## Introduction and rationale

This assessment was carried out with the objective of systematically collecting information regarding the student’s abilities in the topic of Fractions and decimals. It also aims at revealing what the students already knew as well as their progress in learning different concepts in these topics over a period of seven years. The resulting information from the assessment is intended to be an avenue as far as decision making as well as preparation for instructions and learning as a whole. Diagnostic techniques were applied during this assessment and the resulting information becomes very useful when it comes to perform necessary diagnosis the individual needs for each of the students while helping the educators in making necessary adjustments in the manner they relay instructions to students. Math students were made aware of the objectives of a given unit, for this instance, fractions and decimals. They were made aware of what they were expected to be equipped with by the end of the topic while also being made aware of the methods to be used in assessing their progress as well as the means of evaluation, Flockhart, D. (2007)

After the assessment process follows an evaluation step, which involves testing the effectiveness, teaching techniques and the level of curriculum strategies by performing tests on the students’ feedback systems and performance. This step helps in determining the quality of the approach adopted by educators thereby identifying the areas that require adjustments if any. The evaluation adopted here was in such a way that it was in accordance with intended objectives of the teaching systems adopted in teaching math as well as the approach used in passing knowledge to the students.Other factors such as culture, gender and socio-economic activities of the students were also factored into the design of this assessment technique in order to ensure that the mode of evaluation adopted is as sensitive and inclusive as possible. This way the students were set free to express their degree of abilities, knowledge and skills with regard to the topics of fractions and decimals. The evaluation approach that this system opted for is an important avenue for delivering progress related reports to parents and guardians of the students in question. It is also a very important decision making guide with regard to students’ progress in arithmetic skills.

## An overview of the knowledge of fractions and decimals

According to *Clarke*, D. M., *Roche*, A., & *Mitchell*, A. (2008), fractions and decimals form a very important aspect of mathematics and therefore should be given close attention. It has also been found out that the process of teaching and learning fractions and decimals is quite hectic and difficult due to many reasons which include problems arising from interpretation, representation and coding difficulties. It has also been understood that learners and teachers of fractions and decimals find themselves misapplying operations meant for whole number operations to fractions and decimals. It is important to appreciate the importance of fractions and decimals in mathematics when delivering this knowledge to learners. This is so especially to the with regard to the mid ages in the mathematics curriculum. Elsewhere the knowledge of fractions and decimals can be used to partition data, whether continuous or discrete. The aspect of mathematics is made up of scenarios that cannot be relied upon to give whole numbers. Situations that are given mathematical descriptions in the field of mathematics do not always give absolute figures. Even the figures in terms of variables, functions and constants that are used in giving mathematical definitions are not always given as whole number figures.

## Background information

The evaluation documented in this report was conducted with the intension of keeping check on the abilities, strengths and weaknesses of the child as far as fractions and decimals are concerned. The study was followed by an evaluation procedure which involved interviewing the child based on the topics covered during seven-year training period. The answers given by the child were then used to approximate his abilities in this particular topic. The interview was divided into different sections with each section being based on the sub-topics taught and learnt during a certain year of study. These sections were interviewed at different points in time (weeks) and spread throughout the entire course so that the resulting responses would reflect the assessment performed during the entire program. The assessments lead to the need to state the areas where the child could comfortably handle and those he couldn’t and hence the need to recommend additional training techniques and new approaches in teaching the child in future.

## The test conditions

The assessment interviews were conducted at a point when the student was confirmed to be in a perfect mental and physical state. This was taken into consideration due to the fact that illnesses and mental disturbances are known to interfere with the manner in which learners respond to assessment questions. Necessary medical tests were carried out to ensure that the child was perfectly fine both mentally and physiologically. The psychological state of the child was also inspected such that it was ensured that he was psychologically ready for the assessment interview. The performance was boosted through motivation as this is well known to have a huge influence on the learners’ performance. A promise of being rewarded should he reach and or pass the pass mark was made with an intention of increasing his aggressiveness. The final condition that was put in place was the perfect setting of the place of interview.

