It is no longer enough for higher education to focus just on the transmission of information and the retention of facts. Rather, we now expect students to come out of higher education equipped with a range of high-level skills and abilities such as:

- critical thinking skills to enable them to handle and interpret concepts, evidence and ideas
- the ability to think and act as experts
- innovation and creativity to enable them to produce original insights and valuable knowledge for the benefit of society (Imperial College, 2019).

In addition to adopting learning and teaching techniques that foster the development of these high-level skills, we also need to transform higher education assessment practices as well. Within engineering mathematics, where the end of year exam still predominates, this requires careful selection and design of exam questions to ensure students are assessed for competence in both the basic and high-level mathematical skills. In this piece, I look at two taxonomies that aid the development of such exam questions. These two are the Mathematical Assessment Task Hierarchy (MATH) taxonomy (Smith et al., 1996) and Galbraith and Haines’ Mechanical-Interpretive-Constructive taxonomy (Galbraith and Haines, 2000).

## The MATH taxonomy

The MATH taxonomy groups mathematical skills into three groups, A, B, and C.

### Group A skills

Smith et al defined Group A skills as the standard, routine procedures taught to students along with the factual information they have to recall. They further divided this group into three categories:

*Factual knowledge*– the ability to recall previously learned information in the way in which it was given*Comprehension*– the ability to use standard techniques to solve a problem. This includes the ability to recognise symbols in a formula and to substitute into the formula using information learnt previously.*Routine procedures*– the ability to use routine procedures to solve a given problem in the same way previously learnt to solve similar problems. This requires students to have worked on similar problems beforehand.

### Group B skills

Group B skills focus on the ability to use mathematical information in new ways, for example, applying routine procedures to new situations. These skills fall into two categories:

*Information transfer*– includes the ability to transform mathematical information from one form to another, for example from verbal to numerical (or vice versa), or from algebraic to graphical etc. This category also includes the ability to explain the relationships between component parts of a mathematical problem, the ability to recognise whether or not a particular formula or method can be used in a new context. In addition, this category also includes the ability to explain mathematical processes, including summarising in non-technical terms for non-mathematical audiences.*Application in new situations*– the ability to choose appropriate methods and information and apply them to new situations. This includes modelling real life situations, extrapolating new procedures to new situations, and proving previously unseen theorems and results.

### Group C skills

Group C skills are the skills that enable students to apply previously learned concepts to the analysis and solution of mathematical problems for which no routine procedures have been provided. These skills fall into three categories:

*Justifying and interpreting*– the ability to justify and/or interpret a given result, or a result derived by the student. This includes proving a theorem in order to justify a result, the ability to find errors in reasoning, recognition of unstated assumptions, and recognising the limitations of a model and being able to decide on the appropriateness of a model.*Implications, conjectures, and comparisons*– the ability to make comparisons, with justifications, in different mathematical contexts, as well as the ability to draw implications and make conjectures, with justification, when given or having found a result.*Evaluation*– the ability to judge the value of material for a given purpose based on definite criteria. This includes the ability to make judgements, the ability to select for relevance, the ability to argue coherently on the merits, or otherwise, of an algorithm, organisational skills and the ability to rearrange information and draw previously unseen implications from it.

## The Galbraith & Haines’ taxonomy

Galbraith and Haines arrived at their taxonomy independently of the Smith et al MATH taxonomy. However, after reviewing both taxonomies they suggested that their taxonomy could be interpreted as a summary of the MATH taxonomy (Galbraith and Haines, 2000).

Like the MATH taxonomy, the Galbraith & Haines taxonomy also has three levels, although they have a different terminology. The three levels are the *mechanical*, *interpretive* and *constructive* levels.

### Mechanical skills

These skills refer to the routine use of mathematical procedures as cued by the wording of the question. They are equivalent to Group A skills in the MATH taxonomy.

### Interpretive skills

These skills refer to the ability to retrieve and apply conceptual knowledge. They are equivalent to Group B skills in the MATH taxonomy.

### Constructive Skills

These skills refer to the ability to arrive at a solution or conclusion using a range of mechanical and interpretive tasks without the necessary guidance for doing so. This essentially involves the construction of a solution rather than simply selecting between given alternatives. These skills are equivalent to the Group C skills on the MATH taxonomy.

## Concluding remarks

Most mathematics exams only assess Group A skills, with only a few assessing Group B skills as well, and virtually none assessing Group C skills (Brown, 2010). However, Pounteny et al (2002) suggest that we should be teaching undergraduate students up to the level of Group C as this is the skill level at which students are able to demonstrate “understanding to the point of justifying and explaining knowledge, being able to evaluate actions, and the development of new knowledge”. These, according to Pounteny et al, are the skills that we associate with being a mathematician and a problem solver. Moreover, it is clear that these are the skills expected of higher education graduates in the 21^{st} century. Hence, to ensure that the end-of-year exam remains relevant in the 21^{st} century, we have to ensure that a significant proportion of the exam questions is pitched at the Group C / Constructive level.

## References

BROWN, R. G. 2010. Does the introduction of the graphics calculator into system-wide examinations lead to change in the types of mathematical skills tested? *Educational Studies in Mathematics,* 73**,** 181-203.

GALBRAITH, P. & HAINES, C. 2000. Conceptual mis (understandings) of beginning undergraduates. *International Journal of Mathematical Education in Science and Technology,* 31**,** 651-678.

IMPERIAL COLLEGE. 2019. *Project Xeper -The future of engineering teaching *[Online]. Imperial College,. [Accessed 28/07/2019 2019].

POUNTNEY, D., LEINBACH, C. & ETCHELLS, T. 2002. The issue of appropriate assessment in the presence of a CAS. *International Journal of Mathematical Education in Science and Technology,* 33**,** 15-36.

SMITH, G., WOOD, L., COUPLAND, M., STEPHENSON, B., CRAWFORD, K. & BALL, G. 1996. Constructing mathematical examinations to assess a range of knowledge and skills. *International Journal of Mathematical Education in Science and Technology,* 27**,** 65-77.