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Using mathematics strategies in earlychildhood education as a basis forculturally responsive teaching in IndiaSmita Guha aa St. John’s University , USAPublished online: 22 Jan 2007.
To cite this article: Smita Guha (2006) Using mathematics strategies in early childhood educationas a basis for culturally responsive teaching in India, International Journal of Early Years Education,14:1, 15-34, DOI: 10.1080/09669760500446374
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International Journal of Early Years EducationVol. 14, No. 1, March 2006, pp. 15–34
ISSN 0966-9760 (print)/ISSN 1469-8463 (online)/06/010015–20© 2006 Taylor & FrancisDOI: 10.1080/09669760500446374
Using mathematics strategies in early childhood education as a basis for culturally responsive teaching in India
Smita Guha*St. John’s University, USATaylor and Francis LtdCIEY_A_144620.sgm10.1080/09669760500446374International Journal of Early Years Education0966-9760 (print)/1469-8463 (online)Original Article2006Taylor & Francis141000000March [email protected]
The objective of this small study was to elicit responses from early childhood teachers in India onmathematics learning strategies and to measure the extent of finger counting technique adopted bythe teachers in teaching young children. Specifically, the research focused on the effective ways ofteaching mathematics to children in India, and examined teachers’ approach to number counting.In India, children were taught by their parents or by their teachers to use fingers to count. Thequalitative study conducted by the researcher further enriched the topic with first-hand commentsby the teachers. Although the finger counting method was not the only process that teachers wouldadopt, it was embedded in the culture and taken into consideration while infusing mathematicsskills. The teachers confirmed adopting the Indian method of finger counting in their teachingstrategy; some specified that the method helped children to undertake addition and subtraction ofcarrying and borrowing, as counting by objects could not be available all the time. Although thestudy is limited by its small sample to the unique mathematics learning experience in India, itprovides readers with a glimpse of culturally responsive teaching methods and an alternative math-ematics teaching strategy.
The National Council of Teachers of Mathematics (NCTM, 2000) suggests thatteachers can make connections to children’s real-world experiences to stimulatestudents’ interest in mathematics. According to Hatfield et al. (2000), culturallyrelevant teaching in mathematics provides two important aspects: (1) a recognitionthat mathematics has been present in every culture since societies have hadrecorded histories, and (2) the effect of mathematics on any culture and its people.Culturally relevant teaching embeds student culture into the curriculum in order tovalidate that culture and to transcend the negative effects of the dominant culture(Ladson-Billings, 1994). However, using children’s cultural experience as a basisfor teaching is easier said than done. How can culturally responsive teaching be
*St. John’s University, 8000 Utopia Parkway, Queens, NY 11439, USA. Email: [email protected]
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used to engage all students in mathematics? Culturally responsive or culturally rele-vant teaching requires that teachers incorporate elements of students’ culture ininstruction, moving beyond cursory examples of food, festivals, and holidays (Irvine& Armento, 2001).
Understanding how children in a given culture construct concepts of, and proceduresfor, addition and subtraction may require knowing how that culture uses fingers to shownumbers, as well as knowing the structure of its number-word sequence. Exploring howteaching children finger or number-word practices that support their construction ofuseful concepts and procedures also seems likely to prove fruitful. (Bideaud et al., 1992)
In India, children learn mathematics in a way that differs from the methods adoptedby children in the USA. It starts with simple number counting, then goes on to theprocess of addition, establishes subtraction methods, and so on. While there arelimitations to this system, it allows teachers and students to explore different possi-bilities for learning.
In India, children are taught by their parents or by their teachers at schools tomake good use of their fingers when it comes to number counting. It starts withcounting fingers on one hand, and then the counting shifts to fingers on the otherhand. However, the counting does not end there; with palms of the hands facing up,the joints on each finger are also taken into account (see Figure 1). Therefore, thelimited counting experienced by using only 10 fingers now extends to a highernumber. As each finger represents a number, likewise, each line of the finger isassigned a number in the counting process. The base and tip of each finger is alsotaken into account and represents the bottom and top line, respectively. In between,there are two visible line marks. Therefore, each finger represents four lines, whichinvolves the base, the two subsequent lines (joints), and the tip of the finger; the
Figure 1.
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Using mathematics strategies 17
base and the tip being the lowest and highest of the four numbers, respectively.Number counting starts with the little finger, and the thumb is used as the pointer.The thumb is moved (up and down) counting the lines on the rest of the fourfingers. Normally, the base of the little finger is considered as number one. As thethumb (being used as the pointer) moves up, it counts two, three, and four (seeFigure 2). Next, the thumb goes to the base of the ring finger and resumes thecounting process. Now, it starts with five, six, seven, and eight. This continues untilthe thumb comes to the tip of the index finger, where the count ends at 16. At thispoint, the index finger becomes the pointer and counts the four lines of the thumb,starting from the base. Therefore, a number count of 20 can be achieved with eachhand, as compared to five when only fingers, and not the lines on each finger,are used for counting purposes. When finger lines of both hands are used, then, atotal count of 40 can be achieved. This constitutes the first round of counting.Alternatively, if we are to use fingers only, then the maximum number that we cankeep count of is 10. In some cases, this poses limitations. For example, if we giveeight apples and four oranges to a child and ask him/her to count the total numberof fruits, then the child may face some difficulty as he/she has only 10 fingers tocount with. In this case number counting with finger lines is helpful, as the child can
Figure 2.
