“Engaging students” might be in the top five of most common phrases in mathematics teaching discussions, journals, and professional learning. Strategies to engage students have included cooperative learning, questioning strategies, and classroom discussions. What do these ideas have in common? Each is a way to position students as *active* learners. Students might be active through physical or social activity. In this blog post, we share *why *physical and social activity are important, including a collection of ideas for each, and then we share a model that combines such active learning with cognitive demand. Together, these two dimensions**, active learning** and **cognitive demand**, will help us “coordinate” effective mathematics learning experiences.

**Physical and Social Engagement**

To think about physical and social engagement mathematically, let’s look at it as a continuum on a number line:

For any topic we are teaching, being thoughtful about how to incorporate physical and social activity shifts the student experience to the right on the number line, increasing their physical and/or social engagement in the learning.

**Physical Engagement**.

Physical activity elevates the brain chemicals that affect thinking and learning (Erikson, Hillman, & Kramer, 2015; Jensen, 2008). Wow! So, how might we incorporate physical movement into mathematics teaching? Try these three simple shifts:

**1. From shoulder partner to ***Learning Partner. *

*Learning Partner.*

Rather than talk to the person they are sitting next to, assign a pre-arranged partner from somewhere else in the room. When you have an important question to pose, or a task ready to be compared, ask students to get up and have a stand up conversation with their learning partner.

**2. From table groups to ***Line-Up Groups. *

*Line-Up Groups.*

This is another way to get up and move, but in this situation students have a random group (or partner). Give students a fun way to line up. For example, from oldest to youngest, minutes to get to school, state/country they most want to visit (in alphabetical order), etc. Once lined up, you can form pairs, trios, or groups of four. This group can stand up, or find seats, and they can discuss a problem already solved (e.g., with their table group) or work together on a new task.

**3. From graphing using paper and pencil to ***Human Graphing*.

*Human Graphing*.

For activities involving number lines and the coordinate axis, consider going life-sized. A number line can be created with a long rope, clothes pins, and cards for values. The number line can be used in kindergarten, for physically placing numerals between 0 and 10, and grow to placing numbers between 0 and 1,000,000 and beyond. Fractions, decimals, and percentages can be hung illustrating equivalencies and comparisons. And, imagine placing variable expressions. Once *x *is placed, where might *2x *go? *2x* – 1? (See Bay, 2001 for more ideas). A coordinate axis can be taped down in the classroom, gym/cafeteria, or outside. Students can physically find their location on the axis. Hand students a rope and they can physically show examples and non-examples of functions (see Bay & Wasman, 2000 for more ideas).

*Please use the comment box below to share your ways to get students physically active during math class!!*

**Social Engagement.**

For decades, we have been working on effectively using cooperative groups and whole-class discussions. But, did you know that brain research has found that though *positive* social classroom interactions can *positively* influence the brain, *negative* social classroom interactions can have the reverse effect (Jensen, 2008; Lieberman, 2014). Here are some ideas for making social interactions a positive experience:

**1. Explicitly teach appropriate and inappropriate social interactions. **

This is more than generating a list of “Norms.” This includes role playing and coming up with examples and non-examples of the norms.

**2. In all small group efforts, make sure both ***individual accountability* and *shared responsibility* are addressed.

*individual accountability*and

*shared responsibility*are addressed.

Ways to address accountability include assigning roles within the group, having each student record their group’s work in their own journal/recording sheet, and establishing norms where anyone in the group may be asked to present. This third idea also works towards shared responsibility. Other ways to ensure shared responsibility is to have the whole group share in a presentation and to establish a norm where students must first ask each other for help before asking the teacher.

