Addition and subtraction are important components of children's early mathematical cognitive abilities. With the development of cognitive psychology, research was not limited to only understanding the accuracy and age-difference in the arithmetic process, but further exploring the psychological process of children's addition and subtraction arithmetic and factors that affect it. As the executive and controlling part of the arithmetic process, it is an important index of the individual's problem-solving ability, and research has paid much attention to the selection and application of strategy since the 1980s. In particular, research on addition strategy has been especially avid and productive. Research on children's early addition strategy has discussed the type, quantity, structure, and level of arithmetic strategy. Analyzing information processing has provided a new view for further understanding the characteristics and regularity of the development of children's early mathematical cognitive abilities, and yielded scientific evidence for the education policies and instructional methods of addition and subtraction arithmetic. Many well-known foreign mathematics education programs, such as Cognitively Guided Instruction, Math Trailblazers and so on, put forward some important principles for teachers on addition and subtraction arithmetic education based on research on addition and subtraction strategy development, for example, evaluating children's understanding of strategies for addition and subtraction, providing extensive materials for manipulation, creating suitable problem settings to encourage children to use multiple strategies, encouraging children to speak about and discuss their strategies and so on. This kind of educational method that puts importance on children's problem-solving process and encourages them to construct it themselves has had a great effect on school education and human resource development in America. For teachers, understanding children's arithmetic strategy and observing their thinking process will not only advantageous in scientifically and objectively evaluating cognitive level, but will also help teachers to use actively children's initiative and informal mathematics experiences to give appropriative instructions to realize "teaching for understanding" and "meaningful mathematics learning." For children, during the process of active and interesting problem-solving, they can learn and apply problem-solving strategies, construct and develop mathematics concepts, have confidence and a positive attitude to mathematics learning.
Based on the above, let us review addition and subtraction arithmetic education in kindergartens in China where the accuracy and speed of addition and subtraction seems to be the only aim. In these circumstances, the concept of the arithmetic strategy is often lacking in kindergarten education, not correctly recognized and evaluated. Some teachers are familiar with some of the strategies, but this knowledge is not systematic. Some teachers and parents have a mistaken understanding. Thinking that a quick answer in addition and subtraction is the sole correct way, they neglect the various strategies children create themselves. Some teachers and parents reject or forbid the choice of arithmetic strategies that coincide with children's cognitive developmental characteristics, such as finger-counting. The author has often seen common behavior among children during research: some children cannot stop putting their hands behind back when doing arithmetic; some obviously count with their fingers but deny it; some do not try alternative effective strategies to solve problems even if they cannot recall a specific strategy; some insist on using a certain acquired skill to solve all kinds of problems, lacking flexibility and diversity in the problem-solving consciousness and ability; others are sick of counting; and others clearly want to avoid it.
Seeing and hearing these attitudes, we cannot help asking: is addition and subtraction merely a sequence of stimulation and reflection, only requiring children to recite without understanding of the quantity relationship? Is it much better to get children to quickly memorize the correct strategy taught by teachers without having them understand the relation of numbers than to let children spend some time and energy solving problems in their own ways and experiencing the "re-discovery" process in mathematics? Will those children who solve problems using hand-counting or mouth-counting embarrass adults more than those who cannot use any kind of those strategies? As for arithmetic education in kindergartens, should we have children do a great deal of arithmetic exercises or skill training that have only temporary effect, or create rich materials and meaningful events to allow children to effectively select and apply arithmetic strategies, actively engage into the problem-solving process, and gain confidence in his/her own arithmetic ability? To find the answer to the above questions, we need to re-recognize children's expressions of early arithmetic strategy (taking the addition strategy as an example in this article), and consider effective measures to balance and bridge the relation between conceptual understanding and procedural skills to reach a proficient arithmetic level.
1. Developmental characteristics of children's early addition strategy
1) Types of children's early addition strategy
According to the research by Siegler et al., before entering primary school to receive a normal mathematics education, children are already richly equipped with concepts and experiences regarding numbers and arithmetic, and use multiple strategy types to solve easy addition problems. Generally, there are five types of strategy as below:
|1.||Counting on the fingers strategy. Children use fingers as two addends to calculate the sum, usually accompanied by oral counting. Furthermore, this category includes three sub-types (with 3+5 as an example):
All-counting strategy. Children sequentially count all of the fingers that represent the two addends from 1, extending fingers one by one while counting "1, 2, 3, (stop) 4, 5, 6, 7, 8."
Maximum strategy. Children count fingers from the smaller addend, extending fingers while counting "4, 5, 6, 7, 8."
Minimum strategy. Children count fingers from the bigger addend, extending fingers while counting "6, 7, 8."
