Antecedentes Teóricos

With regard to Numbers, Operations and Relationships, most of the learners in this study struggled with the concept of computation with whole numbers, concept place value and rounding off, addition and subtraction operation on fractions, comparing and ordering decimals, number theory, equivalence relation between fractions, decimals and percentage, and number line.

7.2.1.1 Computation with whole numbers

Computational fluency with addition, subtraction, multiplication and division is an important part of mathematics education in the elementary grades (Reys et .al 2012; Jorgensen & Dole, 2011). However, operations on multidigit whole numbers seemed to be always a challenge for some learners in this study. Various reasons are attributed to be main factors for learners’ difficulty in computation. Researchers Lin, Youn & Lai (2016); points lack of understanding of the meaning of operations and number sense to be to be the main contributing factor. Of all the four basic arithmetic operations, division operation with whole numbers was found to be the greatest challenge with 90% of the sample committing errors, followed by multiplication operation with whole numbers which was found to be a challenge with 62% of the sample committing errors. Addition and subtraction operations with whole numbers were found to present learners with least challenge. The study found that with regard to addition operation with whole numbers only 40% of the sample committed errors and with regard to subtraction operation with whole numbers 46% of the sample committed errors.

From 90% of learners’ erroneous responses, 78% of the sample wrote the dividend and divisor on division symbol and thereafter learners wrote the numbers with no discernible link on top of the division symbol as quotient with some numbers under dividend which were also hard to link to the whole numbers dividend and the divisor as in the following examples of learners’ written work;

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and

Research (Mooney, Briggs, Fletcher, Hansen & McCullouch, 2012) has shown that unless learners understand the algorithms they use, errors creep in their work and their lack of understanding means they also found it difficult to detect errors in their work. Literature points that traditional division algorithm is most difficult to master for most learners (Reys et al. 2009). Amongst the several reasons mentioned the reasons amounting to the difficulties, according to the literature is that division algorithm involves not only the basic division facts but also subtraction and multiplication. Some learners (thus 12% of the sample) resorted into writing the dividend and divisor in column with respect to place value for the two whole numbers and intended to either add, subtract or multiply as in the following examples of learners written work;

and

One key to successfully teaching division algorithm or any algorithm according to Tucker et al. (2013) is to help learners to understand that written algorithm is nothing more than an orderly recoding of what is being done with the model and to accomplish such, they further recommend teachers to model the process one step at a time when teaching computation to the learners.

With regard to multiplication of whole numbers, Jorgensen & Dole (2011) considers multiplication algorithm extremely difficult to teach meaningfully with materials. That could be one of the reasons 62% of the sample committed errors which were found in their work. This study found that 26% of the sample resorted into writing responses which are hard to interpret as they show no link to the whole numbers they were required to multiply as in the following example of learner’s written work;

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, and

. Such errors according to Dole (1993) reflect learners’ lack of meaningful understanding of the computational procedures. Some learners (4% of the sample intended to use the column method to multiply but due lack of procedural fluency coupled with to lack of knowledge of place value such learners failed decompose and align the two whole numbers 3 107 and 35 in column as in the following example of one of the learner’s work;

This group of learners’ errors were classified as both procedural errors and conceptual errors. From the learners’ responses, procedural errors ensued when learners misaligned the two whole numbers. Such errors stemmed from lack conceptual understanding of place value. In one of the study sought to examine the kinds of misunderstanding causing errors which children make in doing multiplication, Booth (1987) also found that some errors where due to lack of conceptual grasp of place value. According to Haylock & Cockburn (2013), too early introduction of vertical layout could possibly lead to such error.

Some learners (16% of the sample) correctly aligned the two whole numbers in columns as in the following example of one of the learners’ work;

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with the intention to apply the column method to multiply the two whole numbers, such learners only multiplied the unit product correctly with correct regrouping of tens digit to the next column but errors owing to lack of basic multiplication facts and conceptual understanding of the procedure were committed when they proceeded multiplying resulting into such learners writing response with no discernible link to the whole numbers they were multiplying.