The setting of the interview venue was in such a way that the student was fully comfortable with the surrounding. A peaceful environment that is free from noise and any other forms of interferences was selected. A very important factor that was prioritized was the level of comfort that the student felt during the interview. This was achieved by letting the child familiarize himself with the venue by visiting it several times before the day of the interview.The venue was selected in such a way that it is situated in an area well known to the child. This was important in reducing phobia and accelerating the rate at which the learner gets familiarized with the venue of the interview. A series of interactions that occurred during the interviews and the corresponding responses obtained are discussed below. The discussions are grouped in accordance with the yearly sub contents of the topic that were covered by the learner during the seven-year program. The assessment process was spread through a period of seven weeks with each week representing a year of study.

### Year 1 assessment

First year assessment was performed during the first week and involved investigating the student’s abilities in elementary and introductory knowledge about fractions. This involved finding out how much he knew about the aspect of one-half as one of the two equal parts that forms a whole of any object. It was expected that the student should have been fully contented with the knowledge of one-half as this is usually considered as the simplest and the most elementary fraction. The first concept under investigation was whether the student was fully aware that a fraction means a portion of a whole object. It was then followed by a test o find out whether he was conversant with ‘half’ and whether he was aware that there can only be two halves of one whole item with each half being of the same size as the other.

Greenes, C. E., Dacey, L. S., Spungin, R. C., & Dale Seymour Publications. (1999) argue that it is expected of students to be capable of creating halves and re-joining halves into whole objects such as fruits and other items at this stage of study. The first questions that the student was asked was to define a fraction and possibly give a physical description or definition using learning aids that were provided. Unsurprisingly enough he gave a correct definition which covered up 90% of the total definition of fractions. The only point that was missing in the definition was the fact that the fractional components of a whole object are of equal sizes. The student was then asked to elucidate his understanding of the term one halfas this assessment question was intended to investigate whether he was acquainted with the elementary foundation of this topic. The first year assessment wound up when the student was asked to use the learning aids provided to create demos about his knowledge of fractions with the aspect of halves being prioritized,

### Year two assessment

Here, the knowledge of fractions was expanded beyond just halves and the aspect of quarters and eighths was introduced. This interview was done during the second week and the interviewer’s expectation was that that by this point, the learner was well aware of other sub-units of fractions and hence was conversant with smaller fractions such as quarter and eighth. Another concept that the learner was expected to have grasped as per the Acara document was the ability to understand how shapes of representing halves, quarters and eighths. With the assumption of the above fact, the interview performed here was intended to evaluate the learner’s abilities in understanding the three fractions and their use especially in shape forms. The assessment questions included questions such as definitions of half, eighth and quarter and using cylindrical and non-cylindrical objects to create these fractions as well as using them in shapes.

### Year three assessment

According the Acara plan, the third year is characterized by extended knowledge about fractions. Here, other fraction forms which include odd fractions and extended even fractions are introduced as additional knowledge. The introduced fractions together with the ones learnt in the previous years include ½, ¼, 1/3 and 1/5 together with their multiples. The main learning activities expected of the students here as documented in Kalra, A., Stamell, J., & Murray, M. (2006) include activities such as modelling the fractions; ½, 1/3, ¼, and 1/5 using objects, identifying the multiples of the above mentioned fractions and assembling different fractions mentioned above alongside their multiples into corresponding whole numbers. The questions asked when performing the assessment interview here were therefore intended to test the student’s ability to perform the tasks described here and included questions such as’;

- Can you create different models of the fractions; ½, 1/3, 1/4 and 1/5?
- Mention two multiples of each of the above fractions
- How many of each of the above fractions and their multiples can be assembled into a whole?
- Assemble each of the modelled fractions and their multiples into whole figures

