The Magic of Mistakes: Are Correct Answers Really Necessary?

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Tony Jones, Mahomet, IL  anthonyjones@cpm.org

Mathematics and science are two areas of academic concentration where the correct answer is a major focus for those involved. These courses seem to have the least subjectiveness of all the academic pursuits in k–12 education.

Allow me to state that I do believe we all, as educators, want to see our students be successful. We want them to achieve at a high level and demonstrate a thorough understanding of concepts and skills. Unfortunately, this focus leads us to do things that we feel maximize student performance because it is easier to discern and measure. To put it bluntly, we too often focus on the correct answer. We make increasing a student’s performance an acceptable proxy for learning. And in doing so, we can lose sight of the beauty in the dance of education in general, and in mathematics in particular.

I recently was introduced to a dichotomy that I had often experienced but never named explicitly: mimicking versus thinking. Too much of what we see in classrooms is mimicry as students memorize multiplication tables, algorithms, steps for solving problems, rules, and theorems. We ask students questions about what we have just been teaching, and then, when they hesitate, we say, “Talk to your partner,”  “Look in your books,” or “Check your notes.” Other times, we simply point to clues we have written on the board or placed on the walls throughout our room. We prop up performance with prompts and then take mimicked responses as evidence of learning.

I am not advocating for a world with only wrong answers where we have no regard for the correct answers. I am, however, advocating for a world where students (and, even more so, educators) are less obsessed with being correct and much more focused on the process of understanding what it means to think through and solve problems.

Learning something new can be confusing and unsafe for many students, so students often resort to mimicry; it is a refuge. When we mimic, we find a place we can occupy where we don’t look foolish. Many students are skilled at mimicry. They are good at knowing what we want and presenting it to us in a way that looks like learning.

However, we must realize that critical thinking is THE pathway for students. Colin Sale sums it up in his book, Thinking Like a Lawyer: A Practical Framework to Teach Critical Thinking to All Students. He says, “When we don’t give our students opportunities to engage in critical thinking at all levels, we are systemically leaving brilliance on the table.”

Critical thinking is perhaps the main cog in the educational wheel for students (dare I say, it is critical!). When we choose to take the focus off the correctness, we make equity real at the classroom level; we give students the opportunity to lead, innovate, and take risks with novel ideas, which is a core part of their educational experience. We must help students see (and experience) that mistakes are a part of learning. This will never be possible in a world where students are afraid of getting the wrong answer

One caveat: I think it is important that we do not make a blanket statement that correct answers do not matter. We understand that people — administrators, parents, students — could hear that in unintended ways. But, we make it clear that if our only focus is the correctness of an answer, then we lose out on the reasoning that helps students get more sophisticated in their mathematical thinking and thus deepen their understanding.

So, how do we capitalize on the “magic of mistakes” to help students and make our classroom a place of critical thinking and mathematical reasoning? My colleagues are fans of lists, as it often helps us see things more clearly. Here are practical ideas wrapped up in a nice, concise list:

  1. Create a classroom where mistakes are magical.
  2. Anticipate “good” mistakes ahead of time and prepare responses for even the most baffling of mistakes.
  3. Use probing questions for those good mistakes in real-time.
  4. Use error analysis regularly. However, at times, do not tell them there is an error but rather allow them to see it themselves. Instead of “Who is correct?” perhaps change the question to “Who do you agree with?” or “Do you agree with any of these answers?”
  5. Continually ask students to explain their reasoning to help them understand the focus is on their thinking.
  6. Ask students to create their own “good” mistakes.
  7. Give a few examples of wrong answers and then ask students which of the wrong answers is more right.

What is on your list? What do you do to support the power and magic of mistakes?

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Algebra Tiles Session

  • Used throughout CPM middle and high school courses
  • Concrete, geometric representation of algebraic concepts.
  • Two-hour virtual session,
  •  Learn how students build their conceptual understanding of simplifying algebraic expressions
  • Solving equations using these tools.  
  • Determining perimeter,
  • Combining like terms,
  • Comparing expressions,
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  • Use an area model to multiply polynomials,
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Foundations for Implementation

This professional learning is designed for teachers as they begin their implementation of CPM. This series contains multiple components and is grounded in multiple active experiences delivered over the first year. This learning experience will encourage teachers to adjust their instructional practices, expand their content knowledge, and challenge their beliefs about teaching and learning. Teachers and leaders will gain first-hand experience with CPM with emphasis on what they will be teaching. Throughout this series educators will experience the mathematics, consider instructional practices, and learn about the classroom environment necessary for a successful implementation of CPM curriculum resources.

Page 2 of the Professional Learning Progression (PDF) describes all of the components of this learning event and the additional support available. Teachers new to a course, but have previously attended Foundations for Implementation, can choose to engage in the course Content Modules in the Professional Learning Portal rather than attending the entire series of learning events again.

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Building on Instructional Practice Series

This series contains three different courses, taken in either order. The courses are designed for schools and teachers with a minimum of one year of experience teaching with CPM curriculum materials. Teachers will develop further understanding of strategies and tools for instructional practices and assessment.

Building on Equity

In this course, participants will learn how to include equitable practices in their  classroom and support traditionally underserved students in becoming leaders of their own learning. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for implementing Chapter 1 that creates an equitable classroom culture and curate strategies for supporting all students in becoming leaders of their own learning. Follow-up during the school year will support ongoing implementation of equitable classroom practices.

Building on Assessment

In this course, participants will apply assessment research to develop methods to provide feedback to students and to inform equitable assessment decisions. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

Building on Discourse

This professional learning builds upon the Foundations for Implementation Series by improving teachers’ ability to facilitate meaningful mathematical discourse. This learning experience will encourage participants to adjust their instructional practices in the areas of sharing math authority, developing independent learners, and the creation of equitable classroom environments. Participants will plan for student learning by using teaching practices such as posing purposeful questioning, supporting productive struggle, and facilitating meaningful mathematical discourse. In doing so, participants learn to support students collaboratively engaged with rigorous, team-worthy tasks with all elements of the Effective Mathematics Teaching Practices.