Let’s say the goal of a math lesson on adding multi-digit numbers with regrouping is for students to regroup ones and tens. The solution has already been stated. There’s no room for student discovery, taking away the “ah-ha” moment when they figure out how to regroup through trial and error. There’s no room for experimentation, making mistakes and revisiting strategies to find a solution.
Dr. Dixon said teachers should target the learning goal as an instructional guide to support students as they muddle through a process, and think about the questions students should answer at the end of the lesson based on the goal. But they should not give away the answer from the get-go.
2. Gradual release of responsibility
“Sometimes,” Dr. Dixon explained, “we talk about gradual release of responsibility as I do, we do, you do, and so if that’s expected in every lesson every day, you should be asking this important question: Is that appropriate in mathematics? And your answer should be no, it’s not in every lesson every day.”
Take the regrouping lesson. If the process is modeled, then the teacher is telling students how to solve the problem. They are not doing the sense making necessary to take on the challenge.
Gradual release of responsibility is not a no-no across the numeracy board. Dr. Dixon said that it is productive for teaching a procedure, such as long division. It’s what learners will then use to solve problems and make sense of the process, allowing them to build their discovery as they test different solutions.
Dr. Dixon recommended rethinking the purpose of scaffolding, typically used to differentiate learning. She explained that typically, it is a just-in-case measure to ensure students limit errors. She argued this is counter to what mathematical reasoning requires: the opportunity to make sense of context and determine the operation to be performed. Making errors is crucial to that inquiry.
Just-in-case scaffolding creates issues of access and equity, Dr. Dixon explained in her blog post Providing Scaffolding Just in Case, “When scaffolding is provided before students have the opportunity to make sense of a challenging task without the extra help, students are inhibited from developing productive perseverance. If this sort of scaffolding is provided for students who struggle, then these same students are denied access to cognitively demanding tasks. When access is denied, equity becomes an issue.”
Teachers should instead provide just-in-time scaffolding when students need it, not to limit errors. Actually, they should elicit student errors to determine where they need support and guidance.
Just-in-time scaffolding helps to develop productive perseverance, explained Dr. Dixon, by allowing students to engage in demanding tasks. Teachers then support them to maintain their engagement even when they struggle.
4. Academic vocabulary
Here’s a math problem for you: Brandon shared four cookies equally between himself and his four friends. He started by giving each person, including himself, half a cookie, and then he shared the rest equally. How much of a cookie will each person get?
Yes, this is an exercise in fractions. Maybe it’s not recognizable because specific terms are absent: numerator, denominator, top number, bottom number. That’s how teachers typically work with fractions, introducing academic vocabulary that takes away from the conceptual problem solving that is truly at the heart of mathematical reasoning.
Using everyday language, explained Dr. Dixon, allows students to describe in their words what the process is to demonstrate understanding. She advised allowing students to first do the sense making by solving tasks and then naming the concept with academic vocabulary. This approach makes it easier for teachers to recognize errors students make and how they arrived at them, presenting just-in-time instructional opportunities.
5. Neglecting to connect concepts and procedures
Dr. Dixon noted that it is essential to teach concepts before procedures. But concepts alone do not make for mathematically minded students.
Once teachers introduce conceptual practices, procedures must follow. It’s crucial to be explicit and intentional about connecting these, urged Dr. Dixon, pointing to multi-digit addition with regrouping as an example. What’s critical is connecting the regrouping concept to the standard algorithm that students will need for future problems.
Revisit the cookie allocation exercise using everyday language, explained Dr. Dixon. “We had students [explaining] what they would do using their language. To connect to the procedure, we would talk about what’s happening when we break these cookies all up into the same size pieces.” Ultimately, what the students did was name the common denominator without academically labeling it.
This example highlights how teachers encourage students to move their conceptual understanding to core mathematical procedures.
6. Small groups
Having students work in small groups is standard practice. Simply dividing students into teams of three to four is not sufficient. Groupwork should be purposeful.
If structured and facilitated appropriately, small groups can substantially strengthen students’ problem-solving skills. Small groups, heterogeneously constructed, should give students opportunities to talk about mathematics, enhance understanding, support each other in making sense of problems, explain and justify their answers to each other and share when they do not understand or disagree with what others state.
The remedy? TQE
Dr. Dixon recommended the strategy Tasks, Questioning, and Evidence (TQE) that frames mathematics instruction in ways that help teachers avoid unproductive strategies. The method helps teachers effectively plan as they select tasks aligned with learning goals, prepare questions to elicit common math errors and understandings and use evidence of student understanding to guide instruction and approaches to it.
Follow Dr. Juli Dixon on Twitter @thestrokeofluck.
About the presenter
Juli Dixon is Professor of Mathematics Education at the University of Central Florida. Dr. Dixon is focused on improving teachers’ mathematics knowledge for teaching so that they support their students to communicate and justify mathematical ideas. She is a prolific writer who has published numerous books, textbooks, and articles. Dr. Dixon delivers keynotes and other presentations throughout North America. She is co-author of Houghton Mifflin Harcourt’s Into Math and Go Math for K-8 Mathematics and AGA and Integrated Mathematics for High School. She is also co-author of Solution Tree’s Making Sense of Mathematics for Teaching book and video series. Especially important to Dr. Dixon is the need to teach each and every student. She often shares her personal story of supporting her own children with special needs to learn mathematics in an inclusive setting. Dr. Dixon published A Stroke of Luck: A Girl’s Second Chance at Life with her daughter, Jessica Dixon.
About the host
Denise Singleton has over 25 years of experience as a mathematics educator and educational content marketer. Her conversational style and enthusiasm will engage and entertain you!
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Building Understanding in Mathematics is a free professional learning community on edWeb.net that provides a platform, advice and support in helping educators learn methods that help students build understanding in mathematics.
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