### Year four assessment

The fourth year contains quite a wider coverage as compared to each previous years of study. The elements of coverage here include the knowledge of fractions and their equivalents as well as carrying out simple computations (simplifications) to determine the equivalents of different fractions. It is also expected that the learners are capable of counting in the form of selected fractions in this case, quarters, thirds and halves as well as the ability to identify different fractions and locate them using number lines. The previous knowledge of the learner as far as the place value concept is concerned was assumed to have gone as far as ones only and hence the need to expand it by involving tenths and hundredths. The final aspect expected here was the ability to create a relationship between fractions and decimals and this involves the inter-conversion between fractions and decimals, Kennedy, L. M., Tipps, S., & Johnson, A. (2008) are of the same idea. The assessment questions here included the following;

- Simplify the following fractions; 4/16, 7/35, 9/12, 75/625
- Simplify the fractions, 9/45, 16/64, 7/21, 8/16 and tell which ones are equivalents of ¼, ½, 1/3 and 1/5.
- Convert the fractions in question 1 into decimals,
- Convert the following into fractions; 0.025, 0.8, 1.25, 6.25
- Identify the place values of 4 and 8 in the number 2345.84,

### Year five assessment

The assessment for this year of study was conducted during the fifth year of the study and the items under consideration were those concepts that are expected to have been grasped by the learner based on the Acara fraction schedule. Amongst the concepts that the learner in this year of study is expected to learn include the following; firstly, the learner is expected to be capable of creating comparisons as well as ordering alongside the ability to effectively achieve the location and representation of these fractions on a number line. It is during this stage in learning where the students learn how to solve simple arithmetic computations that involve fractions with common denominators, Merttens, R., & Kirkby, D. (2000). The problem solving is supposed to cover arithmetic operations such as addition and subtraction. Such a learner is capable of achieving the comparison, ordering and representation of decimals on a number line besides being able to appreciate the fact that the place value system can be broadened beyond just the hundredths level. The corresponding assessment questions were as follows;

- State the order of each of the following fractions, ½, 1/124, 7/1000,
- Arrange the following fractions in (a) ascending order and (b) descending order.
- Represent the following fractions on a number line, 1,5, 1/3, ½, and 1/10
- Solve the following; (a) ½ + 1/3 (b)1/2 – 1/3
- State the place value of 8 in each of the following decimals; (a) 12.238 (b) 6.0948

### Year six assessment

During the sixth week of the assessment, the learner was subjected to an assessment for the sixth year syllabus. It is during this year that the learner is capable of performing comparison and hence locating fractions with related denominators on a number line, add and or subtract fractions whose denominators are either similar or related as well as manually or otherwise determining fractions of certain quantities. The learner should manually or otherwise be capable of performing arithmetic operations, mainly addition and subtraction on decimal numbers and perform assessments on how reasonable the answers are by the use of rounding and estimation. Multiplication of whole numbers by decimals as well as division of decimals by whole numbers that are non-zero is also a concept that considered during this year of study, Mills, S., & Koll, H. (1999). The product and quotient operations involving fractions at this level was restricted o within fractions whose results of such operations are non-recurring. The final sub-chapter that was covered during this year of study were multiplication and division of decimals by powers of 10 as well as the design of the relationship between equivalent percentages, decimals and fractions. With respect to the areas of coverage described above, the student was asked the following assessment questions during the interview.

- Evaluate the following; (a) ½ + ¼ (b) 1/3 – 1/9
- Calculate (a) 1/3 of 18 and (b) 5/12 of 72
- Work out 0.987 X 10
^{4} - Convert the following into percentages; (a) 0.95 (b) 1/5
- Express 48% into decimal

### Year seven assessment

This marked the final year of the study as far as fractions and decimals are concerned. The subtopics that were covered during this year of study include the following. Firstly, the learner was made conversant with the knowledge of using equivalence in the location of and representation of both positive and negative fractions as well as mixed numbers on a number line. He should also be capable to perform arithmetic operations (addition and subtraction) on different types of fractions with the extension into fractions whose denominators are neither similar nor related. In addition to subtraction and addition of fractions, another important aspect that is considered the learner is expected to be able to perform multiplication and division of decimals and fractions using either manual means or electronic technologies.

