The Many Faces of TPACK/Math Teacher Education

=A Brief History of Integrating Technology into Math= by Murat Kol

Technology in our life is like the oxygen in the air. It became an essential part of our life even we did not think of it as before. Almost in all aspects of life there are dramatically important technological changes affecting our life. However the effect of technology in education system is not as overwhelming as in the other areas of life. This lack of change is not a result of insufficient technological opportunities for education. Kaput states the reason for this statement as “major limitations of computer use in the coming decades are likely to be less a result of technological limitations than a result of limited human imagination and the constraints of old habits and social structures” (Kaput, 1992, p.515). Besides the opportunities of educational technologies, the creativity of human mind is another important source for technology integration. There are many researchers believe that a vast amount of opportunities are possible with technology and it promotes the acquisition of concepts quickly and easily (Hitt, 2011). NCTM specially emphasizes the importance of technology in principles as ‘‘Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning’’ (NCTM, 2000, p. 11). 21st century technologies are mainly centered on the computers. Within these computer environment many different software are trying to be integrated into mathematics education such as dynamic geometry environments (DGEs), computer algebra system (CAS), spreadsheets, graphing calculators, statistics packages and graphing software for almost 25 years. For example, there are tens of DGE’s in the market and six DGE’s are listed as original DGE’s in PME group whereas the others are considered as clones of these DGE’s (Laborde et al., 1996). Calculators, especially graphing calculators have many effects on the way we teach mathematics (Waits & Demana, 2000). These devices can do complicated arithmetic computations easily, graph even uncommon functions, perform hard integrations, solve equations and so on. Traditional calculators have the ability to deal with equations numerically. When mathematicians need to deal with equations symbolically, they use CAS which enables the quick manipulation and calculation of algebraic routines. The main aim of a CAS is to automate tedious and sometimes difficult algebraic manipulation tasks.

Although the capabilities of software used in mathematics education evolved rapidly, their progress is not fast enough as desired and they are mainly used for drill and practice for topics which are previously developed in the classroom (Niess et al., 2009; Bowers & Stephens, 2011). Besides the importance of the integration of technology in education a careful implementation is needed while using it. A well-balanced implementation is required for successful instruction between the use of technology and traditional methods like paper-pencil activities (Hitt, 2011). Another important aspect of integration is the access to the technology to be used. If a change the way we teach math using technology is planned, we have to prepare the opportunities to be used by everyone (Waits & Demana, 2000). All teachers, students and other actors should have the access to use them. When we talk about the integration devices, like graphing calculators, into math lessons all classrooms and students should be equipped with these machines. Otherwise, change can occur only in math curriculum with no application.

=TPACK in Math Teacher Education=

Although technology was a great “extra credit” for a teacher, it is nowadays not an “extra” but a “must ability”. A teacher without technology knowledge may not survive in futures’ education system. At this point an important challenge for teacher educators arises here. How the teachers should be prepared so that they are equipped with technology literacy? How the technology should be integrated into math education? The researchers developed TPACK as a total package for this “wicked problem”.

Most of the math curriculum of precollege and undergraduate education does not enfold the technology. They traditionally include course(s) about technology (Niess, 2005). However integrating technology should not be about technology. Instead of the amount or type of technology used, Earle used the terms how and why it is used (Earle, 2002, p. 8). Therefore the emphasis should be on learning and on curriculum designed for that aim. As Niess stated “learning subject matter with technology is different from learning to teach that subject matter with technology”(Niess, 2005). In order to equip the teachers with the ability to teach the subject matter with technology, teacher education programs should be redesigned according to this need. Many of them do not prepare the pre-service teachers to use technology effectively in their teaching practice since they offer no or a limited number of courses teaching how to teach mathematics using technology (Baki, 2000; Leatham, 2006). Albion et al. indicate the main reason as the lack of confidence of pre-service teacher educators (Albion, Jamieson-Proctor, & Finger, 2010). When a pre-service teachers enters a teacher education program they most probably prone to teach mathematics in the way that they learned. Their preconceptions about mathematics are closely related to their prior roles as students (Niess, 2009; Grootenboer, 2008). Hence the teacher educators have a great responsibility to put the important piece to the puzzle. Since integrating ICT into mathematics education can challenge “signature pedagogies” pre-service teacher educators should develop a new approach (Larkin, K., Jamieson-Proctor, R. & Finger, G., 2012). One of the legs of the difficulty underlying the ICT integration arises here. In order to prepare technologically literate teachers preservice educators must take on responsibility by changing their way of teaching. By using ICT elements like software, wikis, forums, media, etc. in the courses offered in education faculties, preservice educators give students a chance to tread in their footsteps in later teaching practice. Hammond et al. (2009) point out this strong influential factor. This is a sort of sequential effect that needs a predecessor. Today’s preservice educators should choose to be the locomotive of this train so that their students follow them.

