The development and arithmetic foundations of early functional thinking

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The development and arithmetic foundations of early functional thinking

Ulises Xolocotzin, Teresa Rojano

To cite this version:

Ulises Xolocotzin, Teresa Rojano. The development and arithmetic foundations of early functional thinking. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Educa- tion, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic.

pp.488-494. �hal-01286955�

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The development and arithmetic

foundations of early functional thinking

Ulises Xolocotzin and Teresa Rojano

Research and Advanced Studies Centre, Ciudad de México, Mexico, uxolocotzine@cinvestav.mx

Functional reasoning is a key strand of early algebrai- zation. This paper presents a cross-sectional study that analysed functional thinking in a sample of 94 elemen- tary school students. Aspects such as following and iden- tifying covariation rules showed dramatic differences between Grade 2, Grade 4, and Grade 6, whereas increas- es in the abilities to command verbal and symbolic rep- resentations were much smaller. After controlling for the influence of nonverbal reasoning, overall functional reasoning was found to be strongly associated with cal- culation skills, but not with skills such as counting, the understanding of numerals, and arithmetic problem solving. These results are discussed in terms of the na- ture of functional reasoning and its relationships to the arithmetical skills learnt during elementary education.

Keywords: Functional thinking, early algebra, arithmetic, quantitative methods.

INTRODUCTION

There is an increasing interest for understanding how algebraic ideas might be introduced during elementa- ry education. Studies of early algebraization suggest that young children are capable of acquiring algebraic competences such as understanding the relationship between two quantities x and y, figuring out a co-var- iation rule (Carraher, Martinez, & Schliemann, 2008), and explaining and symbolically representing such rule (Cooper & Warren, 2007). However, the extent to what children’s cognitive development permits their learning of algebraic notions remains to be fully un- derstood (Carraher, Schliemann, Brizuela, & Earnest, 2006), and the interest for linking algebraic ideas with children’s existing arithmetic knowledge are fairly recent (Russell, Schifter, & Bastable, 2011). This paper adds to the current literature by studying the acqui- sition of algebraic ideas such as that of function, by exploring the ways in which this might be linked to

the arithmetical skills that children acquire during elementary education. This paper reports a cross-sec- tional study designed with two aims. First, the study aimed to identify the progression of a range of func- tional thinking aspects comparing children in Grade 2, Grade 4 and Grade 6. Second, the study analysed the extent to what overall functional thinking might be related to non-verbal reasoning and arithmetical skills, including counting, numerical understanding, calculation and problem solving.

Functional thinking in the early grades

The functional approach to teaching algebra is based on situations involving the simultaneous variation of two quantities and a rule governing such variation.

Functional reasoning used to be absent in the elemen- tary mathematics curriculum. However, during the last 15 years the development of the function concept gained recognition as a strand of early algebraization and, consequently, education programs addressing the mathematics of variation are increasingly com- mon. For example, the Principles and Standards for School Mathematics introduce the subject ‘Analyse change in various contexts’ from Pre-K2 to Grade 12 (http://www.nctm.org/standards/content.aspx-

?id=26853). This is in part due to the recognition amongst researchers of the importance of introduc- ing young pupils to mathematical representations of everyday situations (Blanton & Kaput, 2011), and responds to the evidence that technological environ- ments effectively support the manipulation of dynam- ic mathematical representations of variation without algebraic representations (Rojano, 2008). The current study addressess two research questions.

How do functional thinking aspects progress across elementary grades?

Function tables are powerful tools for scaffolding and

studying functional reasoning in the early grades

(Martinez & Brizuela, 2006), since they help chil-

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The development and arithmetic foundations of early functional thinking (Ulises Xolocotzin and Teresa Rojano)

489

dren to figure out relationships between quantities

(Blanton & Kaput, 2011) and to elaborate algebraic conceptions such as co-variation and generalized cor- respondence (Tanışlı, 2011). Function tables are used in the current study for assessing a set of aspects of children’s early understanding of functions, namely following explicit covariation rules, identifying and using such rules without an explicit definition, and understanding and generating verbal and symbolic representations of such rules (McEldoon & Rittle- Johnson, 2010).