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do the mathematics using only one hand. Sometimes it may be necessary to gothrough two or three rounds to complete the required counting. For example, if thechild needs to count 60, he/she may use finger lines on one hand and repeat threetimes to reach the total count of 60.Figure 1.Figure 2. It is interesting to note that for over 1000 years a common counting method inIndia involved the use of seashells. In fact, as history reveals, some specificseashells were used as currencies. Those shells were considered precious sincedivers had to go deep down to retrieve them from the seabed. The size and qualityof the seashells would then determine the value, as the larger ones were traded fora multiple of smaller shells. Although the currency has changed, in some villagesthe tradition of using seashells still continues to be one of the prime elements innumber counting. Such activities are also culturally relevant for children livingnear beaches and shores. If seashells are not available, then pebbles are used as asubstitute.
Empirical evidence indicates that counting strategies adopted by children in Indiaare also born out of necessity. For example, it is important for people in the ruralareas of India to teach mathematics to their children at an early age as these childrenaccompany their parents to the market to sell goods, and often get involved withrecord keeping. The parents also give responsibilities to these young children tohandle some of the monetary matters (e.g. counting change, running to the nextshop to change the paper money into coins, etc.). Therefore, for these children,learning mathematics at an early age is essential. After becoming familiar withnumber counting, children are exposed to the next steps of simple mathematicaloperations. This step-by-step learning (while helping parents in the business)prepares children to do mathematics mentally, and finger counting becomes reallyhelpful. These children do not depend upon external tools or any kind of mathemat-ical apparatus (e.g. calculators). Children brought up in this environment would usefamiliar objects like rice, segments of orange, pebbles, or seashells to learn basicmathematics for their own survival. Whereas, children brought up in urban areaswould use toys or pictures and blend the traditional method of finger counting withabacus or other manipulatives in learning mathematics.
In some cases, in order to get the child involved in learning mathematics, twoobjects are used to keep the child occupied, one being the finger lines and the otherbeing an appropriate manipulative, such as seashells or pebbles. As indicated above,other constructive ways of adding numbers include using marbles or the cloves ofan orange. It is to be noted that these objects are relevant to the children, and there-fore are used by teachers as their mathematics teaching tools. However, in general,edible objects are avoided as teaching/learning tools in that food is expensive andmay get spoiled during such activities. To teach subtraction, teachers would use thesame objects and, in general, the process is reversed, meaning that once childrenhave added two things they are then taught the inverse function of subtracting themby doing the opposite action. Teachers and parents of Indian children would oftenuse the finger counting method and other manipulatives interchangeably so as toreaffirm mathematics learning. The techniques, while culturally relevant to Indian
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Using mathematics strategies 19
children, can be viewed as alternative mathematics learning strategies that could beexplored for possible adaptation in any primary classroom teaching.
Bideaud et al. (1992) cite Saxe (1982), stating that counting is clearly a sociallytransmitted activity and that people count in very different ways in different cultures.Furthermore, Box and Scott (2004) comment that the idea of counting on fingerswas extended by some cultures to reach higher numbers than 10 and this was aneffective method of counting that is still used worldwide today. Wright et al. (2002)write that ‘Children’s use of finger patterns in early number contexts is extremelywidespread. Fingers are used in a range of ways and with varying levels of sophistica-tion.’ Gelman (1982) argues that ‘counting contributes to later mathematical-cogni-tive developments’. Once counting has gained a cardinal meaning, children can useit to explore the numerical effects of conservation transformations to learn additionand subtraction facts (Siegler & Shrager, 1984) and to support basic arithmeticproblem solving (Carpenter & Moser, 1984; Bideaud et al., 1992).
It would be interesting to investigate the Indian method of finger counting, andspecifically how the Indian finger counting strategies are introduced and subse-quently used with children between the ages of three and seven. Qualitative analysesof teachers who routinely use this finger counting strategy, and its effects on chil-dren’s learning, are presented throughout this article.
Method
The study’s objective was to elicit responses from early childhood teachers in Indiaon their mathematics learning strategies and to ascertain to what extent the fingercounting technique is adopted in teaching young children. Specifically, the researchquestions focused on the frequency and duration of mathematics classes, the effec-tive ways of teaching mathematics to the children, and examined teachers’ approachto number counting.
