## Abstract

Classroom observations have become an integral part of research related to mathematics education. In this qualitative study, we describe the current state of the mathematics education field with regard to the use of classroom observation. The research question was: *How is classroom observation being used to measure instructional quality in mathematics education research*? In all, 114 peer-reviewed manuscripts published between 2000 and 2015 that involved classroom observation as part of an empirical study were examined using a cross-comparative methodology. Seventy (61%) did not use a formalized classroom observation protocol (COP), 21 (18%) developed their own COP, and 23 (20%) used a previously developed COP. Of the implemented COPs, 44% have published validity evidence in a peer-reviewed journal. We perceive the great variety of research approaches for classroom observation as necessary and potentially challenging in moving mathematics education forward with respect to research on instructional contexts.

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## Acknowledgements

We would like to share our sincere appreciation to Timothy Folger, Maria Nielsen, and Davis Gerber at Bowling Green State University, and Dan Chibnall at Drake University for their assistance throughout this project.

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## Appendices

### Appendix 1

Classroom observation protocol | Construct measured | Indicators | Typical study population | Validity evidence | References |
---|---|---|---|---|---|

Instructional Quality Assessment (IQA) | Academic rigor and accountable talk | Instructional tasks, task implementation, explanations of mathematical thinking and reasoning | K-12 mathematics instruction | Content, response processes, internal structure | Boston (2012a, b), Boston et al. (2015a, b), Boston and Smith (2009), Jackson et al. (2013), Schlesinger and Jentsch (2016), Schoenfeld (2013), Wilhelm and Kim (2015) |

Reformed Teaching Observation Protocol (RTOP) | Reform-oriented mathematics and science teaching (i.e., standards-based teaching, inquiry orientation, student-centered teaching practices) | Lesson design; lesson implementation; content; classroom culture | K-12 mathematics instruction | Content, response processes, internal structure | Boston et al. (2015a, b), Jong et al. (2010), Marshall et al. (2011), Peters Burton et al. (2014), Sawada et al. (2002), Schlesinger and Jentsch (2016) |

Mathematical Quality of Instruction (MQI) | Rigor and richness of mathematics present | Common core-aligned student practices; working with students and mathematics; richness of mathematics; errors and imprecision; classroom work is connected to mathematics | K-9 mathematics instruction | Content, response processes, internal structure, relationship to other variables | Boston et al. (2015a, b), Hill et al. (2012), Kapitula and Umland (2011), Schlesinger and Jentsch (2016), Schoenfeld (2013) |

UTeach Observation Protocol (UTOP) | Effective STEM teaching | Designing lessons that are inquiry based, Use real-world connections and involve active participation; Modifying instruction (using questioning, responding to student needs and classroom contexts); content knowledge in the work of teaching | K-12 mathematics instruction | Internal structure | Schlesinger and Jentsch (2016), Schoenfeld (2013), Wasserman and Walkington (2014) |

Teaching for Robust Understanding (TRU) Framework | Attributes of equitable and robust learning environments | Content; cognitive demand; equitable access to content; agency; ownership and identity; formative assessment | K-12 mathematics instruction | Content | Schlesinger and Jentsch (2016), Schoenfeld (2013) |

Oregon Teacher Observation Protocol | Reform-oriented teaching | Habits of mind; metacognition; student discourse; challenged ideas; student misconceptions; conceptual thinking; divergent thinking; interdisciplinary connections; pedagogical content knowledge; multiple representations | K-16 mathematics instruction | Content | Morrell et al. (2004), Wainwright et al. (2004) |

### Appendix 2

Classroom observation protocols | Construct measured | Indicators | Sample in the cited study | Validity evidence | References | |
---|---|---|---|---|---|---|

1 | Dyadic teacher–student contact observational system (Good and Brophy 1994) | Student–teacher interactions (negative and positive) | Interactions around academic work classroom procedures & behavior | 61 at-risk youth grade 3–5 | Internal structure | Baker (1999) |

2 | Classroom Assessment Scoring System (CLASS) | High quality teacher–student interactions | Classroom organization, instructional and emotional support | 440 preschool teachers | Content, response processes, and internal structure | Hamre et al. (2012) |

3 | Classroom Observation of Student–Teacher Interactions-Mathematics (COSTI-M) | Explicit Instructional Interactions | Teacher demonstration, student independent practice, student errors, and teacher feedback | 129 kindergarten classrooms | Internal structure | Doabler et al. (2015) |

4 | Levels of Engagement with Children’s Mathematical Thinking from CGI | Teachers’ attention to student thinking | Extent to which student thinking is elicited and used in instructional (decisions) | 26 elementary teachers (grades 1–5) who had participated in CGI PD | None provided | Franke et al. (2001). |

5 | High Quality-Teaching of Foundational Skills in Math and Reading (Valli and Croninger 2002) | High quality-teaching in upper elementary schools | Small group work and high-level questions | Three instructors of elementary education course | None provided | Newton (2009) |