What arithmetic skills are related to functional thinking?

It is well documented that children rely on their arithmetic knowledge to face their first encounters with algebraic problems (e.g., Van Amerom, 2003).

Disregarding these intuitive arithmetical attempts and urging children to use formal algebraic meth- ods might be misleading (Smith & Thompson, 2008).

Therefore, there has been advocacy for grounding early algebraization on arithmetical knowledge (e.g., Russell, Schifter, & Bastable, 2011). This study adds to this research line by describing the relationship between functional thinking and a set of arithmeti- cal skills that children usually learn during primary school.

METHOD Design

This descriptive study relied on a cross-sectional design to compare aspects of functional reasoning across elementary school grades.

Participants and context of the study

The final sample for this study consisted of 94 stu- dents (58 girls) attending Grade 2 (n = 29, Mean age

= 8.0 years, SD = 0.7), Grade 4 (n = 33, Mean age = 10.0 years, SD =0.5), and Grade 6 (n = 32, Mean age = 11.9 years, SD =0.5). The sample was randomly taken from the morning and evening shifts of a school located in a middle-size city in central Mexico. There are two considerations to make about the context of the study.

First, at the moment of the study the mathematics per- formances of the two shifts in the participant school were fairly close to the national average in the na- tion-wide assessment ENLACE (SEP, 2014), which is similar to national standardized tests such as UK’s SATs. This might be a helpful reference to understand how the participants in this study might compare

to the rest of the Mexican population. Second, it is known that Mexico tends to rank at bottom positions in international mathematics assessments such as PISA (OECD, 2013). This is helpful to understand the relevance of this study for an international audience.

Measures

Functional Thinking Assessment

We wanted to link the progression of functional reasoning across elementary school grades with children’s arithmetical knowledge. Therefore, we employed one instrument designed to do that, name- ly McEldoon & Rittle-Johnson’s (2010) Functional Thinking Assessment (FTA). The items in the FTA are based on function tables that, as mentioned above, are adequate for approaching functional thinking.

The FTA covers four aspects of functional reasoning:

― Apply rule: The student can use a given rule de- scribing the relationship between the numbers in two table columns in order to determine new values of a table.

― Recognize rule: The student recognizes an x – y correspondence rule in a table, and uses it to de- termine the next y value.

― Generate and use a verbal rule: The student writes a correspondence rule verbally, e.g., “you add 4 to the number in the x column to get the num- ber in the y column”.

― Generate an explicit symbolic rule: The student writes a correspondence rule using algebraic symbols, i.e., letters.

These aspects and their order were defined on the basis of both a review of existing educational ma- terials involving function tables, and an empirical analysis of FTA validity and reliability. Item Response Theory measures showed that for all items, the prob- ability of correct response implicated greater ability.

Classical measures indicated high internal consisten- cy (McEldoon & Rittle-Johnson, 2010). These proper- ties suggest that the FTA possess adequate construct validity since it reliably identifies progression across ages, and the grouping of its items is consistent with the aspects that it is intended to assess.

A pilot study (Xolocotzin & Rojano, 2014) confirmed

the appropriateness of the FTA, but it was also discov-

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ered that it might be beneficial to increase the number of items in the FTA, in order to give children a better opportunity to demonstrate their skills. Therefore, we added 4 items that required the generation of a ver- bal and a symbolic rule, without having to use it. These items added to the existing ones assessing the identi- fication, generation, and usage of a covariation rule, which combination indexed the ability to command the verbal and symbolic representation of relation- ships between numbers. Other 4 items were added to balance the operations, for instance, to equalize the number of items involving additions and subtractions.

Exploratory analyses (not reported) showed that these added items had similar difficulty and discrimination capacity as the original ones.

