**Source:** Asia-Pacific Journal of Teacher Education, 47:4, 361-382

*(Reviewed by the Portal Team)*

**Theoretical framework**

The authors point out that all models of knowledge include Subject Matter Knowledge (SMK) and Pedagogical Content Knowledge (PCK).

In each of the mathematics specific models, SMK is defined as the content knowledge held by a teacher and their depth of understanding. PCK is the knowledge required to transform a teacher’s SMK into a form that is understandable by students.

Whilst acknowledging that other categories of knowledge are important all models recognize SMK and PCK as the cornerstones of effective mathematics teaching (Ball et al., 2008; Rowland et al., 2005; Schulman, 1986).

The authors noted that Usiskin offered five dimensions to understanding a concept in mathematics: Skill-Algorithm, Property-Proof, Use-Application (modeling), Representation-Metaphor and History-Culture.

The Skill-Algorithm dimension is knowing how to get the required answer or how to carry out a procedure, often in the absence of knowing why.

The Property-Proof dimension is synonymous with relational understanding, the understanding required to know why a mathematical process works.

Use-Application, the third dimension of understanding involves knowing when to apply the process or concept.

Usiskin states that knowing how to do a procedure and knowing why this procedure works does not truly demonstrate a complete understanding unless a person knows where or when it is appropriate to apply that knowledge. The fourth dimension is called the Representation-Metaphor dimension.

Usiskin stated that true understanding of a mathematical concept is only demonstrated when a person is able to illustrate it in some manner.

These four dimensions can be, and often are, learned independently from each other but together represent true understanding (Usiskin, 2012).

Usiskin recognized the interlinked nature of the different dimensions of understanding that should be developed simultaneously.

Due to the alignment between Usiskin’s viewpoint, models of teacher knowledge and the new post primary level mathematics curriculum in Ireland, it was deemed fit by the authors for purpose as a tool for measuring pre-service teachers’ understanding of mathematics.

The following research questions were developed by the authors and underpinned this study:

● What understanding do pre-service mathematics teachers possess on entry to initial teacher education (ITE) in relation to the solving of linear equations?

● What impact does a workshop underpinned by Usiskin’s framework for understanding mathematics have on pre-service teachers’ understanding of linear equations?

**Research design and sample**

The authors of this paper, all mathematics teacher educators, wanted to tackle the superficial understanding of core concepts among pre-service teachers by challenging them to grapple with the reasoning behind many of the steps and algorithms they had simply learned by rote at school, without any enquiry.

To that end, this study was undertaken across one semester in an Irish university with preservice teachers.

The study comprised a series of mandatory “Maths Thinking Classes”, with a view to evaluating participant progress quantitatively.

Both the classes and the assessment instrument were designed utilising Usiskin’s framework for understanding.

The sample comprised 23 participants who were completing a one-year professional diploma in education to attain a post primary level mathematics teaching qualification.

All participants had studied mathematics to degree level at university prior to participating in this study.

A series of classes were held on topics typically learned by rote.

Students were encouraged to use Usiskin’s five dimensions of understanding to approach and explore each mathematical concept, with a view to enhancing their conceptual understanding and thus improving their SMK and PCK.**Instrument design - **A ten-item assessment instrument, based on the content covered within the classes, was designed by the researchers to align with Usiskin’s framework for understanding.

The pre- and post-test were identical in order to allow the researchers to analyse the change, if any, which occurred as a result of the ten-week intervention.

**Results and discussion**

It was found by the authors that there was a statistically insignificant difference between the pre- and post-test marks in any dimension for the equation section of the assessment.

The data collected by the researchers suggest that these pre-service teachers hold a superficial understanding of linear equations. All 23 students were able to correctly perform the procedure in the pre–test, thus demonstrating proficiency in terms of their procedural understanding.

Despite this strong performance in the Skill-Algorithm dimension, it was found that many students in this study were unable to adequately explain the procedure behind solving a linear equation.

The second research question aimed to investigate if a workshop, underpinned by Usiskin’s framework for understanding mathematics, could impact on pre-service mathematics teachers’ conceptual understanding of linear equations. While the framework allowed the authors to see improvements and declines in the standard of students’ answers, the statistical analysis carried out by the authors indicated that, overall, there was no statistically significant difference in the performance of participants in the elementary algebra section of the assessment, nor within any of the 4 dimensions tested.

Such findings indicate that the progress (or lack thereof) that students made over the course of the intervention was not significant and students, on completion of the intervention, still had poor relational understanding. A large number of students could not think of a method to visualize the problem or the solution nor could they think of concrete materials that could help them in this regard.

**Conclusion**

The authors conclude that this intervention was too ambitious in terms of the amount of content covered and the assumptions made in relation to students’ prior knowledge.

As a result, the pre-service teachers were not exposed to each of the four dimensions of the framework for a sufficient amount of time and the authors firmly believe that this was the reason behind the lack of improvement for many students in three of the four dimensions.

A single one-hour class on each topic was insufficient to impact on participants’ understanding in a significant way.

The authors suggest that capstone modules in which school mathematics is revisited from an advanced perspective need to be developed and taught in tertiary institutions to allow pre-service teachers time to develop a comprehensive understanding of key mathematical concepts.

They note that many aspects of the intervention are beneficial and yield positive results as can be seen by some of the improvements reported, but trying to cover ten different concepts in ten weeks is not realistic and will not give the pre-service teachers the time necessary to develop the levels of understanding required for teaching.

**References**

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?. Journal of Teacher Education, 59(5), 389–407

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The Knowledge Quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255–281.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Retrieved from http://lchc.ucsd.edu/mca/Mail/xmcamail.2015-04.dir/ pdfpRSc5p4oW_.pdf

Usiskin, Z. (2012). What does it mean to ‘understand’ some mathematics? Paper presented at the 12th International Congress on Mathematics Education. Seoul, Korea. doi:10.1007/978-3-319- 17187-6_46