This study involved exploring the thoughts and practice of 12 teachers by
spending time in their classrooms to become familiar with the teacher, the students in the classroom, the teacher’s practice, and the teachers’ thoughts and opinions about their teaching. After an initial interview, the researcher spent time in each teacher’s classroom before conducting two structured observations followed by an in depth interview with the participating teacher. Data consisted of field notes, interview transcripts, structured observation notes, lesson plans, and sample assessments.
The practice of logging time in the classroom was used in order to more
accurately determine classroom practice. Logging time prior to conducting two structured observations provided a more valid assessment of typical classroom practice. While the practice of logging time with classroom visits was maintained throughout the study, there was very little, if any, difference in classroom behavior on the part of the teacher or the students from the first day to the last day of the researcher’s time in the classroom.
Using Observations to Determine Practice
The Mathematics Classroom Observation Instrument (MCOI) was used as an organizational tool for recording information helpful in categorizing teacher practice as traditional or standards-based. While the instrument was too general to offer definitive information, it did serve to organize characteristics of classroom practice in a way that helped the researcher to look for commonalities between the teachers’ actual practice and how research defined traditional and standards-based practice. The instrument was divided into eight sections, each dealing with a different aspect of what was observed in
the classroom during the structured observation. Each of the criteria from the eight sections, and the subsequent categorization is discussed. For more information on the MCOI form, see Appendix D.
Physical Setting/Classroom Environment
This section of the MCOI solicited two pieces of information. The first considered overall student seating and how it facilitated classroom learning, and the second
concerned the materials readily available to students. Student seating was disregarded as an indicator of practice because it became apparent early on in the study that teachers, for the most part, did not have control of student seating. In most cases, student seating was determined by furnishing the new school buildings most schools had recently acquired. The teachers were not necessarily consulted as to their choice of furniture or seating arrangements. The second criterion, classroom environment, was indicative of practice as it included the availability and accessibility of mathematics tools such as manipulatives, calculators, computers, and informational displays.
Lesson Overview
The lesson overview section of the MCOI recorded information on (a) major instructional resources, (b) content delivery, (c) the lesson placement within the
instructional sequence, (d) seating arrangement for this particular lesson, and (e) content focus. Two of the criteria, lesson placement and content focus, were mostly insignificant in terms of this study for determining classroom practice. Another, content delivery, was useful in assessing accuracy in teacher knowledge as far as content taught and teacher
knowledge of grade-level indicators of Standards. Content delivery, however, contributed little to distinguishing between traditional and standards-based instruction.
The criterion, major instructional resources used in the lesson, was suggestive of classroom practice especially when coupled with field note observations of typical classroom activities during classroom observations. For example, if the textbook was the major instructional resource most of the time, it was suggestive of traditional instruction. In the case of the 12 teachers observed, sole and consistent textbook use was highly indicative of traditional instruction.
Seating arrangement for the lesson was suggestive of classroom practice. The concept of seating had more to do with what students were expected to do than with the physical placement of seats. Small group arrangements, whether students are doing the same or different tasks, are indicative of student-centered instruction.
Instructional Overview
The instructional overview section required looking at two different criteria. The first, primary instructional strategy, focused on what the teacher was doing throughout the lesson. Teacher led activities such as lecture, discussion, and demonstration were suggestive of traditional instruction. The second criterion, student activity, focused on what students were doing throughout the lesson. While it is rare that students will be doing only one type of activity throughout a class period, the predominant activity was recorded and secondary activities were noted to provide a more complete assessment of what was happening in the classroom. Listening to or observing the teacher, completing a
practice worksheet, reading the text, or working on an assignment from the textbook characterized traditional practice.
Questioning
The two criteria regarding questioning, quality of questions and questioning techniques, were very important indicators of classroom practice. Traditional instruction was more likely to include either no questions or questions that were mostly narrow and focused on factual recall or required one word responses. However, in standards-based instruction questioning is key. According to Ross, McDouggal, and Hogaboam (2002), student discourse is a key indicator of standards-based instruction.
Classroom Atmosphere
Student involvement and classroom culture/learner attitudes are two criteria associated with Classroom Atmosphere. Student engagement was an indicator of
standards-based instruction to the extent that students were either sitting listening or they were actually doing something that involved them in the lesson. As for atmosphere, while cooperation with the teacher and/or other students is expected in any classroom, the attitudes on the MCOI protocol beyond cooperation are indicative of standards-based instruction.
