Replacing persistent counting strategies with cooperative learning

with cooperative learning

Uta Häsel-Weide

University of Siegen, Siegen, Germany, [email protected]

Persistent counting strategies often come along with a weakness in arithmetic. Due to this, the central objec- tive is to replace counting strategies by teaching how to realize, recognize and use structures. Within the con- text of the project ZebrA1 (Zusammenhänge erkennen und besprechen – Rechnen ohne Abzählen) cooperative learning settings were developed to encourage children to use different interpretations of patterns and struc- tures instead of counting. Results show that working in pairs of children with heterogeneous competences is productive for both partners. Especially activities like “comparing” and “sorting” are suitable to make children interact about mathematics and help them become aware of mathematical structures.

Keywords: Counting, cooperative learning, difficulties in learning mathematic, design research.

INTRODUCTION

Meta-analysis verified that forms of cooperative and peer-assisted learning at elementary schools go hand in hand with stronger effects in content knowl- edge than in traditional forms of teaching (Rohrbeck, Ginsburg-Block, Fantuzzo, & Miller, 2003). The learn- ing performance of children with learning difficul- ties improves especially in well-structured, heterog- enous and tutorial learning environments (Gillies

& Ashman, 2000). Research also seems to show that children with difficulties in learning mathematics can profit from cooperative learning. For that reason cooperative learning is chosen for an intervention intended for children who use counting strategies as a persistent strategy. It is well known that persistent counting is one of the central symptoms of weakness in arithmetic. It is thus necessary for further success

1 Recognizing and speaking about connections – calculating without counting on

in mathematics that children replace counting strate- gies with mental computation and the use of adaptive strategies.

This paper intends to show how learning environ- ments could be of such design that all children profit from cooperative situations. On the one hand chil- dren with a weakness in arithmetic get a first under- standing of mathematical structures and relations which are important to replace counting strategies (Häsel-Weide & Nührenbörger, 2013). On the other hand children who already use mental computation and adaptive strategies can deepen their knowledge and improve their competences in verbalising and reasoning. Working together in pairs of children with heterogeneous competences may be a chance for both of them.

THEORETICAL UND EMPIRICAL STARTING POINTS

Replacing persistent counting strategies

The central symptom for a weakness in arithmetic is the persistent use of counting while problem-solving.

Children who use counting as a main strategy in grade 2 and beyond often develop wide problems in math- ematics since counting is not a calculation strategy that can be built up to work in higher number spac- es. Besides, persistent counting often comes along with a mechanical, non-reflected procedure as well as isolated problem solving. There is a risk that the missing insights evolve to comprehension problems in mathematics education.

To replace persistent counting strategies children

have to develop alternative strategies like mental com-

putation or the use of derived-fact strategies. Most

importantly an understanding of numbers and opera-

tions has to be built up which than can be followed by

exercises to memorize the central problems. Fostering

children in replacing persistent counting strategies should enable them to represent numbers and opera- tions and imagine them as well as to recognize and use the relationship between numbers and problems. The central objective is also a comprehensive structural view on numbers and operations. This means in effect that those children who have problems in recognizing structures need to be enabled to see them. In practice children with persistent counting strategies should be enabled to realize, decompose, represent and describe numbers as structured quantities. The following four aspects characterise the comprehensive objective:

(1) The ordinal conception of numbers – which often comes along with counting procedures – needs to be complemented by a conception of quantities; especial- ly the part-whole-concept needs to be understood. (2) Counting in ones needs to be extended with counting in steps and furthermore be used to identify quanti- ties. (3) Children need to figure out that addition and subtraction come along with changes of quantities.

In a fourth step based on this, (4) basic problems can be memorized and first relations could be used in cal- culation (Häsel-Weide, Nührenbörger, Moser Opitz,

& Wittich, 2014). All mentioned aspects are not “ex- traordinary” ones as they contain competences that are essential for all pupils to be achieved during the first years in elementary school. The competences are so fundamental that they are taught not only in grade 1 but are continuously taught and extended on in the next grades of primary school. The respective lessons are suitable for cooperative learning because children with heterogeneous competences can work on the same contents but on different levels of un- derstanding. In this setting, children with persistent counting strategies can profit from the work of their partner that might probably be different, more elab- orated and structure-focused.

Therefore the learning environments need to be de- signed in a way which allows working at various lev- els. In addition the children should be motivated to recognize and talk about the mathematical structures.

Cooperation and interaction in mathematics education

As mentioned above, cooperative learning seems to be a successful teaching method. This is true espe- cially for forms of cooperative learning which are strongly structured and use clear and common meth- ods (Tarim & Akdenzi, 2008). Nonetheless the way children interact in cooperative situations is vitally

important. The success of cooperative learning seems to depend on interaction activities like verbalising, defending, asking and arguing (Pijls, Dekker, & van Hout-Wolters, 2007). Learning opportunities occur for children if they interpret actions and comments of others.

Children need to learn to communicate about math- ematics, to describe their solution process, to defend their ideas or to question the idea of another child. In classroom interaction as well as in cooperative learn- ing (young) pupils may need the moderation of the teacher. Teachers are invited to ask open questions and give children the opportunity to show their in- sights and perceptions. The best way to support the cooperation process of students is to give them help in structuring the cooperative process instead of sup- porting the finding of the (right) solution (Dekker &

Elshout-Mohr, 2004). Teacher activities should lead children to stay and participate in discussions and support them to disagree with each other (Wood, 1999).

All in all for the designing of cooperative learning environments it is also important that the tasks and the methodical setting initiate different perceptions which will be communicated between the children.

The methodical setting of cooperative learning needs to be complemented by a content structure in the tasks encouraging different views. In order to replace counting strategies and focus on relations between numbers and operations cooperative learning envi- ronments need to initiate different perspectives of numbers, number representations and operations.

DESIGN OF THE STUDY

The present study is a part of the project ZebrA (Zusammenhänge erkennen und bespre- chen  –  Rechnen ohne Abzählen). It combines the design of learning arrangements and the empirical research of the interaction and learning processes that can be reconstructed when children work in their environments. In that way the design of the study fol- lows the idea of mathematics research as design sci- ence (Wittmann, 1995) and comes along with many el- ements of design research (Gravemeijer & Cobb, 2006).

According to the idea of design science 20 lessons

– cooperative learning environments – have been

constructed to foster children in replacing persis-

tent counting strategies (Häsel-Weide et al., 2014). The learning environments focus on an understanding, demonstration and imagination of numbers and op- erations as well as on realizing the relations between them. The children learn the cooperative methods by working in the learning environments.

The environments are used at the beginning of grade 2 in primary school (ages 7 and 8) or grade 4 in special education schools. This period seems to be a proper time for two reasons: (1) In Germany the children in grade 1 work with numbers up to 20, figure out rela- tions between those numbers and add and subtract with numbers up to 20. Obviously, children are al- lowed and encouraged to make experience with high- er numbers, but according to the German curriculum the number space is opening up to one hundred at the beginning of grade 2. Here children orientate them- selves in the new number space, they work with rep- resentations of numbers and focus on the relations between them. These contents run in parallel to a deeper understanding of numbers and operations in the lower number space. (2) The limits of counting strategies become very obvious when children oper- ate with numbers up to one hundred. If the familiar counting strategies become exhausting, children can easily be encouraged to try an alternative strategy (Gaidoschik, 2012).

The study was realised from September to December 2010. All children of the class are taking part in the lessons. Each child who uses counting as its main computational strategy works with a partner who uses different strategies. The pairs are put together by the teachers considering their own estimation of children`s competences and strategy utilization, as well as the results from a test that was developed, re- alized and analysed in the scope of further researches of the project ZebrA.

The project is accompanied by two empirical studies which allow focusing the replacement of persistent counting strategies from different empirical points of

view. Whereas the quantitative study researches the effects of cooperative fostering the study presented in this paper focused on the interpretations of children dealing with the problems and discussing with the partner. We are interested to explore if and how chil- dren using counting strategies can be encouraged to consider and use mathematical structures. Therefore the work of five children using counting strategies as persistent computation (three boys and two girls) and their partners – belonging to four different classes and three schools – was video-graphed in ten lessons of the ZebrA-project. Corresponding transcripts have been made and interpreted by a group of researchers.

The analyses has been compared in an interactive way with empirical findings of other studies and theoreti- cal approaches (Häsel-Weide & Nührenbörger, 2013).

Only the results are presented in this paper.

SELECTED RESULTS Learning environments

In the tradition of mathematics education as design science the designed arrangements are central part of research output (Wittmann, 1995). So the design principles are pointed out below and it is shown ex- emplarily in which way they become apparent in the learning environments.

Structure-focused view on numbers and operations To replace counting strategies children need – among other things – to figure out that operations change quantities. Subtraction needs to be understood as taking away or determining the difference. To help children build up this idea a transparency is used (Figure  1). The transparency allows covering an amount of dots at one go. So subtraction in the model of taking away is linked with the action of “covering”

and not by “counting” (back).

The transparency can be used to cover concrete dots or dotfields which allows finding the difference by a quasi-simultaneous determination of the uncov-

Figure 1a & b: Representation of 15-5=10 and 35-5=30

ered dots. The children are taught in two learning environments to handle the transparency. First, in a dotfield only the ones are covered, so that problems, e. g. 25 – 5 = 10, are represented and solved. In this environment called “subtraction to tens” we focus on the decimal structure of the number systems. The children may realize that these types of problems are easy to solve and that they can manage them without counting – even in higher number spaces. In a sec- ond learning environment other “easy” subtraction problems, e.g. “minus 1”, “minus 10”, “minus tens” and

“minus ones” are focused on.

Sophisticated tasks

The contents of the developed learning environments for replacing counting strategies are designed for all children of a class. Children with persistent counting strategies have the possibility to work on a central understanding while other children deepen their competences. But therefore the given task has to be so complex that different levels of understanding and working are possible. Learning environments according to this so-called “natural differentiation”

(Krauthausen & Scherer, 2013) are characterised by tasks that are (partly) open and/or include descrip- tions, explanations and arguments. Open problems allow the children to pick up on their own mathemat- ical ideas, to deepen their understanding of contents or problems and sometimes to extend their own limits.

In the learning environment “subtraction to tens”

each pair of children gets a couple of cards. There are some cards with quantities until 20 and some with quantities up to hundred (Fig 2). The children are free to choose which cards and how many. They are asked to cover the ones and note the corresponding subtrac-

tion problem. If children manage already without the transparency, they are free to do so.

Cooperative settings and discursive tasks

According to the research result that successful co- operation is based on structured settings, two coop- erative are developed. The cooperative setting “path fork”

starts with a period of individual work which is followed by a period of working together on a further task. This task is build upon the individual work and focused on interaction about the insights and strate- gies. In the second cooperative method “seesaw”

the children work together on a problem from the begin- ning. They undertake different activities which are referred to each other. In the consequence they pick up the interpretation of their partner and carry on.

The tasks for the cooperative method need to allow dif- ferent interpretations so that a discussion can follow.

The learning environment “subtraction to tens” fol- lows the cooperative setting “path fork”. First, the chil- dren work with the cards on their own as described above. Then they collect their problems with the difference 10 on a worksheet (Figure 3). In the cards not all numbers from 10 until 19 are represented as

2 Weggabelung 3 Wippe

Figure 2: Material for sophisticated learning Figure 3: Cooperative worksheet

quantities, so the children are challenged to find the missing ones. They could use the discovered struc- ture or find other problems such as 30 – 20 = 10 or 45 – 35 = 10. The freedom in the interpretation of the formulation on the worksheet allows and probably stimulates interaction.

In some environments “discursive tasks” are used to strengthen the difference and the discussion. Children of a pair are not given the same tasks but tasks which refer to each other (Figure 5). After each of them has solved their card they should compare and describe common and different features. Moreover “sorting”

is used as an alternative method to induce different interpretations and an interaction about them (Häsel- Weide & Nührenbörger, 2013).

Results of the analyses of cooperative situations

The interaction of the children is analyzed to figure out in which way the interpretations of the children are affected by the cooperative interaction with their partners. Especially the interpretation of relations between numbers and problems in the interaction are analyzed. As pointed out above, it was of interest if and how the interpretations of the children with counting strategies differ from the interpretation of their partner and if these children modify the interpretations in the course of interaction. In this paper selected results of the analyses are presented and illustrated with examples. For a deeper insight into the interpretation process see Häsel-Weide and Nührenbörger (2013).

All children of the study were able to work and to co- operate in the learning environments. Children with persistent counting strategies bring in own ideas and act as real partners.

Example 1: After Kolja and Medima have noticed the problems according to the cards in the learn- ing environment “subtraction to ten” they are now asked to find new problems. Kolja, a child with per- sistent counting strategies, suggest the problems 20 – 10 = 10 and 30 – 20 = 10. It is not clear if he wants to create a new, different pattern of problems or if he ex- presses common problems with the result ten. In the interaction his partner Medima notes the problems without questioning and the children find a struc- tured column of problems in turns (Figure 4). On the basis of Kolja’s idea they find a structured sequence of problems. Even if Kolja was not aware of the new structure he suggested, he could pick it up and find the next problem. Since Medima seems to realize the structure immediately nobody feels a need to para-

Figure 5: Discursive tasks with reconstructed results of Thomas and Max

Figure 4: Document of Kolja and Medima

phrase the action. Both children are pleased to note the problems and also do not notice the mistake.

The example as well as the results of the analyses made clear that children need reasons to negotiate different interpretations (Nührenbörger & Steinbring, 2009).

First of all comparing and sorting as formal coopera- tive settings function as reason to interact. In informal situations some interactions come up by mistakes.

Example 2: Asked to compare the cards (Figure 5) in the second period of the cooperative setting “path fork”, Thomas said:

Thomas: Hey Max, Max, I have noticed something.

Here I have ten above [points at the 10 on the left card] and you have ten here below [points at the 10 on the right card].

Max: Oh. I have notice something, too. I have something different, too. There must be nine [rubs out the result 11 and notes 9]. Nine. Look, because nine plus nine equals eighteen.

Thomas notices that the difference ten can be found on both cards on different positions. He focuses on a concrete mathematical sign and describes its physical position. But it seems as if his interpretation causes a process in Max’s thinking. Perhaps Max realizes that there are two other equal results, but not two other equal problems. His statement is not clear at that point.

However, it seems to be a consequence of Thomas’s comment that Max now realizes that he has made a mistake. He corrects it and explains to his partner why the new result should be correct now. Here Max uses the inverse relation between the addition problem 9 + 9 = 18 and the given problem 18 – 9 = 9.

In this episode, relations between problems are real- ized and described in the cooperative setting, proba- bly initiated through the discursive tasks. It is open how far Thomas, the child with persistent counting strategies, understands Max’s argumentation and realizes the relations and differences for himself, be- cause the interaction ends at this point. But there is a chance that he picks up this argumentation later on.

Thomas’s interpretation which focuses on corre- sponding signs was one of the typical interpreta- tions of children using persistent counting strategies.

According to results from Gray, Pitta and Tall (1999)

the analysis show, that children with persistent count- ing strategies focus often on similar signs. Furthermore the children focus on relations between numbers in- stead of relations between problems as Thomas did in comparing the cards as well.

Thomas: Hey, here is nine [points on the “9” in the problem 18 – 9 = on the right card] and there it is eight [point on the “8” in the problem 18 – 8 =]. It is one more and there [point on the problem 18 – 7 =] it is one less.

Here, Thomas describes the relations between the subtrahends. He seems to formulate their relation based on a cardinal concept of numbers.

It becomes clear that the children see correct aspects, however they do not seem to realize the relations between problems as whole. As a consequence it is difficult for them to use the relations for deriving.

Realizing corresponding signs and first relations be- tween numbers is essential but not sufficient for de- veloping derived fact strategies. It needs to be pointed out that children who already solve problems without counting, formulate relations between problems only in few episodes. Mostly they also focus on concrete objects and relations between numbers (Häsel-Weide, 2013). Analyses shows that even teachers are pleased if children do so and do not ask them to figure out the consequence of the relation between numbers for the relations between problems. Probably children with counting strategies benefit less than possible from the cooperation because the interpretation of their partner did not differ essentially from their own.

CONCLUSION AND OUTLOOK

The presented study aims at developing and research- ing cooperative learning environments which ena- bles a structure-focused view on relation between numbers and problems for all children in grade 2. The design results and the results of qualitative research show that it is possible and productive for children with heterogeneous competences to work together.

Children with counting as persistent strategy act as real partners in the cooperation and bring in own interpretations and solutions. Although informal cooperation processes are observed the formal coop- erative settings, especially the activities “comparing”

and “sorting”, are suitable to make children interact

about mathematics.

The children’s interpretations show that the children focus on relations between numbers. In the next step children need to focus on relations between problems.

This could only be observed in very few episodes of the study. Since even children without problems in mathematics focus mainly on the relation between numbers, further research has to show if another com- bination of pairs (perhaps with partners of higher age) or a more direct moderation by the teacher make all children recognize and describe structures between problems. Besides, it has to be figured out how far an explicit focus on relations between problems goes along with non-counting procedures (Gaidoschik, 2012).

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