As a mathematics teacher I have often struggled to help my students feel inspired and motivated to do mathematics. Some students are unmotivated because traditionally they have not been successful in mathematics. This lack of motivation often contributes to behavioral problems and a decrease in academic achievement.

In the schools where I have worked, on-grade level courses are for "non-advanced" students and tend to be designed in a more traditional manner, one in which students have few freedoms and typically do not have a voice in the ways in which they are learning. Self-Determination Theory (SDT) states that a person will display more signs of happiness and increased self-motivation if they have a perception of support for their autonomy as opposed to being controlled (Chirkov & Ryan, 2001).

With a more controlled environment taking place in many public schools in the United States, there is a power struggle that must be addressed to move towards a more autonomous-supportive classroom (Cook-Sather, 2002). In an effort to support students' autonomy and increase their motivation I decided to look at ways to allow students to have input in their learning. My goal was to examine students' perceptions of various mathematics lesson formats.

#### Lesson Formats

To gain these insights, I posed these questions following the implementation of each of three different lesson formats:

- How well do you feel you understood today's task?
- Did you enjoy learning through tiered instruction (or inquiry or direct instruction)?
- If this lesson was taught again, how could it be improved to make it more interesting or to help your learning?
- What is your favorite way to learn in mathematics class?

Each of these prompts were given to all six of my eighth grade classes after teaching lessons using the tiered, inquiry, and direct instruction lesson formats. I also engaged students in whole-class discussions to reveal their perceptions. Let's take a look at the three different lesson types implemented: tiered, inquiry, and direct instruction.

#### Tiered

A tiered lesson is a differentiated lesson based on students' needs. The tiered lesson I taught was aligned to the Common Core State Standards for Mathematics (CCSSM) (CCSSI, 2010, ) standard for eighth grade that focuses on using informal arguments to establish facts about the interior angle sum and exterior angles of triangles (8.G.5).

Students were given a pre-assessment, which was used to form four student groups (i.e., levels 1-4) Each group consisted of four to five students and each student was provided their own laptop. Student groups were instructed to complete four tasks related to the standard.

The level four group was comprised of students who demonstrated an understanding of the learning goal, "Students will be able to use informal arguments to establish facts about the angle sum and exterior angles of triangles." Therefore, the level four group's work was designed as a quick review followed by an extension. The goal for these students was to create their own examples and work through proofs of both the Exterior Angle Theorem and the Triangle Angle Sum Theorem.

Students grouped in level three took guided notes from a video and completed practice problems that focused directly on the learning goal with an extension on the Exterior Angle Theorem. Level two students were given examples and practice problems that were scaffolded to provide an opportunity to gain understanding of the angle relationships of a triangle. Level one students were placed in a group where the teacher provided the instruction, providing opportunities for students to work one-to-one with teachers.

#### Inquiry

The second lesson format presented to students was an inquiry lesson which used a more student-centered approach. Inquiry was described to students as a learning process where students pose and investigate questions to discover new ideas. This lesson focused on solving real-world and mathematical problems involving volume of cylinders, cones, and spheres (CCSSM 8.G.9) using "The Coca Cola Problem" by Dan Meyer (see http://blog.mrmeyer.com/).

Students began by watching a short video clip in which someone fills a pool with soda, and students then posed different questions about the video. From the questions created, students were tasked with determining the number of soda bottles it would take to fill a swimming pool. Students worked collaboratively to determine the information needed to solve the problem. After brainstorming ideas, students were given the diameter and height of the pool and how much one bottle of soda could hold. Students used this information and worked in teams to figure out how many bottles of soda could fill up the pool.

Students discovered that they needed to find the volume of the cylindrical shape pool, so a discussion on how to find the volume of a cylinder ensued. Eventually students decided to use the volume formula for a cylinder to solve the problem.

#### Direct Instruction

The direct instruction lesson can be described as a teacher-centered approach in which students are presented with facts and steps, and they used this information to answer prescribed questions and problems. The direct instruction lesson focused on solving real-world and mathematical problems involving volume of spheres (CCSSM 8.G.9). This lesson began with a video that helped students relate the volume of a sphere to the volume of cones and cylinders.

While I modeled finding the volume of a sphere in two different ways, students took guided notes. After a few examples students tried some problems on their own and then with their teams.

