# Building Mathematics Learning through Inquiry Using Student-Generated Data: Lessons Learned from Plan-Do-Study-Act Cycles

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## Abstract

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## 1. Introduction

- How do PDSA cycles support pedagogical innovation in the classroom?
- How can reform-based teaching be transferred from theoretical ideas to classroom practice during full-time student teaching?

## 2. Background

#### 2.1. The PrimeD Framework: A PD Framework for Teacher Preparation

#### 2.1.1. Phase I Design and Development

#### 2.1.2. Phase II Implementation

The fundamental difference between an amateur and a professional in any field is not one of intelligence or willingness to work hard. Rather, it is that professionals are trained at accessing their own research field, and therefore are much less likely to spend time repeating the others’ prior mistakes. Educational reforms seem to have a less-than-glorious tradition of replicating major aspects of previous failed efforts.(p. 197)

#### 2.1.3. Phase III Evaluation

#### 2.1.4. Phase IV Research

#### 2.2. Reformed Teaching, Inquiry, and Constructivism

Asking questions; Developing and using models; Planning and carrying out investigations; Analyzing and interpreting data; Using mathematics and computational thinking; Constructing explanations; Engaging in argumentation from evidence; Obtaining, evaluating, and communicating information.(p. 42)

## 3. Methods

#### 3.1. PDSA Cycles in the NIC

#### 3.2. Data and Measures

## 4. Results

#### 4.1. Teaching Mathematics through Inquiry

#### 4.1.1. Example PDSA Lesson 1: Inquiry into Trigonometric Ratios

#### 4.1.2. Example PDSA Lesson 2: Exploration of Population Density

#### 4.2. Participation in Lessons Affected by PDSA Change Ideas

#### 4.3. Teacher Candidate Growth in Reformed Teaching

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Secondary Mathematics Challenge Space

**Figure A1.**Model of outcome relationships in a teacher education program [42]. Note: “Student” is used to refer exclusively to children in PreK-12 classrooms; “Teacher” refers to teacher candidates as well as PreK-12 classroom teachers.

#### Appendix A.2. Vision

#### Appendix A.3. Goals

- Subject Matter Knowledge. Teachers have a robust knowledge of mathematics, understanding how concepts and procedures are interrelated and how to frame mathematics knowledge in a meaningful way to help students learn (Mathematics Knowledge for Teaching).
- Pedagogical Content Knowledge. Teachers develop robust pedagogical knowledge to support deep mathematics learning in their classrooms, including the use of tools for teaching mathematics (Knowledge for Teaching Mathematics).
- Knowledge of Orientation. Teachers understand and respect the relevance of the affect of each member of a learning community (e.g., attitudes, culture, beliefs, values, confidence, and anxiety) in learning mathematics.
- Knowledge of Discernment. Teachers understand that discernment encompasses the connections between cognition, metacognition, and learning and decision-making processes. Knowledge of discernment includes understanding developmental processes and the socio-emotional and sociocultural components of learning.
- Knowledge of Individual Context. Teachers understand that learning and decision-making processes take place within the context of the intersectionality of social categories.
- Knowledge of Environmental Context. Teachers understand the importance of building an inclusive and equitable environment to support a robust learning community.

