Assessing Mathematical Knowledge: Fractions, Decimals, And Multiplicative Thinking Essay.

The Importance of Learning about Fractions, Decimals, and Multiplicative Thinking

The four major NAPLAN questions of style to assess and evaluate the mathematical content of knowledge of decimals and fractions by students have been drafted in this portfolio. Each and every question in this assessment has been explained by the different aspects of conceptions it evaluates, misconception points and how it is linked to the Curriculum of Australia (Australian Curriculum, Assessment and Reporting Authority, 2016). In addition to that, a set of questions has also been provided, these are comprehensively reviewed interview questions that assess how students comprehend fractions. The questions also highlight how this questions are vital to designing teaching and learning experiences. To wrap up, the mathematical three key areas are thoroughly looked into, these areas are fractions, decimals and multiplicative thinking. The conceptions of different aspects will always explain the importance of learning, what should be learned and how is it to be learned, while identifying how this key areas are related.

There is a deep exploration of the two content knowledge of both the elementary teachers and the primary teachers at different levels of experience. These content knowledge includes the Mathematical and pedagogical (Turnuklu, 2007). The content knowledge and how it is linked to the effective teaching skills would be improved by looking at the different contents of mathematics from the various outlooks of the main ideas in mathematics especially the numbers. This will then enable the teachers to use the links and connections of the main ideas to explicitly explain these ideas to the young kids. Most of the teachers will always view that the content they have to offer especially in terms of what the documents about curriculum prescribes is always of that particular level of study of the students. This should not be the case since there need to be a wider outlook of the content, out of the box, and a very powerful knowledge of how this conceptions are constructed at different levels. There should also be a focus on the big ideas that will enable teachers to cope up with the demands of the different mathematical crowded curricula. This article looks at the big ideas of number, putting into look at the count, multiplicative thinking, place value, and the multiplicative partitioning and also investigates the micro content that aids in their enhancement and development. 

Question one:

By the sixth year the students should have a deep understanding of how to represent the fractions in different ways. The questions will therefore involve only two conceptions, the addition of the fraction and identifying the correct answer from the choices provided. The first step by a sixth year student will be to solve the above question by adding fractions with the similar denominator then the second step will rely on the ability of the student to make a connection between equal fractions.

The Conceptions of Different Aspects of Fractions, Decimals, and Multiplicative Thinking

These students will always hold wrong ideas about the fractions which includes the addition and equivalent rules of different fractions (Cueto, 2016).  Option two is provided as a test to those students who have the notion that addition of fractions will always involve the adding of the denominators and the numerators separately. There is also another option three for those students who will always correctly add the numerators but will make an incorrect simplification of the correct fraction. Lastly, there is another option four where the students will always make a wrong multiplication of the numerators and are not able to see the how the statements are not logical.

Answer: 48

By the sixth year of learning, a student should be able to correctly put into use the different operations and also be able to calculate a simple fraction of a quantity where the result is a whole number, with no digital technologies. Most students approach will likely be different from the expected as they can either multiply the fraction by a whole number, multiply the whole number by the fraction or divide the whole number by third and the add to obtain two thirds. Crossing over their methods may make the students obtain an incorrect answer and thus waste a lot of time on the question than expected.

Wrong ideas may arise when these students are unable to multiplicatively think and even find it difficult to apply the rules. Additionally, difficulties may also arise when the students are unable to deal with two and three numbers when multiplying, adding or dividing ( Remillard, 2011). The first option on the choices provided is that if the student forgets that there is a two and just go on multiplying a third of 72. The second option provided is that when the students multiply the fraction and wrongly assume that they calculated a third of the whole number and doubled it, nevertheless that student will realize that it is a mistake to obtain 96 marbles. The fourth option provided since it is just half of the total marbles and should be eliminated. 

Answer: b

By the sixth year of learning, a student should correctly convert the decimals and fractions by using the operations used in the fractions, percentages and decimals (Walshaw, 2016). The above question evaluates the ability of the students to identify how the fractions, decimals and the percentages are connected. Basically, the student must comprehend how fractions (equivalent) are calculated and then identified as either decimals or percentages.

Content Knowledge of Elementary and Primary Teachers

The first option has been provided to assess and set the status quo of the correct values, an easy decimal, a fraction and an equivalent percentage. The second option, which is the correct answer in the above question has been drafted in such a way that the students see the 2 and 5 and assume that the values are equivalent which is not true in this case (Wu, 2005). The third option that most students may rule out has the fraction using the digits 4 and 5 as compared to the decimal counterpart digits 8 and 0 and hence have the notion that the answer is correct since the digits are not equivalent. The fourth option has been provided since some students may find it difficult to determine whether 12/20 is equal to decimal and percent provided in that option by either not being able to simplify the fraction or have calculation difficulties. 

