The general public consumes financial products such as loans that are administered in the realm of nominal and effective interest rates. It is debatable if most consumers really understand how these rates function. This article explores the conceptions that student teachers have about nominal and effective interest rates. The APOS theory illuminates analysis of students’ levels of conception. Seventy second-year mathematics students’ responses to Grade 12 tasks on effective and nominal interest rates were analysed, after which 12 students were interviewed about their mathematical thinking in solving the tasks. The findings varied. While some students could not do the tasks due to erratic use of formulae (algebra), I ascertained that some students obtained correct answers through scrupulous adherence to the external prompt of formulae. Most of those students remained stuck at the action and process stages and could not view their processes as mathematical objects. A few students had reached the object and schema stages, showing mature understanding of the relationship between nominal and effective interest rates. As most students remained at the operational stages rather than the structural, the findings accentuate that when teaching this topic, teachers ought to take their time to build learners’ schema for these notions. They need to guide their learners through the necessary action-process-object loop and refrain from introducing students to formulae too soon as this stalls their advancement to the object and schema stages which are useful in making them smart consumers of financial products.

This article is about mathematics major student teachers’ conceptions in the area of financial mathematics with particular reference to effective and nominal interest rates. Second-year mathematics major students’ responses to Grade 12 financial mathematics tasks are analysed using the process of object theories (Dubinsky,

Financial mathematics is not only an enriching mathematics topic in its own right, but it is also a topic of mathematics with practical applications in daily life for everyone. Yet many mathematics teachers do not understand the basics of this topic let alone how to teach it (Pournara,

The nominal interest rate is the annual interest rate without any reference to compounding. The effective interest rate is derived from the nominal interest rate and yields the actual return on investment over a compounding period, which is often more than once per year; it could be half-yearly, quarterly, monthly or even daily. It is unusual to call a rate of interest a nominal rate unless it is compounded more (or less) frequently than once per annum.

Exploring students’ epistemological difficulties through script analysis and interviews is an essential component of quality teaching (Makonye,

Over the last three decades, many articles have been written on the misconceptions that learners show when learning mathematics (e.g. Cockburn & Littler,

In relation to process-object theories of constructing mathematical concepts (Dubinsky,

What are mathematics teacher students’ conceptions of nominal and effective interest rates in relation to APOS and process-object theories?

Since the onset of democracy, the average performance of South African learners on periodic international comparative mathematics tests has been consistently under expectation (Howie,

In this study, the process-object theories of forming mathematical structures inform the exploration of teacher students’ conceptions about effective and nominal interest rates. According to Gilmore and Inglis (

Dubinsky (

An action is a physical or mental transformation of a mathematical entity in response to outside stimuli. Actions may require initiation or mediation by a teacher or peer to direct steps that are explicitly taken towards a goal. It is the beginning stage of learning and making sense of a mathematical situation. Thus, an action conception leads to an operation external to a learner’s mind.

When an individual reflectively repeats an action they may interiorise it into a process. Interiorisation occurs when someone can carry out an action mentally. When someone reflects upon an action without actually engaging with it, they are said to have interiorised that action into a mental process (Aineamani,

When a mathematical entity is ‘seen as an object’ it is seen as if it were ‘a real thing that exists in space and time’ (Sfard,

According to Cottril et al. (

When actions, processes and objects are revisited and a learner has a bird’s eye view of them, they form a schema of the mathematical entity. This schema is organically linked to other schemas. Analysis and reflection on a schema can generate yet another cycle of actions and processes so that new, more advanced mathematical objects and schemas can be formed.

If individuals can carefully compare financial products such as Mashonisa loans, Ponzi get-rich-quick schemes, bank loans, car loans and so on and make an informed decision, they have the schema on objects that are governed by nominal and effective interest rates. Individuals without this schema are in danger of losing out on their life’s savings to schemers. The schemers do their mathematical calculations using nominal and effective interest rates very carefully to hide the disastrous financial effects for their unwary clients.

Dubinsky (

Similarly, Sfard (

Gray and Tall (

The nominal-effective interest rate formula.

But what really is the difference between mathematical processes and objects?