Using these very means, the learner was also expected to be able to give an expression of a single quantity as a fraction of another besides being able to round off floating numbers to the nearest certain decimal places. The relationship that exists between decimals and percentages is also reviewed here as well as the practice of interconverting one to another. It proceeds to cover the determination of percentages of certain quantities while expressing certain quantities as percentages of others using both manual and digital technologies and the ability to obtain solutions to problems that involve simple ratios. The ideas described by Rose, C. M., Minton, L., & Arline, C. (2007) were used to design the corresponding assessment questions which included the following;

- Arrange the following fractions in (a) an ascending order (b) descending order; 5/25, 8/32, 1/8 and 2/6.
- Solve the following problems, (a) 1/7 + 5/8 (b) 3/5 – 1/12
- Divide 1/8 by ¼
- Multiply 1/7 by 0.95
- Divide 0.6 by 1/3
- Express 4 as a percentage of 64
- Round off 0.0649 to the nearest thousandths
- Calculate 25% of 400
- Express 55 as a percentage of 880
- In a class of 120 pupils, there are 40 boys. Calculate the ratio of (a) boys to girls (b) girls to the total population (c) boys to the total population.

## Results and discussions

In general, the student’s response to the questions regarding the year one syllabus was quite satisfactory. His definition of the term half was correct except the fact that he failed to mention the fact that he never mentioned that both halves of a whole object are usually of equal sizes. He defined half as two parts of a whole that can be re-joined together to form the whole object. The questions corresponding to the second, third, fourth and fifth years of study were answered quite well. This is a clear indication that the students understanding as far as the concepts taught in these years of study were quite well understood by the student. Most questions covering the fifth year of study were answered correctly. However, there were little difficulties experienced when it came technically advanced concepts such as multiplications of decimals with powers of 10 as well as the determination of the relationships that occur between equivalent decimals, fractions and decimals. The student seemed to have had challenges or difficulties in certain areas in the above mentioned this problem was however not very destructively intense. The seventh year related assessment was characterized by difficulties in handling problems involving ratios.

## Recommendations

On average, it was concluded that the students overall performance of the student was fairly good. Most of the assessment areas were well tackled and the student seemed confident in answering the related questions. However, he struggled in answering the questions that are mentioned under challenges. It was then recommended that the student be made to understand the fact that parts of a whole are usually equal, he probably knows this from the answers he gave and the models he created but it should be made clearer to him. It is also recommended that the student be walked through the multiplication of decimals with powers of 10 as well as be familiarized with problems involving ratios. These ideas should be taught using methods different from the ones that were used before and ensuring that the student clearly grasps the concept this time round.

## References

*Clarke*, D. M., *Roche*, A., & *Mitchell*, A. (2008). Ten *practical*, research-based *tips for making fractions come alive (and make sense*) in the middle years.

Flockhart, D. (2007). *Fantasy football and mathematics: A resource guide for teachers and parents*. San Francisco, CA: Jossey-Bass.

Greenes, C. E., Dacey, L. S., Spungin, R. C., & Dale Seymour Publications. (1999). *Hot math topics: Problem solving, communication, and reasoning*. White Plains, New York: Dale Seymour Publications.

Kalra, A., Stamell, J., & Murray, M. (2006). *Connections maths 10: Stage 5.3/5.2/5.1*. Glebe, NSW: Pascal Press.

Kennedy, L. M., Tipps, S., & Johnson, A. (2008). *Guiding children’s learning of mathematics*. Belmont, CA: Thomson/Wadsworth.

Merttens, R., & Kirkby, D. (2000). *Abacus 4: Assessment book*. Oxford: Ginn and company.

Mills, S., & Koll, H. (1999). *Fractions & decimals: Differentiated photocopiable activities*. Oxford: Ginn.

Rose, C. M., Minton, L., & Arline, C. (2007). *Uncovering student thinking in mathematics: 25 formative assessment probes*. Thousand Oaks: Corwin Press.