Round the answer to the nearest ten

405.37 x 10 =

(a) 40540       (b) 4070        (c) 4060        (d) 4050.7

Answer: c 

This question assesses the ability of the students to effectively multiply a decimal to the power of ten and then round the answer to the nearest ten (Harel, 2016). It looks at some wrong ideas that the student may have when rounding the digits, especially when the question has a zero in it. In addition, it also looks at the student’s knowledge of the place value.

The first option is provided to show the option that students obtain when they perform an incorrect multiplication or just don’t understand the idea of place values (Thérèse Dooley, 2014). The second option shows those students who make a correct multiplication of the decimal but then make an incorrect rounding off to the nearest ten. The fourth option shows the answer where the students perform a correct multiplication but then make an incorrect rounding off by rounding the tens column alone then forget to eliminate the 0.6. 

Explanation

The assessment on the diagnostic interview assesses the understanding of the students of the First steps in Mathematics key understanding number 5: The students are able to perform an effective comparison and ordering of the fractions and then be able to efficiently place the fractions on a number line. Additionally, the assessment also addresses the ACARA sixth year content evaluator that compares the fractions with their denominators then they are appropriately located and placed on a number line. The assessments help to have a better understanding about the mathematics of fractions and how it is efficient in the effective comparison of the denominators and the fractions (Santiago Cueto, 2016). When the student is able to correctly answer the question provided and also shows adequate knowledge of the content then key understanding six of the First steps in Mathematics can be well looked at. The key understanding six shows that a fractional number can be calculated as either a decimal or a division. What the interviewee will then need cards having the fractions;

The Big Ideas of Number

1.Read each of the following cards of fractions aloud; .

It should be noted if the student is correctly reading the fractions, example is it one over three or one third?

2. On the number line from zero to one, place the fractions below where they are supposed to be and then explain yourself.   Also, explain why one third is larger than one eighth.

It should be noted if the student is able to explain how the denominators are relevant to the total size of the fraction.  It should also be noted if the students are able to see that one eighth is quite small as compared to one half basing on how they are visually represented.

3.  From the number line, identify and remove 1/5 and 1/8 then give the students the 2/3 and 3/4. They should read them loudly and then again placed on a number line. Explain how you placed them on a number line. Explain why 3/4 is greater than 2/3.

It should be noted if the cards lie in the correct position they were placed and if the student is breaking up the number line into thirds or quarters and then placing the fractions (Turnuklu, 2007). Are the students able to compare the fractions of 1/4 and 1/3? Are the students able to provide an equivalent decimal and use it in comparison? Are the students able to compare the denominators and the numerators?

4. The teacher should place 3/5 fraction card on the number line correctly. The teacher should then ask if the fraction card is correctly placed. The student should then explain his/her answer.

It should be noted if the students are effectively comparing the fractions (Handal, 2017). It should also be noted if the students are able to convert the decimals and then able make comparisons. Lastly, it should be noted if the students are able to mentally compare the given fractions.

Multiplicative thinking ability

This is among the major concepts of numbers and also a very significant stage in the understanding of mathematics by the students since it is the next step just after additive thinking and paves way for the algebraic thinking. It links the different aspects of division and multiplication in such a way that they are a good basis where the development of the decimals, ratios, place values, fractions, algebra, higher mathematics and proportions can be built from. Multiplicative thinking has been explained as that capacity to work appropriately with different numbers, being able to solve all sorts of division and multiplication problems and having the methods to communicate ideas of multiplication effectively. To be able to enhance multiplicative thinking into the students then division and multiplication should be taught to the student as one (Australian Curriculum, 2016). When teaching the basic array model in the multiplicative thinking then the terminology should access the facts of multiplication. Teaching students to adjust from additive thinking to multiplicative thinking can be well achieved by doing symbolic, pictorial and concrete representations in addition to the authentic activities that enable them come up with defined meanings and also developing prerecorded and mental processes that entails the ten base system. It is good to note that it is also the responsibility of the teacher to give effective and appropriate opportunities that will enable the students increase their thinking multiplicatively and also comprehend those factors that support mathematics and will be experienced in further studies.

