Derrick, C. L. (2005). Improving student retention through articulation and reflection. Instructional Technology Monographs 2 (1). Retrieved <insert date>, from http://itm.coe.uga.edu/archives/spring2005/derrick.htm

 

Improving Student Retention

Through Articulation and Reflection

by

Connie L. Derrick

University of Georgia

Abstract

This study was conducted to determine if articulation and reflection are effective instructional strategies to improve student retention and application in the mathematics classroom. Sixth grade math students participated in the writing and sharing of prompted journal writings while studying a unit on fraction computation. Quantitative data such as pretest, posttest, and delayed test scores were collected to determine if students retained and applied their knowledge. In addition, the journals were evaluated using a rubric to determine the quality of the journal writing in order to explore a relationship between the quality of the journal writing and their test results. Eighty percent of the participants who submitted journal responses scored 85% or higher on the first posttest.

Literature Review Methods Results and Discussion Conclusions References

 

Introduction

Testing and accountability are part of the educational structure today. From elementary school to graduate level instruction, tests are used as a convenient way to assess knowledge for a large group of students. Elementary students must participate in state testing, such as the Georgia Criterion Reference Competency Test (Georgia Department of Education, 2004). High school students are faced with the pressure of performing well on the Scholastic Aptitude Test in order to be granted admission into many colleges and universities. Many graduate programs require the Graduate Record Exam as part of the application process. Whether we agree with it or not, testing is an assessment tool that will probably remain a part of the educational system. Students must be able to retain and apply their knowledge in testing situations and in the real world. Research indicates students are not able to transfer their knowledge between the real world and the classroom (Chaney-Cullen & Duffy, 1999). In order to accomplish this, students need to be given the opportunity to explore concepts, develop an understanding of fundamental mathematical processes and algorithms and use the combination of the two to solve authentic problems. Memorizing the algorithm without understanding may not be helpful when faced with an unscripted problem. Likewise, a lack of procedural knowledge may hinder ones ability to focus on higher level problem solving. I think that a blend of instructional theories, such as constructivism, cognitive apprenticeship, project-based learning, and the instructivist approach will facilitate learning that transcends the retention and testing gap.

My problem is based upon the need to improve student retention and application by incorporating articulation and reflection in the mathematics classroom in an attempt to answer the following questions:

Will articulation and reflection improve retention and application?

Does the quality of journal responses contribute to student success?

The study will focus on sixth grade math students. By having students write and reflect about mathematical topics and procedures, retention and application will evolve naturally. The purpose of this study is to determine if articulation and reflection increase retention and application.

Piagetâs perspective focuses on how logical reasoning is developed and the role school plays in the process. Questions and examples that lead to further investigation, activities that challenge students, and finding out the ways students are thinking are important for cognitive development (as cited in Green & Gredler, 2002). Vygotskyâs focus on cognitive development revolves around conceptual thinking, categorical perception, logical memory, and voluntary attention. There is a correlation between the written language and the transfer to other situations. Reciprocal teaching is a good example of a constructivist teaching method (Green & Gredler, 2002). This ties into the articulation and reflection strategies used in the cognitive apprenticeship model. One must develop an understanding of the meaning of articulation and reflection and how it fits into the educational setting. Then, determine if this is an effective teaching method.

 

Literature Review

My study is designed to evaluate the implementation of writing and reflecting in the math classroom. A brief overview of cognitive apprenticeship will be presented, followed by a more in-depth review of articulation and reflection. As I embarked on the review of literature, I found limited research on articulation and reflection. Collins, Brown, and Newman were often cited. Most of the available literature was written from an expository perspective on the effects of writing in the mathematics classroom. I located a few studies in Dissertation Abstracts International that looked promising, but were not readily available due to cost and time constraints. Therefore, I will summarize my findings based upon the literature I was able to review at the time.