Non-verbal intelligence

Participants completed the Matrix Reasoning sub- test of the Weschler Intelligence Scale for Children IV (Wechsler, 2007) according to the standard proce- dure. In order to have an age-standardized measure of non-verbal intelligence, the raw scores were con- verted to scaled scores for Mexican population.

Arithmetic ability

General arithmetic abilities were indexed with the raw scores obtained in the Arithmetic scale of the Evaluación Neuropsicológica Infantil (Neuropsychological Children Assessment, ENI) by Matute, Rosselli, & Ardila (2007).

This is a battery for children aged 5 to 15. This instru- ment was selected because it assesses a comprehensive set of arithmetic abilities:

― Counting. This includes items that require num- bering objects with and without interference, e.g.,

“How many stars and bells are in this card?”

― Numerical understanding. This includes items indexing abilities for commanding numerals, such as reading numbers, writing numbers, com- paring numbers, and ordering numbers.

― Calculation. This subtest requires the making of numerical series, both forwards and backwards, as well as the verbal and written solution of arith- metic operations (e.g., 23 + 14).

― Arithmetical problems. This subtest requires the verbal solution of word-problems (e.g.,

“A second-hand motorcycle was sold in $ 8, 700,

which is three-fourths of its original price. What is its original price?”).

Procedure

After selecting the school, the corresponding authori- ties were contacted in order to explain the project in- tentions and requirements and obtain official permis- sion for accessing the students. Parents of the selected groups were required to give informed consent for their children’s participations. After this, complete groups of Grade 2, Grade 4 and Grade 6 were tested collectively with the FTA. Grade 4 and Grade 6 groups answered the FTA in one session lasting up to one hour, whereas children in Grade 2 groups were tested in two sessions of up to 30 minutes each. In total, 196 children were tested with the FTA. A sub-sample of 100 students was randomly selected for this study. Except for few children who required two sessions due to interruptions, the testing of arithmetic ability and non-verbal intelligence was made individually in one session lasting 45 minutes to one hour. Children were made to perform other tasks related to their cognitive development, the results of which are not reported here. The testing took place in a quiet room within the school, with presence of the researcher and the child only. Before each session, children were requested to verbally consent with the activities, after being clar- ified that their participation was anonymous, volun- tary, and that they could stop at any moment. Three children did not complete the testing sessions because they decided to stop, and other three did not complete a second session, leaving a final sample of 94 children.

RESULTS

Comparisons by functional thinking aspect, grade and gender

The percentages of correct responses in each of the FTA aspects were analysed with a mixed ANOVA in- cluding the within-participants factor Aspect (Apply rule/Recognize rule/Verbal representation/ Symbolic representation) and the between-participants factors Grade (Grade 2/Grade 4/Grade 6) and Gender (male/

female). None of the scores was weighted. The as-

sumption of sphericity was violated, and although the

results of the analysis with and without corrections

were virtually equal, here we report the Greenhouse-

Geisser corrected results. There was a significant

main effect of Aspect [F (2.46, 217.037) = 118.323, p <.001,

h

²

= .573], suggesting that children were more able to

apply a rule than to recognize a rule, representing it

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491

verbally, or representing it symbolically, in that order.

The significant main effect of Grade [F (1, 88) = 42.789, p <.001, h

²

= .493] indicated that the functional thinking aspects changed from one grade to another. There was also a significant aspect x grade interaction [F (6, 217.037) = 42.789, p < .001, h

²

= .493].

Post-hoc pairwise tests with Bonferroni adjustments revealed that the aspect apply rule increased signifi- cantly from Grade 2 to Grade 4, but not from Grade 4 to Grade 6. Also, the aspects verbal representation and symbolic representation increased only from Grade 4 to Grade 6, but not from Grade 2 to Grade 4. Neither the main effect of Gender [F (1, 88) = 0.57, ns] or the interactions aspect x gender [F (3, 217.037) = 0.93, ns], and aspect x grade x gender [F (6, 217.037) = 0.11, ns]

resulted significant. Figure 1 illustrates these results.