Sample
The teachers participating in this study were selected from different schools in India.In total, 10 teachers from two cities in India—New Delhi and Kolkata—participatedin the study. All the teachers were female, and confirmed teaching children betweenthree and seven years of age. Of the 10 respondents, seven held Bachelor’s degreesas their highest degree, and three held Master’s degrees. All participants were eithercertified by the State Board of Education or were Montessori trained. (In India)Montessori training is an alternative to Education Board Certification, especially inearly childhood education. When asked about their teaching experience, two teach-ers stated that they had been teaching between zero and five years, three teachershad between 6 and 10 years of teaching experience, and five teachers confirmedteaching for more than 10 years. Each of the participants confirmed teachingbetween 16 and 40 children each day, and their records indicated the average classattendance to be 90%. Table 1 provides a comprehensive description of the subjects.
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A two-part questionnaire was developed by the researcher to collect responsesfrom the early childhood teachers in India (see the Appendix). The first partincluded five questions seeking to gather demographic information relating to eachof the teacher participants (e.g. gender, educational degree, major field of study,respective teaching experience, and the grade levels taught) plus two further ques-tions related to class enrollment and average daily class attendance. The secondpart of the questionnaire contained 12 open-ended questions. These questionswere related to teaching mathematics to children. Specifically, the questionsfocused on the frequency and duration of mathematics class, the effective ways ofteaching mathematics to the children, and the examined teachers’ approach tonumber counting. The questions developed also explored the use of fingers innumber counting, and whether the participants thought it helped children in learn-ing mathematics. Further, questions pertaining to teaching addition and subtrac-tion were presented. Lastly, the teachers were asked to cite any major problemsthey faced during their teaching session, and the steps they took to address andsubsequently overcome them. In addition to their completed questionnaires, eachof the teachers who responded and completed the questions had a focused inter-view session.
Results
The teachers confirmed that on an average the children learn mathematics four tofive days a week, and spend about 30–40 minutes daily in learning mathematics.Only one teacher indicated that she taught mathematics for one hour a day. Some ofthe teachers’ responses are presented here in the form of ‘excerpts’, thereby offeringsome insight into the context within which these teachers work.
Teaching methods
Nine out of ten teachers confirmed teaching children to count with their fingers. Oneteacher responded that she used beads and an abacus to teach children the methodsof number counting. Interestingly, this teacher explained that she was born andbrought up in East Africa and was not exposed to finger counting methods in herchildhood days. Teachers who responded positively about using fingers for numbercounting found the method economical, and mentioned that the ‘traditional’ methodwas easy to use, helped in swift counting, and also assisted children in developing
Table 1.
Highest degreeTeaching experience(average no. of years) Certification/training
Class size (average)
Class attendance (%)
Master’s (7) 7 Board certified (mostly) 30 90Bachelor’s (3) 10 Montessori trained 20 98
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Using mathematics strategies 21
motor skills. They also pointed out that the finger counting method was simple andconvenient in that they did not need additional tools for doing mathematics.
While responding to the question pertaining to the strategies some teachers adoptwhen teaching mathematics, one of the respondents emphasized children’s involve-ment and the influence of familiar objects in their learning process:
We encourage active participation on the part of the children. The concept of numbersone to nine is introduced with the help of pictures, and situations picked from withinthe range of the child’s experience.
The same was echoed when another teacher confirmed
the number concept 1–9 is introduced to the children with the help of beads, rice,match sticks, and some pictures. Then, the concept of zero is introduced and isexplained by giving suitable examples before introducing number 10. We also take thehelp of familiar situations.
Two teachers indicated that fingers on the hands are also used to explain thenumber concept, addition and subtraction.
Some of the teachers made unique comments on their mathematics teachingmethods. Respondents reported that they would have children memorize and recitethe numbers from 1 to 10, and then match the same using blocks. Others reportedthat they had children recite rhymes or would use pictures to count from 1 to 100.Next, they would use objects such as balls to add or subtract, and then use fingersfor counting purposes.
Answering the same question, two teachers confirmed using Montessori methodsto teach mathematics to young children. One of them elaborated the teaching process:
We use the Montessori method in our teaching where certain apparatus is used to teachthe basic concept of numbers to the children. Different apparatus is used to develop theconcept of 1 through 10. Spindles or counters are used to make children aware of quan-tities, bigger and smaller numbers, odd and even numbers etc.
Teachers adopted different techniques such as using visual aids, learning by doing,and setting examples from within the class, and held practice sessions for sums bydoing them on the classroom board, making things bright and colorful, making chil-dren act as objects, and giving day-to-day examples. One of the methods used by therespondent teaching children from lower socioeconomic condition was to make useof flower garlands. Children would count numbers on their fingers, and the teacherwould confirm children’s finger counting by using flowers as objects to make agarland. For example, children would use finger lines to count up to 10, and thenthe teacher would use 10 flowers to make a garland (physical representation). Foractive participation, roles would sometimes be interchanged with the teacher usingher fingers or finger lines to count and children making the garlands. Expanding onthis concept, the teacher would then make two garlands, and ask children to tally thenumbers using their fingers or finger lines. The basic idea is to explore and extendpossibilities to reinforce number counting by using fingers and tallying the same withfamiliar concrete objects.
