Literature Review
Importance
The importance of childhood education is well established, and numerous studies confirm the matter. International experience confirms that the quality of pre-school education affects the later development of children (MacDonald & Carmichael, 2017).
Early acquisition of mathematical skills is crucial for toddlers to facilitate their learning abilities during the school years (MacDonald & Carmichael, 2017). Even though it is traditionally believed that children before seven do not have fully-developed ability to reason logically, a study by Wright, Robertson, and Hadfield (2011) confirms that 5-year-olds are well on the way to developing such skills, while 6-year-olds often demonstrate a full ability to think using logic. In short, early exposure of children to mathematics activity, including length measuring, positively affects future success in the acquisition of mathematical skills.
Key Concepts
Length measuring may become an intricate task for young children since it implies that they understand several important concepts. Bush (2009) states that there are three essential concepts that need to be learned for children to acquire the skill of measuring length, including transitive reasoning, identical units, and iteration. Bush (2009) states that transitivity is based on comparison and explains it as the ability to understand that if A=B and B=C, then A=C.
The idea of identical units is essential for measuring with standard and non-standard scales since units of length for measuring objects must be identical (Bush, 2009). Iteration is the ability to use an object repeatedly to find measurements (Bush, 2009). While the three-concept approach seems appropriate, it does not provide enough details of the thinking process while measuring length.
More detailed approaches identify six or eight basic concepts that children should understand to develop the ability to measure length. According to Clements and Stephan (2004), they are partitioning, unit iteration, transitivity, conservation, accumulation of distance, and relation to number. Partitioning is the ability to mentally divide an object into smaller equal parts, such as centimetres or inches. Conservation refers to the understanding that the length of an object remains unchanged even if observed from a different location. Accumulation of distance is an appreciation that “the number words signify the space covered by all units counted up to that point” (Clements & Stephan, 2004, p. 7).
Comprehending the relation between number and measurement implies the understanding that discrete units can be used to measure continuous objects. Lee and Francis (2016) contribute two other concepts to this list by suggesting the addition of attribute, the appreciation that lengths span fixed distances and origin, the understanding that any point on a scale can be used to start measurements. In short, the number of concepts is inconsistent in the reviewed body of literature.
Learning Progression
The learning progression is based on the nature of the developmental stages of children. According to Sarama, Clements, Barrett, Van Dine, and McDonel (2011), there are seven stages of understanding length. First, children do not recognize length as an attribute and believe that anything that is not straight cannot belong. Second, they start to understand that length is an absolute rather than a comparative descriptor.
During the third stage, children acquire the appreciation that attributes can be directly compared, while the fourth phase signifies their ability to compare them indirectly. The fifth step in learning the matter is gaining the ability to perform end-to-end measurements. In the sixth stage, students can measure an object using iteration of standard and non-standard units. The last stage relates to the full competence in measurements “knowing the need for identical units, the relationship between different units, partitions of the unit, zero points on rulers, and accumulation of distance” (Sarama et al., 2011, p. 670). Teachers should guide their students from one developmental stage to another.
There are several remarks about the transition from one stage of leaning length measurement to another. According to Blevins and Cooper (1986), it is vital to understand that children’s appreciation of transitivity may be fragile and minor alteration in tasks may lead to significant difficulties in performance. Since transitivity is a concept that is acquired gradually, teachers are not to hurry to move to further stages of learning the matter.
Kotsopoulos, Makosz, & Zambrzycka (2015) also suggest that instead of the consecutive introduction of non-standard and standard units, both approaches to measurement should be introduced simultaneously given that children are familiar with numbers. Ultimately, the learning progression can be altered to meet the needs of a particular group of learners.
Suggestions for Teaching
Educators are to apply relevant teaching strategies confirmed by the latest empirical evidence to improve learning outcomes. Even though the measurement of length is taught repeatedly, research by Kamii (2006) shows that more than half of seven-graders are unable to use measurement skills adequately. Colliver (2017) claims that the reason for low performance may be little interest in the learning activities. Curiosity to the subject in pre-school children should be fostered through child-initiated play rather than mathematics activities prescribed by adults (Colliver, 2017).
McLennan (2018) suggests promoting measurement activities in kindergarten during outdoor playtimes to captivate toddlers’ attention. For older children, Kamii (2006) recommends using rulers with explicitly marked zero away from the edge to provoke children’s thinking about units. In short, the use of these findings may improve children’s competence in length measurement.
