Scenario: Umay teaches the limit concept
Grade (student age): K 12 (age 17-18)
Contributed by: Semiha KULA, Esra BUKOVA GÜZEL, University of Dokuz Eylül, Turkey
Context – national, curricular, professional, other
Secondary school lasts four years in Turkey, between the ages 15 and 18. According to the national secondary school mathematics curriculum, the limit concept is thought during the last year of secondary school. Teacher education program for secondary mathematics education is a five year program; the first three-and-half years are undergraduate studies and remaining year and half is a non-thesis masters’ program. In these programs, subject matter knowledge, general pedagogical knowledge, and pedagogical content knowledge oriented lessons are taught. There are also lessons related to school-based placement in the last three semesters. Umay is a pre-service teacher and a senior in her program. Umay gave lessons to 13 students about limit at one of the secondary schools her second semester at the school-based placement. Her entire lesson plan consisted of four lessons. Limit in extended real number set was taught for 23 minutes during the third lesson and 14 minutes during the fourth lesson.
Umay began the lesson on the limit in the extended real number set by asking the questions about infinity. She used the pictures of Escher, which are shown below, to explain and discuss the concept of infinity.
Umay asked the students “What can you say about the images reflected on the big enough plane mirrors which are facing each other?”. She tried to associate what the students learned in physics with the infinity concept. Next, she used an example adapted from Hilbert's paradox of the Grand Hotel by herself. She asked the students what could be done if 5 guests came to a hotel having infinite rooms, in each room of which a guest could stay and all these rooms were full. Then she asked the students discuss and find a solution about the case that instead of 5 guests, any newly arriving guests consisted of infinite number people came to the hotel and needed to be accommodated. In the direction of the students’ answers, Umay determined their concept images about infinity. She gave the graphs of two polynomial functions and and asked the students to find how and values would change in the case that x increased or decreased without bound. Umay started her teaching by using graphical representation for the limit at infinity and by asking questions to the students and by getting answers from them she tried to find how and would change in the case that value was 1, 2 and 3 and then in the case that value increased or decreased continuously.
After these answers, she illustrated how to express what they did algebraically.
After Umay wrote the algebraic representation, she expressed it verbally.
Umay: Then what did we say? While was going to infinity, was and it went to infinity. While was going to infinity, went to infinity. Did you see? I multiplied 2 with infinity and the result is again infinity.
Umay began her 4th lesson by expressing verbally and algebraically the properties related to adding and multiplying a real number with infinity. Until this time, Umay realized her own choices because there were no guides how to introduce infinity. Then by using Derive 5, she drew the graph of . This function and its graph was given in the curriculum to be used in this subject.
By using the features of Derive 5, she showed the students interactively how y changed when changed on the graph and talked about the values y took when approaching to from left and right and when was going to + and . She was able to capture the students’ attention more with the use of the graphical representation by using the software. After the graphical representation, Umay used tables that were not on the curriculum to show the change in when was approached to 0 from left and right hand. She presented the change in in the case that approached to and by using two different tables.
In addition to graphical and tabular representations, Umay often used verbal representation to emphasize points and support the conversions during her lectures. At last, she algebraically expressed the functions’ limit values found for the so-called value as follows:
Knowledge Quartet Coding Commentary
Contributed by: Semiha Kula, Esra Bukova Güzel University of Dokuz Eylül, Türkiye
Knowledge Quartet Dimension: Transformation
Knowledge Quartet Code: Choice and use of representations (CUR)
Scenario: Umay teaches the limit concept
During the preparation of her limit concept lesson plan, Umay examined the mathematics curriculum very carefully and found that there was no explanation on how to instruct the concept of “infinity”. Additionally, while teaching limit in the extended real number, it was only suggested to make an introduction by using the graph of the function . Furthermore, in Turkey, there is no guide for teachers in the mathematics curriculum or teachers’ handbooks about what the representations are and how they can be chosen or can be used. Representation is explained by some researchers as any configuration that can denote, symbolize characters, images, concrete objects or otherwise represent something else (Palmer, 1978; Kaput, 1985; Goldin, 1987, 1998 cited in DeWindt-King and Goldin, 2003). Shulman (1986) states that the most important element of PCK is to use representations appropriately in education (Turner, 2008). Use of the representations is also recommended by the researchers (Ball, 1990; Even, 1990; Stylianou, 2010).
Umay made the decisions by herself about which representations that she would use in her lessons and how she would use them in teaching the limit concept. These decisions were important to make as they were not stated in the curriculum and she needed resources to support learning. It was her own choice to use the pictures of Escher and afterwards made an introduction to the limit in the extended real numbers by using the graphs of polynomial functions. Then Umay used the graph given in the curriculum. While using graphs is supported in the curriculum, using software for drawing graphs were not handled. Umay presented this graph by using Derive. In this example, Umay showed the limit to her students, by taking advantages of the dynamics of this software. Also even though the curriculum suggested using graphical and algebraic representations for the limit in the extended real numbers set, Umay used verbal and tabular representations, too. Without a doubt, her using both different representations and made conversions between these representations were important in terms of contributing to understanding of the concept. Kaput (1992) and Moru (2006) state that the use of more than one representation in teaching the concept helps students to develop better understanding of the mathematical concepts and if the relations between different representations are not made, the understanding of the concept cannot be achieved . Similarly, Elia et. al. (2009) stated that understanding the same concept in multiple systems of representations, the ability to show it with these representations and the ability to convert one representation from one system to another is necessary for the acquisition of the concept.
According to the researches, the teachers had difficulty in making connections among algebraic, tabular and graphical representations (Even, 1990; Norman, 1992; Stein et al., 1990, see in Rider, 2004). Even (1990) stated that algebraic and geometric representations are important in understanding the concept of limit and secondary school student teachers were unable to make connections between algebraic and graphical representations. Contrary to this, our finding showed that Umay made connections between these two representations and so on. Umay also used in her lessons mainly four types of representations for the concept of infinity limit and limit at infinity which are: graphical, tabular, algebraic and verbal. She did not have difficulty by using the representations and converting them to each other in her lessons.
Ball, D. L. (1990). The mathematical understanding that prospective teachers bring to teacher education. Elementary School Journal, 90, 449-466.
Dewindt-King A. M. ve Goldin G. A. (2003). Children’s visual imagery: aspects of cognitive representation in solving problems with fractions. Mediterranean Journal for Research in Mathematics Education. 2 (1),1-42.
Elia, I., Gagatsis, A., Panaoura, A., Zachariades, T., & Zoulinaki, F. (2009). Geometric and algebraic approaches in the concept of “limit” and the ımpact of the “didactic contract”. International Journal of Science and Mathematics Education. 7, 765-790.
Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21, 521-544.
Kaput, J. (1992). Technology and mathematics education. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 515–556). New York: Macmillan.
Moru, E. K., (2006). Epistemological obstacles in coming to understand the limit concept at undergraduate level: a case of the National University of Lesotho. Doctoral dissertation, University of the Western Cape.
Rider, R. (2004). The effect of multi-representational methods on students' knowledge of function concepts in developmental college mathematics. Doctoral Dissertation, North Carolina State University.
Stylianou, D., A., (2010). Teachers’ conceptions of representation in middle school mathematics. Journal of Mathematics Teacher Education. 13 (4), 325-343.
Turner, F. (2008). Beginning elementary teachers’ use of representations in mathematics Teaching. Research in Mathematics Education. 10 (2), 209-210.