**3. Incorporate talk moves.**

In whole group discussions, incorporate talk moves (for example, time to think, say more, who can repeat, explain what someone else means, request for an example or counter-example), that ensure that each student’s contributions are included, understood, and valued (Chapin, O’Conner, & Anderson, 2013). Sample questioning prompts that encourage everyone’s contributions include:

- Do you understand what ___ is saying?
- Explain how [class member/group member] solved this ___ task.
- What do you think about what ___ said?
- Do you agree? Why or why not?
- Does anyone have the same answer but a different way to explain it?
- Which strategy might you want to use on a future problem (and why?)?

Remember that all of these ideas are focused on ensuring that the social engagement is *positive. *That means every child has to be positioned as competent. And, that is how we should reflect on whether or not our strategies for social engagement are working.

*Please use the comment box below to share your ways to support positive social interactions!*

**Cognitive Engagement**

We hope you haven’t stopped reading yet, because the strategies shared so far are just one dimension of learning—*active *engagement. But, engaged in *what*? Clearly we need to also focus on *cognitive* engagement. In mathematics teaching, we commonly refer to this as Levels of Thinking (Bloom’s Revised Taxonomy) (Krathwohl, 2002), Level of Cognitive Demand (Smith & Stein, 1998) or Depth of Knowledge (Webb, 2002). We need a second, vertical dimension for Cognitive Engagement:

Himmele and Himmele (2017) have combined student participation and cognition in their Total Participation Techniques (TPT) Cognitive Engagement Model into a coordinate axis with both of these number lines. We think such a model is just what we need to work towards lessons that will maximize learning for our students. In the graphic below, an adapted, mathematized version of the TPT, notice that Quadrant I is larger. That’s because this is where we want our lessons to be!

Imagine you are studying a lesson you are teaching tomorrow or next week. Ask yourself, ‘In which Quadrant does this lesson land?’ If it is in Quadrant II, ask, “How might I infuse some of the ideas from the physical and social engagement section above (or other ideas)? If the activity is in Quadrant IV, ask, “What ways can I raise the cognitive demand of this task or lesson?” If the activity is in Quadrant III, you may need to scrap it and find something that has more potential to help your students learn! (Questions adapted from McGatha & Bay-Williams, 2018). Of course, doing this questioning as part of a professional learning community, rather than on your own, can be more socially active and cognitively demanding for you! We hope that in either case, independently or collaboratively, you can use this model as a tool for supporting high quality mathematics learning experiences.

*Please share how you might use or have used this tool in your setting! *

**References **

Bay, J. M. (2001). Developing number sense ON the number line. *Mathematics Teaching in the Middle School, 6*(8), 448 – 451.

Bay, J. M., & Wasman, D. G. (2000). Making the coordinate grid come to life with human graphing. *Mathematics Teacher*, *93*(7), 553-554.

Chapin, S., O’Conner, C., & Anderson, N. C. (2013). *Talk moves: A teacher’s guide for using classroom discussions in math *(3rd ed.). Sausalito, CA: Math Solutions.

Erikson, K. I., Hillman, C. H., & Kramer, A. F. (2015). Physical activity, brain, and cognition. *Current Opinion in Behavioral Sciences, 4*, 27-32.

Himmele, P. & Himmele, W. (2017). *Total participation techniques: Making every student an active learner*, 2^{nd} edition. Alexandria, VA: ASCD

Jensen, E. (2008). Brain-based learning: The new paradigm of teaching. Thousand Oaks, CA: Corwin.

Krathwohl, D. R. (2002). A revision of Bloom’s Taxonomy: An overview. *Theory Into Practice**, 41*(4), 212–218.

Lieberman, K. (2014). Difference not deficit: Reconceptualizing mathematical learning disabilities. *Journal for Research in Mathematics Education, 45*(3), 351-396.

McGatha, M. B., & Bay-Williams, J. M. (2018). *Everything you need to know for mathematics coaching: Tools, plans, and a process that works for any instructional leader. *Thousand Oaks, CA: Corwin.

Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. *Mathematics Teaching in the Middle School, 3*(5), 344-350.

Webb, N. (March 28, 2002) “Depth-of-Knowledge Levels for four content areas.” unpublished paper.