Maths — Assessing Fraction Knowledge Essay Example

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Assessing Fraction Knowledge


Diagnosis is the best tool used to examine what students understand about fraction sense and how they think mathematically. It also reveals the strategies, which successful students use in tackling a problem. Assessment can also provide teachers with a deeper understanding of the mathematical ideas and concepts that underlie the rules and procedures frequently used. A good diagnosis gives an insight on the instructions and possible changes in teaching techniques (Phelps, 2005). The diagnostic and curricular information afforded by assessments can provide powerful tools for guiding and informing instruction. This paper will give information on how children complete mathematics tasks and diagnose what they know about fractions.


This is a report on the assessment conducted to year four level students in the field of mathematics. The aim of the assessment was to find out what the students know and what they do not know about fractions. Ideally, the task was used before the start of a set work in order to identify a starting point for planning a focused program. It has been realized that students’ written explanations of what they think may not provide sufficient information on their understanding of fractions (Farrell, & Farmer, 1988). Fractions are a part of a whole when the whole is cut into equal parts (Bright, Behr, Post & Wachsmuth, 2004). It consists of a numerator and a denominator .Fraction sense refers to the ability to understand and comprehend what numbers are, the relationship between themselves, their sizes, as well as the effect of operating on them (Hill, 2004). A student with a sense of fraction is able to manage and respond to the mathematical demands of life. Students ought to know how to make sense of numbers, how to use them, and be confident that their problem solving processes will enable them to arrive at a precise solution (Phelps, 2005). In this context, children should understand fractions as parts of a whole; which can be divided in different ways to represent objects, collections or a given quantity. By making students to be aware of key understandings of fractions and giving them exercises, it will be possible for one to point out what they know and what they do not know. This report will detail information on two junior students and how they think about fractions. Various tasks and activities will be undertaken so as to ascertain the methodologies to be used in teaching them math fractional concepts; identification, comparison and equality (Hill, 2004).

Planning the Task

Planning is the most critical step in succeeding imparting knowledge on fractions to junior learners as it gives guidelines throughout the project. I had to dissect the list of problems I wanted to solve. Then I developed goals or the outcome of the task. It was important to know what students think what fractions are and how they can represent them diagrammatically. Before introducing the students on the activities of the day, I arranged for the materials to be used in the task including pencils and models (Steiner, 1987). After identifying the two students, I went ahead and introduced them to the task of the day because they were unaware. In order to get the best information from them, I asked them to write what they think about fractions and the difficulties they have in recognizing and comparing fractions. The first task was to show whether students can identify which of the two fractions, 2/5 and 3/5 is smaller. I told them to draw what they know for easy explanation. I asked them questions like: Which fraction is bigger and why? Draw them on a piece of paper. Probing questions will be used so as get acquainted of the understanding of fractions that students have. I will make drawings of various fractional representations on paper (Hill, 2004). I will also use oranges; cut them into different fractions which I will use to probe children to state what they know about them. The tasks I will plan prior to their execution concerning fractions will be aimed at evaluating children’s perception of the various sizes of fractions and comparing them. From this information, I will be in a better position to diagnose their weaknesses and make a lesson plan on what they ought to be taught concerning fractions Farrell & Farmer, 1988).

Diagnostic Assessment

In trying to understand what children think of fractions, I came up with a series of tasks which will help them understand what they need to know about fractions. These tasks enable me to diagnose and assess their knowledge in fractions and encompass the following among others:

• What the children do

• Specific question to be asked

• How children will learn in terms of objective

I chose two students in a class, Mary and John. John was to work on the first task of identifying which fraction is bigger. I asked John ‘which of the two fractions is bigger? John tried very hard by drawing two different circles, on one he divided by two and the other one he divided by three. Then he told me 2/5 is bigger. I asked him how he got the answer and he said he only compared the sizes of the divided circles. Planning for learning, John needs to learn the nature of fractions especially if he engages in activities which he can partition objects will be helpful (Steiner, 1987). The emphasis will be put on the fractional sense and comparison of fractions’ sizes. I asked the same question to Mary but she could not figure out what to say but she later said that 3/5 is bigger giving the reason that three is bigger than two. I realized that planning for learning activities encourage John to use a strategy of drawing a circle and later partition and comparing them really helped (Thompson, 1984). Mary used a matching strategy. Mary explained, “Half is about the same size as one third. She does not see any problem in saying that one-third and one-half of the same objects are equal. Mary compared the numerators only.

Since Mary had problems in identifying the nature of fractions, the best way to plan for her teaching is to expose her to the idea of fractions in different contexts (Farrell & Farmer, 1988). She needs to be provided with items like a piece of bread and asked to cut into different pieces and then link it to the idea of fractions and symbols .The emphasis need to be on the relationship between the numerator and the denominator.

The second task was of the broken eggs. The purpose of the task was to check whether students could use written symbols to name fractions within collection of items. After drawing on the board eight eggs, three of which were broken, I then ask this time round starting with Mary if she can identify the fraction of the broken eggs. She was unable to name the fraction of the broken eggs but simply used a whole number instead. Mary had little idea on how to name the fraction in this task (Bright, Behr, Post, & Wachsmuth, 2004). It was important to find a more focused starting point by using other task commonly used in fraction mainly to understand halves and quarters. Such a student will move quickly in the understanding of fraction if a focused curriculum is used to suit their needs. John correctly used written symbols to show that 3/8 is the fraction of the broken eggs. Fraction sense describes the development of concepts such as the meaning of a fraction, ways of representing fractions, relationships among fractions, and the relative size of numbers (Ball, Bass & Hill, 2003).

