R. Kwan et al. (Eds.): ICHL 2011, LNCS 6837, pp. 212–223, 2011. © Springer-Verlag Berlin Heidelberg 2011

A Study on Effective Math Teaching Strategy Design in Hybrid Learning Environment

Liming Zhang1 and Jianli Jiao2

1 University of Macau, Av.Padre Tomas Pereire, Taipa, Macao

lmzhang@umac.mo 2 South China Normal University,

Guangzhou, Guangdong Province, 510631, China jiaojianli@126.com

Abstract. Computer supported instruction can easily facilitate different learning environments, different teaching models and learning styles. In hybrid learning environment, the teachers face a big challenge to find the suitable teaching strategies for a specific content. In this study, three experiments were conducted to explore the different math teaching strategies for different contents and different characteristics of the students in hybrid learning environment. The results indicated that traditional teaching was more suitable for the algebra related topics. Computer supported hybrid teaching was more effective for the graph related topics. Towards different students’ characteristics, medium and low performance students benefited more from the computer supported hybrid teaching. The traditional teaching was more suitable for the high performance students. The student-centered hybrid learning requested significant more teaching hours to facilitate effective learning results.

Keywords: Math teaching strategy, Hybrid learning, Cognitive load theory, Teach-centered, Student-centered.

1 Introduction

Recent advances of information technology have facilitated a convergence between traditional face-to-face and technology-mediated learning environment, which is so-called hybrid or blended learning. The hybrid learning environments try to take advantage of the merits of both learning environments [1]. Hybrid learning usually refers to using multiple approaches in teaching, including combining technology-based materials and traditional print materials, group and individual studies, structured pace study and self-paced study. So besides the computer supported learning environments, hybrid learning also emphasizes the effective combination of different modes of delivery, models of teaching and styles of learning [2].

Mathematics is one of the most difficulty subjects in the secondary studies. Many researchers have been seeking for new pedagogies to improve the student performance in math. Technology based math teaching is one of the choices and

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playing an increasingly important role in teaching and learning. Commonly used hybrid math teaching approaches could fall into two categories. One category is the courseware or online learning environment based teaching and learning, in which the teaching contents have been developed already. Teachers can use it to demonstrate the abstract math concepts in class, and students can use it to further explore the concepts or practice the exercises by themselves in class or after class. Another category is the dynamic math software based teaching and learning, such as the Geometer’s Sketchpad (GSP), in which the teaching contents have not been developed yet and could be easily developed at the same time during the teaching. Dynamic math software is a dynamic construction and exploration tool that enables teachers and students to explore the mathematics in the way of learning by doing that is simply not possible with traditional tools. Teachers and students can construct an object and then explore its mathematical properties by dragging the object with the mouse in class during teaching and learning. It can easily be integrated into traditional teaching by teachers and effectively support the students’ active self-explorations. Both courseware and dynamic math software could be online or offline versions.

To explore the effective hybrid math teaching strategies, we have the following research questions.

♦ What is the effective teaching strategy in using courseware based hybrid

math teaching? ♦ What is the effective teaching strategy in using dynamic math software based

hybrid math teaching? ♦ Which way is more effective between teach-centered and student-centered

hybrid math teaching?

In this paper, a series of three experiments were conducted. Experiment 1 focused on the effectiveness comparison between courseware based and traditional teachings. Experiment 2 was conducted to compare the effectiveness between the dynamic software based and traditional teachings. Experiment 3 concentrated on the dynamic math software based effectiveness comparison between teacher-centered teaching and student-centered learning. The effective math teaching strategy design is suggested based on the experiment results.

2 Hybrid Learning Environments and Cognitive Load Theory

2.1 Hybrid Learning Environments

There were two different kinds of computer supported learning environments used in the experiments in this study. One was interactive courseware on trigonometric functions. It was developed by using GSP. There were seven sections totally about 70 pages in the courseware, including tutorials, examples, interactive graphs, and exercises in each section [3]. It took about one year to develop it. One page of the courseware in its section 2 is illustrated in Figure 1.

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Fig. 1. Example of the developed courseware

Another learning environment was Dynamic Mathematical Laboratory (DM_Lab), which was one of dynamic math software series developed by the experiment school. This tool had been applied in many math courses in the experiment school and proved to be a useful instructional tool. DM_Lab interface is illustrated in Figure 2 [4]. Both the software used in the experiments are CD ROM and are used in class.

2.2 Cognitive Load Theory

Cognitive Load Theory (CLT) was introduced in this paper to evaluate whether computer supported instruction could support more efficient and deeper learning activities so as to increase the learning effectiveness.