Cognitive Apprenticeship

The cognitive apprenticeship instructional model is based on the premise that learning should be authentic and have meaning. In reviewing the literature, several strategies emerged, such as modeling, coaching and scaffolding, articulation and reflection, and exploration (Brill, Kim, and Galloway, 2001). Modeling may take the form of a demonstration or verbalizing the process. Coaching and scaffolding refer to the role of the facilitator and exploration addresses the process of finding knowledge independently. Articulation and reflection focus on the conveyance of ideas through verbalizing or the writing process and the opportunity to revisit these ideas.

Articulation and Reflection

It is important to explore the meaning of articulation and reflection. Collins, Brown, and Newman, (1987) state ãArticulation includes any method of getting students to articulate their knowledge, reasoning, or problem-solving processes in a domainä (p. 18-19). In addition, they also state, ãReflection involves enabling students to compare their problem-solving process with that of an expert other, other students, and ultimately, an internal cognitive model of expertiseä (p.18-19).

Jurdak and Zein (1998) conducted a study entitled, The Effect of Journal Writing onAchievement in Attitudes toward Mathematics. The study looked at middle school students (11-13 years old) in Beirut . The experimental group participated in journal writing while the control group did not participate in the journal writing. The following criteria were taken into account: mathematical communication, conceptual understanding, procedural knowledge, problem solving, and achievement in mean test scores, and the attitudes toward mathematics. The Mathematics Evaluation Test was used, along with a Likert questionnaire and student evaluation essay to measure attitudes. Jurdak and Zein (1998) found that journal writing studentsâ mean score was higher for conceptual understanding, procedural knowledge, and mathematical computation. There was little difference between the two groups in problem solving, attitudes, and school achievement. The researchers were surprised that school achievement test scores were not impacted by the journal writing process (Jurdak & Zein, 1998).

Another study of ninth grade Algebra I students confirms the need to incorporate writing in the math curriculum. This study looked at relationship between writing and metacognition. The studentsâ writing indicated that this framework of metacognition exists. Writing allowed students to share their thought processes and reasoning with others. Writing could support the development of skills needed for problem-solving (Pugalee, 2001). Pugalee indicates a lack of research to substantiate writing activities in mathematics and the hopes that his study will encourage more research in this area.

In an essay entitled, Advanced Math? Write? Brandenburg indicates that through the incorporation of writing, students retained what they learned ( Brandenburg , 2002). She advises teachers to consider the time needed to grade journals and to develop a rubric to facilitate assessment. Brandenburg (2002) was very open to various writing ranging from definitions to self-evaluation and felt the process helped students ãdeepen their understanding and retentionä (p. 67-68). Havens (1989) indicated that her students developed better conceptual understanding and application instead of relying on memorization. Borasi and Rose (1989) focused on incorporating journal writing in a college math class. They concluded the following benefits: writing helps develop a better understanding and articulation and reflection improved learning and problem solving. They confirmed one of my rationales for undertaking this study; students will internalize meaning if they use writing to explain concepts and procedures.

Albert and Antos (2000) wrote about a journal writing project in which fifth graders wrote about how they used math in the real world. Students shared and reflected on their writings, wrote problems and worked in groups to solve them. They contend that the writing process allowed the students to bridge the gap between the classroom and meaningful situations. Similar comments regarding the understanding of math concepts increasing through writing and sharing and providing an avenue to make comparisons were echoed (p. 527). In addition, writing helped students develop confidence and ownership for their learning. (p. 528). Shepard (1993) shares the opinion that the writing process facilitates the development of meaningful conceptual learning.

Another article, written by Johanning (2000) reiterates similar sentiments regarding writing in the mathematics class. Writing provides students a chance to develop and clarify what they have learned. This includes learning from oneâs mistakes. Students were involved in a process of writing their solution, discussion in groups, and revision. The studentsâ writing showed the reasons for approaching a mathematical problem. The process of sharing with others gave students a large knowledge base to apply later.