Analysis of Instruction Leading to the Development of Higher Order Skills
This section of the MCOI requires an examination of the type of instructional activity focusing on three criteria. They are (a) the amount of student investigation or research, (b) the level of student engagement in the activity, and (c) the mathematics skills being developed. The progression of student investigation begins with none,
increases to an activity which focuses on lower level mathematical process skills, and finally to an activity that emphasizes higher level mathematical process skills. In order to categorize instruction, traditional instruction is rated as no investigation or research and a primary focus on only one or more of the first seven mathematical skills listed. The mathematics skills list is a hierarchy of mathematics skills ranging from basic to higher order. There are 17 skills with the first seven suggestive of traditional instruction and the remaining skills being more indicative of mathematics instruction that is standards-based.
Overall Classroom Rating Profile
The seventh category on the MCOI is a statement about the overall effectiveness of classroom instruction. There are five statements about the quality or effectiveness of the lesson with consideration given to student engagement and alignment with standards. The defining characteristics of these statements involve the extent to which the lesson is aligned with standards and requires students to use higher level thinking skills. The first two statements are associated with higher level thinking skills indicative of standards- based instruction. The last two suggest no or minimal alignment with standards-based instruction. The middle statement is indicative of instruction that does not clearly line up with any of the other four statements.
Mathematical Processes Benchmarks from Ohio Academic Content Standards: Mathematics
This section provides a list of the process benchmarks from the Ohio Academic Content Standards for Mathematics in middle school. There are 11 benchmarks and the
evidence of three or fewer benchmarks in a lesson, for the purposes of this study, was considered traditional instruction.
Context: Who Are They?
This study represents teachers throughout a four county area in southern Ohio. While teachers participating in the study may have been casually acquainted through various professional development experiences, none of them were familiar with any of the other participating teachers. The number of years of teaching experience for this group of teachers ranged from seven years to thirty-seven years.
Classroom settings were varied. Of the 12, one teacher’s classroom was housed in an elementary building separate from the high school, one was housed in a separate middle school building, and the rest of the classrooms were housed in a campus setting with a middle school or junior high occupying a distinct section of the high school wing of a campus or, in one case, a sixth grade occupying a distinct section of an elementary wing of a campus. All teachers worked with middle-school students with the majority of them teaching a single grade-level. Teacher certification was primarily elementary with eight teachers certified to teach grades one through eight. Additionally, other
certifications included one teacher certified in each of the following: grades 4-9 mathematics, high school mathematics, grades 4-9 combined with high school
mathematics, and high school mathematics combined with elementary. See Appendix E for more specific information on teacher certification, grade-level, and longevity.
For the purpose of reporting information associated with this study, teachers were each given a pseudonym. Teachers’ pseudonyms reflect the gender as well as the order of
initial commencement of data collection indicated by the alphabetical ordering of the first letter of the pseudonym. For example, as signified by the fourth letter of the alphabet, Ms. Diane Davis, a female, was the fourth teacher interviewed prior to beginning classroom observations.
Meet the Teachers
In order to provide background for the study, a brief overview of each teacher is presented. The overview provides basic information about the classroom and practice of each teacher while identifying the instructional practice as standards-based or traditional as defined in chapter. A summary table of teacher practice derived from structured observations is included as Appendix F.
Mr. Allen Anderson
Mr. Anderson, a teacher of 11years, teaches mathematics to 6th through 8th grade
students. He seems to be a teacher whose primary mode of instruction is traditional. During visits to his classroom, the only instructional strategy observed was lecture with mathematics computation demonstrations on the board. His communication with students required one word responses dealing with math facts or procedures. His lesson plans were printed copies of lesson plans from the textbook publisher with the dates hand-written.
Mr. Anderson considers himself to be a good teacher who teaches using standards- based instruction. He readily admits, however, that he doesn’t attend to the process standards or use manipulatives in his teaching and considers that it might be detrimental to his students’ understanding of mathematics.
I don’t believe I get that [mathematical processing] from each individual student because a lot of that [mathematical process skills] I believe is above some of the students’ heads. Because of not doing some of the group work, not doing some of the hands-on stuff, I think I lose a lot of it [conceptual understanding]. (Anderson) Mr. Anderson has an abundant supply of mathematics manipulatives obtained from various workshops and professional development opportunities in which he has participated. All of the materials were stored in a locked closet in the back of the room and are not used. He cites the homogenous ability groupings of his classes as a reason for not using the manipulatives. Mr. Anderson feels that use of manipulatives with high groups slows them down. He says that low groups are unable to perform using manipulatives and, while some of the middle group might benefit from using manipulatives, he does not think it is fair to students if the teacher treats one group differently than another so he does not use manipulatives with any of his students. Neither does Mr. Anderson have students work in groups or do any type of project that requires application of mathematics although he frequently mentioned in his lectures possible applications of mathematics.