Surprisingly, some learners (10% of the sample) copied the 3 107 as 107 due to carelessness. All the learners who carelessly copied 3 107 as 107 aligned the whole numbers in column as in the following examples of some learners’ written work;

and

showing that they intended to use column method to multiply but due to lack of conceptual understanding standard multi-digit multiplication algorithms coupled with lack of knowledge of basic multiplication facts, such learners wrote responses with no discernible link to the two whole numbers they were trying to multiply. Tucker et al (2013) points out that learners are not ready to learn the standard multi-digit multiplication if they have not mastered their basic multiplication facts. Some learners (6% of the sample) wrote the two whole numbers 3 107 and 35 in the division symbol. Such seemed to confuse multiplication operation to division operation.

When teaching operations on whole numbers, Tucker et al (2013) points three instructional tasks for teachers to complete; developing the meaning of operation, develop basic facts and develop algorithms.

With regard to subtraction operation on multiple digit whole numbers, research (Haylock & Cockburn, 2013) found that subtractions get difficult when one or more digits in the second number are greater than the corresponding digits in the first

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number. In using the column method to subtract, the errors of subtracting larger digit from smaller digit as in the following example of learner’s written work;

, were common amongst 16% of the sample. Such errors occur when learners are using algorithms without understanding (Mooney et al. 2012). Research (Alwyn, 1987;Sadi, 2007; Reccomini, 2005) attributes such errors to overgeneralization of commutativity of addition and multiplication operations to subtraction. Learners should be made aware at an early age of the importance of order in subtraction (Sadi, 2007). Hatfield et al (2008) recommends the use of concrete materials to be used with a take-way model to show that the first number, or minuend is modeled blocks or chips and the second number is removed from that set.

Some learners (16% of the sample) who seemed to have no meaning of subtraction operation resorted into writing responses with no discernible link to the multidigit whole numbers they were required to subtract one from the other as in the following examples of some of the learners’ written work;

and

According to Mooney et al. (2012) learners did not sufficiently remember the algorithms to succeed answering in finding the deference between the two multiple digit whole numbers and such errors are the results. To help learners build a conceptual understanding of the concept behind subtraction operation with multiple digit whole numbers, Haylock (2006) recommends teachers to provide learners with plenty of opportunities to connect manipulation of coins or base ten blocks with manipulation of the symbols supported by appropriate language. The word

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“borrowing” and “carrying” as widely used in some elementary mathematics textbooks and teachers is by no means discouraged when teaching subtraction as it is said to be meaningless and unhelpful (Heddens et al. 2012). Haylock & Cockburn (2013) and Haylock (2006) recommend “exchange” and “regroup” to be appropriate language replete with meaning over “borrowing” and “carrying” when teaching standard algorithm for subtraction. Some learners (8% of the sample), for example, this

learners work; seemed to have

good understanding of the standard algorithm for multiple digit subtraction i.e. exchanging ten in one place for one in the next place to the left and regrouping. However, errors due to carelessness resulted into such some computation errors. It seems that such learners did not check their work before proceeding to answer the next items.

With regard to addition operation of multiple digit whole numbers, two types of errors were found in learners’ work. The error of regrouping and failing to add the regrouped digits as in the following examples of some of the learners’ written work;

and

,was common amongst 12% of the sample. Amongst the typical errors learners commit in addition of multiple digit whole numbers, the error of regrouping and failing to add the regrouped digits amount to common errors in literature (Hatfield et al. 2012). Haddens et al. (2009) attributes the error of regrouping to difficulties with place value. They point out that if regrouping is introduced at abstract level, learners often have difficulty with place value when writing the sum. Some learners seemed to lack an idea of basic addition facts. Such learners resorted into writing numbers with no discernible link to three multiple digit numbers they were required to add as partial sums as in the following examples of some of the learners written work;

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, and

. Regardless of place value involved, Haddens et al (2009) recommends understanding of basic facts to be cornerstones for success in addition. Some of those learners who seemed to lack knowledge of basic addition facts were only able to find the partial sum of the unit digit continue to write the partial sums with no discernible link to the three multidigit number they were intending to add and using the vertical column method as in the following examples of some of the learners written work;

and .

