Working toward a third space in the teaching of elementary mathematics

Abstract

Building on work in the area of third space theory, this study documents one teacher’s efforts to create third spaces in an elementary mathematics classroom. In an attempt to link the worlds of theory and practice, I examine how the work of other theorists and researchers – inside and outside the field of education – can create new lenses for classroom practitioners. In addition, the article provides evidence that third spaces may be more difficult to realize than others have described. Rather than forcing a third space to emerge, what this study finds more important is creating an environment that will allow third spaces to surface more organically as students and teachers engage in the everyday life of the classroom.

Keywords:

• third space theory
• teacher research
• elementary mathematics
• critical incidents

Notes

1. All names are pseudonyms chosen by the participants.

2. I use the term ‘student‐teacher’ to represent any uncertified, pre‐service teacher who is pursuing certification through the university.

3. See Compton‐Lilly (2007) for an in‐depth discussion of alternative discourses and how they apply to teacher research.

4. The district and school names have been changed for the purposes of confidentiality.

5. Scores of three (proficient) and four (advanced) are other evaluative ratings teachers can assign to students on the report cards.

6. Retrieved from district web page, 12 June 2006.

7. For the first week of summer school, Kavon was the eighth member of the second‐period class. After moving to another area of town, a bus service was no longer available for Kavon.

8. Again, all names have been changed for purposes of confidentiality. The names used in this report are pseudonyms chosen by the students themselves.

9. Each problem I created offered multiple number choices with which the students could work. Rather than placing the numbers in the problems, I listed the number choices below for the students to insert into the problem. In doing so, the children were asked to pick ‘just right’ numbers – numbers with which the students, themselves, felt they could be successful. Rather than requiring all children to use the same numbers, this process allowed me, as the teacher, to assess the various ways in which the students were approaching the problem rather than monitoring their work for correct answers. Furthermore, with each set of numbers, I attempted to assess a specific mathematical concept. Students came to realize this as we discussed why certain numbers were chosen. Rather than looking at the size of the numbers, we also discussed the different concepts the various numbers addressed. For example, by choosing the numbers 104 and 87, I was trying to push the students who consistently resorted to the traditional subtraction algorithm but had difficulty ‘subtracting across a zero’ to find alternative ways to approach this problem.

10. The Spanish word ‘disparejo’ roughly translates to the English terms ‘uneven, unequal, disparate, and/or different’ (http://education.yahoo.com/reference/dict_en_es/ [accessed 10 August 2006]).

11. For further information on the upheaval surrounding proposed legislation concerning illegal immigration, please see del Barco (2006), Hendricks (2006) and Swarns (2005).