Cognitive load theory is mainly concerned with how cognitive psychology has influenced pedagogy [5]. It builds on an information-processing view of cognition, defining long-term and working memory as main structures of human cognitive architecture. Learning is proceeding well only when the amount of information to be processed in the task is within the bounds of working memory. Once exceeded, it would lead to a shortage in allocations of cognitive resources to impede learning, which is so-called cognition overload. CLT is concerned with engineering of instructional designs in order to avoid overload of learners’ cognitive systems.

CLT researchers have recognized three categories of load during instruction. They are Intrinsic, Extraneous and Germane cognitive loads [6]. First, Intrinsic cognitive load refers to the load placed on working memory by the intrinsic of the materials to be learnt. It is entirely determined by levels of element interactivity. Simultaneously, it is affected by the expertise levels of leaner [7]. Second, Extraneous cognitive load is the load placed on working memory by the instructional design itself. Unlike intrinsic cognitive load, extraneous cognitive load is imposed by inappropriate instructional procedures. It is under control of the instructor. Last, Germane cognitive load is the load placed on working memory during schema formation and automation. CLT assumes a limited working memory connected to an unlimited long-term memory [8]. How to decrease the extraneous cognitive load, in order to free the working memory for tasks associated with the germane cognitive load is the prime goal of instruction.

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Fig. 2. DM_Lab Interface

There are three classic categories of cognitive load measurement techniques: Subjective, Physiological and Task Performance. Subject techniques use rating scales to report the experienced effort or the capacity expenditure [6]. The direct subject measurement was used in this study to assess the extraneous cognitive load of the subject who was learning in the computer supported environment. The NASA-Task Load Index [9] is a kind of subjective measures which consists of rating scales, providing participants six dimensions to perceive cognitive load: mental demand, physical demand, temporal demand, effort, frustration tolerance, and performance. Because NASA-TLX is a multidimensional scale, it is usually administered only once, at the end of a learning or test phase [10]. There is another un-dimensional 9-point scale developed by Paas [11]. Because it has only one item asking participants to rate their perceived mental effort, it can be used during the process of performance as well. Both questionnaires were used in the experiments in this study.

3 Experiments and Data Analysis

There were three experiments being conducted. Since the students during the experiments could not be randomly assigned, quasi-experiments were used in this study. CLT was introduced into the experiment 2 and 3.

3.1 Experiment 1

Experiment 1 was designed to investigate the effective teaching strategy in using courseware based hybrid math teaching.

An experiment was conducted to compare the students’ achievement in two groups. Trigonometric functions were selected as the instruction topic. One class of students served as control group with 40 students was taught by traditional instruction alone. The teaching contents included trigonometric algebraic properties deduction and trigonometric graphs drawing on the blackboard. A similar class of students served as experimental group with 41 students used the courseware as a supplement to traditional instruction. The courseware was designed and developed by the same

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teacher who gave lectures to both the student groups [6]. He had taught the corresponding topic for three years in traditional way with well-designed lecture notes and good experience. The instructional strategy design in courseware was consistent with his traditional instruction. The GSP was used to develop the courseware.

Two classes of Grade 10 students from the same high school in Macao were selected in the experiment. They were randomly designated as experimental group and control group, respectively. Based on the previous semester’s math averages, each group of students were divided into three clusters: high performance, medium performance and low performance. The experiment lasted four weeks. Pretest and posttest were conducted before and after the experiment to compare the students’ achievements. The test papers came from the Teachers’ Book of the trigonometry textbook used in Macao. The question types in pretest and posttest were similar. The questions themselves basically were same. The flow chart of the experiment procedure is illustrated in Figure 3.

Fig. 3. Flowchart of the experiment 1 procedure

Data Analysis. SPSS ANOVA analysis results indicated that though there was no significant difference on students’ achievement from both groups, the difference between the mean of posttest, however, were noticeably greater in the experimental group than those in the control group. It indicated that the courseware supplemental instruction produced achievement effects superior to the traditional instruction alone in this experiment.

The correct answer rate of the examination papers was further analyzed. There were total 30 questions in the pretest and posttest papers. Of which, there were 19 questions related to the trigonometric graphs and 11 questions related to the trigonometric algebra deduction. The correct answer rate in trigonometric graph related questions was obviously higher in the experimental group than that in the control group. In the meanwhile, the correct answer rate in trigonometric algebra deduction related questions was comparatively lower in the experimental group than that in the control group. The results indicated that courseware supported hybrid teaching design can help the students to understand the principle of the graphs better than the traditional blackboard drawing did. To the algebra deduction, however, courseware was not more effective than the traditional blackboard deduction, though all the same deduction was included in the courseware.