NAPLAN-Style Multiple Choice Questions

The vitality of adjusting from thinking additively to multiplicatively is that those students not well good at the multiplicative thinking don’t have the basic skills and knowledge essentially in performing school mathematics. For instance, an equation with the following properties 2x3, an additive thinker will take it as 2+2+2, which is quite intricate to the studies of such a student when bigger values are provided. The students need to understand that multiplication is not repeated addition but rather it is of more put a quantity into scaling. 2x3 is just three groups of two (Walshaw, 2013). To a more beneficial manner the additive thinking will play a role of self-sacrificial when the property of distribution of multiplicative thinkers is used in arrays, for example 8 eights can be viewed as 3 eights and 5 eights, and when scaling of the numbers is done then additive thinking will be used in quick calculations. However, it is important to note that this adjustment will take several years, since there need to be introduction and evaluation of the ideas which cannot be entirely understood during the initial years but just until the students are into their teen ages. It is therefore very crucial that the elements and properties of multiplicative thinking be enhanced early enough and a thorough research done.

There is always an agreement among different people on the understanding that is needed to enhance the development of thinking multiplicatively, this requires a new approach in the teaching and learning of both division and multiplication at school, appropriate use of the arrays in the studies, particular approach to terminologies and also more than adequate knowledge of the basic facts of multiplication. First and foremost, to enhance the skills of multiplicative thinkers then both division and multiplication should be integrated and then taught as one a concept called the situation of multiplication (Harel, 2016). The importance of teaching this concept is that it focuses on both the above terms (multiplication and division) by looking at three different quantities namely; the size of equal groups, size of each group and the total amount. In a more practical way, if one of the quantity is not known then the other known quantities will either be multiplied or divided in order to obtain the unknown quantity, this is a key conception that is always used in the mathematical algebra. Secondly, there are three quantities encompassed in an array in this concept of multiplicative situation and are very crucial in the enhancement of the skills of multiplicative thinkers, and hence these arrays must be introduced through tangible division and multiplicative situations (Dooley, 2014). For instance, if Bruce has 3 bags of 5 oranges, what is the total number of oranges he has? This is a very great way of visually representing the three quantities in question. The problem above can be symbolically represented as 3x5 and this linked to an array, which will help complete and depict the pictorial, symbolic and concrete model of a teaching termed as constructivist. All this arrays allows the students to develop a more extensive and pliable comprehension of the multiplicative situation and also to fully acknowledge the two-dimensional way of the process in multiplication. Emphasizing of the particular terminologies relating to arrays also have an impact on the gaining of the multiplicative thinking. This specifically factors will include; multiple or product. The language in use will enforce the relationship between division and multiplication and should be introduced early enough during array introduction to help adjust from additive thinking. To ensure maximum efficiency, this language should be aided by large use of things that will necessitate the students learning not through procedures but from standard concepts (Turnuklu, 2007). Finally, it has also been suggested that automatically accessing the facts of multiplication of the standard numbers zero to nine will also play a critical part especially when we were to calculate large numbers. For example, when a student answers a basic fact of multiplication and is also able to prove a substantial reason for the given answer, then his/her willingness and confidence increases as compared to those students who don’t have access to these facts of multiplication and thus inadequate confidence when answering questions and they are not willing to learn new mathematics.

Interview Questions to Evaluate Students' Comprehension of Fractions

The curriculum of Australia main objective is to see that the students are accurate when using division and multiplication to deal with a range of mathematical problems, at the same time understanding how the two are related by the fourth year end. Additionally, the key understanding number 5: repeating of the equal amounts or quantities and then splitting them into equivalent parts will always help relate multiplicative and division properties and also understand the importance of these properties during classroom sessions. For any teacher, it is essential that he/she understands the vitality and theoretical supports of the process. A student will always find it hard to understand the mathematics they encounter in the future especially when there is no authenticity in the activities that are used to develop multiplicative thinking abilities. Some of this activities include the use of charts, objects, arrays, number lines calculators and many others that will help reason and solve the problems. The students will achieve good results when both division and multiplication can be well explained, by looking at the equal groups, partition, combination, comparison and measurement problems (Turnuklu, 2007). The first type of problem involves two known quantities, then either both factors and product is not known or one of the factor and the product are both known while the other factor is not known. A combination problem example includes, how many combinations of three colors are in seven colors. Comparison problems involve two sets of quantities where another set is a complete multiple of the other one. Lastly, measurement problems will involve determining the number of groups if the total amount is known as well as the number in each group. Partition problems will involve knowing the number of objects in every group if the total amount of the groups is known. A thorough temporary teaching system may be required to help develop this process. Students will also help in manipulation of the arrays so that factors of a problem can be easily found, in the process developing their knowledge of the commutative characteristic of the division and multiplication and the inverse relationship they have. Additionally, students will put into use a calculator to look at this relationship as they develop their sense in a quick and more immediate feedback. The distributive property of multiplication can also be highlighted using the arrays and especially when large numbers are involved in the calculation. The equation 24x45 can be written as (20+4) x (40+5). Using of the distributive property helps in the comprehending of the mathematical operations as well as improving mental computations.

