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Promoting Mathematics Learning Through Problem Solving in the K8 Classroom

Improving student success and deepening learning

Learning through problem-solving promotes deep, coherent mathematics understanding. It is a critical tool for creating a highly effective learning environment for students. Through the use of strong routines, students learn how to take an active role in reasoning and sensemaking. Active learning will help students understand new mathematical concepts and relationships as they progress in their school careers.

In this web seminar, a professor of education at California State University, Fullerton who specializes in middle school mathematics instruction discussed how the use of routines has the potential to significantly increase student success and deepen learning.

DANIELLE CURRAN
Product Director, Mathematics
Curriculum Associates

We’re very excited to have Mark Ellis with us. Mark is a leading mathematics educator and professor; he’s known for his scholarship about the history of school mathematics, middle grades mathematics teaching and learning, and equity in mathematics education.
His work has been featured in many prominent publications.

All of us at Curriculum Associates are excited to have Mark as part of our Ready® Math author team, along with Gladis Kersaint, who is at the University of Connecticut. Both of them are focused on ensuring that Ready materials incorporate the best instructional practices and address the needs of teachers and students.

MARK ELLIS
Professor
College of Education
California State University,
Fullerton

I’d like to begin by looking at traditional mathematics teaching and its impact on students' habits of doing school mathematics. I want to make the case for why we must change the routines of mathematics teaching and learning.

There is typically minimal interaction among many students in class; they are trying to quickly get right answers using memorization, and not thinking. They’re following algorithms, they’re not using technology, and there can be a learned helplessness. Students are often mimicking examples, looking for keywords, showing steps but without any explanations or justifications.

If we are honest about it, we wouldn’t characterize these habits in a positive way. But we have to realize that these students have spent years in our classrooms and in our schools, and they developed these habits, for the most part, in those settings.

Moreover, we have to acknowledge that we educators, teachers and administrators need to accept responsibility to change math classrooms so that we create environments that will nurture more productive habits. We need to give careful attention to different routines that will give students new opportunities to develop more productive habits and deeper understandings of mathematics.

Emphasizing the growth mindset
Our students are growing up in a digital world, and they need to be able to analyze and interpret the quality of information and what it means to the question they’re investigating. These are new skills and ones that must be taught. In mathematics, it means students can’t be taught to just mimic algorithms. They must understand concepts and relationships in order to analyze, interpret and apply mathematics to real-world problems.

Furthermore, research tell us that our belief from the 20th century that only some people could do math was sorely mistaken. We now know that mathematics ability is primarily a function of opportunity, experience and effort, not innate intelligence. This is reflected in the idea of the growth mindset—a belief that with effort and productive feedback, all students can learn.

This means that we must see mathematics teaching as cultivating students’ mathematics abilities, not separating them into those who can and cannot do math. Schooling must provide equitable access and support for all learners; it’s about figuring out how to provide the right kind of support for all students to be successful.

We now have standards that focus on students understanding the concepts behind the calculations. Students need to use math flexibly to solve non-routine problems and they must have skill in communicating mathematically. Whenever a learner’s working memory decides that an item doesn’t make sense or have meaning, the probability of it being stored in long-term memory is extremely low. Brain scans have shown that when new learning is readily comprehensible, makes sense, can be connected to past experiences and has meaning, there is substantially more cerebral activity, followed by dramatically improved retention.

Developing new routines
What does a new routine look like that would support learning with understanding and with coherence? Knowing that research on cognition and learning tells us that it’s not about dichotomies of teacher guidance or student-led learning, we shouldn’t polarize those. The key is the most productive sequencing of each—when to ask students to engage in thinking and when to step in as teachers to help them process their ideas. The role of the teacher is still very important, it’s just different. Teaching for understanding is about structuring opportunities for students to learn through problem-solving. This involves intentionally activating students’ thinking about a new situation that extends, refines or builds on their prior knowledge and experiences.

What that means is that learning through problem-solving is a strategy to promote making sense of new concepts or relationships within mathematics, not simply as a process of figuring out answers. And once students generate new ideas and insights from problem solving, they are cognitively ready for teacher-led instruction that will help them to organize, refine and better understand how this new learning connects to prior knowledge.

This process can be supported through a three-phase routine: “think-share-compare.” During the “think” phase, students are given an opportunity to use prior knowledge and intuitive ideas for solving a problem that involves a new concept or relationship. As students “share” their thinking with peers, they recognize new connections, surface incomplete understandings, and begin to bridge gaps in knowledge. The teacher’s role during this time is to probe student thinking, posing prompts that push them to clarify their ideas and/or recognize incomplete understandings related to the new concept or relationship.

During the “compare” phase, the teacher uses intentionally selected examples of students’ thinking to focus attention on specific mathematical features in order to help clarify their understandings and to introduce new notation and terminology. At this time, after having been given time to generate their own thinking, students are cognitively ready to productively process this new information. The sequence closes with students being asked to articulate—orally and in writing—what was learned and how it connects to prior knowledge.

Supporting teacher growth
It is also important to support teachers in shifting their instruction to align with this new routine and the new expectations for student reasoning and sensemaking. The value of trying new teaching methods outweighs the risk of making a mistake. How can we support teachers’ growth mindset toward their own professional practice? How can we make it safe for teachers to take risks and try something new?

From the teacher’s point of view, it’s important to set reasonable, attainable goals for professional growth and to collaborate with others around these. Focus on continuing to improve, not on being perfect. Reflect on your practice often and accept setbacks as part of the learning process.

To watch this web seminar in its entirety, please visit: www.districtadministration.com/ws111516