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When students understand that they are grouped asymmetrically, the ceiling of the task is  lowered without changing a word of text.  The team "expert" tends not to question her work; nor does she inquire for ideas or feedback from her “novice” teammates. The novices do not critically examine the work of the “expert,” and do not push themselves to be involved in the mathematics of the task.

When collaboration is a necessary element of completing a task, student outcomes improve. If students have unique information necessary to the completion of the task, they cannot rely on an asymmetric expert/novice orientation. Although strategic task design will not eliminate all imbalances in group engagement, it supports this goal. Difficulty of tasks can be harmful on either end of the spectrum - too easy and there is no need for reliance upon one another; too difficult and members of the team may automatically disengage regardless of the degree to which students are grouped symmetrically.

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Tasks should use of multiple strategies and flexibility in interpretation so that students with different abilities and strengths can -- and really, must -- all contribute to the success of the group. A task with low cognitive demand is unlikely to be groupworthy if one team member can complete it independently, without input of alternate perspectives or strategies.

Monitor tasks to ensure that one group member with high social status is not dominating the completion of the task and other students merely trusting that their ideas were correct or not holding their team members accountable for justifying their ideas and being equal stakeholders in the task.

Encourage students to see themselves as mathematicians and builders of knowledge, even if they have struggled with math in the past. Groupworthy tasks should be centered on the students, not the teacher.

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As a teacher, it is important to me that the grade-level content is infused into the exploration, but student engagement is highly influenced by the the degree of authenticity with which the math is incorporated into their question. In my mind, it becomes a question of chicken and egg -- is the math a tool to answer the question, or the question a tool (or excuse) to do the math?

"If students were able to work for at least some of the time in the ways mathematicians do - posing problems, making conjectures using intuition, exploring with and refining ideas, and discussing ideas with others - then they would not only be given a sense of true mathematical work, which is an important goal in its own right, they would also be given the opportunity to enjoy mathematics and learn it in the most productive way." -- J. Boaler

When school mathematics so far differs from true mathematics, students are deprived of the opportunity to understand and learn math as it really is.

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Allow opportunities for disagreement while students explore meaning and value.

Build on prior mathematical, community, and cultural knowledge - make connections between the math and what they already understand!

Open the task up to multiple representations - or even require it! Give students the opportunity to see the mathematics in different ways, and to interpret the connections between these representations.

Require generalization - how do the patterns extend beyond the concrete context, or on a larger scale?

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Supporting Problem Solving by Accommodating Differences

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Supporting Problem Solving by Accommodating Differences

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Teaching for equity requires supporting students both in developing their knowledge of mathematics, as well as supporting development of social skills, student skills, and language, and maintaining high expectations across all of these areas of knowledge.

“accommodating differences.” Each student in our classrooms have different backgrounds – both in life and mathematics – that influence their prior knowledge, intellectual strengths, and personal interests, in turn differentiating the amount of support or enrichment they require.

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A student may not remember a formula or algorithm, but with the foundations developed through problem-solving is far more likely to be able to apply their learning beyond the classroom.

"If teachers pose and extend problems of interest to students, they enjoy mathematics more, they feel more ownership of their work, and they ultimately learn more." -- J. Boaler

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"It is important that students both see the need for these models and understand them, which is not to say that students need to discover them without teacher support. It is enough that students should experience how disorganized numbers become without a table to organize them, how opaque those numbers become without a graph to visualize them, and how hard it is to make predictions outside our existing data without an algebraic equation." -- Dan Meyer

Our goal in modeling is to create an appropriate strategy to attempt to answer interesting and meaningful questions, not to use a given procedure within an inauthentic context. Math helps us to understand the injustice that is close to the hearts of our students, and to better understand our world in the hopes of changing it for the better.

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Math is...

Creative - "After making a guess, mathematicians engage in a zigzagging process of conjecturing, refining with counterexamples, and then proving." -- J. Boaler

Collaborative - "[Mathematicians] gave many reasons for collaboration, including the advantage of learning from one another's work, increasing quality of ideas, and sharing the 'euphoria' of problem solving." -- J. Boaler

Lived - "Mathematics is still alive, not something that has already been decided and just needs to be memorized." -- J. Boaler

About Questioning -

"What information matters here?"

"What information do I need here?"

"Now that I have identified the relevant information, what do I do with it to answer my question?"

"Is my prediction correct? How well do my calculations represent reality?"

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"The math that millions of Americans experience in school is an impoverished version of the subject and it bears little resemblance to the mathematics of life or work or even the mathematics in which mathematicians engage." -- J. Boaler

Mathematics is about questioning and sense-making, rather than applying isolated procedures. It allows for analysis and problem solving in the process of identifying the best method to represent a relationship.