|
Pilot Study 1
The Influence of Developmental Levels of Thinking on Understanding of
Advanced Mathematical Concepts in Middle School Students |
|
Statement of the problem:
Data from placement of University of Texas at El Paso declared mathematics, science, and engineering majors shows that 36% of high school graduates and first year students were placed in Pre-Calculus and only 15% of students were placed in Calculus courses. One of the main causes of it is that few high school students take a Calculus course in high school. NAEP (2000) data indicates that only 10-12% of high school students take Calculus courses. This percentage drops significantly for minority students: only 3-4% of Hispanic high school students take Calculus courses in high school
|
Rationale:
In order to increase the number of minority and lower socioeconomic status high school students taking Calculus courses and help them to be successful in Calculus the role and placement of these courses need to be rethought. This project is focused on exploration of possibilities of early development of big Calculus ideas in middle grades.
It is generally accepted that Calculus is a subject for older students, the “elite” and is often considered a “gatekeeper” course, a course that keeps students from progressing towards degrees that require upper level mathematics. It is well documented that minority and lower socioeconomic status students are underrepresented in mathematics related careers. These same students continually score lower on national and international tests and course-taking patterns reveal that they have limited access to upper level mathematics courses. The problem is two pronged as these students work with teachers who may not be qualified to teach upper level mathematics courses and the lack of participation from minority and lower socioeconomic students in mathematics programs prevent them from success in mathematical related careers.
Mathematics standards, along with a shortage among qualified teachers, have brought forth increased interest in the preparation teachers receive in their fields. The teachers’ educational background is the most widely accepted standard of teachers’ understanding of and expertise in their field. Educational backgrounds of mathematics teachers are not consistent. While no one would argue the need for teachers to know their content, many middle school mathematics teachers fall short in meeting recommendations for coursework preparation made by national associations of teachers. Only 7 percent of middle school mathematics teachers have taken courses in all recommended areas and about one-third have completed none of the coursework recommendations (NSF, 1998). Often state certification requirements do not maintain high expectations in mathematics background for teachers. One study found that most secondary students pursuing a minor in education were not required to take courses in pedagogy and that most elementary teachers did not have adequate subject matter preparation (Luft, Buss, Ebert-Mat, & Eslamieh, 1997). 40 per cent of high-school mathematics teachers in high-poverty schools have no training in mathematics (The Education Trust, 1996).
Teacher shortages often force school districts to place teachers in mathematics classrooms with little or no background in this important curriculum area. There is evidence that shows a correlation between out-of-field teaching (lack of strong formal preparation in the content area being taught) and lower student performance (ECS, 1999). It is content area preparation, then, and not certification per se, that may be a relevant consideration. These factors, along with the sociological background of minorities and lower socioeconomic environments influences keep a large percentage of students from participating and succeeding in mathematics courses like Calculus.
|
|
Sample:
Wiggs Middle School in the EPISD has 709 students in the sixth, seventh, and eighth grades. These include 82.1% of which are considered qualifying as lower socio-economic status. The sample will come from Wiggs Middle School. The students involved in this project will be those in the classes of teachers involved with the PETE/Noyce project and working faculty of the MSP RPL work. (n=120)
The pilot project hypothesized that a certain developmental level of maturity was necessary for middle school students to understand advanced mathematical concepts. A pilot study was conducted with a small sample of middle school students with above average GPAs. The pilot sample consisted of two sixth graders, four seventh graders, and six eighth graders from Wiggs middle school. Pre and post content tests were administered and a ten day treatment program was administered using a hands-on visualization approach to learning
“big ideas” in calculus.
|
|
Report from Teachers Involved with Pilot Study 1
The problem we are interested in looking at is whether students must reach a certain level of maturity before they are capable of comprehending higher levels of mathematical thinking. Research shows learning will occur when information is presented at the appropriate level for the student. Students must be capable of learning the information with respect to maturity level and psychological development. http://www.cals.ncsu.edu:8050/agexed/leap/aee535/learning_principles.ppt
One area of research looks at students who are home schooled. Home- schooled children are usually taught at a higher level due to the individualized attention. Do students who are home schooled actually understand the concepts of algebra if they are taught it at the 6th and 7th grade level? According to Bob Hazen, who is an advisor for home schoolteachers. “You want to be cautious about accelerating kids *too* fast mathematically. The reason for my caution here is that even for kids who master algebra early on (say, grades 6-7), there are still development/maturational factors to consider. Developmentally, there is still a maturity that kids come into about age 14 or so that allows them to have a perspective, a range of insight, a depth of understanding, a "with-it-ness" that only arrives at this age and not before. I saw this with my own older son (now 16, taking calculus this fall as a H.S. junior): he mastered algebra as a 2-year course in grades 6-7, then mastered geometry in 8th grade, both times getting grades of solid A to A-, with a depth of conceptual understanding and excellent number sense and problem solving skills. But it wasn't until 9th grade that I saw for the first time a capability in him to reflect-generalize-connect different aspects of what he was studying - "Oh, Dad, what we're doing here is just like what we did over there..." I consider this a maturational development, not an academic-intellectual issue.”
In looking at the problem stated above, we hypothesize that a certain developmental level of maturity is necessary for middle school students to understand advanced mathematical concepts. During the fall semester we designed units aligning them to our curriculum at Wiggs, which covers advanced mathematical concepts of Visual Calculus. These units included topics of optimization, graphing, integration, and differentiation.
In order to test the hypothesis we designed an experimental project that was implemented in different level classrooms at Wiggs. What we were looking for was whether students in 6th, 7th, and 8th grade could comprehend the same level of mathematics. The project consisted of three steps: pre-test, hands-on activities, and a post-test. We gathered data by comparing pre-test knowledge with post-test knowledge.
The optimization unit was used on 6th and 7th grade students. Eleven sixth graders and fourteen seventh graders were part of the project. The sixth graders consisted of humanities students, which mean that they are at a higher level. The seventh graders contained a diverse assortment of students, including humanities, regular, and ESL students. Data from the pre-test indicate that the sixth graders brought more knowledge into the unit. This was one of our expectations considering their humanities background. However at the end of the project the seventh graders showed a bigger gain of knowledge, which equaled or surpassed the sixth graders. |
| Optimization Data |
|
|
The conclusion we can draw from the optimization unit is that even though the gain of knowledge for the seventh graders was greater then that of the sixth graders, both grades final understanding of the mathematical concept was comparable to each other.
All three-grade levels at Wiggs were involved in the graphing unit. The breakdown of students for this project included 17 sixth graders, 12 seventh graders, and 20 eighth graders. The classes involved contained a broad spectrum of mathematical ability. This can be seen in the pre-test which shows that all three grade levels brought approximately the same amount of knowledge to the unit. |
|
Graphing Data
|
| In summary it seems that all three-grade levels gained the same amount of knowledge while working on the graphing unit. Student maturity did not seem to matter in the understanding of the unit, if you look at the graph containing posttest assessment. The data shows that at the conclusion of both units the amount of comprehension was similar for all grade levels.
In order to further the interest in this topic we would consider changing some aspects of the project. One of the changes would be to break the units into smaller and more concise sub-units. This would make more sense because of the time constraints we have in our classrooms. Another change would be to look at and compare females to males. Last but certainly not least; we might consider getting more students involved with this project. |
|