TPACK may help the educators by being “an overarching” conception for the famous integrating technology issue into education. By the help of TPACK, teachers do not only use technology in schools, they also know how to integrate these technological tools to a selected content with choosing the most appropriate pedagogies for the selected content and technology. A math teacher equipped with TPACK is not solely prepared for today’s classrooms but they get the knowledge for navigating within the classrooms of tomorrow (Lee, H., & Hollebrands, K., 2008). Besides the utmost importance of TPACK, it is much more complex than just defining the intersection of the “three circles”. How can TPACK of a math teacher can be defined? Niess (2005) modified Grossman’s (1989, 1990) four components of PCK to characterize the outcomes for TPCK: “(1) An overarching conception of what it means to teach a particular subject integrating technology in the learning; (2) Knowledge of instructional strategies and representations for teaching particular topics with technology; (3) Knowledge of students’ understandings, thinking, and learning with technology; and (4) Knowledge of curriculum and curriculum materials that integrate technology with learning.” Rogers (1995) depicted a sequential process to whether embrace or refuse when a person is faced to an innovation. Deriving from Rogers’ model Niess, Sadri, and Lee (2007) introduced a new model for a teacher learning to integrate a specific technology in teaching and learning mathematics. According to this reframed progress, a teacher trails the following levels: (1) Recognizing (knowledge); (2) Accepting (persuasion); (3) Adapting (decision); (4) Exploring (implementation); (5) Advancing (confirmation). Then Association of Mathematics Teacher Educators (AMTE) prepared a visual for these TPACK levels. Niess et al. (2009) emphasize an important caution on this progress that although the levels are seen to be a linear, it does not need to show a regular increasing pattern. Niess et al. (2009) prepared a development model for mathematics teachers TPACK. This model is constructed I four themes namely Curriculum and Assessment, Learning, Teaching, and Access. In each themes five TPACK levels and different descriptors are given in an overarching manner. In different themes and descriptors a mathematics teacher may belong to different levels.

An important mission for teacher educators is to check their existing programs and redesign them to prepare teachers with TPACK for teaching mathematics by providing experiences supporting knowledge and skills (Niess, 2009). Since the mathematics method courses have a potential to shape pre-service teachers’ TPACK knowledge these courses should include field experiences on teaching particular topics using appropriate technologies. Field experiences are the areas in which pre-service teachers can design, implement and assess the effect of technologies and solve the lack of understanding to anticipate students’ difficulties (Niess, 2009; Lee & Hollebrands, 2008). With the help of these courses pre-service teachers may experience the affordances of technologies for specific contents by designing their lessons (Agyei & Voogt, 2012). Lyublinskaya and Tournaki (2012) recommends collaborative work and guidance for successful lesson planning for better TPACK development. By developing their own lessons, pre-service teachers have the opportunity to think about the curriculum that will probably teach in field.

