Before I started planning math units using the Understanding by Design framework, I frequently found myself stuck on the bells and whistles of day-to-day activities. I worked hard to make math fun and engaging. Sometimes I’d incorporate a cool poster. At other times, I’d bring in an engaging video to hook my students. I even created skits to help them grapple with new concepts. While math was certainly fun, my lessons were still deeply rooted in rote memorization. I could see that merely making a math lesson engaging was not enough to promote deep understanding, but I had yet to find a consistent method to drive understanding. 

A big breakthrough for me came in the form of UbD backwards design, and as I learned and adopted its framework, I reevaluated my intended goals for students as whole mathematicians. I knew that I did not want students merely memorizing steps — this might set them up for success within a single lesson, but not as lifelong learners — I wanted students to feel confident, capable, and brave in taking on new mathematical concepts. Thus, with the guidance of a fantastic co-teacher, I turned to Jo Boaler’s literature on the “growth mindset” to form my enduring understandings. 

For my fourth-grade math unit on large-number division, I started at the end, brainstorming what I wanted students to understand and be able to do as a result of what they were learning in my classroom. Of course, the obvious understanding was decomposing numbers based on place value through terminologies such as quotient, dividend, and divisor. Yet, I believed my students could go deeper. I wanted them to understand what the divisor was doing to the dividend and to remain flexible and open to the many ways division can occur. I also wanted my fourth-graders to feel comfortable exploring their newfound understandings by teaching others, writing their own problems, as well as intentionally and unintentionally making their own mistakes. Thus, my enduring understandings became:

  • There can be more than one way to solve a problem. Understanding and using a range of strategies can help us work more effectively and efficiently.
  • Making mistakes is natural — when we let ourselves learn from them, we can understand more deeply.

With these two EUs in mind, I could design lessons and choose day-to-day activities that weren’t merely fun and engaging but also helped students construct meaning and understand more deeply. For example, I frequently presented students with incorrect division problems to fix. I would make a common mistake based on misunderstandings I’d seen in class and ask them to tell me what was wrong and why. Students were significantly more excited to correct my errors than their own and eagerly participated in helping me to learn. From there, I asked the mathematicians to intentionally make a common mistake on a long-division problem and ask a partner to catch and correct it. This normalizing of mistakes translated quickly into celebrating and learning from real errors as they occurred. As the class began to feel more confident making mistakes, they volunteered more in front of their peers, took bigger risks tackling large numbers, and went back to correct their own work with less shame or embarrassment. Additionally, working with various strategies for solving long division assisted their learning process and allowed students to check their work in multiple ways or move to a new approach if they became stuck. Through Boaler’s emphasis on learning from mistakes, I had restructured my enduring understandings to consistently and intentionally incorporate mistakes and promote flexible thinking, and I saw it pay off. 

Enduring Understandings helped me shift the ultimate goals of my unit from mastery of skills and students producing “correct answers” to their having deep understanding and skills that they could apply in the unit and beyond — the learning became relevant both during the lesson and long into the future. Enduring Understandings broadens the lens of what it means to be successful, and by shaping units around the end goal of flexibility and growth, educators can take the most fact-based curricula (such as operations or algorithms in math) and deepen student learning beyond memorization. As with all backwards design, we must first look at what we want students to take away from our units. For me, becoming a fearless mathematician is about as good as it gets. 

Sawyer Henshaw (she/her) is a freelance writer based in Southern California. She holds a Bachelor’s Degree in English from Scripps College and a Master of Arts in Teaching from the University of San Francisco. Sawyer has worked in education for the past five years, teaching in kindergarten, third, fourth, and sixth-grade classrooms across California. 

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