Relationships between arithmetic and functional thinking

A multivariate regression analysis identified the arithmetic strands that might have been related to children’s performance in the FTA. The dependent variable was a composite score indexing overall func- tional thinking, defined as the summed raw scores of the FTA indexes, namely apply rule, recognize rule, verbal representation and symbolic representation.

Positive and significant correlations (not report- ed) were found between these scores (all rs > .39, all ps < .001), justifying their aggregation.

The independent variables in the regression analyses included two dummy variables that identified Grade 2 and Grade 4 individuals, the scaled scores of the matrix reasoning test, and each of Arithmetic scale

Figure 1: Percentage of correct responses by aspect and grade

Independent variables B SE

β

^p

Grade 2 -14.480 1.231 -.777 <.001

Grade 4 -7.614 1.179 -.422 <.001

Matrix reasoning .776 .233 .231 .001

Calculation .285 .117 .184 .017

Notes: all ps two-sided, only significant results are presented

Table 1: Summary of the multivariate regression analysis including overall FTA score as dependent variable and Grade, matrix reasoning and the indexes of the ENI Arithmetic scale as independent variables

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scores of the ENI, mean-centred by Grade. The results are shown in Table 1.

Children in Grade 2 and Grade 4 scored lower in the FT test than children in Grade 6, which is consistent with the results of the ANOVA presented above. The coefficient of the matrix reasoning score indicated that those who had a more non-verbal reasoning also scored higher in the FT test. As for the arithmetic scores, counting, numerical understanding, and ar- ithmetical problems, were not significantly related to the FT. In contrast, the positive and significant effect of calculation indicated that those who were more able to make numerical series and resolve arithmetical operations were also better at responding the FTA.

DISCUSSION

How do functional thinking aspects progress across elementary grades?

The comparisons by aspect, grade, and gender, are consistent with the results reported with other appli- cations of the FTA (McEldoon & Rittle-Johnson, 2010), suggesting its adequateness for assessing function- al thinking. The absence of gender effects suggests that functional reasoning is accessible for both boys and girls. Performance in all of the aspects increased across grades. However, there were differences in their rates of increase from one grade to another. The ability to apply a rule develops steadily, and approach- es top performance by Grade 4. This seems only natu- ral considering that applying rules is a form of follow- ing instructions, which is something that children get familiar with from early stages of education. Younger children might be unable to apply a given covaria- tion rule because they do not understand it, as is in- dicated by their lower performance recognizing and representing such rules. Performance recognizing a rule seems to progress regularly across elementary grades, although is not fully commanded by older students. The difficulty to command verbal and sym- bolic representations is noticeable. The differences across grades are apparently not regular and rather modest, especially for symbolic representation. Recall that children in Grade 4 perform at the same level of children in Grade 2, and children in Grade 6 failed to reach above 30% of correct responses.

The low performance on symbolic representation might be the result of either low students’ abilities or hard items. This is difficult to disentangle with the

collected data, so the results of this aspect should be interpreted cautiously. Analyses of the items’ discrim- ination and difficulty for the current sample showed that 5 out of the 9 items involved were very difficult, even for top performers. The rest were also difficult but at least 30% of top performers were able to solve them. The items also involved different mathematical relationships, including addition, subtraction, mul- tiplication, and combinations of these. The effects of potential differences in the capacity to handle these operations might be confounded with the capacity to symbolize. Also, the assessment of symbolization as a unitary aspect might not be optimal. This might be ad- dressed by decomposing it in sub-aspects of different difficulty. For instance, symbolizing a covariation rule that has been presented in natural language might be one easier sub-aspect than symbolizing directly from data.

Functional reasoning aspects change differently across grades, suggesting that the progression from one aspect to the other is not necessarily uniform.

Further studies should be designed to replicate these results, especially those regarding symbolic representation aspect, considering the apparent dif- ficulty for assessing this aspect.

