Six Shared Expectations for Math

By: Jeff Heyck-Williams

At Two Rivers we have made a conscious effort to change the way that everyone thinks about, talks about, and engages in math.  We have shifted the expectations to place the emphasis away from just memorizing math facts to also understanding the math.  We have shifted from pushing students to memorize lots of algorithms so that they can solve specific kinds of problems on tests to applying the math concepts they understand to solve any complex math problem.  We have shifted from a school where it was OK to say, “I’m not really a math person” to a place where everyone celebrates working hard to solve a math problem.

The foundations of this shift can be described in six shared expectations for math: 1. Math is fun.  2. All students can learn math.  3.  We have a common approach to problem solving.  4.  We value multiple representations.  5.  We value concise and precise communication. 6.  We value reasoning and proof.  These six expectations, shared publicly with staff, students, and families, codify our beliefs about mathematics and have made the shift in the culture of numeracy at our school real.

1. Math is fun: Time is built into lessons to play with math. 

I don’t recommend to anyone that they ever lie.  In fact, I encourage people to tell the truth.  Math is hard… or at least it can be.  However, being difficult is a long way from being impossible, and there never is a reason to say, “I can’t do math.”  First this statement is a gross overgeneralization.  Everyone can do math.  Second, it only reinforces negative stereotypes about math and math ability that can become a self-fulfilling prophecy for our children.

Instead, I recommend that we start from a place of positivity.  Math is fun.  Spending time working on something that is challenging leads to huge rewards as you discover the underlying patterns and, yes, beauty of the mathematics.  Simply shifting our orientation in the way that we talk about math has huge benefits.

2. All students can learn math: Every student has the capacity to learn deeper conceptual knowledge in mathematics. 

The emphasis here is in two places.  First I mean ALL students when I say all students.  I mean special education students with IEPs.  I mean English Language Learners.  I mean students who qualify for free or reduced price lunches because their household income is below the poverty line.  I mean girls as well as boys.  I mean all students. 

Second is that learning math is not just learning a set of algorithms.  Instead we mean, that students learn to not just compute fluently, but also to understand the mathematics that they are working with and to apply it to appropriate situations.

3. A common approach to problem solving: We use the K-W-I to teach steps to problem solving in and outside of math.

To aid students in developing the strategic competence to solve any problem that they face, we utilize a common approach to problem solving across the school, both in and out of math classes.  This approach emphasizes three key points.  First we work for students to first understand the problem by describing what they know.  Then we have students describe what they need to find out.  Finally, before problem solving they identify ideas for how they might approach the problem.  By slowing students down and using the same process with every problem we face, students develop a habit of mind towards all problems that emphasizes first understanding a problem before searching for answers.

4. Representations:  We value the multiple ways that ideas can be modeled or demonstrated and encourage making connections between various representations.

As students work and produce final products in mathematics, we don’t value one way of doing things over another.  Rather the emphasis is first on understanding.  Students utilize representations of mathematics whether manipulatives, drawings, or numbers to understand and solve the problems.  No one representation is valued over another.

Once a student feels confident in their understanding of their work, we then shift the emphasis to communicating.  Students explore whether or not their representations clearly express their understanding of mathematics for their classmates, teachers, and often an outside audience.  If not, they revise their representations to better express their understanding to others.

Only after we have ensured that students understand the problem and have communicated their solution effectively do we shift our attention to whether their solution method and representation was efficient. 

5. Concise and precise communication: We reinforce precision and brevity during class discussion.

With the question of efficiency, we push students to utilize accurate vocabulary and language to express their ideas.  As students explore ideas as a class, we have opportunities to reinforce concept development and vocabulary learning as we refine the way that we talk about mathematics.

6. Reasoning and proof: Mathematical arguments are weighed on the merits of their logic NOT on the status of the speaker or beauty of the language.

Last but not least, we emphasize that it is logic that is weighed most heavily in our classrooms.  As many of us know, many arguments are won by the popularity of the person speaking or the complexity of the words that he or she uses.  However, we want students to develop a critical ear for arguments that places value on what makes sense.

It is with these six expectations for how students, staff, and families engage in mathematics that we have created a foundation for changing the culture of numeracy at Two Rivers.

2, 4, 6, 8: Social Learning by the Numbers

By Jeff Heyck-Williams

At Two Rivers, we often say that social learning is just as important as academic learning.  This isn't just talk.  We believe that time spent teaching our students about how they can effectively interact and the personal skills needed to persevere and thrive in their work is just as important as the time we spend teaching them academic content.  In the world of high stakes academic testing and accountability, focusing on social learning can seem counterintuitive.  After all, you only have so much time to cover all of the things that kids need to know.  However, we have seen over and over again that an investment in a child's social skills has payoffs far wider than a single test score.  Students who have developed habits of perseverance, commitment, responsibility, and productive collaboration are more successful in and outside of the classroom.  

The questions remain, what does our social curriculum look like?  And how do we help students develop the habits listed above?  What I have found helpful is to think about social learning by the numbers: 2, 4, 6, 8.  We have identified 2 types of character that kids need to be successful.  We have 4 scholarly habits that articulate what those types of characters look like on a daily basis.  We take 6 weeks at the beginning of the school year to introduce the scholarly habits and develop the culture of learning.  This all forms a foundation of 8 hours each day during the school year when we can craft the experiences for students that foster the development of their social learning.  So here is social learning by the numbers. 