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Figure 3b.
Figure 3a.
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Using mathematics strategies 23
In relation to appropriate teaching methods, one teacher commented:
In my opinion there is no specific method to teach mathematics, but one has to takethe help of traditional as well as modern methods to make the children understand thesubject. The teacher has to modify her ways of teaching according to the aptitudes,capabilities of the children.
The teacher’s view reflected ‘flexibility in teaching methods’ and a focus on thefoundation and basic understanding of mathematics.
Teachers’ perspectives
The researcher also investigated teachers’ perspectives on the methods that they, theteachers, felt worked well with children as they learned mathematics. Most teachersnoted the use of pictures and toys as being particularly useful in getting childrenactively involved. Citing her views on children’s learning, one teacher said: ‘A childshould be involved actively in the learning process. Color beads on the abacus arevery helpful in learning math.’ She further extended her comments by saying that inorder to teach basic problems like addition, subtraction, multiplication, and divisionteachers must provide examples through the use of children’s favorite objects such asfood items and toys. In regards to finger counting, she commented: ‘Counting isdone with fingers each standing for one when it is 0–10, or line marks on fingerscounting four including the fingers’ bases and ends (tips)’ (Figure 4).
Figure 4.
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Some teachers justified using various objects for counting purposes. Theyconcluded that children loved counting objects like toys or balls. To them, promot-ing counting methods with the help of different objects works well in learningmathematics, and children enjoyed this method and also seemed to learn veryquickly. Lastly, one of the teachers stated: ‘In my experience, using fingers to countnumbers teaches the children a better way of doing their problems.’ She continued:‘By the time children learn counting they simultaneously learn it as an abstractsubject. For solving problems there is always an established process which they areable to follow where the basics are known.’
A couple of teachers indicated that they use fingers more when they teach additionand subtraction. One of the teachers mentioned that although effective numbercounting with fingers could only work to a certain extent, nevertheless, it helpedchildren understand the basic mathematics.
During the interviews the teachers mentioned their experience as the (main)source of any action they might take during their teaching sessions. Provided beloware the teachers’ responses when asked to comment on the methods they wouldchoose to teach addition and subtraction.
One of the teachers preferred to teach ‘grouping’ before teaching children theprocess of addition and subtraction. She cited the following example to illustrate thisteaching strategy: ‘IIIIIIIIII (10 ones) = (1 item)’. Again, the teacher did not have aconcrete justification other than basing her teaching strategy on her experience.
Another teacher stated: ‘Addition and subtraction are taught with objects likeballs, or dots, and the signs are also taught simultaneously’.
She further quoted that subtraction was taught in a similar way but in reverse order.These are symbolic aids that teachers use to assist children in the mathematicslearning process.
Again, citing children’s preference for familiar items, one respondent confirmedthat addition and subtraction were taught by setting computations/mathematicstasks involving household objects, while another teacher commented that childrenfrom three to four years of age learnt addition and subtraction by increasing thenumber of objects or decreasing the number of objects, respectively. She furtherstated that when the children are between five and seven years old they start usingdifferent numbers, made by their fingers, for addition and subtraction. They wouldalso use the abacus for their mathematical drill. One teacher elaborated on thecounting process of the children:
For the beginners (age 3+), they draw balls and count them to add. For subtraction,they cross the balls out and count the remaining. Next stage is for children over age 5,where they are asked to remember the higher number in their head (memory) and then
added to (+) gives (=) O O O O O O O
O O O
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Using mathematics strategies 25
hand count the remaining numbers. For example, 5 + 4 = 9. They keep 5 in theirmemory and count 4 fingers starting from number 6. (See Figure 5)
Some teachers stated that they found it easier to use lines on the fingers to countbeyond 10, while they would simply use the fingers to count from 1 to 10. Whilenumber counting with fingers or finger lines seemed popular among the teachers,some of them would complement the method by using playful objects for countingpurposes. They found that it reinforced their mathematics skills and also generatedinterest on the part of the children. Other teachers mentioned that beyond normalclassroom learning, older children would often assist younger children in numbercounting, addition, and subtraction. This would normally occur during play time,and fingers as well as objects would be observed being used for these purposes (seeFigures 6 and 7).
Current research suggests teachers’ involvement to be very important in the learn-ing of mathematics, something supported by this study’s respondents who assertedthat children could understand mathematics quite efficiently when they were taughtwith interest and proper care. The teachers in this study also stated that childrenneeded a certain amount of maturity in understanding the subject which they wouldacquire while growing up, and mentioned the three to seven year age group ascritical in developing appropriate learning skills.