Sfard (

In this formula (see ^{(m)} the nominal interest rate and

Piaget (

Constructivists argue that learners are not explicitly taught the misconceptions they have, but make them by themselves (Confrey & Kazak,

Behaviourist learning theories (McLeod,

According to Davis (

The research used a qualitative research design. Eisner (

Students’ responses to tasks were first analysed under the categories of correct, partially correct and incorrect, as well as not attempted. After this, 12 student teachers were interviewed in pairs to elicit the thinking behind their responses, whether the responses were correct or wrong. The students chosen for interview constituted a stratified sample by performance and gender. The interviews were analysed so that the stages where students faltered in concept formation could emerge.

To ensure reliability of the research, data were collected through both written tasks and interviews. This allowed for probing of students in the interviews to see if they stuck to their written answers and to determine their thinking on nominal and effective interest rates. There was also internal consistency reliability to assess the degree to which different tasks involving the same concept produced comparable results (see

Construct validity helps to guarantee that the measure essentially measures the intended construct, in this case students’ conceptions on nominal and effective interest rates. This was the most important form of validity in this research. I selected nominal and effective interest rate tasks from 2012 and 2013 Mathematics matric examinations as well as from textbooks approved by the Department of Basic Education. This increased the face validity of the research.

To be faithful to the theoretical framework, students’ responses in scripts and interviews were analysed focusing on whether the conceptions found were at the action, process, object or schema stages of concept formation or in transition between one stage and another. Thus, teacher students’ conceptions were analysed through the lens of how they constructed knowledge with the process to object constructions.

Data were analysed both deductively and inductively. Deductive analysis was informed by the APOS framework for building mathematics knowledge. Inductive analysis occurred through grounded theory. Grounded theory is a continuous process of ‘constant comparison’ (Glaser & Strauss,

When students were given written tasks on effective and nominal interest rates, the work was marked and their performance is shown in graphs (see

Percentage performance of teacher students on nominal and effective rate tasks.

Boxer and whisker plot for teacher students’ performance.

Some statistics on students’ performance on the tasks.

Characteristics | Number |
---|---|

Population size | 70 |

Median | 58.5 |

Modes | 57. 60 and 62 |

Mean | 57.3 |

Minimum | 31 |

Maximum | 71 |

First quartile | 54 |

Third quartile | 62 |

Interquartile range | 8 |

Standard deviation | 7.9 |

I now briefly analyse some written work.

One of the items was:

Question 2: Mrs Ndlovu invested R10 000 in the bank with interest compounding monthly. After one year, she had R10 750 in the bank. Calculate 2.2 the effective interest rate and 2.3 the nominal interest rate.

For calculating effective interest rate one student wrote:

This emanates from the following formula:

The student used formulaic reasoning. Given that R10 000 grew to R10 750 in one year, there was no need to go the formula way. They should have used the object stage thinking that R750 interest was earned on a principal of R10 000, so the effective interest rate must have been 7.5%. Thus the student was operating at the action level where they used the formula as a way to process external stimuli.

The substitution was quite correct if they wanted to find the nominal interest rate, but then the student divided on both sides by ‘log’ as if ‘log’ was an algebraic variable representing a number so that it can be ‘cancelled’:

This was a conception of equation balancing, prefaced by the rule ‘you do the same thing to both sides of an equation and they are still equal’ (Pimm,

In trying to answer the same item, some students mixed up the periods, for example

Some students seemed to have proper action and process conceptions as evidenced by correct answers found through use of the formula. However, I wished to be convinced whether they had formed in their minds permanent objects or schemas of nominal and effective interest rates. Such doubts could only be laid to rest through interviews. I sat in a classroom and interviewed students (see

Participant N and Participant P.

How do you understand the difference between nominal and effective interest rate?

I do not fully understand this with … effective. Nominal … when you get interest after 12 months. Effective; monthly, daily interest obtained. Effective interest rate you get more.

How is nominal compounded monthly?

I just stick to the formula.

Explain what you did on 2.

I just used the formula but I did not understand. I try to understand it but I can’t lie. I don’t. I am senior primary specialist.

And the difference between effective and nominal interest?

In nominal it’s fixed in a period of time; will remain the same. Effective interest rate accumulates over time directed by the nominal. The accumulated interest is the effective rate.

Is the effective rate fixed?

The effective rate will change, as the years increase the effective rate increases. The knowledge that the effective rate is more than the nominal helps.

For me it was effective [referring to

Effective rate is an accumulation of compounding period. Interest accumulated is the effective. Does not have a fixed period.

The most difficult part of financial maths is defining difference between nominal and effective interest rates.