The results of these tasks indicate that students have problems in stating reasons for their comparison of fractions. This means that they have to be vigorously involved in tasks of identifying fractions in collections, objects or quantities so as to make them understand the fractional sense. Engaging students in various activities on identifying fractions and comparing their sizes is an imperative initial step in imparting fractional knowledge to beginners in mathematics. The lesson plan to be developed ought to have various practical tasks which students are involved so as to make them knowledgeable (Thompson, 1984).

Teaching Fractions

When teaching fractions, I had to come up with a lesson plan that addresses the needs of every student. I selected activities that can be adapted to suit the range of students’ responses. Coming up with questions that can be used during and after the activity to help them focus on the aspect of mathematics they need to learn (Behr, Wachsmuth, Post & Lesh, 1984). Math as a subject requires thorough practice and dedication in order for a student to excel. Young students, however, need help in understanding fractions at an early age. Fractions can be internalized more easily by small kids if presented to them in a practical manner. However, it is important to note that mathematics is not something that can be learned on simple thoughts and beliefs.

As a way of helping students understand fractions, encourage them to draw different shapes of fractions. Learning fractions is mainly about creating a visual mathematical pattern for the child to think. Fractions can be easily understood by drawing charts that represent the same fractions in many ways (Ball, Bass, and Hill, 2003). Real life impression helps the children understand important aspects of math fractions. You can use cards to distinguish fractions that will help the child solve the problem much faster. Provide opportunities for students to engage in mathematical discourse and share and discuss their mathematical ideas, even those that may not be fully formed or completely accurate. Provide opportunities for students to build on their reasoning and sense-making skills about fractions by working with a variety of manipulative tools (Hill, 2004).

Lesson Plan

Mary had difficulties identifying fractions. This lesson extension will be fun and will help Mary become familiar with fractions allowing her to become more comfortable with the process. It can be done as an individual lesson, or can incorporate a full class.

Students need guided practice, when learning new skills, they should be asked questions that prompt them to use previously acquired taught skills and strategies. Work should be checked after each attempt to ensure that the skill is not practiced incorrectly due to lack of understanding (Steiner, 1987).

Observing students through this or any lesson will provide clues about the processes that the students are using as they complete the set task. The lesson plan should be well documented with proper planning, use of appropriate diagnostic tasks and then assess the understanding of the children through use of exercises (Phelps, 2005). The teacher ought to understand difficulties of the children in identifying fractions so that practical examples such as broken eggs, divided oranges and pieces of paper can be used in teaching them. Through application of these tasks, students will understand fractions, compare them and even equate them with ease.



High quality assessments can provide teachers with useful information about student misconceptions related to the role of equal parts in the definition of fractions; using and interpreting representations of fractions that display the whole; and understanding equality. These assessments can provide a window into student thinking, and help teachers think about their own instructional practices and concrete ways in which they might be improved (Phelps, 2005). Good assessments help teachers reflect what it means to understand mathematical idea and how assessing is different from looking at computational fluency. Assessments also give teachers the chance to look across grade levels and see how student thinking is or is not getting deeper and richer over time. Mathematically rich performance assessments provide insight into the strategies used by successful students (Thompson, 1984). It gives the idea of the tools that most students use to solve complicated problems thus enabling teachers to transform these strategies into tools for the whole students. This assertion implies that through proper planning and use of easily understandable diagnostic tasks enables the teacher to understand students’ difficulties and implement an appropriate teaching model.


Ball, D., Bass, H., and Hill, H., (2003). Knowing and Using Mathematical Knowledge in

Teaching: Learning What Matters. School of Education and Department of

Mathematics, University of Michigan.

Behr M.J., Wachsmuth I., Post T.R., & Lesh R. (1984). Order and equivalence of rational

numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323- 341.

Bright G.W., Behr M.J., Post T.R., & Wachsmuth I. (2004). Identifying fractions on number lines. Journal for Research in Mathematics Education.

Farrell, M.A. & Farmer, W.A (1988). Secondary mathematics instruction: an integrated approach. Janson Publications: Rhode Island.

Hill, H., (2004). Content Knowledge for teaching mathematics measures (CKTM

Measures). Unpublished manuscript, University of Michigan, Ann Arbor.

Phelps, G., (2005). Equating and Scoring. Learning Mathematics for Teaching,

Unpublished manuscript, University of Michigan, Ann Arbor.

Steiner, H. G. (1987). Philosophical and epistemological aspects of mathematics and their

interaction with theory and practice in mathematics education. For the Learning of Mathematics, Vol. 7, No. 1: 7-13.

Thompson, A.G. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105-127


Maths - Assessing Fraction Knowledge

School of Education

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Dear Parent/Carer:

As part of their development as teachers, teacher education students studying teaching through OUA and Curtin University in the Bachelor of Education (Primary) program enrolled in the unit EDP245 Mathematics Education are required to work with children to learn about children’s understandings of number related concepts and skills. The teacher education students will work with the children as they complete some short mathematics activities.

The teacher education students will examine the data collected from their work with the children and make judgements about the children’s understandings and skills. They will then submit a report about these assessments and their learning as a teacher as a formal assessment requirement of their mathematics education unit. In this process, and in the teacher education students’ formal assignment submissions, the children will not be identified. That is, your child’s name, image, or other features of the work that might identify your child will not be used.

If you are happy for your child to participate in this small study and for his or her work to be used, please sign the form below and return it to the student. If you have questions about the study or activities please contact Audrey Cooke on [email protected]


Audrey Cooke

Name of Child: Mary Casson, Jack Baker, John Carrasco, Martin Pappadopalous

I agree for my child to participate in the short mathematical activities with a teacher education

student, and I agree that work produced by my child may be used as part of a report about the


Parent/Carer signature: D. Casson, M. Baker, M. Carrasco. A Pappadopalous Date: 16- 7-2011