Non-verbal intelligence and functional reasoning

The significant relationship between matrix reason- ing and the FT scores reflects that perceptual reason- ing, indexed by the understanding visual patterns, is closely linked with the development of mathe- matic abilities in general (Parkin & Beaujean, 2012).

Nevertheless, future studies should investigate if un- dersanding numerical patterns could be specifically related to the understanding of functions.

What arithmetic skills are related to functional thinking?

The results of the multivariate regression analysis outline the arithmetic skills that might be associated with functional thinking. The effects of the Grade 2 and Grade 4 dummies are not discussed since they replicate the grade effects discussed above. The sig- nificant matrix reasoning effect was expected since performance in this test tends to be correlated with general mathematics ability (Jordan, Glutting, &

Ramineni, 2010) and, therefore, it was important to

account for the influence of this variable when as-

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493

sessing the relationships between arithmetic and

functional thinking.

The lack of significant counting effects might reflect the abstract nature of functional thinking. Counting was assessed with tasks in which children numbered objects accessible to visual perception. In contrast, functional thinking tasks required working with re- lationships, either explicit or not, between abstract quantities, i.e., represented by numerals. The lack of numerical understanding effects might be seen as counterintuitive. One would expect that skills for mastering the symbolic number system might be also used for functional thinking. Both of these involve generalizations. However, this seems to be not the case, at least for this sample. Children might command the structure of ten base-10 system, generalizing the relationship between the location of a numeral in a number and its value. For instance, knowing what the numeral 3 represents depending of whether is located first (32) or second (132). However, this sort of gener- alization seems unrelated to generalizations about relationships between different sets of quantities.

The significant effects of calculation suggest that those more able for making sequences and resolving arithmetical operations also resolved more FT items.

Probably they attempted the function tables with iter- ative calculations, or by trial and error, and then they figured out the covariation rules. Another possibil- ity is that they figured out the covariation rule first and then proceeded to do the calculations. The later seems less likely considering that items involving rep- resentation were more difficult than items involving the imputation of missing data, such as following a rule or employing a non-explicit rule. Further studies might shed light on this issue. It is important to men- tion that this result seems consistent with suggestions made about the important role that calculation might have for grounding algebraic reasoning. Russell and colleagues (2011) identified activities that sustain both arithmetic and algebra, namely understanding op- erations, generalizing and justifying, extending the number system, and using notation with meaning.

Whether children might engage with these activities whilst doing function tables deserves attention.

The lack of significant effects of arithmetic problems does not rule out that problem solving is associated with functional reasoning. There was a sizable coeffi- cient and a large standard error (B= .64, SE= .47). This

might be related to the instrument. This score was obtained from a single test with eight items, which is small compared to the calculation score, made with 42 items from 4 tests. Thus, a small increase in the arithmetic problems score could be associated with large increases in the FT score, producing the ob- served large B. However, with standardized scores, calculation’s β remained sizeable, whereas the arith- metic problems one decreased notoriously (.18 and .09 respectively). This might suggest that calculation is more related to functional thinking than arith- metic problem solving. However, this is uncertain considering the differences in their measurement.

Future studies should include convergent measures of arithmetic problem solving in order to make a fair measurement of this aspect and assess its relationship with functional thinking more accurately.

Limitations and further studies

The sample for this study was selected purposfully, and comes from a very particular population. It would be useful to conduct other studies with samples from different populations to observe similarities or dif- ferences in functional reasoning. Also, measurement limitations for symbolic representation and prob- lem solving indicate cautios interpretations of the results related to these aspects. The cross-sectional design does not allow for developmental inferences.

It is highly desirable to collect longitudinal data for studying the development of functional reasoning.

This study is part of a larger project aimed to identify the cognitive underpinnings of functional reasoning.

In future, studies will be carried out to confirm the presented findings, and to investigate a wide set of cognitive capacities in addition to arithmetic.

ACKNOWLEDGMENT

This research was generously funded by the Mexican Council of Science and Technology (CONACYT), Grant No. 168620.

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