Formulae do not allow you to think outside the box. We think that the formula that are given must be used for all the question … sometimes you forget the ± sign in a formula … minus should be there … sometimes it should be +.

Does effective interest rate change over years?

I think it changes.

Participant N, female student; Participant P, male student.

Participant N is confused as she thinks that if the interest is stated once per year then it must be nominal. She is at the action level of conception as she clearly states that she sticks to the formula. Her interpretation that the nominal interest rate is 7.25% is purely directed by her faith in the formula. To her, the formula seems to be the ‘object’ – the end in itself – which really is not the case. Participant N clearly shows that her mathematical thinking on these concepts is governed by formulae – ends in themselves – but she is clearly unhappy because she complains that any mistake in the substitutions leads to

Participant S.

What is the difference between effective and nominal interest rates?

Effective interest compounded monthly rate will be better as opposed to nominal interest rate of a year, because you get more. Nominal is over one year, and effective is compounding monthly. We distinguish between the two because they are different rates. They are different because there are two different formulas for that.

So…

Nominal means like no change. Effective means impact, better impact.

Yes…

Well I can’t calculate the interest rate. [Looks for a book in the bag to get a formula from a past matric textbook in her bag. The student would have rather figured the answer.] [After working] Nominal is 12% and effective is 12.7%.

And why are there different?

I do not know why they are different. I used formula from high school. I do not know the difference between the two.

Participant S, female student.

Participant S thinks the interest rates are different because they have different formulae. That means that she is at the action conception level. She has not yet interiorised the actions into processes. To her they are different ‘objects’ because they look different not because they essentially would earn different amounts of interest at the end of the year (see

Participant Q and Participant R.

What is the difference between effective and nominal interest rates?

I don’t really remember.

Why do we need the two?

With effective you compound every month but nominal it’s once.

Nominal does not depend on compounding. When we had research on interest rates, I wasn’t sure of the difference between the nominal and effective interest rates. I do not know the difference. We just work it out … we just work it out.

Look at the question.

Nominal is 12%. I don’t remember the formula. Effective is 12%. I guessed the answer.

Nominal is 12%, effective is 12% [here the student used a formula].

[Comment: I see here that Participant R understands the question.]

So which investment is better between; 12% p.a. nominal versus 12% effective annual?

12% effective is better because its more reliable.

Participant Q, male student; Participant R, male student.

While Participant Q’s responses are not much different from the other students, Participant R said the ‘effective interest rate is better because it is more reliable’. This he said even though the effective interest rate yielded the same interest amount as the nominal interest rate. Clearly his view is a belief. At the end of the interview with this pair, he said, ‘Most banks give nominal, others give effective interest rate. The reason is to protect them from liability.’ This was an important remark, but it was clear that he could not say

Participant T.

What is the difference between effective and nominal interest rates?

Effective is the actual interest you will be given. Nominal is the stated. If nominal is 12% stated, the effective could be a little more than 12%.

12% p.a. nominal versus 12% effective annual, which is better?

Hoo … the effective is the same … will be the same as the nominal. If it’s more than one year, then the effective will be more.

Participant T, male student.

Clearly Participant T had achieved the object and schema conception level I was looking for (see

Participant Y and Participant K.

What is the difference between effective and nominal interest rates?

The effective I think is compounded many times.

I can’t remember the difference, but one is done monthly and the other quarterly.

Monthly, you pay interest every month.

The effective is better, I am thinking of the equation. The effective gives more interest because it’s compounded more.

You said it could be 10%. The calculation seems right.

12% p.a. nominal versus 12% effective annual, which is better?

I have no idea but you get the same interest at the end of the year, so they are the same.

Participant K, female student; Participant Y, female student.

Participant Y and Participant K did not rely on formulae; they used common sense as we were speaking to each other. They never, unlike Participant Q and Participant R, clutched for their bags to look for formulae from books or calculators when asked a question. To me they had reached schematic conception of nominal and effective interest rates. This is because they had interiorised the actions to processes and encapsulated the processes to objects. They were willing to revisit their conceptions and did not regard them as fixed; this is exemplified by Participant Y saying ‘I have no idea but you get the same….’ Her conceptions came out as appropriate even when she said she was not too sure.

Ethical approval of the research was obtained from the Ethics Committee in Education of the Faculty of Humanities acting on behalf of the University Senate. After obtaining approval each participant gave their written informed consent to participate in the research. They also gave informed consent for the publication of the research in a research journal. In particular the students were keen to see the research published so that it would help them in preparing to teach the topic of nominal and effective interest rates.