6 | Robust Mathematical Discussion (RMD) protocol | Quality of mathematical discussion | Mathematical and discursive strength of discourse | Two 8th-grade math classes | None provided | Mendez et al. (2007) |

7 | Comprehensive School Reform Classroom Observation System (CSRCOS) | Instructional practice at scale | Instructional opportunity, student activities, and teacher–student relationships | 145 3rd through 5th-grade classrooms | None provided | McCaslin et al. (2006) |

8 | COS-1, 3, and 5 (Classroom Observation System for First, Third, and Fifth Grade) | Quality of classroom supports | Quality of emotional and instructional interactions and amount of exposure to literacy and math activities | 791 children at grades 1, 3, and 5 | None provided | Pianta et al. (2008) |

9 | Observing Patterns of Adaptive Learning (OPAL) | Promoting mastery goals in the classroom | Task, authority, recognition, grouping, evaluation, and time | 28 elementary education majors | None provided | Morrone et al. (2004) |

10 | Classroom Implementation Framework | Lesson quality | Tasks, role of teacher, social culture, mathematical tools, and equity | 26 in-service secondary mathematics teachers | None provided | Arbaugh et al. (2006) |

11 | Growing Awareness Inventory: (GAIn) protocol ** derived from Culturally Responsive Instruction Observation Protocol (CRIOP) | Culturally responsive pedagogy | Classroom relationships, discourse, and sociopolitical consciousness | 19 secondary math and science preservice teachers (PSTs used GAIn to code cooperating teachers lessons, i.e., it was an instructional tool) | None provided | Brown and Crippen (2016) |

12 | TIMSS 1995/1999 Video Study procedure | Mathematics lesson structure and presentation | Organization of classroom interaction, instructional activities, and organization of math content | 39 8th-grade classrooms in Italy | None provided | Santagata and Barbieri (2005) |

13 | Science Learning through Engineering Design (SLED) ** derived from Inquiring into Science Instruction Observation Protocol (ISIOP) | Design-informed pedagogical methods for STEM instruction | Engineering design-informed pedagogical methods | 35 Grades 5 and 6 STEM teachers | None provided | Capobianco and Rupp (2014) |

14 | Mathematics Integrated into Science: Classroom Observation Protocol (MISCOP) | Quality of science lesson when math is integrated | The degree to which mathematics is integrated into student-centered learning of science | 54 secondary STEM teachers | Content validity, Response processes, Internal structure | Judson (2013) |

15 | Classroom Video Analysis (CVA) | Usable knowledge for teaching mathematics | Teachers’ ability to analyze authentic teaching events | Nationally recruited sample of 676 elementary and middle school teachers | None provided | Kersting et al. (2016) |

16 | Electronic Quality of Inquiry Protocol (EQUIP) | Quality of inquiry-based instruction in math and science | Categories include instruction, discourse, assessment, and curriculum | 52 classrooms (35 teachers) middle school science teachers | Internal structure | Marshall et al. (2011) |

17 | School Observation Method (SOM) and Rubric for Student-Centered Activities (RSCA) (Ross et al. 1998) | Student-centered classroom instruction | Instructional orientation, classroom organization, instructional strategies, student activities, technology use, and assessment | 45 observations across 4 different STEM education programs | None provided | Hall and Miro (2016) |

18 | Cases of Reasoning and Proving (CORP) | How teachers use the proof tasks during a lesson | Context and nature of the lesson, cognitive demand of the tasks, and proof schemes | Three geometry teachers | None provided | Sears and Chavez (2014) |

19 | Classroom Observation Instrument (COI) | Advancing student thinking | Teacher lesson planning, Classroom practices, and “on-the-fly” decision making | 18 first grade teachers using Everyday Mathematics curriculum | None provided | Fraivillig et al. (1999) |

20 | Classroom Observation Inventory (COI) | Culturally relevant pedagogy | Dimensions of CureMap: teaching mathematics for understanding; centering instruction on students’ experiences; developing students’ critical consciousness about or with mathematics | 14 high school teachers | None provided | Rubel and Chu (2012) |

21 | Mathematics Scan (M-SCAN) | Standards-based mathematics teaching practices | Structure of lesson, multiple representations, students’ use of tools, cognitive depth, discourse community, explanation & justification, problem solving, and connections & applications | 88 3rd grade teachers—43 of whom taught at schools receiving Responsive Classroom (RC) training | Content, response processes, relationship to other variables, and internal structure | Ottmar et al. (2013) |

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Bostic, J., Lesseig, K., Sherman, M. *et al.* Classroom observation and mathematics education research.
*J Math Teacher Educ* **24, **5–31 (2021). https://doi.org/10.1007/s10857-019-09445-0

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### Keywords

- Classroom observation
- Instruction
- Qualitative
- Validity