7.2.1.2 Concept of rounding off

Rounding integrates understanding of approximate values with place value and naming numbers (Reys et al. 2012). Taylor & Harris (2014) emphasize a need for learners to know when they should round up or down. Rounding off whole numbers to the required place value seemed to be a challenge for some of the learners in this study. The error of rounding down instead of up as in the following example of learner’s written work;

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, was found to be common amongst 6% of the sample.

Tylor & Harris (2014) recommends number line as a good model to provide a good representation to support learners in determining which tens or hundreds number is nearest to the number to be rounded. Some learners (10% of the sample), erroneously re-wrote all the digits after they correctly rounding up to the required place value as in the following example of one of the learner’s work;

Reys et al. (2012) attribute such errors to lack of the importance and meaning for rounding off. Some learners, 22% of the sample seemed to have no idea of the concept of rounding off. Some learners wrote the responses as follows;

and . Such

learners have not developed the rounding skill and the rules associated with rounding off the numbers. When teaching the concept of rounding off the numbers, learners must be taught in a way which will enable them to view rounding as something that not only make numbers easier to handle, but more important make sense.

7.2.1.3 Operations on fractions

Fractions are commonly and pervasively used in everyday life but amount to one of the elusive concept most of the learners. Most learners view fractions as meaningless

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symbols, for example, the following are some of learners’ work on concept of

operations on fractions; and

. Such errors, according toresearchers Fazio & Siegler, (2011) stems from lack of conceptual understanding of fractions. With regard to addition and subtraction operations on mixed fraction, this study found that the errors of writing responses with no discernible link to the mixed numbers to be computed as the previous two examples of some of the learners’ written work, were common amongst 26% and 28% of the sample in addition and respectively. Literature attributes various reasons to learners’ difficulties in understanding fraction concepts. One of the main reasons according to Hatfield et al. (2008) is that fractions are interpreted from several perspective i.e. part-whole region (area of a model), measure, set, ratio and division. Some of these interpretations are the source of misconceptions. Research (Fazio & Siegler, 2011; Newstead & Murray, 1998) point out that learners who are taught to only consider fractions as part-whole have limited meaning of fractions. Part- whole interpretation according to researchers, is important but fails to convey the vital information that fractions are numbers with magnitudes. Some learners in this study wrote;

and . Such

errors were found to be common amongst 6% of the sample in addition and 2% in subtraction. Learners erroneously added the numerators and the denominators separately and that was also seen in one of the learners’ work wen subtracting due to overgeneralization of whole numbers schema to fraction schema (Tucker et al. 2013; Siegler et al, 2010; Fazio & Siegler, 2011). Newstead & Murray (1998); Ndalichako, (2013); and others also found same kind of errors. Teachers’ overdependence on procedures for manipulating fractions with little or no time for demonstrating their

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conceptual meaning is also pointed out to be amongst the reasons contributing to learners’ difficulties with fractions concepts (Moss & Case, 1999).

Fazio & Sigler (2011) recommends teachers to strive for development of both conceptual understanding with procedural fluency through the use of manipulates and visual representations of fractions.

One of the most striking findings in this study was that most learners were puzzled when required to find the “fraction” of “quantity” 2

5 of 300. Most learners 40% of the sample

failed to consider 2

5 of 300 to be 2

5 × 300. Due to lack of conceptual knowledge of fraction of

the fraction of a quantity, different interpretations were made as in the following example one

of learner’s erroneous response; . Some learners, 38% of the sample knew that “of” was for the multiplication operation and were able to write 2

5 of 300 in the form of 2

5 × 300 as shown in the following examples of some of the

learners’ work;

and

Due to lack of knowledge of procedure and lack of knowledge of basic multiplication and division facts such learners wrote erroneous response.

