Pre-service Middle School Mathematics Teachers' Understanding of average
Author(s):
Nadide Yılmaz (presenting / submitting) İ. Elif Yetkin Özdemir
Conference:
ECER 2017
Network:
Emerging Researchers' Group (for presentation at Emerging Researchers' Conference)
Format:
Paper

Session Information

ERG SES D 04, Mathematics and Education

Paper Session

Time:
2017-08-21
13:30-15:00
Room:
W2.05
Chair:
Meinert Arnd Meyer

Contribution

Statistics plays an important role in daily life (Jacobbe & Carvalho, 2011). Hence, developing knowledge and skills to do statistics is one major goals of mathematics curriculum. In grades 5-8, students in Turkey are expected to formulate questions, collect and analyze data, and interpret results (MoNE, 2013). In this scope, students are expected to understand measures of central tendency (i.e., mean, median, and mode) as tools to represent data sets. Average, as a tool for summarizing and describing a data set, is a useful concept that is constructed within the given context in data handling (Watson & Moritz, 1999, 2000). Sometimes it refers to the mean or arithmetic average, sometimes it refers to the value the most frequently observed, that is mode. That is, measures of central tendency are specific types of averages (Van De Walle, Karp, & Bay-Williams, 2010). 

Research on the concept of “average” mostly focused on students’ understandings and cognitive development (Mokros & Russel, 1995; Watson & Moritz, 1999, 2000). For instance, Mokros and Russel (1995) asked open-ended problems to fourth, sixth, and eight graders and examined how they construct and interpret the notion of average. Findings indicate that young students refer to typical, usual or middle, whereas older students also use mean, median, or mode in order to represent average of a data set. Watson and Moritz (1999) focused on students’ ideas of average in different real life contexts. They interviewed with 94 students from grades 3 to 9. Answers were analyzed by means of the levels described by The Structure of Observed Learning Outcomes (SOLO) as, prestructural, unistructured, multistructural, and relational. Also students’ answers were categorized in terms of which measures of central tendency they refer as the average of a data set. They determined six levels, namely preaverage, single colloquial usage for average, multiple structures for average, representation with average, and application of average to one complex and two complex tasks (Watson and Moritz, 2000). These studies revealed that students’ understanding of average are mostly procedural and restricted to the mean, even though median and mode could also be used to represent the average of a data set.

Research also focused on pre-service and in-service teachers’ understanding of “average.” Some studies focused on only the concept of mean (arithmetic average) (Gfeller, Niess, & Lederman, 1999), whereas others examined all measures of central tendency but not relating with average (Groth & Bergner, 2006; Jacobbe, 2007; 2008). On the other hand, some focused on the concept of average (Begg & Edwards, 1999; Callingham, 1997; Russell & Mokros, 1991). For instance, Begg and Edwards (1999) worked with in-service and pre-service elementary school teachers and found that their definition about average prone to answer “was in the middle.” Also they tended to use mean (arithmetic average) rather than median or mode and applied algorithm for computing mean when they were asked to interpret measures of central tendency (Callingham, 1997; Russell & Mokros, 1991). Research also showed that teachers didn’t think data set as a whole when they made interpretation about average and lack of conceptual knowledge (Jacobbe & Carvalho, 2011). These findings, however, are based on limited research, in accessible literature. Hence, this study aims to examine how pre-service middle school mathematics teachers interpret the concept of average in real life situations. In this way, it aims to contribute literature by analyzing pre-service teachers’ understanding of average. This study seeks to answer the following research question: “In what ways do pre-service middle school mathematics teachers interpret the concept of average?”

Method

Participants were 59 pre-service teachers attended in a statistics course on their third year of the undergraduate program in a state university in Ankara. Survey research design was used (Fraenkel & Wallen, 2006). Three open ended tasks were asked in three separate sessions. Each session lasted approximately 45 minutes. Task 1 and 2 were adapted from Watson and Moritz’s (2000) study and Task 3 was developed by the researchers. In Task 1 and Task 2, pre-service teachers were given two real life situations involving average and asked to explain what “average” means for each context. In Task 3, they were also given a real life situation, but the term “approximately” were used instead of “average”. This task was developed in order to compare whether participants’ responses would differ, because the term used for “average” in Turkish (i.e., ortalama) is very similar to the term used for “mean (arithmetic average)” (i.e., aritmetik ortalama).” Participants were also asked to calculate the average of a data set, and to create a data set for a given average. Data were analyzed with respect to SOLO (Structure of Observed Learning Outcomes) model. Responses were categorized by Solo levels, namely prestructural, unistructured, multistructural, and relational (Watson & Moritz, 1999). Prestructural level refers not using to any measures of central tendency or the aspects of the context when describing the average. Unistructural level refers to associating only one aspect of average (e.g., mode or mean) or of the context of the task. Multistructural level refers to considering multiple aspects of average (e.g., considering two or more measures of central tendency to represent the data set but some inconsistencies may occur). Relational level refers to considering average as a way to represent the whole data set in a given context (i.e., considering the given context when determining the value that is representative of the data set).

