To date, no empirical international study based on probability samples has analyzed future mathematics teachers’ learning outcomes in teacher education, as well as little research has focused on teachers’ conceptions about the nature of proof as a mathematical argument. Therefore, the focus of this study is to examine future secondary teachers’ conception of proof. 

Recent research has tended to focus on individuals’ proof-judgments, but it has neglected individuals’ views regarding the nature of proof as a mathematical argument (e.g., Sowder & Harel, 2007; Morris, 2002; Martin & Harel,1989). As a representation of written argumentation, proof-writing shows the process of proving and individuals’ conceptions of proof may possibly be represented in writing their own mathematical argumentations; hence analyzing how proof is written provides us the opportunities to detect and study individuals’ own conceptions of proof.

Research Questions

The goal of this study is to investigate future teachers’ ability to evaluate mathematical arguments and the mode of representation used when evaluating or writing arguments, and our research questions are:

1) What is future secondary teachers’ knowledge of proof in evaluating whether an argument is a proof or not? How does this knowledge vary across countries?

2) In writing proofs, what are the representations of written argumentation used by future lower secondary teachers? How does this preference vary across countries?

Drawing from the work of Stylianides (2007), our use of the term mode of argument representation means how argumentation is presented to justify an argument. We propose three modes of argument representation: narrative (stating a complete argument with/without calculation), algebraic (stating by using formulas and algebraic manipulations), and graphical (stating by using diagrams).

Method

As the first cross-national assessment of teacher education and development, the Teacher Education and Development Study in Mathematics (TEDS-M) was conducted in seventeen countries in 2008. The target population of lower secondary teachers primarily consisted of students who would be certified to teach grade 8 (13-14 year olds), as well as other grade levels depending upon the certification standards of each country. In the overall study, 8,207 future secondary teachers were included. We drew the analysis of U.S. and other participated countries in present study. Specifically, in this presentation we will present analyses of international averages and comparisons between USA and Chinese Taipei regarding reasoning items. As a representative of other participated countries, we selected Chinese Taipei which is the highest achieving country overall reasoning items.  

The TEDS-M study used three assessment formats: multiple choice (MC), complex multiple choice (CMC), and constructed response (CR). Our analysis focused on six released reasoning items. TEDS-M reasoning items focused on understanding justification for the truth or falsity of a statement by reference to mathematical results or properties (Mullis et al., 2007).

To answer the first research question, we analyzed four CMC items that asked respondents to judge whether arguments are mathematically correct proofs for a statement, such as, “If the square of any natural number is divided by 3, then the remainder is only 0 or 1.” To answer the second research question we analyzed two CR items that asked future teachers to write a proof, such as “write a proof that the graph of the sum of two linear functions goes through the intersection point of them.” Responses to these two items were categorized for mode of representation used: narrative, algebraic, or graphical representation.  

Summary of Findings

Since the entire data set is not publicly available, these findings derive from analyses of responses to the released items. The following results are only a small sample of the full results to be presented at the session.

Proof-judgments. Overall, 35% of the future teachers in the TEDS-M study did not recognize the difference between the justification of a specific case and a proof. Also, only 46% of future teachers judged the inductive argument as a mathematically incorrect proof. Almost 51% of U.S. future teachers failed to evaluate the inductive argument correctly and this result is consistent with the research results from Morris (2002) and Martin & Harel (1989). In contrast, only 15% of Chinese Taipei teachers failed to evaluate the inductive argument.

Representations of proof-writing. Based on the analysis, most the future teachers preferred to use algebraic representation in writing mathematical argumentation. In the item to write a proof of the sum graph of two linear functions, only 7% of responses earning full credit provided an argument using the given formulas of the two linear functions (algebraic representation) instead of using graphical and/or narrative arguments. This preference for algebraic representations was most dramatic in responses from Chinese Taipei participants (algebraic representation: 62% vs. narrative/graphical representation: 6%), whereas U.S. future teachers didn’t show significant difference (algebraic representation: 2% vs. narrative/graphical representation: 2%).

Discussion and Conclusion

This observed preference for the algebraic representation raises a question about the nature of proof, whether representing the procedure of calculation or algebraic manipulation is sufficient as proof-writing, which should address ‘the generality of a proof’ (Knuth, 2002). An argumentation has to carefully lay out the steps of the proof in a general way, such as hypothesis, theorem, and conclusion (Balacheff, 2002). It is possible to write a general proof using a narrative and/or graphical representation.

Data analysis on this sample suggests that further study is needed to investigate whether teachers’ preferences in proof-writing might relate to patterns in their conceptions of deductive reasoning shown in proof-judgments. Future study in this area should be careful to characterize mode of argument representations in proof-writing because the classification of the mode makes it possible to take more reliable measures of different preferences in proof-writing.

Because TEDS-M is a cross-national comparative study, we can learn from comparing performance of future teachers across the world on these proof-related items that we might not learn if the study had just been conducted in the U.S. TEDS-M could address various research questions. After the TEDS-M International Data Base releases, we will analyze the data for patterns between individuals’ proof-judgments and their preferences of representation mode in proof-writing and how the patterns vary across countries.