Language is compared to the acquisition of knowledge and concepts, both are evolving based upon new experiences. This supports the logical connection between the articulation and reflection to increase the retention and amount of knowledge acquired within the time limits of the educational setting. Students need to have declarative and procedural knowledge for math computations in order to move on to the more complex problems. Students become focused on the computational skill, instead of the problem-solving aspect. A common method of instruction for diverse students is to teach the concepts, operations, and the problem-solving strategies using a direct approach (Bottge, 2001).

One aspect of this study is to look at inert knowledge and ritual knowledge, which is part of the math classroom, and how to help students retain and apply this information. Inert knowledge is something that is not called upon very often, used primarily to pass a test and then not utilized. Ritual knowledge consists of the steps needed or the routine to follow in order to arrive at a response. Knowledge does not have to be one or the other. Educators need to make ritual knowledge lessons more meaningful for students and inert knowledge lessons need to include ways for students to be more active in their learning process (Perkins, 1999). Brown and et al. share the opinion students may be able to pass a test but not be able to apply the concept in the real world. Students would benefit from journal writing because it provides the opportunity to connect the concept and algorithm with the real world. Teachers benefit because they can prompt further reflection and follow the thought processes of students and guide when necessary.

Based on their perceptions of their intelligence, children may see problem solving as an opportunity for learning or an activity that promotes failure (McGuinness, 1993). Writing and reflection might provide students with an opportunity to be active participants in a risk free environment and change their negative perceptions about themselves. Through articulation students share thought processes, make comparisons, and may discover new ideas. The teacher can facilitate the transfer of knowledge by helping students make other connections (McGuinness, 1993).

D.C. Phillips depicted the roles of the learners as active, social, and creative. The idea being that active learning is acquired through discussion and investigation. The social learner relies on discussion and communication with others and the creative learner may look at knowledge as a process of being created. Active engagement of the learner may foster retention and application of knowledge (as cited in Perkins, 1999). Overall, the literature indicates the process of articulation and reflection is a worthwhile strategy to implement in mathematics instruction.

Again, the minimal amount of information and research on articulation and reflection in mathematics opens the door for further research in this area. In general, writing and reflection are considered to be worthwhile endeavors, but formal research is difficult to find.

My study will attempt to find the connection between articulation and reflection and the ability to retain and apply basic knowledge and heuristics. The assessment instrument for this study consists of a fraction computation assessment. This does not mean that conceptual learning and higher level problem solving are not taking place throughout the learning process. This study represents a narrow aspect of the learning process, retention and basic application. Additional or more in-depth research would be needed to fully evaluate conceptual learning and problem-solving abilities through the use of articulation and reflection.

 

Methods

The purpose of this study is to determine whether articulation and reflection improve retention and application in the mathematics class.

Participants

The study took place in a suburban middle school located in the southeastern United States. Most of the students were from middle to upper middle socioeconomic backgrounds. In order to efficiently manage the collection of data, sixth grade students in one class were invited to participate in this research. Eight students elected to participate in the study.

Instruments

A fraction computation test consisting of twenty problems was developed to evaluate basic skills. The same test was used as a pretest, posttest, and surprise test. A rubric was created to assist in the evaluation of studentsâ journal writing. The journal evaluation instrument was designed using a component point system to decrease subjectivity. All responses were evaluated using the same rubric during the same time period. The criteria for the fraction computation entries were: 1.) Explain in your own words the mathematical procedure you used to solve this problem. 2.) Write an example problem and include the solution. 3.) Explain in detail how or why you might use this skill in the real world. 4.) Write down any questions or ideas you have that you would like to share.

Design

In order to increase validity and reliability, the fraction computation assessment was designed to represent all four computations in equal amounts. In addition, the same assessment was used every time to eliminate format as a variable. Objectivity was minimized by using a point system to evaluate the journal responses and responses were evaluated during the same time period.