Statistical data on performance on nominal and effective interest rate tasks (see

On investigating and assessing students’ written and interview responses to effective and nominal interest rates tasks, four categories of students’ conceptions emerge. The first category consisted of students who failed to get correct answers because they could not scrupulously use the formula in order to obtain correct answers. Thus these students failed to operate at the processes stage; they had not interiorised the actions (Dubinsky, Weller, McDonald & Brown,

In this category, blind and erratic substitution into financial mathematics formulae was common. Participant N and Participant S fall in this category. For example, Participant N said ‘sometimes you forget the + sign in a formula … minus should be there … sometimes it should be plus’ and added ‘the most difficult part of financial maths is defining the difference between nominal and effective interest rates.’ One would assume that students like Participant N would have difficulty in using the formula and as a result get wrong answers. These students are still at the activity level (Dubinsky et al.,

These are the students who got correct answers for wrong reasons. These students were satisfying the ritual of getting the correct answers to get the approval of their teachers and getting good grades but without understanding the gist of what they were doing. Many of these students got correct answers through using formulae they did not understand. Examples of these students are Participant R and Participant Q (see

These were the few students who actually had achieved the object of learning conception. They had already encapsulated the processes into objects ready to incorporate them in their schemas. For example, on item 2:

What is the difference between effective and nominal interest rates?

Effective is the actual interest you will be given. Nominal is the stated. If nominal is 12% stated, the effective could be a little more than 12%

So Participant T operated at object level. He completely understood and gave reasons for his stance.

Participant Y said, ‘The effective is better, I am thinking of the equation. The effective gives more interest because its compounded more.’ Participant Y also had arrived at this level. Participant P also understands as he said, ‘In nominal … it’s fixed in a period of time; will remain the same. Effective interest rate accumulates over time directed by the nominal. The accumulated interest is the effective rate.’

Students like Participant T and Participant Y had clearly arrived at the schema stage of conception. This was shown in their brightly lit eyes, and their exclamations of ‘Hoo’. They knew what was happening and the traps that lay in the questions.

The research question was: What are mathematics teacher students’ conceptions of nominal and effective interest rates in relation to APOS and process-object theories?

I report that most teacher students are at the action conception on these concepts and seem unaware that their conceptions are unsatisfactory and incomplete. They need to advance their conceptions to the desirable object and schema stages. The fact that formulae help them to get correct answers seems to stall their efforts to learn more. Thus, most teacher students in this research showed that their understanding is operational (at action or process stages) rather than structural (at object or schema stages) (Sfard,

Some students said to me: ‘at school we were just given formulae’. This suggests that some of the students’ incomplete conceptions result from teachers who want learners to obtain answers quickly, and are not concerned about developing the lasting and more powerful object and schema conceptions. The implication for teaching mathematics of this is that teachers must not rush to introduce calculators and formulae to learners when they are teaching nominal and effective interest rates. This hinders learners from constructing the object and schema conceptions which are the ultimate goals for teaching these topics, in order for learners to be financially literate. Rather, teachers must encourage their learners to learn about these concepts inductively, through engaging in numerical investigations and exercises that help to make the APOS constructions. Practical real-life street investigations on loans and loan products (e.g. Mashonisa loan sharks versus bank loans, Ponzi schemes, fixed deposit savings and others) go a long way in developing mature conceptions related to nominal and effective interest rates in mathematics teacher students.

The recommendations are that more studies be done on how the APOS theory may be used for research in financial mathematics education.

My special thanks go to my colleagues Marie Weitz and Bharti Pharshotam for encouraging me to do the research and for reading drafts of this manuscript.

The author declares that he has no financial or personal relationships that may have inappropriately influenced him in writing this article.

James invested R50 000 in the bank for 1 year. The nominal interest rate was 8,5%. Find the effective rate. (3)

Mrs Ndlovu invested R10 000 in the bank with interest compounding monthly. After one year, she had R10 750 in the bank.

What can you say about the effective and nominal interest rates here?

Calculate:

the effective interest rate (2)

the nominal interest rate (3)

What is nominal interest rate?

What is effective interest rate?

Why do we need to distinguish between the two?

What is the difference?

Someone invests R1000 for a duration of one year and is awarded an interest of R120.

What is the annual nominal interest rate?

What is the annual effective interest rate?

In the above case, the R1000 was deposited at