Review of Trends
Researchers had many studies on TPACK many of which are self-report surveys and rubrics aimed to measure the TPACK knowledge of teachers. Apart from these studies mathematics education researchers contributed to literature in different perspectives. One of the important contributions is developed a model for TPACK levels of teachers developed by Niess et al. (2009) as discussed before in this study. The number of studies trying to measure the impacts of software environments on TPACK dramatically increased in the last years (Shafer, 2010; Zee & Gillow-wiles, 2010; Martinovic & Karadag, 2012; Meng & Sam, 2013). Almost all of the studies show that the designs with software integration into lesson studies contributes teachers TPACK levels. Besides the different software, the impact of hardware components are studied in some researches (Jang & Tsai, 2012; Lyublinskaya & Tournaki, 2012). The upward trends in education like interactive white boards and graphic calculators are investigated in these studies. Similar to studies about software, the studies related to hardware components have also positive impact on teachers’ TPACK knowledge.

Jang and Tsai (2012) report that there is no significant difference in teachers’ TPACK levels according to gender. They also state teachers who have more teaching experience shows higher TPACK than those who have fewer years of teaching experience. However their one of the most important contributions is the study of the effect of teachers’ TPACK levels on students’ achievement scores. Although they cannot find a significant difference in students’ scores, students who are taught by a higher TPACK level teacher have greater average score in exams. They claim that this medium size effect can be a significant difference with a large sample size future study. After that they tentatively conclude that “possibly a teacher’s TPACK level can predict his/her students’ passing rates” (Lyublinskaya & Tournaki, 2012, p.316). This study may be the unique one trying to relate teachers’ TPACK level and students’ achievement in the literature.

=Math Teachers’ TPACK=

Many mathematics teachers are aware of technological opportunities like interactive whiteboards, graphing calculators, dynamic mathematics software, graphing programs, computer algebra systems etc. and use them in their lessons. However the quality of ICT usage is not just using technology itself, but how the selected technology is integrated into a particular content with well-selected activities in the classroom settings. A teacher with efficient TPACK knowledge is assumed to know how to integrate these key technologies to a specific content with specific objectives applying the most suitable pedagogies. 21st century teachers are expected to know how to integrate the technology in every aspect of education like curriculum designs, implementation, management and evaluation (Jang S. J., Tsai, M. F., 2012). Hence it has the utmost importance for a teacher to be equipped with TPACK knowledge to survive in future’s education system.

There are many countries started initiatives involving technology integration projects in their education systems like FATIH Project in Turkey. Many of those projects are based on only providing the technologies. Although they are aimed to have big impact on the education systems, they do not have one as the professional development of in-service teachers are ignored. Since today’s teachers main problem is that they learned mathematics in the past and mathematics can be taught as they learned (Niess, 2009), providing only technology cannot be enough for the desired integration. In order to want teachers to change their teaching ways using technology, professional development opportunities must be supplied to them (Waits & Demana, 2000; Bos & Lee, 2012). There should be ongoing support for teachers during the academic year. Throughout the ongoing support teachers should help each other by sharing their ideas about efficient technology integration (Niess, Lee, Sadri & Suharwoto, 2006). Since integrating technology in mathematics lessons is a kind of evolution of teaching mathematics, it needs time and experience for teachers to believe in mathematical power of technology (Bos & Lee, 2012). Niess et al. (2006) suggested school support and encouragement from others, access to computers, and more practice for successful support for teachers to improve their integration of technology.

Teachers should be careful while preparing and designing lesson plans. They should build up a lesson focused on the content not the technology itself. Because the main goal of a mathematics teacher is to teach the mathematics not the technology. By considering TPACK as the interaction of three components, the challenge here is the designing the lesson like a recipe of a soup. What should be the order and amount of three ingredients (T, P, C) to have a delightful soup? Is the order important during the recipe? The answers to those questions are important for a well-designed TPACK lesson plan. Let us think about the three components technology (T), pedagogy (P), and content (C) and assume that there are a, b, and c many different alternatives respectively. If we want to list all the possible lesson plans, there are a.b.c many lesson plans available theoretically by fundamental counting principle. However many of these lesson plans are waste. Therefore the important skill for a teacher is to decide which alternative(s) is appropriate for a particular content by selecting technology and pedagogy.