Figure 5.
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Concluding discussion
In India the use of fingers in number counting has been in practice for many genera-tions, irrespective of socioeconomic conditions. While children from the privilegedclass would use fingers for counting in addition to different objects such as the abacusand other manipulatives, underprivileged children would use day-to-day objects andmore finger counting methods in their learning process. From her findings, theresearcher concluded that mathematics teaching could be made more interesting tochildren if themes could be constructed around the mathematics topic (addition,subtraction, etc.). The selection of appropriate themes and use of common objectscan make learning enjoyable and extend beyond cultural barriers. For example, whileassisting children in learning a unit on the sea, teachers can infuse mathematics usingseashells. This is supported by Johnson (1995) who, in her article ‘Curriculumproject: India. Fulbright Hays summer seminar abroad’, explains how elementary-level thematic units on India could be designed as a stepping-stone towards learningamong children of different backgrounds. Literature reviews explored many facets ofnumber counting and the use of fingers in mathematics, all well supported by theresearch findings. The current qualitative study further enriches the topic with first-hand comments by a small sample of teachers, and how they use fingers for variousmathematical applications. Although the finger counting method was not the only
Figure 6.
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Using mathematics strategies 27
strategy/process that teachers would adopt, as a strategy it is embedded in the cultureand taken into strong consideration while infusing mathematics skills in the minds ofyoung children. Almost all teachers (9 out of 10) confirmed using the Indian methodof finger counting in their teaching strategy. While some would use fingers or fingerlines more than the others, they all commented on the positive aspect of this preva-lent counting method, as they find it culturally relevant.
This study explored various methods that teachers felt worked well with childrenin learning mathematics. Responding to the questionnaire, the participants revealedsome of their teaching strategies; views which were further expanded during theinterview sessions. The research focused on mathematics teaching strategies thatteachers found appropriate for optimal learning; all teachers agreed that a childshould be involved actively in the mathematics learning process. The areas of discus-sion began with number counting strategies and led on to teaching methods thatteachers adopt and subsequently methods that were learned by the children.
Number counting
Counting strategies have been identified as a strategy in the literature in relation tomathematics cognition. Younger children are prone to count concrete objects,
Figure 7.
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whether it is with their fingers or Unifix cubes (or other manipulatives), because theydo not yet have a strong sense of number (Saxe, 1991). The current research resultssuggest that the teachers in this study felt children learn mathematics well when theyare exposed to using objects and when they use their fingers for counting purposes.This group of teachers further asserted that with active participation and hands-onplay it is easier to help children understand basic mathematics, whereas specifyingonly numbers did not appear to provide any link between the children’s activity andtheir understanding of the number concept. Research by Franke and Kazemi (2001)has identified the counting strategy as building on the direct modeling strategy as thechild continues to represent the action in the problem but does so by physicallykeeping track of the number of groups and counting the number in each group usingskip counting.
Research by Wright et al. (2002) suggests that instruction in early number mustaccord with and take account of children’s spontaneous finger-based strategies.Expanding upon number counting with fingers, Bideaud et al. (1992, p. 55) state:‘… each new finger allows a child to make a one-to-one correspondence between thenumber word spoken and the current finger symbol set’. They further state that theone-to-one correspondence between fingers and objects directly enables quantifica-tion. They do not consider counting on one’s fingers or using other backup strate-gies as a sign of inferior ability, but instead mention that counting on one’s fingerscan indicate adapting one’s strategies to one’s knowledge base and to the demandsof the task at hand in order to reduce the probability of committing a retrieval error.Their study indicated that children who count on their fingers are not necessarilyless knowledgeable but only more cautious. Citing the same study, Kerkman andSiegler (1997, p. 13) indicate that the research results provide additional support forthe position that should children choose to count on their fingers then it is importantthat they do it correctly. Bideaud et al. (1992) also state that when a child follows thepathway from finger symbol set to number, he or she starts out having two systemsof signs available for representing quantity: a gesticular analog system which emergesfirst; and a verbal system, which is built up like an analog labeling system.
In the current research, the researcher gathered teachers’ responses on fingercounting and held in-depth interviews to gain an insight into their teaching methods.All teachers except one indicated using finger counting as one of their methods ofteaching mathematics to the children. They commented that they would use bothfingers and the finger lines in appropriate situations. Some teachers extended theircomments by stating that inclusion of finger lines in counting practices can beconsidered innovative, unique, and different. To them, it served the purpose ofcounting without the aid of any external tools. They mentioned that early childhoodstudents are both concrete and visual learners and that this particular counting tech-nique can be used in any classroom and could be playful learning for the children.