Curriculum Links
Curriculum Overview
The topic of length measurement is adequately addressed in the Australian Curriculum from Foundation to Year 3. According to the Australian Curriculum, Assessment and Reporting Authority (ACARA, n.d.), during the Foundation Year, students are expected to learn a direct and indirect comparison of measurements, including length. Consequently, before starting formal education, children are supposed to have an understanding of length as an absolute attribute.
The activities focus on comparing the lengths of familiar objects and using appropriate everyday vocabulary to describe the findings. By the end of Year 1, children are expected to be able to measure and compare two objects using uniform informal units (ACARA, n.d.). Teachers suggest using hands, feet, and toothpicks to measure the lengths of familiar objects and compare the attributes pairs of items.
In the course of Year 2, students should gain the skill of selecting appropriate informal units to measure various objects (ACARA, n.d.). Additionally, they are to compare sets of objects, draw adequate conclusions, and discuss their findings. During this year, educators show cartoons and offer to read various stories about the utility of length measurement (ACARA, n.d.). During Year 3, formal units are introduced through appropriate media and conversation (ACARA, n.d.).
By the end of the year, children are expected to measure length using a ruler and compare measurements based using conventional scales (ACARA, n.d.). In summary, the Australian Curriculum aims at gradually developing length measurement skill by repeatedly addressing the matter between Foundation and Year 3.
Connection of the Suggested Pedagogies
The Australian Curriculum is consistently structured in accordance with the findings of the literature review presented in this paper. During the first four years of formal education, students are gradually familiarized with key concepts suggested by Clements and Stephan (2004) through appropriate activities. Additionally, ACARA (n.d.) adheres to the developmental stages described by Sarama et al. (2011) and gradually introduces new competencies according to students’ age. Every year, students review their knowledge learned in the previous years, which is vital according to Blevins and Cooper (1986) since such complicated concepts as transitivity are learned gradually. However, despite the logical and comprehensible structure, the literature review suggests that the Australian Curriculum can be improved.
First, more outdoor activities should be introduced to facilitate student-initiated mathematics activities. Colliver (2017) and McLennan (2018) provide excellent examples of activities and parks and forests that arouse children’s interest without a need for adults to set tasks. Second, ACARA should consider introducing formal and non-formal length units simultaneously since Kotsopoulos et al. (2015) provides significant empirical support for the matter. Since children in Year 1 are already familiar with the numbers, rulers can be used together with other objects to perform measurements.
Finally, ACARA (n.d.) does not provide any recommendations about what rulers should be used during Year 3. According to Kamii (2006), it would be beneficial if ACARA insisted on using rulers with explicitly marked zero away from the edge. In summary, even though, according to empirical evidence, there are minor alterations that can be made to the Australian Curriculum, its overall structure is logical and consistent with recent research findings.
Example Activity
Activity Description
In my teaching experience, I designed an outdoor activity that aimed at comparing the lengths of sticks. During the activity, children were encouraged to look for a stick they like for 3 minutes and then show it to the class. Then the children were instructed to split into pairs and decide who has a long stick by directly comparing the objects. After that, the pairs had to team up with another one and determine who has the longest stick and who has the shortest.
After that, the groups would have to stand in line where the first person would have the longest stick, and the last person has the shortest stick. I asked students to draw conclusions about the length of the stick if student A stands between students B and C. The activity helped the children to compare objects directly and indirectly and learn the concept of transitivity.
Justification
The activity described above has proven to be effective in helping children acquire crucial mathematics skills. The primary reason for the matter is that it was an outdoor play-based activity, which is vital, according to McLennan (2018). Children seem to express more interest and become more adherent to instructions when they are introduced as rules of a game. Second, direct and indirect comparison, together with the concept of transitivity, could be learned by the students of Year 2.
The activity adheres to the Australian Curriculum and the learning progression described by Sarama et al. (2011). Third, stick attributes comparison is an innocuous and inexpensive intervention, which central according to Colliver (2017). I was not afraid to try the activity since even if children found it unexciting, the risks are minimal. In short, the proposed activity is stimulating, cost-efficient, and consistent with the latest empirical evidence and Australian curriculum suggestions.
References
Australian Curriculum, Assessment and Reporting Authority. (n.d.). F-10 Curriculum: Mathematics. Web.
Blevins, B., & Cooper, R. (1986). The development of transitivity of length in young children. The Journal of Genetic Psychology, 147(3), 395-405. Web.
Bush, H. (2009). Assessing children’s understanding of length: A focus on three concepts. APMC 14(4), 29-32.
Clements, D., & