the global economy depend on a broad foundation of math and science learning, as do our hopes for preserving a vibrant democracy and the promise of social mobility for young people that lie at the heart of the American dream.”

It is imperative that the adults of today are able to be sufficiently innovative, and to think with enough creativity and flexibility to  allow educators to lead the way to the development of a transformed educational system.  Educators must be empowered to lead and to develop solutions which are responsive not only to test data, but also to the unique needs, challenges and strengths of individual schools, districts and communities.  The collective knowledge of millions of highly qualified education professionals must be allowed and enabled to lead our schools forward so that all children, in all schools and from all backgrounds will have the opportunity to live their lives free of poverty doing work that contributes to a vibrant and healthy economy and society.

Rewrite ESEA, and on behalf of every American child, it’s time to “get it right.”


The Carnegie Corporation of NY and The Institute for Advanced Study. “The Opportunity Equation: Transforming Mathematics and Science Education for Citizenship and the Global Economy.” 2009. http://carnegie.org/fileadmin/Media/Publications/PDF/OpportunityEquation.pdf

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.


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