While the respondents in the current study did not specify a particular method inteaching mathematics to be better than others, a study by Uttal et al. (1997) indi-cates that some kinds of manipulatives are likely to be more effective than others.Research by Box and Scott (2004) indicates that people in some cultures use
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Using mathematics strategies 29
collections of shells, pearls, pebbles, elephant teeth, sticks or even coconuts to keep atally. Use of unique objects for counting purposes seems to be culturally relevant, asshown by the researcher in the current study quoting teachers who have adoptedindigenous methods such as having children use flowers to make garlands as a wayto keep a tally. The current study also indicated that the teachers in this samplefound familiar objects to be helpful in teaching mathematics to the children. Theseteachers also reported using daily objects for teaching mathematics. Therefore, itwould be reasonable to assume that culturally relevant manipulatives would also aidchildren in their learning process. Children living near the seashore would be likelyto use pebbles or sea shells as opposed to plastic containers or other objects thatwould be commonly used by children from urban areas.
In research findings on counting methods, Box and Scott (2004) report the use oftally sticks to be an alternative way of counting a group of objects. They found it tobe an efficient way of keeping track of the number of objects being counted as oneneeded only to carry around a small piece of wood or bone. People in the rural areasof India often use sticks made from branches of trees or may even use wooden sticksfor counting purposes (see Figure 3b). The use of wooden sticks also helps in identi-fying addition, subtraction, multiplication or equal signs so that children can learnthe mathematical operations as well as the symbols. In some cases, people even usegrains of rice to show children the counting process (as seen in Figure 3a). Rice isconsidered as an important staple food, it is affordable, and is used as a tool to aidchildren’s counting practices. The children’s literature book One grain of rice (Demi,1997), which tells the story of an Indian girl’s ability to trick the Raja (the king) outof 230 grains of rice, illustrates this concept. Since rice is readily available in every-body’s home in India, children can easily relate to it. Children pick up each grainone at a time and place it in a small container, creating their own banking world.Using rice for counting is culturally specific, and requires eye–hand coordination.The teachers in this study said that the use of rice involves fine motor skills whilelearning mathematics. Although this particular item may not benefit the learningobjectives among children from other cultures, it definitely has a broader impact onthe object utilization process. This particular example shows the indigenousapproach in the learning environment and that even unassuming objects can beutilized for learning purposes. However, teachers must consider the fact that manip-ulating smaller objects requires fine motor skills, and not all children have thoseskills. Therefore, teachers must evaluate and consider all aspects before choosingmanipulatives or tools to enhance children’s learning.Figure 3a.Figure 3b. In reviewing the teachers’ responses to the survey questionnaire, and throughprobing questions during the interviews, the researcher analyzed various strategiesthat teachers stated they used to encourage children in number counting. Most ofthe teachers confirmed having used fingers for number counting. Some teachersfurther commented that they used a variety of other objects and pictures, and thatchildren would also count how many hands, how many legs, and how many eyesthey have. While some teachers pointed out that children learned counting from 1 to10 by using their fingers, and sometimes they were asked to draw, color and count to
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fill in the missing numbers, other teachers stated that counting numbers was donefirst with counting any object and then slowly the children would learn counting ontheir fingers. Further, they stated that the abacus helped children in number count-ing, and gradually, when they had learnt numbers more than 10, they would thenstart using their own fingers for counting.
Most of the teachers involved in this study stated that the traditional Indianmethod of using fingers in number counting was very helpful. They suggested thatby learning to count the lines in the finger, the children could develop coordination.The teachers confirmed that they had taught children to count numbers with theirfingers but also mentioned that counting numbers with fingers comes at a later stagewhen the concept of numbers is clear and also when they are adding numbers orsubtracting. Some teachers indicated that counting with fingers was an easy and fastway of counting, children could move their fingers frequently, and, furthermore,finger counting made it easier to count when they did not have any apparatus avail-able to help with counting.
Most of the teachers participating in this study suggested that the use of fingersfor counting purposes eliminated the need for additional tools, but they alsocommented that children loved to count with objects such as toys or balls. Theyconfirmed having taught children to count with fingers where each finger repre-sented one when working within the range 0–10, or line marks on fingers whencounting four, including finger bases and tips. The teachers did not find anyspecific method to be best in teaching mathematics to young children, but statedthat it would be better if the traditional as well as modern methods were blended tohelp children understand the subject. They further argued that teachers shouldmodify their methods of teaching according to the aptitudes and capabilities of thechildren.
The socioeconomic structure in India leaves a wide gap in education and learningbetween the ‘haves’ and ‘have-nots’. Therefore, in the current study, the researchergathered responses from teachers from both sections of society in order to provide ageneral view in mathematics learning among young children and investigate thecultural trait, if any, that existed. The teachers mentioned that some children fromunderprivileged sections of society are good at mental mathematics. They used moreof the finger counting method as compared to learning by object manipulation.Teachers who taught children from more privileged backgrounds found that acombination of the counting method with the help of different objects worked wellfor them in learning mathematics. Children enjoyed this method and also learntquickly. Children used finger counting and also loved to do mathematics in activitybooks containing large numbers of pictures. Teachers mentioned children using anabacus and noted that the color beads on the abacus seemed to be very helpful tochildren in learning mathematics.