Designing Teaching and Learning Experiences

After six years and coming the seventh year, the students are anticipated to be using the mathematical operations involving fractions correctly and are able to represent the fractions in different ways. This will help in the studying of advanced mathematics, enhance decimal study understanding as well as percentages and probability (Remillard, 2011). A good understanding of the fractions will at large proportions rely on the type of aspects of fractions a person is exposed; and this will include the different constructs, ordering, modelling, equivalency, operating and the relationships they have with decimals.

Five different constructs of fractions can be used to give the fractions their meaning; quotient, ratio, part-whole, operator and measurement. The quotient interprets the remainder in a problem involving division when the problem requires equal sharing or partitioning which is expressed as fraction or just a decimal. For instance, 1/4 is the same as 0.25. The ratio construct can be explained as follows; a fraction shows a ratio of two parts to one thing or just one part to a whole (Walshaw, 2016). The part whole construct which is the most used since it is easy, the whole of an object is partitioned into equal parts, example ¼ shows that the whole object has been partitioned into 4 parts and one object is being observed. An operator construct is used to calculate just a segment of the entire object, example ¼ of 40=10.In the measurement construct, for example if only ¼ of a size is required then ¾ can be calculated then used to find ¼. It has been suggested that the ratio, operator and the measurement constructs are not emphasized in the studies which is some of the reasons as to why the students find it difficult to deal with fractions. Therefore, it will be best to expose the students to the multiple interpretations.

Several models will be efficient in showing how fractions are easy to deal with. The easiest models to adapt will include the area, region, length and set. The area and region models are quite the same since both are a whole, while the parts in the region are congruent the parts in the area are not congruent but equal. The two models can use different shapes to show fractions. The rectangle is the easiest to use. The students will then color or draw the different segments of different shapes to show fractions. The length model shows how any unit of length can be well partitioned then shown as fractions (Walshaw, 2013). Once the students understand the length model then they can start representing the fractions using number lines. The number lines can also be used to highlight the idea of a large number of fractions that can be found between two distinct fractions. Set models will include arrays and can also be used to represent fractions. By partitioning a set into equal parts, fractions will be introduced as representations of each group in a symbolic manner and thus developing the meaning of fractions to the student.

The Misconception Points in Fractions

Students also ought to have a well understanding of the equivalent fractions before they start using operations to solve problems involving fractions since the understanding enables them to have an idea that a fraction is like any other number and can be multiplied or added (Australian Curriculum, 2016). By having pictorial, symbolic and concrete representations then students will be able to order and also rename the fractions (equivalent). Pictorial models will be used in the ordering of fractions by the supplied material that help determine the equivalence of fractions. Symbolic representations in the same way will help since its manipulation will help compare different fractions with their denominators. Concrete representations such as papers, pattern bars and fraction blocks are also very much effective.

The students should also have a well understanding of the fractions and the equivalent fractions as this helps check for reasonableness in an answer. The pictorial models of subtraction and addition fraction problems are the best to start with as this will help create ideas of the importance of denominator. Problems of subtraction will always cause difficulties in the learning experience of the student if the student does not understand how to regroup the fraction (Remillard, 2011). Problems of multiplication are the easiest to deal with since the student should just multiply the numerators and the denominators separately. Nevertheless, it should be noted that multiplication will always give a smaller number but this is not the case with the division of fractions since a larger number is obtained. By the use of the pictorial or the concrete representations, dividing of the fractions can be well understood and when the divisor is symbolically inverted then a similar answer will be given when the fractions are multiplied. 

To comprehensively understand the ideas behind the fractions, a student should try to first understand the ideas of the decimal numbers as they always support each other. At the different stages of learning, a student should try to develop the fractions alongside the decimal numbers and by that a person can consider himself/herself fit for arithmetic problems (Handal, 2017). A student should be able to accurately convert the fraction numbers into decimal numbers and vice versa, since this helps strengthen the conception that fraction are actually numbers.

The students should be able understand how decimals are used to describe probabilities, how they can be represented and the different connections in the decimal numbers while being able to accurately convert and operate between decimals and fractions. Decimals are used in the following; base ten system, real world applications, many occupations and other several aspects of life. Developing the understanding of the decimal world can be achieved by first understanding fractions, place values and being able to compare and order decimals through the various models as well as using the accurate techniques to read the decimals.