#### Appendix A.4. Teacher Orientation

#### Appendix A.5. Teacher Practice

- Culture. Teachers establish a culture of access and equity through classroom structures and culturally relevant pedagogy to support each and every student in learning and participating in mathematics deeply. These classroom structures empower students to value diverse perspectives by elevating their voices, providing leadership opportunities, and developing a strong learning community. Teachers model vulnerability, viewing mistakes as learning opportunities. Varied approaches are visible and valued.
- Active Engagement. Teachers actively engage students in learning mathematics and/or science with meaning.
- Conceptual Understanding. Teachers explicitly foster, model, and insist upon conceptual understanding and coherence for all learners at all levels as a primary means for promoting procedural understanding in mathematics. Teachers insist that all teaching activities and learning experiences embrace the development of conceptual understanding as the fundamental core of learning and form the foundation for peer discussions.
- Connections. Teachers structure lessons through a phenomena-first approach, recognizing that authentic contexts are the foundation of the lesson and frame the content to be learned. Contexts are not simply enrichment that happens after the “real” lesson if at all.
- Reasoning. Inquiry-based projects are incorporated in every unit. Quantitative reasoning is modeled as scientific inquiry (claim, evidence, rationale).
- Questioning. Questioning is purposefully crafted to foster higher-order thinking and alternative modes of thinking about mathematics. Teachers pose questions of their students and encourage their students to ask deep, rich questions about their mathematical reasoning and that of their peers.
- Assessment. The ability to provide students feedback through formative (ongoing) and summative (reflective) assessment is differentiated from and valued more than grades. Assessments are ongoing, are aligned to standards, and (in)form teacher practice. Teachers understand that assessment can take many forms including formative (ongoing) and summative (reflective) assessment. Teachers incorporate a variety of assessments to ensure that each and every student has an opportunity to express their current understanding, including, but not limited to, observations, student-to-student and student-to-teacher dialogue, projects, performance tasks, interviews, portfolios, presentations, exit slips, and dynamic technology-based activities. Teachers recognize that understanding develops over time and leverage opportunities to reassess throughout the learning process.

#### Appendix A.6. Student Outcomes

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**Figure 1.**Condensed Model of the PrimeD Framework (adapted from [16]).

**Figure 5.**Screenshot of the Trigonometric Ratio Geogebra app [54].

**Figure 7.**Changing the slider changes the measures of Angle A and Angle C and the trigonometric ratios [54].

**Figure 9.**Teacher candidate RTOP-E scores. Time 1 = Phase I internship, mid-November. Time 2 = Beginning of Phase II internship, late January. Time 3 = Second half of Phase II, late March.

Prompt | Response |
---|---|

Challenge or goal of this PDSA cycle. | Collaboration with data collection |

Context (e.g., grade level, course, topic) | 10th grade, geometry, density |

Expected duration of this PDSA cycle. (e.g., 10/15 min). | One lesson, modeling data collection will be at the beginning of the lesson |

Change idea or strategy for meeting your challenge | Model data collection before the students collect their own data |

Prediction(s)/hypotheses (What you think the change idea/strategy will accomplish?) | Modeling the data collection will show students how to complete procedures. Avoid confusion when starting the collaboration and data collection |

Evidence to collect | Student work and student ability to complete data collection on their own/with minimal help from the teacher |

PDSA Lesson (Inquiry through Trigonometric Ratios) | Comparison Lessons: No (and Percent) That Engaged in Any Work (at Least One Question) | |||||
---|---|---|---|---|---|---|

Class | No. Students | No. (and Percent) That Completed Independent Work | No. (and Percent) That Engaged in Any Work (at Least One Question) | Lesson 1 before PDSA Lessons | Lesson 2 after PDSA Lessons | Lesson 3 after PDSA Lessons |

1 | 22 | 8 (36.4) | 13 (59.1) | 2 (9.1) | 2 (9.1) | 3 (13.6) |

2 | 27 | 18 (66.7) | 21 (77.8) | 9 (33.3) | 12 (44.4) | 18 (66.7) |

3 | 29 | 15 (51.7) | 16 (55.2) | 3 (10.3) | 10 (34.5) | 5 (17.2) |

4 | 31 | 19 (61.3) | 20 (64.5) | 3 (9.7) | 12 (38.7) | 15 (48.4) |

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## Share and Cite

**MDPI and ACS Style**

Rakes, C.R.; Wesneski, A.; Laws, R.
Building Mathematics Learning through Inquiry Using Student-Generated Data: Lessons Learned from Plan-Do-Study-Act Cycles. *Educ. Sci.* **2023**, *13*, 919.
https://doi.org/10.3390/educsci13090919

**AMA Style**

Rakes CR, Wesneski A, Laws R.
Building Mathematics Learning through Inquiry Using Student-Generated Data: Lessons Learned from Plan-Do-Study-Act Cycles. *Education Sciences*. 2023; 13(9):919.
https://doi.org/10.3390/educsci13090919

**Chicago/Turabian Style**

Rakes, Christopher R., Angela Wesneski, and Rebecca Laws.
2023. "Building Mathematics Learning through Inquiry Using Student-Generated Data: Lessons Learned from Plan-Do-Study-Act Cycles" *Education Sciences* 13, no. 9: 919.
https://doi.org/10.3390/educsci13090919