In actual calculating, children count from the first addend instead of the all-counting strategy. When the smaller addend is in front, such as in 3+5, counting from the first number is the same as the maximum strategy. While children realize the smaller number and count from the bigger one, it is regarded as the minimum strategy.
|2.||Oral counting strategy. Children say the numbers out loud and count when calculating the sum. Sometimes this is performed as silent counting with subtle mouth motion. It also can be divided to three types like the above: all-counting strategy, maximum strategy and minimum strategy. It differs from the finger-counting strategy in whether or not it involves finger action.
|3.||Finger representation strategy. Children represent two or more addends with the fingers, and get the answer without counting.
|4.||Breakdown strategy. It means children break down one addend into two smaller numbers, add one to the other to get a familiar answer, and then add the remaining number. For instance, in 3+4, if the child has remembered that 3+3=6, then he/she can break down 4 to 3 and 1 to get 6, and then add 1 to get 7. In case of addition where sum is 10 or over, a particular breakdown by 10 is used, for example, 6+5, the child can break down 6 to 5 and 1, then 5+5=10, and 10+1=11. This is a special type of breakdown strategy of grouping by 10.
|5.||Retrieving strategy. Children directly derive an answer by remembering the results of some calculations they learned before.|
What needs to be pointed out is that although both the finger-counting strategy and finger representation strategy use fingers, there is a difference both in children's outer behavior and in the inner cognitive process. For instance, when calculating 3+5, if a child counts out 5 fingers before he/she counts out 3 fingers, and then counts all of the fingers showed, that will be a finger-counting strategy. In contrast, if a child extends 3 fingers on the left hand and 5 fingers on the right simultaneously, and gets the answer by looking at the fingers without counting, that is finger representation strategy. From the outer behavior, it is not difficult to understand that, when children use finger representation strategy, they have formed a corresponding finger diagram to one addition, such as five fingers corresponding to 5, and five fingers of one hand and two fingers of the other hand corresponding to 7. It is not necessary to count all fingers corresponding to the addends one by one, so it is better than the finger-counting strategy in calculating speed and accuracy.
In addition, a strategy using familiar objects (such as operating blocks or sticks, or drawing lines or circles in paper) has the same cognitive function as the finger-counting strategy. That is to say, children use objects (regardless of whether they are natural tools for counting like fingers or external objects) to represent abstract numbers, and control the calculating process while counting objects to avoid mistakes. Many researchers did not provide extra operating objects for children when observing their arithmetic strategies, so the visible object strategy is not been listed among the types of arithmetic strategies. Considering children's cognitive developmental characteristics and interest in learning, we suggest providing such visible objects for children to help them better understand number concepts and check the calculating process.
2) Levels of children's early addition strategy
The five strategies make significantly different demands on cognitive resources and time, and appear during the early phase of children's arithmetic learning in a certain order, which reflects the process that changes in stages in accordance with the learning level.
First, compared with the other four "supportive strategies," the retrieving strategy mainly searches the answer from memory and does not require much checking of the problem-solving process of working-memory resources, so the response speed is the fastest. Meanwhile, the accuracy of retrieving strategy depends on children's previous experience with utilizing supportive strategies. In other words, using supportive strategies can help to establish the correct and stable link between two addends and answer, so that they can retain the correct answer in long-term memory. Secondly, the finger-counting and oral counting strategies rely more on the knowledge of the concept and procedure of counting. When executing such kinds of strategies, children need to use more cognitive resources to engage, supervise and adjust counting behaviors, and need external behaviors or external language to process, so the thinking process is comparatively externalized. In contrast, the finger strategy, breakdown strategy and retrieving strategy are based on children's existing presentation or internal language to conduct arithmetic. The long-term memory system is utilized in the arithmetic process more and more, so thought is characterized by gradual internalization. Third, there are different levels within the finger-counting strategy and oral counting strategy, that is, it develops from all-counting strategy, to a maximum strategy, and then to a minimum-strategy. In the first stage of arithmetic, all children count from 1, but as their concept of counting develops, they discover that they can count anywhere within the sequence of numbers. Furthermore, after several tries, children will find that changing the place of two numbers does not affect the arithmetic result. In this way, the minimum strategy has become the predominant for its quick, accurate and easy execution.
3) Developmental characteristics of children's early addition strategy
There are two remarkable developmental characteristics of children's early addition strategy. First, the five basic strategies appear in the early stage of children's addition arithmetic learning in sequence, and the high-level strategies are based on the mature application of low-level ones. Second, children's application of strategies develop in waves, that is to say, children use low- and high-level strategies simultaneously for a certain time. With time and development of cognition, these strategies, which differ in level of maturity, will show considerable change in frequency of use.