2 Types of Character Development

Building on the work of the Character Education Partnership (CEP) in their position paper, "Performance Values: Why They Matter and What Schools Can Do to Foster Their Development," we define our social curriculum with an understanding of two types of character that people need to be successful: moral character and performance character. 

Moral character is defined by the traits that "enable us to be our best ethical selves in relationships and in our roles as citizens."  It includes character strengths like empathy, fairness, trustworthiness, generosity, and compassion.  Developing strong moral character is essential to living and working in productive communities whether they be our schools, workplaces, or neighborhoods.  Generally, moral character can be understood as interpersonal skills, or how individuals relate to each other.  

In addition to moral character, performance character is the type of character that "enable[s] us to achieve, given a supportive environment, our highest potential in any performance context."  The qualities of performance character include traits like effort, initiative, diligence, self-discipline, and perseverance.  These personal character traits are what individuals need to reach their potential and accomplish personal goals.  Where moral character can be understood as interpersonal skills, performance character generally encompasses intrapersonal skills, or what individuals believe about themselves and how they act on those beliefs. 

4 Scholarly Habits

At Two Rivers, we believe that both of these two types of character are essential for lifetime success, but people aren't just born with these character traits and that they can be fostered in everyone.   We believe that character must be taught and fostered intentionally in our interactions with children and the experiences that we create in our classrooms.  Which brings me to our 4 scholarly habits.  Across our school we have identified 4 habits that embody the traits we want all students to develop in both moral and performance character.  The habits stated as “I statements” for students are: I work hard.  I am responsible and independent.  I am a team player.  I care for our community.  In the broadest sense, the first two, working hard and being responsible and independent, encompass the performance character traits that we teach.  The second two, being a team player and caring for our community, encompass the moral character traits.

Beginning with performance character, by teaching students the scholarly habit of I work hard, we focus students on the concepts of a growth mindset, perseverance, and risk taking.  We want to engender in all students a personal belief that through their own effort they can improve both their understanding and abilities in any context.  This comes with developing a willingness to attempt difficult tasks and having a set of coping skills to deal with situations when they become more challenging. 

The second scholarly habit of I am responsible and independent focuses students not just on the outward manifestations of responsibility like being prepared for class and using self-control, but also the inward manifestations like being attuned to both the quantity and quality of the work that they produce every day.  Through these first two scholarly habits, we work to make performance character concrete traits that every student can improve upon.

Shifting to our focus on moral character, we define the scholarly habit of I am a team player by talking to students about how their actions in a community directly influence the accomplishments of the whole community.  Consequently we name for students that their actions need to help everyone learn and they need to take ownership for their share of the work.  More importantly, we talk about the difficult work of collaboration and how cultivating a set of skills to negotiate differences, develop shared understanding, and ultimately build trust are essential in developing moral character.

The fourth and final scholarly habit that we teach to students, I care for my community, speaks to how we develop habits of compassion.  This means that we look for and name acts of kindness explicitly at our school.  In addition, it means we talk about the types of actions that students can take to make their school community a better place from keeping our shared space clean to treating each other with care.

By naming these scholarly habits and teaching them explicitly to students, we make performance character and moral character concrete and real.  They have examples of traits that exemplify working hard, being responsible and independent, being a team player, and caring for their community.  They can draw on these examples as we work through the course of a school year.

First 6 Weeks

To develop these habits, we recognize that teaching anything well takes time.  If we are serious about developing students' social skills, then we have to spend the time to help kids learn these skills.   With this in mind we begin to teach towards the 2 types of character through the 4 scholarly habits in the first 6 weeks of school.  In focusing on social learning during the first six weeks of school, we draw from the work of the Northeast Foundation for Children's Responsive Classroom approach to education.  

Thinking about how we teach towards social learning in those first 6 weeks, it is useful to think about the twin priorities of social learning and academic learning on a balance scale.  As the school year opens social learning is raised to a priority and is high on the scale.  This doesn't mean that we don't address academic learning, but it is recognized that it is the secondary goal at the beginning of the year and is relatively lower on the scale.  As the first six weeks progress, the scales slowly shift from a priority on social learning to a priority on academic learning.  The goal being that at the end of the first 6 weeks the scale has shifted to academic learning as the priority and social learning as the secondary goal.  Intentionally dedicating the first month and a half of school to the moral and performance character of our students pays off as the year progresses and students are able to draw on the skills and concepts that they learned in the first 6 weeks as work gets more difficult.  

So the first 6 weeks functions as a time to explicitly name, model, and explore the 4 scholarly habits and how they will live in the lives of our students over the course of the year.

8 Hours a Day

Which brings me to my last number of social learning, 8.  Our school is designed to work with kids in approximately an 8 hour day.  Within those 8 hours we craft experiences for students to optimize their learning.  In 8 hours each school day, we have the opportunity to provide experiences that help kids learn reading, writing, math, social studies, science, PE, Spanish, and visual and performing arts.  We also have the opportunity to provide experiences where kids learn the skills that embody the 2 types of character articulated by our 4 scholarly habits in the first 6 weeks of school.

I end with the number 8, because school doesn't end with the first 6 weeks.  Rather the 8 hour days continue, and within those time frames just as we craft amazing lessons in social studies and math, we also are mindful of crafting amazing lessons in social learning.  In fact, only when students are challenged by the level of work and demands of living in a vibrant learning community are they ever able to develop scholarly habits.  While the first 6 weeks introduces them to the traits of character, it is the work that happens during the 8 hour days for the rest of the year in which students develop and master their own scholarly habits.