At reDesign, our approach to math education is rooted in a set of research-supported theories and practices that best prepare students to understand enough math to be able to critically engage with numbers in the world and to be in a position to pursue advanced math if they choose to do so. We developed a Discovery-based Mathematics Learning Experience Planner that pulls these theories and practices together.

**Why Discovery-based? **

There is not a clear consensus in the field that defines the difference between discovery-based learning and inquiry based learning. In some instances, they are used interchangeably, sometimes discovery is defined as a form of inquiry and others inquiry is defined as a form of discovery. Some have described inquiry as more formalized or structured while discovery is more open and less structured. As we thought about reDesign’s approach to mathematics it was important for us to settle on a definition and a distinction between the two learning theories. Hanum (2018) wrote, “Learning activities using discovery method (discovery) is actually similar to inquiry (inquiry). Inquiry is the process of answering questions and solving problems based on facts and observations, while discovery is finding concepts through a series of data or information obtained through observation or experimentation” (p. 4). Ultimately, we settled on defining our planning tool as discovery-based because a key component of our approach to mathematics is conceptual.

**Relationship Between Concepts and Procedural Fluency**

Traditional math instruction consisted of the teacher modeling a procedure (process) for students and then students practice using that procedure. As our society has changed, it became clear that the future “requires individuals to be able to critically think, problem solve, and adapt to new environments by utilizing transferability of ideas” (Wathall, 2016, p.2). This transferability of ideas in math is conceptual. There is a false dichotomy that math is either procedure or conceptual. Our approach doesn’t replace procedural with conceptual, but through discovery-based math students are able to see how procedures flow from the understanding of the concept. Wathall (2016) explains, “Mathematics *can *be taught from a purely content-driven perspective. For example, functions can be taught just by looking at the facts and content; however, this does not support learners to have complete conceptual understanding. There are also processes in mathematics that need to be practiced and developed that could also reinforce the conceptual understandings. Ideally it is a marriage of the two, which promotes deeper conceptual understanding” (p.4). Designing math instruction that is conceptual and discovery-based, rather than solely procedural, better prepares students for higher-level math.

**Role of Modeling and Explicit Skill Instruction **

Critics of discovery-based learning wrongly assume that students are left all on their own to figure everything out. While discovery-based mathematics does ask teachers to open up tasks to allow multiple pathways, methods, or representations (Boaler, 2016) it does not abandon students without scaffolds and supports necessary to help when they get stuck. Discovery-based mathematics is designed to allow for productive struggle, where teachers pose the problem first before teaching a procedural method (Lynch, et al, 2018). Teachers act as diagnosticians, identifying what is causing students to be stuck and select from the varying degrees of scaffolds to best support students while also ensuring they continue to do the heavy lifting of critical thinking and problem solving.

Unlike traditional math teaching where teachers often modeled a process or procedure, modeling and explicit skill instruction in discovery-based mathematics is more often around a set **Math Habits of Mind** that prepare students to think like mathematicians. Similar to literacy strategies for reading, these math habits of mind provide students with reference points and language to discuss their thinking process. “A curriculum organized around habits of mind tries to close the gap between what the users and makers of math *do* and what they *say. * Such a curriculum lets students in on the process of creating, inventing, conjecturing, and experimenting.” (Cuoco, Goldenberg, & Mark, 1996, p.376). These habits and essentially, the metacognitive reflection on the use of them help students to develop genuinely mathematical ways of thinking.

**Structure of the Mathematics Learning Experience Planner**

The Discovery-based Mathematics Learning Experience planner is aligned with the Learning Cycle (Rudenstine, et. al., 2017) with a launch, investigation, synthesis, and reflection. Within that cycle, there are seven segments (engage, question, explore, instruct/model, extend, synthesize, and reflect). The flexibility of the instruct/model segment to shift based on the task and student needs is an essential feature of this planning tool.

