Investigating the Development of Prospective Mathematics Teachers’ Pedagogical Content Knowledge of Generalising Number Patterns through School Practicum

Jun. 01, 2012

Source: Journal of Mathematics Teacher Education, Vol. 15, No. 3, p. 207–226, June 2012.
(Reviewed by the Portal Team)

This case study was conducted to explore the development of pedagogical content knowledge (PCK) through a school practicum course.
The pedagogical content knowledge (PCK) included two components:
(1) knowledge of students’ understanding and difficulties and
(2) knowledge of topic-specific strategies and representations.

The case study also examined how observation of number pattern lessons in schools and discussions of these observation contribute to prospective teachers’ PCK.

Research questions
The following research questions were explored:
• To what extent do they take students’ understanding of patterns into account, and how do they address difficulties that students might encounter in the processes of abduction, transforming abduction and deducing the formula?
• How do they use pattern-specific strategies such as arithmetic generalisation, algebraic generalisation and naıve induction, and multiple representations of patterns to apply these strategies?

The participants of the study were three female prospective elementary mathematics teachers, who enrolled in a 4-year teacher training programme in a university in I˙zmir, Turkey.
For this study, the authors focus on prospective teachers’ observations in schools and their micro-teaching lessons in the School Practicum (SP II) course.

Data sources
Data sources are the three prospective teachers’ lesson plans, micro-teaching lesson videos, interviews conducted after their teaching episodes, videos of pattern lessons taught by mentors in schools and videos of whole-class discussions during the university component of the SP course.

Discussion and conclusion

The analysis of data indicated a development in prospective teachers’ PCK of patterns throughout the SP course.
One aspect of this development reveals itself especially in the way prospective teachers take students’ understanding of and difficulties with patterns into account, as well as in the way they use pattern-specific strategies during the abduction phase.
Prior to their practicum experience, participants did not use the commonality to provide a direct expression for any term of the sequence.

Furthermore, after the SP course, the prospective teachers tended to notice commonality to provide a direct expression.
This improvement may have emanated from their observations of the first component of PCK: knowledge of students’ understanding of and difficulties with particular topics.

During their second micro-teaching lessons, the prospective teachers emphasised that one needs to find a relationship between the order of terms and consecutive terms to be able to find.
With this approach, students can realize why they need to focus on such a relationship instead of on the relationship between consecutive terms, which might prevent them from seeing a general structure of the pattern.

Another improvement is related to the use of variable in deducing the formula.
The prospective teachers' reflections on the mentors’ and their own lessons indicate that they discovered students’ difficulties with variables and they addressed these difficulties during their lessons.

In addition, observing students in the classroom helped them discover students’ understanding of patterns, the difficulties encountered by them and pattern-specific strategies used by the mentors in a more realistic context.
For this improvement, the mentors played an important role.
Observing the lessons of the mentors contributed to the development of the prospective teachers’ knowledge in many aspects, such as content knowledge, PCK, curriculum expectations and different teaching approaches.

The study also indicated that discussions of the observations in the SP course resulted in an improvement of the prospective teachers’ PCK.
Another improvement of the participants’ PCK is related to the students’ difficulty with finding the rule, which was an important issue handled during the discussions of the observations in university component of the SP course.

Despite these improvements, the prospective teachers’ PCK fell short in two aspects.
The first is related to the use of representations and links between them.
Although their knowledge of pictorial representations improved, the prospective teachers preferred to discover local commonalities using table representations.
The prospective teachers did not have opportunities to observe effective use of pictorial representations in the mentors’ lessons.

The second aspect that the prospective teachers’ PCK fell short is related to the knowledge of topic-specific strategies.
Development was observed in the way they noticed commonalities.
However, when it comes to dealing with more complicated patterns, it was observed that their approach changed direction towards induction from generalisation.

With regard to the issue of effectiveness of the activities in the SP course, it can be concluded that observations in real classroom settings and discussions of these observations in the SP course resulted in an improvement of the prospective teachers’ PCK.

Updated: Jun. 22, 2014