#### Students' Perceptions of the Lessons

Student feedback on the three lesson formats was interesting. The majority of students stated that they preferred to learn on individual laptops because they liked the opportunity to work at their own pace and pause the video. One student said "Tiered (lessons) may be my new favorite way to learn. It was very helpful with this lesson."

Many students who enjoyed working through the tiered lesson liked the opportunity for collaboration as well. Students shared that they enjoyed discussing ideas with their team when they were stuck. Generally, students preferred lessons that included technology and opportunities to work as a group.

However, there was a portion of students who shared that they would rather experience direct instruction and have the opportunity to work independently or closely with the teacher.

Overwhelmingly, students preferred lessons that are engaging in some way such as using whiteboards, watching videos, or simply incorporating music. During the inquiry lesson, students were challenged to think critically and some students "got it" right away, but many did not. While students wanted to be rescued, I instead posed questions to help them persevere through the lesson.

#### Lessons Learned and Tips for the Classroom

In trying out different lesson formats, I discovered that students enjoyed using laptops for instruction. Doing so allowed me to differentiate lessons and work one-on-one with students. I also know now I need to include more tasks that encourage students to work as a team while also using computers. Asking these questions of my students helped me to learn about the lesson formats students prefer, what they find helpful, and how they feel they learn best. Below are some tips for teachers interested in incorporating student choice in the classroom:

- Allow students to provide input on your instructional design during the beginning of the school year when getting to know students.
- Design questions that focus on different lesson formats so you can get meaningful feedback from your students.
- Select tasks for your lessons that provide an entry point for each and every student—otherwise students may become frustrated and little productive struggle or success will occur.
- When seeking student input, present different forms of lessons that are proven effective in the classroom.

#### Conclusion

Allowing students opportunity for choice in the classroom empowers them to make decisions regarding their learning. Doing so increases their feeling of autonomy. As Brooks and Young (2011) suggest, teachers hold a vital role in supporting students' motivation to complete a task, which demonstrates the need for a more autonomous supportive classroom.

Obtaining candid feedback from my students has allowed me to reflect on what went well and what did not go well in my lessons. Students made great recommendations on ways that I could improve. Moving forward I will follow-up with students and begin formatting lessons to include students' preferences, while maintaining quality and rigor.

I also learned the importance of giving students the opportunity to provide input on their learning, which helped them realize that they play an active role in their own learning. With a voice, students seemed to be more motivated to complete a task. This aligns with the idea that intrinsically motivated people are found to exhibit an increased level of perseverance, creativity, vigor, self-confidence, and success (Deci & Ryan, 2001).

Providing opportunities for students to participate in the decision-making process in regards to their learning can have a great impact on student engagement and motivation (Ferguson & Braxton, 2011, p. 55).

In closing, I advocate that it is just as important to differentiate based on student learning preferences as it is to differentiate based on achievement because sometimes students' needs are not just about scaffolding ideas but also about how the topic is presented.

#### References

Brooks, C.F., & Young, S.L. (2011). Are choice-making opportunities needed in the classroom? Using self- determination theory to consider student motivation and learner empowerment. *International Journal of Teaching and Learning in Higher Education, 23*(1), 48-59.

Chirkov, V.I., & Ryan, R.M. (2001). Parent and teacher autonomy-support in Russian and U.S.: Adolescents common effects on well-being and academic motivation. *Journal of Cross-Cultural Psychology, 32*(5), 618-635.

Cook-Sather, A. (2002). Authorizing students' perspectives: Toward trust, dialogue, and change in education. *Educational Researcher, 31*(3), 3-14.

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. *Common Core State Standards (College- and Career-Readiness Standards and K–12 Standards in English Language Arts and Math)*. Washington, D.C.: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org.

Deci, E.L, & Ryan, R.M. (2001). The "what" and "why" of goal pursuits: Human needs and the self-determination of behavior. *Psychological Inquiry, 11*(4), 227-268.

Ferguson, D., Hanreddy, A., Draxton, S. (2011). Giving students voice as a strategy for improving teacher practice. *London Review of Education, 9*(1), 55-70.

*Michelle Burroughs is an eighth grade mathematics teacher at Millennium Middle School, Orlando, Florida.*

Michelle_burroughs@scps.k12.us

Published January 2019.