One learner did not write any response for item that assessed knowledge and skills in finding the fraction of quantity. Blank spaces imply that learners found the item difficult to answer to such an extent that they even failed to figure out how to start answering the item.

7.2.1.4 The concepts of decimal fractions: operations of decimal fraction, comparing and ordering decimals

Developing decimal concepts forms one of the most important strands for equipping learners with number sense. Research (Moloney & Stacey, 1997; Lai & Tshiang, 2009) point that learners acquire operation skills on decimals by merely rote learning and devoid of any meaning. In this study, most of learners (64% of the sample) committed errors on item which required them to compare and order the decimals from the smaller to the biggest with relatively less learners (only 34% of the sample) committing

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errors on item that required which required them to subtract a decimal number from other decimal number. When required to order the decimal numbers from smallest to the biggest, the study found that most learners (26% of the sample), erroneously wrote responses as in the following example of one of the learner’s written work;

.

Researchers (Monoley & Stacey, 1997; Lai & Tshiang, 2009; Roch, 2005; Deniz, 2014) also found learners treating the decimal numbers in the same way some learners in this study did and attributes such errors to overgeneralization of whole numbers schema to decimal numbers i.e. a whole number with more digits is larger than a whole number less digits. attributes learners’ difficulties in comparison of decimals to lack of appropriate mental imagery for the numbers being compared. Tucker et al. (2013) recommends integration of decimal numbers with models that will allow learners to literally see when one decimal number is greater than another to develop meaningful rules and procedures for comparing decimal numbers. Jorgensen & Dole (2011) recommends teachers to use appropriate language support structures in which the meaning of the values of decimals are emphasized e.g 3, 4 “three and four tenth” rather than “three point 4”.

With regard to operation on decimals, the study found that 34% of the sample was unable to subtract 25,8 from 59,3 . Literature points out that the most common error pattern in addition and subtraction of decimals comes from a rote rule learned when the students were working with whole numbers (Tucker et al. 2013). The study found that the common error of subtracting larger digit from smaller digit as with whole number subtraction was also common for some learners in subtracting the decimals. 10% of the sample wrote the same response as in the following example of one of the learner’s written work;

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. According to (2006), difficulties with decimals arise only if learners forget the principle of place value. This study also found that the errors of writing responses with no discernible link to the decimals numbers involved in problem as in the following examples of some of the learners’

written work; and

, was common amongst 22% of the sample. According to literature, Learners are not ready to handle operation with decimals until they understand the relationships among decimals by comparing decimals through modeling the decimal numbers with place value blocks (Hatfield et al. 2008).

7.2.1.5 Number theory: concept of multiple of a number

Multiples of any given number are obtained by multiplying that specific number by each of the natural numbers, 1, 2, 3, 4, 5, and so on. Being able to recognize multiples and having and awareness of some patters and relationship within them help to develop a high level of confidence and pressure in working with numbers (Hylock, 2006). Most learners in these study seemed to lack an idea of the concept of multiple of a number.

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Some learners (38% of the sample) was unable to find the multiples of 7. Some learners (32% of the sample) erroneously responded by writing numbers with no link to the 7 as in the following examples of some of the learner’s written work;

, and

. Some learners seemed to have knowledge of multiples of numbers, however, such learners even wrote multiples of 7 which are over 56 without considering the restriction as in the following example of some of the learners written work ;

, and

. The errors of failing to consider restriction as in the previous examples were common amongst 22% of the sample. Such errors may be due to carelessness. According to literature such errors may attributed to difficulty in reading and comprehending (Haddens et al. 2009). Haddens et al (2009) points that sometimes children have difficulty in reading and comprehending the problem and this