The researcher also probed into teaching issues or any problems that teachersfaced during mathematics instruction. Teachers found teaching ‘number sequenc-ing’ to the children to be one of the major problems, and they would use pictures orobjects to address the issue. This relates to identifying ‘numerals’, as teachers would
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Using mathematics strategies 31
use fingers to relate ‘count’ with the specific quantity. For example, they would closetheir fist and as they counted ‘1’, ‘2’ or ‘3’ they would open one finger at a time torelate the number to quantities.
Teachers claimed that in order to teach basic problems such as addition andsubtraction they would set examples using the children’s favorite objects such asfood items, toys, etc. Furthermore, teachers observed repeated practice to be aneffective way to learn mathematics and by setting examples with inanimate objects orproviding children with different objects and also associating numbers with theirquantities. Teachers would often create a link as they taught mathematics, such asenabling children to handle the quantities so that they could see, feel, and make outthe difference between units, tens, hundreds, and thousands.
Addition and subtraction
Kamii (1985) mentions number to be a mental structure that each child constructsout of a natural ability to think. He continues by saying that since any number isbuilt by the repeated addition of 1, its very construction can be said to include addi-tion, and further points out that addition grows out of the child’s natural ability tothink. The current research findings indicated methods that teachers would adopt inteaching addition and subtraction. In this respect, teachers found counting withfingers helpful, and one of the teachers voiced her opinion by saying that the fingercounting method made counting easy for difficult problems for the higher classes.She further stated that the method helped children to do addition and subtraction ofcarrying and borrowing as counting with objects or the abacus could not be availableall the time. In rural India, children also used culturally specific objects such asseashells, pebbles, and wooden sticks as deemed appropriate, and as available asmanipulatives/counting tools.Figure 5.Figure 6.Figure 7. Teachers often make use of manipulatives to teach children the process of addi-tion and subtraction. However, the use of culturally relevant objects empowersstudents intellectually, socially, emotionally, and/or politically (Ladson-Billings,1994) since such objects are not only familiar to the students but also carry specialmeaning as well. The research findings of Ladson-Billings are supported empiricallyin that the responses gathered from teachers in the current research also suggestedthat teachers used familiar objects to help children learn.
The research study also highlighted some interesting ways in which teachersapproach the teaching of addition and subtraction. While each approach is unique,the teachers did not confirm one approach to be better than the others. Teachersadopted their unique strategies as the situation demanded and as deemed appropri-ate. However, since 9 out of 10 teachers confirmed Indian counting strategies to betheir principal tools in teaching mathematics to young children, it provides statisticalsignificance to the finger counting method. With unique research findings, the smallsample size of the current research will entail further study on this topic.
Teachers’ responses showed a pattern in their views that demonstrated familiarobjects to be effective in teaching mathematics to young children. This supported
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earlier findings that indicated use of pebbles and sea shells by children living near theshore, or counting numbers with rice by children where this item is in abundance.
In their article on perspectives on using concrete objects in the learning of mathe-matics, Uttal et al. (1997) state: ‘Many of the attempts to improve mathematicsinstructions have called for greater use of concrete objects. Both teachers andresearchers have suggested that concrete objects allow children to establish connec-tions between their everyday experiences and their nascent knowledge of mathemat-ical concepts and symbols.’
Piaget (1970), Bruner (1966), and Montessori (1917) have all demonstrated thatchildren learn best through the use of concrete objects. From Piaget, educators haveadopted the notion that elementary school children’s thinking is concrete. Uttal et al.(1997) cite Bruner (1966), stating that elementary school children’s thinking focuseson concrete properties that can be actively manipulated. The authors furthermention that Bruner specifically calls for the use of concrete objects in instruction,suggesting that using many different concrete objects could help to move childrenbeyond their focus on the perceptual properties of the individual objects. Theyfurther indicate that concrete objects can be an effective aid in the mathematicsclassroom, but also caution to use them effectively, and that teachers must take intoaccount how children do (or do not) understand symbolic relations. In the currentresearch teachers indicated the importance of conceptual development among chil-dren, and found a link between the numbers and the objects being used. This drawsa similarity with the findings from earlier research, thereby establishing an outline ofan effective mathematics learning strategy.
The study is limited to the Indian setting, and focuses on cultural traits in themathematics learning process; nevertheless, the study provides readers with an alter-native teaching style when additional resources are limited, yet the learning processmust continue.