=Future Recommendations=

Plenty of studies related to TPACK can be found in the literature. However we need more content based specific studies situating the impacts of specific pedagogies and technologies on selected content areas. There is an obvious need for longitudinal studies to discover the effects of teacher education programs on teachers’ TPACK development. Is technology of very high worth for investment? In order to answer this question the impact of teachers’ TPACK knowledge on students’ achievement must be determined. Therefore there is clear necessity for studies examining the relationship between teachers’ knowledge and students’ attainments.

=References=

Agyei, D. D., & Voogt, J. (2012). Developing technological pedagogical content knowledge in pre-service mathematics teachers through collaborative design. Australasian Journal of Educational Technology, 28(4), 547–564. Retrieved from http://www.ascilite.org.au/ajet/ajet28/agyei.html

Albion, P. R., Jamieson-Proctor, R., & Finger, G. (2010). Auditing the TPACK confidence of Australian pre-service teachers: The TPACK confidence survey (TCS). In C. Maddux, D. Gibson, & B. Dodge (Eds.), Research highlights in technology and teacher education (pp. 303–312). Chesapeake, VA: Society for Information Technology in Teacher Education.

Association of Mathematics Teacher Educators. (2009). Mathematics TPACK (Technological Pedagogical Content Knowledge) Framework. Retrieved from http://www.amte.net/sites/all/themes/amte/resources/MathTPACKFramework.pdf

Baki, A. (2000). Preparing student teachers to use computers in mathematics classrooms through a long-term pre-service course in Turkey. Journal of Information Technology for Teacher Education, 9(3), 343–362. doi:10.1080/14759390000200090

Bos, B., & Lee, K. (2012). In-service Teachers’ ICT-TPCK Development in an Elementary Mathematics Master Teacher Program. In International Conference on Online Learning (pp. 1–23).

Earle, R. S. (2002). The Integration of Instructional Technology into Public Education: Promises and Challenges. ET Magazine, 42(1), 5–13. Grootenboer, P. (2008). Mathematical belief change in prospective primary teachers. Journal of Mathematics Teacher Education, 11(6), 479–497. doi:10.1007/s10857-008-9084-x

Grossman, P. L. (1989). A Study in Contrast: Sources of Pedagogical Content Knowledge for Secondary English Journal of Teacher Education, Vol. 40, No. 5. (1 September 1989), pp. 24-31, doi:10.1177/002248718904000504

Grossman, P. L. (1991). Overcoming the apprenticeship of observation in teacher education coursework. Teaching and Teacher Education, 7(4), 345-357.

Hammond, M., Crosson, S., Fragkouli, E., Ingram, J., Johnston‐Wilder, P., Johnston‐Wilder, S., … Wray, D. (2009). Why do some student teachers make very good use of ICT? An exploratory case study. Technology, Pedagogy and Education, 18(1), 59–73. doi:10.1080/14759390802704097

Hitt, F. (2011). Construction of mathematical knowledge using graphic calculators (CAS) in the mathematics classroom. International Journal of Mathematical Education in Science and Technology, 42(6), 723-735.

International Society for Technology in Education. (2000a). National educational technology standards for students: Connecting curriculum and technology. Eugene, OR.

International Society for Technology in Education. (2000b). National educational technology standards for teachers. Eugene, OR.

Jang, S.-J., & Tsai, M.-F. (2012). Exploring the TPACK of International Society for Technology in Education. (2000a). National educational technology standards for students: Connecting curriculum and technology. Eugene, OR: Author. Taiwanese elementary mathematics and science teachers with respect to use of interactive whiteboards. Computers & Education, 59(2), 327–338. Retrieved from http://dx.doi.org/10.1016/j.compedu.2012.02.003

Kaput, J. (1992). Technology and mathematics education. In. D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515-556). New York: MacMillan

Laborde, C., Kynigos, C., Hollebrands, K., & Strässer, R. (2006). Teaching and Learning Geometry with Technology. Handbook of research on the psychology of mathematics education: Past, present and future (pp. 275–304). Rotterdam, The Netherlands: Sense Publishers.