Though there was no significant difference on students’ achievement between high performance clusters, the average score of high performance cluster in the control

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group was higher than that in the experiment group. It indicated that the traditional teaching was more suitable for the high performance students.

On the contrary, the average scores of medium and low performance clusters in the experiment group was higher than that in the control group, though there was no significant difference on the final achievement. Moreover, the medium performance cluster in experiment group made greatest progresses among all the students. It indicated that the courseware supported hybrid teaching could help the medium and low performance students to better understand the math concepts. The medium performance students benefit most from courseware based hybrid teaching.

3.2 Experiment 2

Experiment 2 was designed to explore the effective teaching strategy in using dynamic math software based hybrid math teaching.

An experiment was conducted to compare student achievements and cognitive load from two groups of students [12]. The participants consisted of 71 F2 students in a secondary school in Macao. Class A with 36 students was selected as the control group. This group was taught by the traditional teaching alone. Class B with 35 students was selected as the experimental group and was taught by the dynamic math software based hybrid teaching. Based on the previous semester’s math averages, each group of students were divided into three clusters: high performance, medium performance and low performance. The Parallelogram Unit in elementary geometry was selected as the content which was to be taught to the students in the experiment. The content primarily consisted of two sections: Basic Properties and Determinants of Parallelogram, Basic Properties and Determinants of Rectangle, Rhombus, and Square.

The measurement tool for student achievement was the school-based test papers “Parallelogram Unit Quiz I, II, and Ⅲ”. The Self-reporting Questionnaire of Cognitive Load (CL) was also used in the experiment to measure the perceived cognitive load. The questionnaire, which consisted of a single question with a 9 points scale, was adopted in this study. The experiment took place over two weeks during the routine hours of the school day. Both the experiment and control group took 14 geometric lessons. Each lesson lasted 40 minutes. The topic “Basic Properties and Determinants of Parallelogram” was the stage one and was taught in 7 lessons; while the topic “Basic Properties and Determinants of Rectangle, Rhombus, and Square” was the stage two and was taught in further 7 lessons. The same math teacher delivered the entire courses to both groups. The instructional software used in the study was DM_Lab. The flowchart of the experiment procedure is illustrated in Figure 4.

The data of the study came from test papers of pre-test, Parallelogram Unit Quiz I, II, III and Self-reporting Questionnaire of Cognitive Load. Quiz I, Quiz II, and Quiz III were regarded as the mid-test, final-test, and post-test, respectively. The former two tests were conducted during the experiment period with the cognitive load questionnaire together. The post-test was conducted one month after the experiment with no cognitive load questionnaire assessment. All data was analyzed by independent sample t-tests on SPSS 15.0.

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Fig. 4. Flowchart of the experiment procedure

Data Analysis. The result showed that there was no significant difference on both math achievements and students’ self-reported CL between the experiment and control groups. Figure 5 illustrates that in the pre-test, the mean of the control group was higher than that of the experiment group. After the stage 1 teaching, the mean of the experiment group became higher than that of the control group. The means of the control group, however, turned to be higher in the final and posttests than those in the experiment group. To explore the reason, further analysis was conducted to the different performance clusters – high performance, medium performance and low performance students in both the control and experiment groups, which are illustrated in Figure 6.

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Based on the test scores, the dynamic math software based teaching method had negative effect to the high performance students. The traditional teaching method was more suitable for them. In a short term, the dynamic math software based teaching was more suitable for the medium and low performance students to improve their learning achievements, where the low performance cluster made greatest progresses among all the students after the first stage. In a long term, however, there was no advantage.

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Fig. 8. Students’ math achievement of high, medium, and low performance clusters

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Though there was no significant difference on CL between the experiment and control groups, all high, medium and low performance students in the experiment group claimed lower CL than those in the control group as illustrated in Figure 7 and 8, respectively. It indicated that the dynamic math software based teaching method could help the students to reduce their CL in understanding the math concepts.

In the second stage, the traditional teaching outperformed the dynamic math software based teaching method. Interview was conducted after the experiment to explore the reason. The students in the experiment group complained too much learning materials and tight time schedule in using software in the second stage. It indicated that dynamic software based hybrid teaching requested more teaching time than the traditional teaching did.

3.3 Experiment 3

Experiment 3 was designed to investigate effectiveness between teach-centered and student-centered hybrid math teaching.