Using culture as bait to ‘hook’ students who tend to shy away from mathematicseducation, or students who are considered less interested or even weak in mathemat-ics skills, the teacher can potentially engage all students regardless of race, ethnicity,caste, or socioeconomic status to learn mathematics in a playful way. Althoughculturally familiar manipulatives or tools may be more interesting for one group ofchildren as compared to others, they can also generate curiosity among the entiregroup that can assist in their learning endeavor. Teachers must learn how to providestudents with culturally relevant scaffolds to build students’ informal and basicknowledge in mathematics, and at the same time be sensitive to others who may nothave the similar cultural experiences. Thus, it could seem that all students couldbenefit from multicultural experiences and learn to value the ideologies and perspec-tives of others as well as their own.
References
Bideaud, J., Meljac, C. & Fischer, J. P. (Eds) (1992) Pathways to number: children’s developingnumerical abilities (Hillsdale, NJ, Lawrence Erlbaum Associates).
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Box, K. & Scott, P. (2004) Early concepts of number and counting, Australian MathematicsTeacher, 60(4), 2–6.
Bruner, J. S. (1966) Toward a theory of instruction (Cambridge, MA, Belknap Press).Carpenter, T. P. & Moser, J. M. (1984) The acquisition of addition and subtraction concepts in
grades one through three, Journal for Research in Mathematics Education, 15, 179–202.Demi (1997) One grain of rice: a mathematical folktale (New York, Scholastic Press).Franke, M. L. & Kazemi, E. (2001) Learning to teach mathematics: focus on student thinking,
Theory into Practice, 40(2), 102–109.Gelman, R. (1982) Accessing one-to-one correspondence: still another paper about conservation,
British Journal of Psychology, 73, 209–220.Hatfield, M. M., Edwards, N. T., Bitter G. G. & Morrow, J. (2000) Mathematics methods for
elementary and middle school teachers (4th edn) (New York, John Wiley).Irvine, J. & Armento, B. J. (2001) Culturally responsive teaching: lesson planning for elementary and
middle grades (New York, McGraw-Hill).Johnson, N. L. (1995) Curriculum project: India. Fulbright Hays summer seminar abroad (ERIC
Document Reproduction Service No. ED 401167).Kamii, C. K. (1985) Young children reinvent arithmetic: implications of Piaget’s theory (New York,
Teachers College Press).Kerkman, D. D. & Siegler, R. S. (1997) Measuring individual differences in children’s addition
strategy choices, Learning & Individual Differences, 9(1), 1–18.Ladson-Billings, G. (1994) The dreamkeepers: successful teachers of African American children (San
Francisco, Jossey-Bass).Montessori, M. (1917) The advanced Montessori method (New York, Frederick A. Stokes).National Council of Teachers of Mathematics (NCTM) (2000) Principles and standards for teaching
school mathematics (Reston, VA, The Council).Piaget, J. (1970) Science of education and the psychology of the child (D. Coltman, Trans.) (New
York, Orion Press).Saxe, G. B. (1982) Culture and the development of numerical cognition: studies among the
Oksapmin of Papua New Guinea, in C. J. Brainerd (Ed.) Progress in cognitive developmentresearch (vol.1), Children’s logical and mathematical cognition (New York, Springer-Verlag),157–176.
Saxe, G. B. (1991) Culture and cognitive development: studies in mathematical understanding(Hillsdale, NJ, Lawrence Erlbaum Associates).
Siegler, R. S. & Shrager, J. (1984) Strategy choices in addition and subtraction: how do childrenknow what to do?, in C. Sophian (Ed.) Origins of cognitive skills (Hillsdale, NJ, LawrenceErlbaum), 229–293.
Uttal, D., Scudder, K. & DeLoache, J. S. (1997) Manipulatives as symbols: a new perspective onthe use of concrete objects to teach mathematics, Journal of Applied Developmental Psychology,18, 37–54.
Wright, R., Martland, J., Stafford, A. K. & Stanger, G. (2002) Teaching number: advancingchildren’s skills and strategies (London, Paul Chapman).
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Appendix
Survey: teachers’ demographic information
Using counting strategies in early childhood education as a basis for culturallyresponsive teaching in India
Gender (circle one): Male Female
Highest degree earned: ________________________________
Major/Honors: _______________ __________________Undergraduate Graduate/postgraduate
How long have you been teaching?0–5 years 6–10 years more than 10 years
Grade level you currently teach: _________________
Class enrollment: _____________________________(present average class size if more than one)
Average daily class attendance: _____________________________
Questionnaire
1. How often (no. of days per week) do you teach mathematics to the children(3–7 years)?
2. For how long each day do you teach mathematics to the children (3–7 years)?
3. Name and explain some of the ways you teach mathematics to the children.
4. What methods do you feel work well with children in learning mathematics?
5. How do you teach counting numbers?
6. Do you teach the children to count numbers with fingers?
7. How does this method help children to learn mathematics?
8. How do you teach addition and subtraction?
9. How do you teach multiplication and division?
10. Is there any major problem that teachers face while teaching mathematics toyoung children? If so, what are the problems (briefly explain)?
11. How do you overcome these problems among the children you teach?
12. Any other comments.
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