Larkin, K., Jamieson-Proctor, R., & Finger, G. (2012). TPACK and Pre-Service Teacher Mathematics Education: Defining a Signature Pedagogy for Mathematics Education Using ICT and Based on the Metaphor “Mathematics Is a Language.” Computers in the Schools, 29(1-2), 207–226. doi:10.1080/07380569.2012.651424

Leatham, K. (2006, January). Characterizing the preparation of preservice secondary mathematics teachers to teach mathematics with technology. Paper presented at The Association of Mathematics Teacher Educators, Tampa, FL. Slides retrieved from http://www.mathed.byu.edu/%7Ekleatham/ResearchProjects/AMTEpresentation2006.html

Lee, H., & Hollebrands, K. (2008). Preparing to teach mathematics with technology: An integrated approach to developing technological pedagogical content knowledge. (G. L. Bull & L. Bell, Eds.) Contemporary Issues in Technology and Teacher Education, 8(4), 326–341. Retrieved from http://www.editlib.org/p/28191

Lyublinskaya, I., & Tournaki, N. (2012). The Effects of Teacher Content Authoring on TPACK and on Student Achievement in Algebra: Research on Instruction with. In R. N. Ronau, C. R. Rakes, & Margaret L. Niess (Eds.), Educational Technology, Teacher Knowledge, and Classroom Impact: A Research Handbook on Frameworks and Approaches (pp. 295–322). IGI Global. doi:10.4018/978-1-60960-750-0.ch013

Martinovic, D., & Karadag, Z. (2012). Dynamic and interactive mathematics learning environments: the case of teaching the limit concept. Teaching Mathematics and its Applications, 31(1), 41–48. doi:10.1093/teamat/hrr029

Meng, C. C., & Sam, L. C. (2013). Developing Pre-Service Teachers’ Technological Pedagogical Content Knowledge for Teaching Mathematics with the Geometer's Sketchpad through Lesson Study. Journal of Education and Learning, 2(1), 1–8. doi:10.5539/jel.v2n1p1

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA.

Niess, M. L. (2005). Preparing teachers to teach science and mathematics with technology: Developing a technology pedagogical content knowledge. Teaching and Teacher Education, 21(5), 509–523. doi:10.1016/j.tate.2005.03.006

Niess, M. L., Lee, K., Sadri, P., & Suharwoto, G. (2006). Guiding Inservice Mathematics Teachers in Developing a Technology Pedagogical Knowledge (TPCK). In Society for Information Technology Teacher Education International Conference Proceedings of SITE 2006. Orlando, Florida.

Niess, M. L., Ronau, R. N., Shafer, K. G., Driskell, S. O., Harper, S. R., Johnston, C., … Kersaint, G. (2009). Mathematics Teacher TPACK Standards and Development Model. Contemporary Issues in Technology and Teacher Education, 9(1), 4–24.

Niess, M. L. (2009). Mathematics Teacher TPACK Standards and Revising Teacher Preparation. In L. Paditz & A. Rogerson (Eds.), 10th International conference “Models in Developing Mathematics Education” (pp. 445–449). Dresden. Retrieved from http://math.unipa.it/~grim/21project.htm

Shafer, K. G. (2010). The Proof is in the Screencast. Contemporary Issues in Technology and Teacher Education, 10(4), 383–410.

Waits, B. K., & Demana, F. (2000). Calculators in mathematics teaching and learning: Past, present, and future. In M. J. Burke and F. R. Curcio (Eds.), Learning mathematics for a new century (pp. 51-66). Reston, VA: National Council of Teachers of Mathematics.

Zee, E. H. Van, & Gillow-wiles, H. (2010). Knowledge Growth in Teaching Mathematics/Science with Spreadsheets: Moving PCK to TPACK through Online Professional Development. Journal of Digital Learning in Teacher Education, 27(2), 42–52.