We can find the sequence in which the strategies appear from the typical behavior of addition arithmetic: for the same question "what is 3 plus 2?" in the early stage, children will extend the fingers representing 3 and 2 respectively from 1, then count all of the fingers from 1. Later, children will extend three fingers first, and then count "4" and "5" one by one while extending the fingers one by one. Later, children will count out loud from 3 to 4 and 5, and after some time, will answer "5" directly. This indicates that addition strategies appear in fixed sequence: the strategy use gradually shifts from the externalized addition strategy depending on objects to internalized strategy based on symbol representation. The reason for the step-up in strategies is that, with the mature and accurate use of the previous strategy, the demands made on cognitive resources for executing the strategy are greatly decreased. Therefore, the surplus cognitive resource is then used to search for excess process elements in the present strategy, to elicit the discovery and exploration of new strategy and drive strategies to adjust and change to quickness, accuracy and energy-saving continuously. Meanwhile, through correctly executing the supportive strategies such as counting, showing fingers, etc., it will not only promote children's understanding of the quantitative relation between addends and answer, but also form the link between the two, and the formed link will help children use the strategy more quickly and more accurately. Finally supportive strategies will disappear to promote the transition process, and retrieving-strategy will appear and be used frequently. The evolution process of addition strategies has two implications for us. First, the supportive strategies such as finger and oral counting can effectively help children to form the link between addends and the correct answer to promote the appearance and use of retrieving strategy. Second, the way of mechanically reciting addition problems easily leads to the wrong answer due to incorrect memory, and causes children to become unable to use effective strategy to solve new tasks that cannot be accomplished by retrieving an answer because they lack quantitative understanding and experience in supportive strategies. This will limit the functioning and application of children's thought.
In addition, children tend to use multiple addition strategies, and the change in use frequency of each strategy will display a wave pattern with time. As problem solvers, children use multiple addition strategies, not only one. Even younger children do not solve problems only with finger-counting. For some easy problems, e.g., the addends are smaller numbers or double numbers, children will try to retrieve the answer from memory. For some difficult problems or the answers not methodically reached, children will change to finger-counting or other supportive strategies to get the correct answer. Older children also use these strategies, but use the retrieving and breakdown strategy more often, and the use frequency of finger and oral counting declines. Therefore, the development of strategy does not advance in a step-like fashion so that high-level strategies totally replace low-level strategies, but in a wave-like pattern in which each strategy reaches a climax and valley at a different time. This means that children can select a suitable strategy to solve problems successfully and arrive at the correct answer according to the difficulty of problems and the actual effect of each strategy for themselves.
2. Suggestion for arithmetic strategy education in kindergartens
Kindergartens have always stressed training to master arithmetic skills so that when children begin to acquire mathematics knowledge that includes easy calculating steps, such as multiple digits arithmetic, sentence problems, etc., in primary school, they can answer easy arithmetic problems quickly and correctly. By doing so, they invest more cognitive resources into other areas of problem-solving that will be helpful to solve these more high-level mathematical problems. However, the solution of high-level mathematical problems not only relies on children's abilities to do simple arithmetic, but it is related more with comprehension of quantity, awareness of problem solution and meta-recognition level. The present research also found that proficiency in arithmetic must find a balance between conceptual comprehension and skill practice. Placing a disproportionate emphasis on either one will have a negative effect on flexibility, speed and accuracy of problem-solving. When children are encouraged to apply, explain and discuss arithmetic strategies, their understanding of number and arithmetic is enhanced so that they solve mathematical problems more flexibly and better than peers, and meanwhile, they acquire proficient arithmetic abilities. During pre-school, as the preparatory phase for mathematics study, mathematics education (including arithmetic education) in this period should help children to effectively construct mathematical knowledge, solve problems flexibly and positively experience a sense of accomplishment in mathematics. Consequently, it is necessary to introduce and develop arithmetic education strategy in kindergartens.
1. Pay attention to and understand the performance and characteristics of arithmetic strategies by children
Studying and understanding the performance and characteristics of arithmetic strategies will help teachers to deeply understand children's cognitive characteristics and thinking process, and set, implement and adjust education activities in accordance with development of children. On the other hand, because strategy can acutely reflect different processes for the same answer, teachers can appraise the level of children's thinking more exactly, and it will provide scientific grounds on which teachers provide appropriate and individual instruction.