Embedded within multiple segments are key beliefs and practices around mathematical mindsets, educational equity, and metacognition. The planning tool provides a structure for designing lessons but also lays a theoretical foundation for the shifts in teacher beliefs and practices around their role in a discovery-based mathematics classroom.

**References & Resources**

*An Inquiry Maths lesson – Inquiry Maths*. (2020). Inquiry Maths.

http://www.inquirymaths.com/home/an-inquiry-lesson

Artigue, M., & Baptist, P. (2012). *Inquiry in Mathematics Education: Background*

*For Implementing Inquiry in Science and Mathematics at School* [E-book]. The Fibonacci Project. https://www.fondation-lamap.org/sites/default/files/upload/media/minisites/action_internationale/inquiry_in_mathematics_education.pdf

Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). Does discovery-based instruction enhance learning? *Journal of Educational Psychology, 103*(1), 1–18.

Boaler, J. (2016). *Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. *Jossey-Bass.

Bruner, J. (1961). The act of discovery. *Harvard Educational Review, 31, 21-32. *

Cuoco, A., Goldenberg, E.P., Mark, J. (1996). Habits of mind: An ongoing principle for mathematics curricula. *Journal of Mathematical Behavior. *

Dweck, C.S. (2006). *Mindset: The new psychology of success. *Ballantine Books.

Department of Agriculture Education and Communication, Warner, A., & Myers, B. (2017, February). *Implementing Inquiry-Based Teaching Methods* (No. AEC395). Institute of Food and Agricultural Sciences. https://edis.ifas.ufl.edu/pdffiles/WC/WC07600.pdf

Elliott, S.N., Kratochwill, T.R., Littlefield Cook, J. & Travers, J. (2000). *Educational psychology: Effective teaching, effective learning (3rd ed.)*. Boston, MA: McGraw-Hill College.

Flavell, J. H. (1979). Metacognition and cognitive monitoring: A new area of cognitive–developmental inquiry. *American Psychologist, 34*(10), 906–911

Freire, P. (1972). *Pedagogy of the oppressed*. New York: Herder and Herder.

Hammond, Z. L. (2014). *Culturally Responsive Teaching and The Brain: Promoting Authentic Engagement and Rigor Among Culturally and Linguistically Diverse Students* (1st ed.). Corwin.

Hanum, N. (2018). The difference student learning outcomes using discovery learning and inquiry learning in elementary school. *International Journal of Education, 6*(1), 1-9.

Katz, J. D. (2014). *Developing Mathematical Thinking: A Guide to Rethinking the Mathematics Classroom*. Rowman & Littlefield Publishers.

Lynch, S.D, Hunt, J.H. Lewis, K.E. (2011). Productive struggle for all: Differentiated Instruction. *Mathematics Teaching in the Middle School, *23 *(4) *195-201.

Ozsoy, G. (2011). An investigation of the relationship between metacognition and mathematics achievement. *Asia Pacific Educational Review, *12, 227-235.

Rosa, M. & Orey, D.C. (2011) Ethnomathematics: the cultural aspects of mathematics. *Revista Latinoamericana de Etnomatematica, *4(2), 32-54.

Rudenstine, A., Schaef, S., & Bacallao, D. (2017, June). *Meeting Students Where They Are*. National Summit on K-12 Competency-Based Education. http://www.aurora-institute.org/wp-content/uploads/CompetencyWorks-MeetingStudentsWhereTheyAre2.pdf

Sagi, Y., Tavor, I., Hofsetter, S., Tzur-Moryosef, S., Blumenfeld-Katzir, T., & Assaf, Y. (2012). Learning in the Fast Lane: New Insights into Neuroplasticity. *Neuron, 73*(6), 1195-1203.

Wathall, J. (2016). *Concept-Based Mathematics: Teaching for Deep Understanding in Secondary Classrooms (Corwin Mathematics Series)* (1st ed.). Corwin.

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