Procedures

Students were given a pretest prior to the teaching of fraction computation in their class. After the completion of the computational section of the fraction unit, students took the same test as a posttest. They were told in advance to prepare for the test. Approximately one month later students were given the same test, but were not told about the test in advance.

Students were encouraged to compile their journal writings in a composition notebook or folder. All journal responses were assigned after the pretest but prior to the first posttest. Some class time was provided for journal writing and students were instructed to complete some entries for homework. Students were encouraged to write in a timely manner. Teacher assigned journal writings were required to explain the procedures and application of addition and subtraction, multiplication, and division of fractions.

 

Results and Discussion

The quantitative test data was collected using a written test format. The pretest provided an initial starting point or baseline. Confounding factors such as instructional strategies should be taken into account. All the participants had the same instructor. Students were encouraged to write and share their journal responses for fraction computation. Time was allotted during class to write and articulate their responses. In addition, students were encouraged to complete their writings or discussions at home.

The data indicated that 100% of the participants increased their score on the first posttest; which ranged from 25-95 percentage points. Approximately 30 days later the second posttest was administered. In order to use this as a tool to evaluate retention, students were not told in advance. All three assessments were exactly the same.

Comparison of the posttest and second posttest assessments indicated split results. Half of the participants showed an increase in their second posttest score, while the remaining 50% showed a decrease. Interestingly, both groups were relatively close in their range of percentage points. The range of decrease was 10-25 percentage points, while the increase was 5-25 percentage points.

Another component of the research included an evaluation of the journal responses to determine if a correlation existed between the quality of journal responses and assessment scores. The results have been compiled in the table below.

Journal Response Assessment Table

Participant

Adding and Subtracting Fractions

 

Multiplying Fractions

Dividing Fractions

Journal Response Points Total

(maximum 12 points)

 

First Posttest Score Second Posttest Score

Amount of Increase or Decrease

 

 

 

1
4
4
0
8
100%
80%
- 20
2
0
0
0
0
70%
45%
- 25
3
4
4
4
12
85%
95%
10
4
3
2
0
5
45%
35%
- 10
5
0
0
0
0
60%
85%
25
6
0
0
0
0
75%
85%
10
7
3
0
0
3
95%
85%
- 10
8
2
3
0
5
90%
95%
5

 

 

A score of 85% or higher on the first posttest was demonstrated by 80% of the participants who submitted some journal responses. The participants who did not submit any journal responses scored 75% or below on the first posttest. The second surprise posttest represented a decrease in 50% of the scores and an increase in 50% of the scores. Of the two highest journal response scores, one showed a decrease of 20 points and the other an increase of 10 points. Two participants who did not earn any journal points scored 85% on the second posttest.

Conclusions

Based upon the assessment data comparing the pretest and first posttest scores, the articulation and reflection of fraction computation appeared to be beneficial to all participants. Even though all participants did not submit journal entries, they may have benefited by the sharing of the journal entries during class time. Students were encouraged to share even if they did not have their written journal response with them.

It is difficult to isolate articulation and reflection as the sole reason for an increase in achievement on the first posttest. Certainly, other factors such as motivation, study habits, and instructional strategies have an impact on learning. The data from the pretest to the first posttest supported the implementation of articulation and reflection as an effective avenue for learners to construct meaning and validated articulation and reflection as a component of best practices for educators.

The retention of fraction computation was explored even further by looking at the results of data from the second posttest score and the quality of journal responses. While the data indicated the same number of participants increased as decreased, it should be noted that greatest decrease occurred for a participant who did not earn journal points and the greatest increase occurred for a participant who also did not earn any journal points. Overall, 75% of the students, scored an 80% or higher on the second posttest. Again, other factors that are difficult to assess may have played a role in this. Therefore, the data does not substantiate whether the quality of journal writing had a direct impact on retention as noted by the second posttest scores.