Data is important to Mr. Anderson, and he frequently mentioned using the state achievement data reports generated by the district central office to determine what he needs to focus on when planning for future instruction. Mr. Anderson reports his students’ regularly achieve high test scores, and he takes pride in those testing results. When asked if he was doing standards-based instruction, he referred to the textbook, which according to the supplementary material from the textbook publishers, is aligned
with Ohio Standards. He then referred back to test results which break down students’ scores into individual areas of difficulty for each student. Mr. Anderson exhibited a comfortable working knowledge of state testing data generated from his students’ test scores.
Mr. Anderson views the Standards as an exact guideline of the mathematics content he is required to teach each year regardless of students’ prior knowledge. To clarify this assertion, Mr. Anderson said while waiting for his 7th grade class to begin, “If one of these students goes down and tells Mr. Jackson [the principal] that I am teaching 6th grade standards in here, I get in trouble. I am not supposed to teach 6th grade standards in here.” (Anderson)
Mr. Anderson’s knowledge of the Standards was drawn from the textbook lesson planning software tool that allowed one to click on a lesson in the text and the software will display the lesson plan that lists the state indicators for that lesson. Mr. Anderson was unfamiliar with the actual Ohio Academic Content Standards for Mathematics book although he reported owning a copy which he said was somewhere in his classroom. To summarize his feelings about the Standards, Mr. Anderson said, “Teachers, we know what kids need if we’re doing what our job is. If they’re truly a teacher, truly here for the kids, they’re going to do their job. We don’t need standards to go off of. However it’s nice to have those Standards to see where the kids need to be at when we’re done.” (Anderson)
Mr. Ben Brown
Ben Brown teaches eighth grade mathematics. The school district requires all eighth grade students to take Algebra 1 for high school credit. Having taught seven years as a mathematics teacher, Mr. Brown uses direct instruction as his primary teaching method and asserts that consistency is the driving force behind his teaching practice.
The driving force for me is probably consistency…. I found that if I tried to get fancy with this stuff and show them some of the stuff I had learned in college that they just got confused. And, I am probably one of the most boring teachers here because I am so methodical about everything. But I want to make sure they understand and if I’m boring, I’m boring. But, if I go through material in the same way and I’m consistent with it, it gives them a steady platform to work from. If I use the same concepts and same terminology every time we do something then they don’t feel like I’m throwing new material at them. They just think they’re getting the same thing with something a little extra on the end. I give them something that they can work from basically. (Anderson)
This sense of consistency prevailed in Mr. Brown’s classroom throughout every visit and observation. Instruction, as well as the classroom routine, remained the same. Instruction from one class period to the next was, as a general rule, the same. Mr. Brown used the same mathematical examples, and he presented them in the same order. In addition, his wording, timing, and extemporaneous conversation was almost identical from one period to the next.
Mr. Brown professed knowledge of the Standards. With this in mind, he said that in the month leading up to the state achievement test he has to move away from his Algebra 1 curriculum in order to review the indicators that are required for eighth grade but are not covered in the Algebra I coursework. Although he complies, Mr. Anderson does not seem to view this practice as his primary responsibility.
I hate the standardized tests. I always tell my kids, I’m here to get you ready for high school. That’s my number one job. If you do well on the standardized tests, great. But my job is to get you ready for algebra II and that’s the way I push myself during the year. (Anderson)
Ms. Carla Case
Ms. Case has been teaching seventh grade mathematics for all but the first year of her nine-year teaching career. She is very student-oriented and reportedly participates in all the school-sponsored motivational events such as hat day and pajama day, both of which were activities to encourage students to attend school during “Count Week.” (Count Week is the one week selected in October of each year on which school district funding is determined based on average daily attendance for that week.) Based on conversations and interactions between Ms. Case and students entering and visiting her classroom, she seemed to be a popular teacher with the students. She is animated and declared a passion for teaching mathematics and for teaching mathematics in a non- traditional manner.
I feel this is my passion for why I teach math. I feel a lot of students struggle with math, and it’s their hardest subject. Most of the time when they come in at the
beginning of the year, they hate it. And, so, I feel that my job is to get them to enjoy it and I just want them to know for one year that math doesn’t have to be hard. It can be fun. There are projects and things we can do that make math exciting. It doesn’t have to be notes, bookwork, notes, bookwork. And that’s why I teach this, and that’s why I teach that way. (Case)
According to the MCOI analysis and field notes, Ms. Case is most characteristic of a non-traditional teacher in practice. (See Appendix F.) Students have mathematics textbooks they use but Ms. Case does not rely on the textbook nor does she work through the textbook systematically. “There’s some things I’ve got to leave out because I’ve got to get to this. I’ve got to do this, and that’s what the state tells me. You know, so some things I do leave out.” (Case) As a general rule, Ms. Case reports that she uses the text book as a resource to build activities she uses to present the material in the classroom.
As a result of the many professional development activities in which she has participated, Ms. Case reports having a large repertoire of activities and teaching ideas