An experiment was conducted to compare students’ achievement and cognitive load between two groups [13]. The dynamic math software was used in both groups. One group was teacher-centered, in which the teacher used the software in supporting the traditional teaching. The other group was student-centered, in which the students used the software to explore the math concepts by themselves guided by the teacher. The participants in this study consisted of 83 Form 5 students in a secondary school in Macao. Class A with 42 students was selected as the control group. This group was taught by the teacher-centered hybrid teaching. Class B with 41 students was selected as the experiment group and was taught using the student-centered learning method. Based on the previous semester’s math averages, each group of students were divided into three clusters: high performance, medium performance and low performance.

The Ellipse Unit was selected as the content which was to be taught to the students in the experiment. The content primarily consisted of three sections: Ellipse’s formula, Ellipse’s geometric properties and Ellipse’s parameter equation. The instructional software used in the study was DM_Lab.

The measurement tool for student achievement was the school-based final test paper. Experiment cognitive load scale used in the study was a Chinese edition revised from NASA-Task Load Index. This scale has 6 dimensions: (1). mental demand; (2). physical demand; (3). temporal demand; (4). effort; (5). frustration tolerance; (6). performance.

The experiment took place over two weeks during the routine hours of the school day. Both the experiment and control group took 12 Ellipse lessons. Each lesson lasted 40 minutes. The same math teacher delivered the entire courses to both the experiment and control groups.

Pre-test was conducted before the experiment to investigate participant’s mathematics ability in Ellipse. There was significant difference in expertise experience between the experiment group and control group. The reason was that there were several students in the experiment group failed to go up to the next grade and stayed in the same grade for an additional year. To exclude the pre-test effect, the repeated-measures were used in the data analysis in this study. The final math test scores in last semester were also analyzed by SPSS. There was no significant difference between two groups.

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Besides the test, the tasks were also conducted on lessons 2, 3, 6 and 11. The task topics were usually one question and related with the corresponding lecture. The students were requested to finish the task individually within 10 minutes. The 9-point scale Self-report Cognitive Load Questionnaire was conducted directly after the task paper collected. The question was “How difficult was this task?” The flowchart of experiment 3 is illustrated in Figure 9.

Data analysis. The data were analyzed on SPSS. UNIANOVA was used to measure the students’ achievement based on the final test performance. There was no significant difference between the control and experiment group in students’ achievement. The mean in the control group was slightly better than the one in the experiment group. However, there was significant difference in the final question in the test paper between the control and experiment group. The control group achieved significantly higher scores in that question than the experiment group did (t=-3.396, Sig=.001). The question was graphic related question.

Fig. 9. Students’ self-reported CL

There was no significant difference on the sum of CL dimensions between the control and experiment groups. However, there was significant difference in Temporal Demand dimension between the control and experiment groups. The experiment group claimed more time required in learning during the experiment. Based on the higher marks in final test and lower time required in learning, the control group performed better than the experiment did.

In the view of students’ Task, and Task CL, the experiment group performed significantly better in most of the situation, especially in the graph related questions. Considering the combination results of the Task performance and Task CL, the student-centered hybrid learning method could facilitate more efficient learning in non-test performance.

The students’ performance in graph related problems was much better during the non-test performance than those in the test. It indicated that DM_Lab could help the students understanding better the graph related problems; meanwhile it might cause the students’ dependence on the software tool which resulted in worse performance during the test without the software support.

There was no significant difference on the sum of CL dimensions between different performance clusters; however, the medium and low performance clusters in experiment group claimed significant high CL in Temporal Demand dimension. It indicated that teacher-centered method was more effective to the medium and low performance students for the final test. Meanwhile, student-centered method was

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better to them for the non-test results. Teacher-centered and leaner-centered dynamic software based hybrid math teaching methods had the same effectiveness to the high performance students.

4 Conclusions and Suggestions

Experiment 1 demonstrated that courseware based hybrid math teaching method was more effective in teaching graph related topics, though the significant difference between two groups did not reach. Meanwhile, the traditional teaching was more effective in teaching algebra related topics. It seems that courseware might distract the students’ attention from the algebra deduction. It is recommended that teachers should spend more time in class on algebra related topics when they would use courseware supported teaching. If it could be possible to use different teaching methods towards different performance students, traditional teaching would be recommended for the high performance students and courseware supported hybrid learning would be recommended for the medium and low performance ones.

The results of experiment 2 showed that the dynamic software based hybrid teaching method had negative effect to the high performance students. The traditional teaching was more suitable for them. If there could be sufficient teaching hours, the dynamic software based hybrid teaching method was more suitable for the medium and low performance students to improve their learning achievements and reduce their CL in understanding the math concepts.