The Misconception Points in Decimals

To connect the ideas between the decimals and fractions it is essential that we first look at the fractions that can easily be represented such as the tenths and the hundredths. When wanting to introduce a decimal then it should be linked to a fraction, the model and the language that is used (Remillard, 2011). A student should be keen on using his language for example, 0.3 should be read as three tenths and not zero point three.  They should also be able to write the decimals for the hundredth and the tenth based fractions. Nevertheless, such fractions such as ¼ should be viewed as quotients that is 1 divide 4 and equivalent fractions should also be easily identified. Use of a calculator will be the most effective tool that can easily show the connection between the decimals and the fractions.

Another important thing is to understand the connection between the decimal values and the place values as this helps in the computation of the decimals. A student is required to revisit all the properties of the place values (Handal, 2017). The decimal role must be well emphasized in different systems. Some of the most practical examples of the decimal role will be found in the metric system that includes the length, money, weight and volume.

Last but not least, the ordering and rounding off of the decimals is tasked upon the student, who should be aware of the decimals and also have the ability to order and round off the whole numbers. Many wrong ideas have been brought up about comparing and ordering of the decimal numbers which is because of lack of understanding the decimal numbers (Harel, 2016). This may be due to the following; internal zero in the decimal number, a decimal number that seems less than zero, a short number being larger, reciprocal is hard to work out and many other factors. Therefore, a student should have enough exposure of these activities of ordering and comparing decimals through pictorial, symbolic or concrete representations.

Conclusion

One of the main reasons as to why Australian curriculum was developed was to ensure that Mathematics made the curriculum more extensive and deeper rather than being wide. It should be noted that multiplicative thinking, decimals and fractions are one the major conceptions in the field of mathematics during primary schooling. So it will be best for a teacher to best understand the entire knowledge of mathematics, how the different conceptions are related to each other and how these methods can be used to develop each other. A teacher should therefor possess adequate knowledge of the content that is essential in teaching these concepts efficiently.

The common core state standards were also needed to address the curriculum and were seen as quite wide and deeper. If the contents above were to be dealt with one by one, then the teacher will experience difficulties in his/her teaching experience. The teacher should thus think in a wider perspective rather than dealing with the contents individually. To enable the teachers, do this tough task, then adequate professional acknowledgement should be given to the teachers that will help motivate them understand the importance of this contents. In this article, a deeper insights of the bigger number ideas have been provided to help the teachers on how to develop the skills. The final main ideas as we had indicated earlier were the proportional thinking and the generalization thinking which are more practical in the secondary school education.

References

Australian Curriculum, A. a. (2016, October). Australian Curriculum, Assessment and Reporting Authority . Retrieved from www.acara.edu.au: https://www.acara.edu.au/docs/default-source/corporate-publications/acara-2015-16-annual-report.pdf

Australian Curriculum, Assessment and Reporting Authority. (2016). Australian Curriculum, Assessment and Reporting Authority. Australia: Australian Information Commissioner.

Elif B. Turnuklu, S. Y. (2007). The Pedagogical Content Knowledge in Mathematics: Preservice Primary Mathematics Teachers’ Perspectives in Turkey. The Journal,(Content Knowledge), 2-6.

Handal, B. (2017). Philosophies and Pedagogies of Mathematics. Sydney: The University of Sydney.

Harel, G. (2016). What is Mathematics? A Pedagogical Answer to a Philosophical Question. Carlifonia: University of Carlifonia.

Janine T. Remillard, L. R. (2011, April). A Comparative Analysis of Mathematical and Pedagogical Components  of Five Elementary Mathematics Curricula. Retrieved from www.gse.upenn.edu: https://www.gse.upenn.edu/icubit/sites/gse.upenn.edu.icubit/files/ICUBiTCurriculumanalysis.pdf

Santiago Cueto, J. L. (2016). Teachers’ pedagogical content knowledge and mathematics achievement of students in Peru. Educational Studies in Mathematics, 5-20.

Thérèse Dooley, E. D. (2014). Mathematics in Early Childhood and Primary Education (3-8 years). Charles Sturt University.

Walshaw, G. A. (2016). Effective Pedagogy in Education. The International Academy of Education.

Walshaw, M. (2013). Explorations Into Pedagogy Within Mathematics Classrooms: Insights From Contemporary Inquiries. Curriculum Inquiry, 71–94.

Wu, H. (2005). Must Content Dictate Pedagogy in Mathematics Education?? Carlifonia: University of Carlifonia.