Given that, at present, teachers' perceptions of children's arithmetic strategy are inconsistent and even lacking, improvements should be made from theoretical studies and education observation. First, teachers can understand the general rules and characteristics of children's arithmetic strategy development and form correct recognition of their strategy development through related professional books or journals on children's cognitive development in mathematics. Second, teachers should observe children's performance of arithmetic strategies during education activities and in daily life. This will advance the understanding of theoretical knowledge and facilitate the effective application of the knowledge of children's strategy development to educational practice. Teachers can employ concrete ways to observe strategies according to their energy and needs of children. For instance, they can choose some arithmetic problems with different difficulties according to the objective of mathematics education, give individual tests to children, record children's responses, language and behavior during the test in detail, and ask children how they arrived at the answer to make a final adjustment and analyze the type and frequency of strategy-using by each child. Furthermore, during daily life and education activities, teachers can ask the children about their methods and understand their thinking process quickly; communicate with colleagues and exchange opinions on children's strategy development performance; or make continual observation and a record of children's strategy-application to reflect their progress in mathematical ability with detailed and real data, so as to help parents to understand their children's arithmetic abilities and eliminate some of their misunderstanding.
2. Give children proper instruction on the application of arithmetic strategies
Based on an understanding of the development of children's arithmetic strategy, teachers should actively adopt effective education measures to guide children to apply strategies, induce children to consider ways of problem-solving, enhance the efficacy of children's strategy-application, and facilitate their development of arithmetic abilities based on their conceptual understanding. The following can serve as reference.
First, during arithmetic education activities, teachers should leave enough time for children to think by themselves, rather than just be satisfied with getting an answer quickly, and ask children to report the detailed solution way. This can help children to recognize the necessity of adopting some ways to solve problems, and give children the chance to arrive at the answer in their own way to avoid the phenomenon of just copying other's results.
Second, teachers should encourage children to employ multiple strategies to solve arithmetic problems. By asking children if there are other ways to solve the problem, teachers can help them understand different arithmetic strategies.
Third, teachers should provide necessary and proper materials during the course of strategy application, such as blocks, sticks, paper, pen and other tools, so that children can choose proper materials to present the quantity relationship and control the arithmetic process, which is very necessary at the first stage of arithmetic learning and in cases of difficult problems.
Fourth, teachers should provide proper instruction to lead children to use upper-level strategy in accordance with the development of arithmetic strategy and present strategy level. For instance, we know that the minimum strategy is characterized by its quickness, accuracy and ease when solving problems. Can we, however, just tell children 'keep the big number in mind, and then count forward'? The effect of strategy instruction depends on the following three factors: 1. whether the strategy is among the child's proximal development zone or not; 2. whether the child has understood the advantage and operations of the strategy or not; 3. whether the child has had proper strategy practice or not. If the child has applied the strategy of counting from the first number, it is timely to introduce the minimum strategy, because the child has understood the counting principle regardless of order, but the point is that teachers should remind children to compare the large and small numbers. If the child's prior strategy is counting from 1, then s/he has not understood that the beginning position does not make a difference in the counting result, so we should help her/him to understand the counting principle first. Therefore, we should select the proper strategy type to give instructions according to the child's strategy development level. Furthermore, strategy instruction is very important to the selection and correct application of strategies for children. Taking the minimum strategy as an example, when we display the problems in which the first number is much smaller than the second one, e.g., 2+9, we can guide children to use minimum strategy, tell them the operation of this strategy, then have them practice some challenging problems, such as 3+22, 4+33, tell them the advantage of this strategy, and ask them to explain the strategy by themselves at the end. Such instructions can strengthen their understanding and memory and enhance their frequency of strategy use.
3. Create opportunities for children to express and communicate arithmetic strategies
Creating opportunities for children to express and communicate arithmetic strategies will not only help them to understand and reflect on their use to control the executing process more effectively and enhance strategy efficiency, but will also help them to understand the diversity of problem-solving strategies and develop an attitude of respecting for others' opinions through discussing and observing others' strategies.
First, we can create proper problem situations, for example, daily mathematics problems, to encourage children to adopt various arithmetic strategies and ask them to explain or perform adopted strategies that will help them to gradually recognize and reflect their thinking process. Due to the limitation of expression and meta-cognition of pre-school children, this kind of strategy reporting activity is usually difficult at the first stage. During this time, teachers should take flexible and gradual measures to help children consciously think about the strategies used, such as giving an example of thinking out loud, allowing children performing with behavior, and so on. Second, organizing activities like reporting and communicating strategies can guide children to recognize the variety of arithmetic strategies and promote learning and other strategies via observation, imitation and other ways. Through sharing and communicating, children can understand and compare multiple strategies, which will stimulate their awareness of divergent thinking, and they can learn their peers' problem-solving strategies via observation, imitation and other ways. Meanwhile, children can learn how to listen to and respect others' opinions from this.
Taking addition strategy as an example, the study states the author's thoughts and suggestions on arithmetic education in kindergartens, with the aim of promoting active thinking about how mathematics education should effectively integrate the developmental characteristics of children's mathematics cognition, and value the thinking process and problem-solving abilities of children, so as to improve the quality of mathematics education in kindergartens, give full play to kindergarten's function as preparing education, and promote the long-term development of children's mathematical abilities.