His mohawk flopped side to side as he pointed rapidly at his scrawling mathematics work. My 6th grader was having a hard time explaining his thought process. “It’s like, I started with this idea, and tried this formula with these numbers, but then the math didn’t work out, and I wasn’t sure what this number was for, so I just gave up.”

Sound familiar? My students routinely struggle to verbalize their thinking in the mathematics classroom. As a result, one goal of mine is to assist students in revealing their thought process. I’ve grown in this practice by using thinking routines to elevate discussions, reflections, and formative assessments. Thinking routines normalize the practice of making thinking visible, growing student capacity to name “thinking moves” (Harvard Graduate School of Education, 2022a). Here are three thinking routines I regularly use in my mathematics classroom to elevate student thinking:

- Notice? Wonder? Next?
- I used to think… Now I think….
- Parts, Purposes, Complexities

**Notice? Wonder? Next?**

The Notice? Wonder? Next? thinking routine asks students to share, “What do you notice? What do you wonder?” These questions provide students a starting point for exploring their curiosities about a picture, word problem, problem-solving task, graph, and so forth. The National Council of Teachers of Mathematics (NCTM, 2023) provides a resource page with articles, examples, and teaching aids to support the “Notice and Wonder” routine.

In my classroom, I often need one more step to help students fully utilize the Notice and Wonder routine, and that is by asking students, “What’s next?” Now that we have noticed and wondered, what should we do mathematically? How do we solve the problem or question we have posed? This prompting of “What’s next?” places students in the driver seat of their problem-solving capabilities.

**I used to think… Now I think…**

As we continue to emphasize growth mindset in the classroom, the I used to think… Now I think… thinking routine normalizes change in our understanding of topics (Harvard Graduate School of Education, 2022b). In this thinking routine, students spend time reflecting on their initial conceptions of a topic, naming how their understanding has shifted or grown with deeper exploration.

During a lesson exploring polygons, middle school students shared these reflections:

- I used to think polygons had all equal angles. Now I think polygons can have angles of different measurements.
- I used to think I needed to memorize formulas to find the area of every polygon, which I didn’t think I could do. Now I think I can break a polygon into triangles and rectangles to find the area.

Having students examine how their thinking has changed creates flexibility and adaptability in learners, an important skill for any mathematician.

**Parts, Purposes, Complexities**

The Parts, Purposes, Complexities thinking routine encourages students to carefully observe an object or system to identify connections between the parts and the whole (Harvard Graduate School of Education, 2022c). This might include a detailed graph, data set, table, function, formula, or word problem. This routine asks students to examine each of the three questions:

- What are its parts?
- What are its purposes?
- What are the complexities?

A high school Algebra I teacher used this routine when introducing the point-slope form of the equation of a line {*y *– *y*_{1} = *m*(*x *– *x*_{1})}. Students noticed “the parts” or the variables in this equation, examining the operations occurring between variables to determine the purpose of the equation. One student shared, “What’s making this complex for me is the difference between ‘*y*’ and ‘*y*_{1}’. I don’t understand how you know where to put the number and where to just leave the *y* alone. Does it matter?”

By exploring this thinking routine, students examined the *why* and *how* behind this formula, leading to deeper mastery of this topic versus surface-level memorization. This routine also teaches the general skill of analysis, which in general means to break a complex topic down into simpler parts to gain a sense of the whole.

**Putting Thinking Routines to Work**

Using thinking routines in the mathematics classroom promotes a deeper relationship with the content at hand.

Thinking routines center the learning on student construction of knowledge, developing a classroom culture where students drive their learning. Thinking routines serve as a scaffold to help all students achieve success. As you move toward more student-centered classrooms, consider how you might use a thinking routine as part of a problem-solving task, performance task, or project-based learning (PBL) experience (McHugh, 2023).

**References **

Harvard Graduate School of Education. (2022a).* Project Zero’s thinking routine toolbox.* Project Zero. https://pz.harvard.edu/thinking-routines

Harvard Graduate School of Education. (2022b). *Thinking routine: I used to think… Now I think..*. Project Zero. https://pz.harvard.edu/resources/i-used-to-think-now-i-think

Harvard Graduate School of Education. (2022c). *Thinking routine: Parts, purposes, complexities*. Project Zero. https://pz.harvard.edu/resources/parts-purposes-complexities

McHugh, Maggie. (2023). *Bring Project-Based Learning to Life in Mathematics, K-12. *Thousand Oaks, CA: Corwin Press.

National Council of Teachers of Mathematics (2023). What is Notice and Wonder? Classroom Resources: https://www.nctm.org/noticeandwonder/