Page 232 - Children’s mathematical development and learning needs in perspective of teachers’ use of dynamic math interviews

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Appendices
Appendix C: Analytical Framework
To explore the adequacy of a dynamic math interview, the following aspects of the
dynamic math interviews can be analyzed.
1. Ratio open to closed questions posed by teacher. Open questions are assumed to elicit greater information and therefore preferred over closed questions. At the start of the dynamic math interview, closed questions may nevertheless be more suitable for the purpose to establish trust or to check the teacher has understood the child correctly. By asking in-depth questions (e.g., What did you mean by that?), the teacher can gain more information or clarity (Delfos, 2001; Ginsburg, 1997). The proportion open questions should be higher than the proportion closed questions.
2. Questions focused on child’s math experiences, beliefs, and emotions. With the intention of a wider scope of a dynamic math interview, the teacher can also ask questions focused on child’s math experiences, beliefs, and emotions. What kinds of mathematical problems do you find easy/hard? What in the mathematics lesson should change/ not change? How does it feel when you cannot solve a problem? (Allsopp et al., 2008; Bannink, 2010; Ginsburg, 1997). The proportion of total number of questions with focus on child’s math experiences, beliefs, and emotions is counted.
3. Questions focused on child’s thinking and problem-solving processes. These questions help gain insight into what the child understands and they do not understand. How did you solve this problem? Tell me. The teacher can obtain an explanation for why the child does not understand things or cannot complete the problem correctly (e.g., Allsopp et al., 2008; Ginsburg, 1997, 2009). The proportion of total number of questions with focus on child’s thinking and problem-solving processes is counted.
4. Questions to check the child knows the right answer. With these questions the teacher can gain information about mathematics achievement levels and mastery of skills. The attainment of process information as opposed to product (i.e., outcome) information should nevertheless prevail for the dynamic math interview to have added value near standardized tests (Franke et al., 2001; Van Luit, 2019). The proportion of total number of questions with focus on checking the child knows correct answer is counted.
5. Questions to identify math learning needs by actively eliciting ‘student’s voice’. By posing questions with a solution-focused character the teacher can help the child begin moving towards solutions and future regarding mathematics learning. Have you ever had great math help? What did the person who gave you that do? What is your next math learning goal? What do you need to reach that goal? are examples of questions that elicit student’s voice (Bannink, 2010). Also increasing waiting time after posing a question can maximise the chances of gaining insight into the child’s own thinking, the child’s ideas, the promotion of commitment, and increased ownership (Black et al., 2004). The proportion of total number of questions with focus on identification of child’s math learning needs by actively eliciting student’s voice is counted.
6. Support given. The teacher can provide support during a dynamic math interview. We distinguished: a) stimulating the child to write down steps in thinking, b) verbal support (e.g. hints), c) verbal support provided by notes by the teacher, d) material support (e.g. manipulate with imitation money), e) use of concrete representations of abstract models, f) use of representations of concrete mathematical actions and situations, g) clear structuring of problem/ task, h) reduction of complexity, i) demonstration, and j) modelling. Support provided four times or more is indicated as most frequently provided support. The tool we developed for the conduct of a dynamic math interview contains the