Expected Outcomes

Results showed that most of the pre-service middle school teachers’ responses level are unistructural (Task1:65%, Task2:60%, Task3:58%). Responses, coded as multistructural level, are less than 20% of the total (Task1:15%, Task2:20%, Task3:18%). On the other hand, responses coded as relational level, corresponded to nearly 10% of the total (Task1:10%, Task 2:12%, Task3: 9%), which shows that just a few pre-service teachers consider the context when deciding which measure of central tendency could be used to represent the data set. Also, pre-service teachers who are at the prestructural level are relatively less (Task1:10%, Task2:8%, Task3:15%). It is worth noting that pre-service teachers’ responses which are at the lowest level and at the highest level are observed the least compared to other levels. To sum up, participants tended to describe average in terms of mean (i.e., arithmetic average). They mainly didn’t consider that mode or median can also be used to represent the average of a data set. Parallel to this finding, they tended to calculate the mean when they were asked to find the average of a data set. Likewise, when they were asked to create a data set for a given average, they used the algorithm for calculating the mean and create the data set accordingly. Even when we used the term “approximately” instead of “average” their perception of viewing average as mean didn’t change. Hence, findings showed that pre-service teachers’ understanding of average was limited to the concept of mean, which is also supported by literature (Begg & Edwards, 1999; Callingham, 1997; Russel & Mokros, 1991). Findings supported that pre-service teachers should be provided with rich opportunities to discuss the role of average as a way to represent of a data set and its relation with measures of central tendency.

References

Begg, A.,& Edwards, R. (1999). Teachers’ ideas about teaching statistics. Proceedings of the 1999 Combined Conference of the Australian Association for Research in Education and the New Zealand Association for Research in Education, Melbourne: AARE. Online: www.aare.edu.au/99pap/. Callingham, R. (1997). Teachers’ multimodal functioning in relation to the concept of average. Mathematics Education Research Journal, 9(2), 205-224. Fraenkel, J. R., & Wallen, N. E. (2006). How to design and evaluate research in education (6th ed.). New York, NY: McGraw-Hill. Gal, I., Rothschild, K., & Wagner, D. (1990, April). Statistical concepts and statistical reasoning in school children: Convergence or divergence? Paper presented at the Meeting of the American Educational Research Association, Boston, MA. Gfeller, M. K., Niess, M. L., & Lederman, N. G. (1999). Preservice teachers’ use of multiple representations in solving arithmetic mean problems. School Science and Mathematics, 99(5), 250-257. Groth,R. E. & Bergner, J. A. (2006). Preservice elementary teachers’ conceptual and procedural knowledge of mean, median and mode. Mathematical Thinking and Learning, 8, 37-63. Jacobbe, T. (2007). Elementary school teachers’ understanding of essential topics in statistics and the influence of assessment instruments and a reform curriculum upon their understanding. Online: www.stat.auckland.ac.nz/~iase/publications. Jacobbe, T. (2008). Elementary School teachers’ uınderstanding of the mean and median. In C. Batanero, G. Burrill, C. Reading & A. Rossman. Jacobbe, T. & Carvalho, C. (2011). Teachers understanding of average. In C. Batanero, G. Burrill, C. Reading, & A. Rossman (Eds), Teaching statistics in school mathematics: Challenges for teaching and teacher education (pp. 199–209). New York: Springer. Ministry of National Education [MoNE] (2013). Mathematics curricula program for middle grades (5, 6, 7 ve 8. Grades). Retrieved from http://ttkb.meb.gov.tr/www/guncellenen-ogretimprogramlari-ve-kurul-kararlari/icerik/150 Mokros, J., & Russell, S.J. (1995). Children’s concepts of average and representativeness. Journal for Research in Mathematics Education, 26, 20-39. Russell, S. J., & Mokros, J.R. (1991). What’s typical? Children’s ideas about average. In D. Vere-Jones (Eds.), Proceedings of the Third International Conference on Teaching Statistics (pp. 307-313). Voorburg, Netherlands: International Statistical Institute. Watson, J. M & Moritz, B. J. (1999). The Development of Concepts of Average. Focus on Learning Problems in Mathematics, 21 (4), 15-39. Watson, J. M., & Moritz, J. B. (2000). The longitudinal development of understanding of average. Mathematical Thinking and Learning, 2(1&2), 11-50. Van de Walle, J.; Karp, K., & Bay-Williams, J. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (eighth edition). Boston: Pearson

Author Information

Nadide Yılmaz (presenting / submitting)
Hacettepe University
Faculty of Education- Department of Elementary mathematics
Ankara
İ. Elif Yetkin Özdemir
Hacettepe University, Turkey

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