The results of experiment 3 indicated that the teacher-centered hybrid teaching performed slightly better than the student-centered hybrid learning method in the average marks in the final test. However, the student-centered hybrid learning method performed better than the teacher-centered hybrid teaching in non-test marks. It showed that the normal test paper might not be a good evaluation tool for the student centered instruction. Regarding the experiment CL, the student-centered hybrid learning method requested significantly more time to facilitate the teaching and learning. Considering the combination results of the students’ final test and the experiment CL, the teacher-centered hybrid teaching could be more effective.

In general, the three experiment results indicated in one aspect that only if the teacher could make the same effort in teaching, there would be no significant difference among different teaching methods on overall students’ achievements. In other aspect, however, different teaching methods did have different effects on students’ learning time as well as their understanding towards graph and algebra related topics.

Based on the experiments results, there are following suggestions towards hybrid math teaching strategy design.

Computer supported hybrid leaning, no matter courseware based or dynamic

software based, is certainly more effective in teaching graph related math topics. It is highly recommended to be used in teaching.

Computer supported hybrid leaning, no matter courseware based or dynamic software based, has negative effect in teaching algebra related math topics. It is suggested to use traditional teaching method on those topics. Furthermore, if the algebra related topics were taught within the hybrid teaching period,

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the teacher should pay extra effort to attract the students’ attention to the algebra related math topics for better understanding.

Computer supported hybrid leaning, no matter courseware based or dynamic software based, has slightly negative effect on high performance students. Traditional teaching would be recommended for them.

Teacher-centered hybrid teaching, no matter courseware based or dynamic software based, has positive effect for the medium and low performance students.

Dynamic software based student-centered hybrid learning requests significant more teaching hours. It would have positive effect for the medium and low performance students if sufficient learning time could be provided.

References

1. Graham, C.R.: Blended learning systems: definition, current trends, and future directions. In: Bonk, C.J., Graham, C.R. (eds.) Handbook of Blended Learning: Global Perspectives, Local Designs, pp. 21–33. Pfeiffer Publishing, San Francisco (2006)

2. Heinze, A., Procter, C.: Reflections on the Use of Blended Learning. Education in a Changing Environment. University of Salford, Salford, Education Development Unit (2004)

3. Zhang, L.M., Ho., P.K.: A Case Study on the Necessity of the Teaching Method Design Associated with CAI Implementation in Mathematics Teaching. In: Proceedings of the World Conference on Educational Multimedia, Hypermedia & Telecommunication, Florida, USA, June 26-30 (2006)

4. Wai, F.L.: DM_Lab and experimental geometry instruction. Macao Foundation, Macao (2002)

5. Sweller, J.: Cognitive load during problem solving: Effects on learning. Cognitive Science 12, 257–285 (1988)

6. Sweller, J., van Merrienboer, J.G., Paas, F.: Cognitive architecture and instructional design. Educational Psychology Review 10(3), 251–296 (1998)

7. Kalyuga, S., Ayres, P., Chandler, P., Sweller, J.: The Expertise Reversal Effect. Educational Psychologist 38(1), 23–31 (2003)

8. Kirschner, P.A.: Cognitive load theory: implications of cognitive load theory on the design of learning. Learning and Instruction 12, 1–10 (2002)

9. NASA-TLX. Task Load Index, http://humansystems.arc.nasa.gov/groups/TLX/ (retrieved on February 7, 2011)

10. van Gog, T., Pass, F.: Instructional efficiency: revisiting the original construct in educational research. Educational Psychologist 43(1), 16–26 (2008)

11. Paas, F.: Training strategies for attaining transfer of problem-solving skill in statistics: A cognitive load approach. Journal of Educational Psychology 84, 429–434 (1992)

12. Ouyang, S.Y., Zhang, L.M., Chan, N.H.: A Study on Effectiveness and Cognitive Load of Secondary Math Teaching Using Dynamic Geometry Software PG_Lab. In: Proceedings of International Conference on Information and Communication Technologies in Education (ICICTE 2009), Corfu, Greece, July 9-11 (2009)

13. Zhang, L.M., Chan, N.H., Chu, Y.L.: Cognitive Load Theory Based Effectiveness Evaluation on Dynamic Math Teaching. In: Tsang, P., Cheung, S.K.S., Lee, V.S.K., Huang, R. (eds.) ICHL 2010. LNCS, vol. 6248, pp. 427–438. Springer, Heidelberg (2010)