Some of the inconsistencies discovered when comparing the journal responses to achievement gains warrant the need for further studies in the area of articulation and reflection in all classrooms. This study does not provide data to support that articulation and reflection is the best way or 100% effective in developing long term retention. Maybe different results would be derived if another study included more participants and or a longer period of time. Perhaps others will continue to research this aspect of education in the quest for more effective classroom strategies to meet the needs of all learners.

 

References

Albert, L., & Antos, J. (2000). Daily journals connect mathematics to real life. MathematicsTeaching in the Middle School, 5(8), 526-531.

Borasi, R., & Rose, B. (1989). Journal writing and mathematics instruction. Educational Studies in Mathematics, 20, 347-365.

Bottage, B. (2001) Reconceptualizing mathematics: Problem solving for low- achieving students. Remedial & Special Education, 22(2), 102-112. Retrieved from Ebsco Host database on June 28, 2004 .

Brandenburg , M. (2002) Advanced math? write!. Educational Leadership, 60(3), 67-68. Retrieved from Ebsco host on June 28, 2004 .

Brill, J., Kim, B., & Galloway, C. (2001). Cognitive apprenticeships as an instructional model. In M. Orey (Ed.), Emerging perspectives on learning, teaching, and technology. Available website http://www.coe.uga.edu/epltt/CognitiveApprenticeship.htm.

Chaney-Cullen, T. & Duffy, T. (1999). Strategic Teaching Framework: Multimedia to support teacher change. Journal of the Learning Sciences, 8(1), 1-40.

Collins, A., Brown, J., & Newman, S. (1987). Cognitive apprenticeship: Teaching thecraft of reading, writing, and mathematics (Tech. Rep. No. 403). Urbana-Champaign: University of Illinois : Center for the Study of Reading . 1-37

Georgia Department of Education (2004). Available website: http://www.doe.k12.ga.us

Green, S., & Gredler, M. (2002) A review and analysis of constructivism for school-based practice. School Psychology Review, 31(1), 53-70. Retrieved from host database on June 20, 2004 .

Havens, L. (1989). Writing to enhance learning in general mathematics. Mathematics Teacher, 82(7), 551-554.

Johanning, D. (2000). An analysis of writing and postwriting group collaboration in middle school pre-algebra. School Science and Mathematics, 100(3), 151-160. Retrieved from Ebsco Host on July 2, 2004 .

Jurdak, M., & Zein, R. (1998). The effect of journal writing on achievement in and attitudes toward mathematics. School Science and Mathematics, 98(8). Retrieved from Ebsco host database on June 25, 2004 .

McGuinness, C. ( 1993) Teaching thinking: New signs for theories of cognition. Educational Psychology, 13(3/4), 302-311. Retrieved from Ebsco host on June 30, 2004 .

Perkins, D. (1999). The many faces of constructivism. Educational Leadership, 57(1), 6-11.

Pugalee, D. (2001) Writing, mathematics, and metacognition: Looking for connections through studentsâ work in mathematical problem solving. School Science and Mathematics, 101(5), 236-245. Retrieved from Ebsco Host on July 2, 2004 .

Shepard, R. (1993). Writing for conceptual development in mathematics. Journal of Mathematical Behavior, 12, 287-293.

Appendix B

Journal Evaluation Rubric

This evaluation tool will be used to evaluate the fraction computation assigned topic journal writing entries. The requirements are as follows:

1. Explain in your own words the mathematical procedure you used to solve this problem.

2. Write an example problem and include the solution.

3. Explain in detail how and why you might use this skill in the real world.

4. Write down any questions or ideas you have that you would like to share.

 

Score of 1: Entry not attempted or did not attempt to explain, except for an example problem or minimum participation.

Score of 2: Entry included an explanation of the process but was unclear in some parts. Left out one or more of the requirements.

Score of 3: Journal included three out of the four requirements and is free of major mathematical errors or unclear thoughts.

Score of 4: Journal entry included all four components. It is free of significant mathematical errors or unclear thoughts.