As teachers, we are employed to teach students knowledge with the intention that they will be able to understand it and even apply it at a later time without the supervision of the teacher. The biggest problem teachers face with regard to this idea is how to identify whether understanding has occurred. However, to know if there is understanding, we must first know what understanding is.

Let us have a look at a few scenarios:

- If a learner says “Ok, I got that” after you have explained something, does that mean that they understand it?
- If the learner can repeat processes well, does that mean they understand the work?
- If the learner can
**repeat**back to you what you have explained, does that mean that understanding has occurred?

According the *Concise Oxford Dictionary*, the term *understanding* is defined as “the ability to comprehend something; the power of abstract thought”. This definition does not immediately guide us to the answer we need. There are many mentions of definitions of understanding. Barmby, Harries, Higgins & Suggate (2008) refer to several definitions of understanding from other writers, such as:

- “The process of learning relational maths is building up a conceptual structure” (from Skemp)
- “The richer the conceptual context in which one can embed a new fact, the more one can be said to understand the fact” (from Nickerson)
- “Reasoning is the establishment of the links between representations” (from Christou & Papageorgiou)
- “Relative differences between systems allow students to actively develop meaning by contrasting and comparing” (from Steinbring)

To summarise, the learning of maths requires the construction of a structure for concepts. This learning will be of better quality if the new information can be strongly linked to previous knowledge. By making these links and seeing different representations, mathematical reasoning will improve, and development of the reasoning process will occur (through questioning and contrasting of different representations). With this elucidation of definitions, the next step is to examine the ideas of understanding from the teacher’s perspective and the learner’s perspective.

Simon (2006) introduces the concept of *Key Developmental Understandings* (KDU) and defines it as “a conceptual advance that is important to the development of a concept”. Thus, KDU’s are ideas/concepts needed in the successful development of other mathematical concepts and to set the correct learning trajectories. A concept such as knowing the difference between tens and units when learning about numbers is a KDU. If there is a flaw or area that is lacking in this section of knowledge, it will hinder the future development of more complex concepts, such as addition or multiplication of numbers. These concepts rely heavily on understanding. A teacher will aim to get this KDU across to the learner but may not be able to. Through assessment, be it formative or summative, the teacher expects to be able to judge if the learner has attained the KDU’s involved. This should be the basis for the notion of understanding from a teacher’s perspective.

Barmby et al (2008) show us through a Piagetian Constructivist framework that when new information is received, every child in the class will have varying degrees of understanding. This is because they are active participants in the formation of that understanding. This links to Simon’s concept of KDU’s since without completely attaining a specific KDU, full mathematical ability will not be present. The learner will not be able to reach the full “personal understanding” that Godino (1996) speaks of.

Godino (1996) also mentions “Meaningful Prototype Practices” (MPP) which are actions that a person carries out to attempt to solve their problem. With varying degrees or understanding and incomplete KDU’s, these MPP’s will not be completed uniformly. Thus to assist in this, not only the KDU’s and relevant mathematical tools need to be present, but a reason for the problem to be solved is crucial to the learner wanting to work and understand. In Bloom’s Taxonomy, understanding is key to the further steps of Applying, Analysing, Evaluating and Creating. To move forward with a mental process, these steps are imperative.

A problem raised by Hiebert & Carpenter (1992) is that “Understanding usually cannot be inferred from a single response on a single task; any individual task can be performed correctly without understanding”. Thus, if the learner can repeat Pythagoras’s Theorem back to the teacher, does it mean that the learner understands the theorem? To the learner, his knowing it will allow him to think he does understand it. To the teacher, being able to apply the theorem will show his understanding.

To further strengthen this argument, we will look at the results from Brown (1999). He examined student teachers who were not strong at maths when they were in school but were now on a teaching practical as maths teachers. The results found that the learners of these student teachers wanted algorithmic methods, not deep understanding and that the ‘ideal’ notions of teaching from their courses was a far cry from the ‘real world’ classroom they walked into. If the learners in these classes only wanted algorithms to solve their maths, they would be eliminating most strands of Mathematical Proficiency, according to Kilpatrick, Swafford & Findell (2001), and only focusing on the *Procedural Fluency* strand.

In my personal teaching, I have come across a huge difference between my expectations of the learner’s understanding and their own. There is, however, a strong correlation between the “mathematical ability” of the learner and how much they want to understand the maths. There are learners in my current Checkpoint class who do not want to take maths at IGCSE level. They are purposefully choosing to not do it due to their future career choices. In this context, do I fight and teach at a level that will hopefully provide them with everything that they need for understanding where understanding was not before, or do I teach for the context of where this learner is going and teach algorithmically to help them just pass?

When teachers teach, their aim is normally to teach for understanding; however, that aim is not always realised: “Understanding is never complete; for we can always add more knowledge, another episode, say, or refine an image, or see new links between things we already know” (White & Gunston, 1992). Understanding means more than just being able to reproduce a procedure.

The above ideas demonstrate that the concepts of understanding between teachers and learners are often very different. We must be cognisant though that for every rule, there is at least one exception. There are learners in the classes who want a deeper understanding and they will be the ones to have the same idea of understanding as the teachers; just as there will be teachers who think that if the learners can repeat an algorithm, they have taught correctly for understanding. We are unable to discount the context of where the learner wants to go in life. This will influence the validity of the reason for solving the maths. While notions of understanding are different for the majority of learners and teachers and the question of “Do you understand?” becomes superfluous, it is imperative that we never stop trying to determine if there is understanding and always make the utmost effort to provide what is necessary for that understanding to develop.

#### References

Allen, R., Fowler, H., & Fowler, F. (1990). *The Concise Oxford dictionary of current English*. Oxford: Clarendon Press.

Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2008). The array representation and primary children’s understanding and reasoning in multiplication. *Educational Studies In Mathematics*, 70(3), 217-241.

Brown, T., Mcnamara, O., Hanley, U., & Jones, L. (1999). Primary Student Teachers’ Understanding of Mathematics and its Teaching. *British Educational Research Journal,* 25(3), 299-322

Godino, J. D. (1996). Mathematical concepts, their meanings and understanding. In *PME conference* (Vol. 2, pp. 2-417). THE PROGRAM COMMITTEE OF THE 18TH PME CONFERENCE.

Hiebert, J., & Carpenter, T. (1992). Learning and Teaching with Understanding. In D. Gouws, *Handbook of Research on Mathematics Teaching and Learning* (pp. 65-97). New York: MacMillan.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). *Adding it up*. Washington, DC: National Academy Press. (pp. 115-155)

Simon, M. (2006). Key Developmental Understandings in Mathematics: A Direction for Investigating and Establishing Learning Goals. *Mathematical Thinking And Learning*, 8(4), 359-371.

White, R., & Gunstone, R. (1992). *Probing understanding*. London: Falmer.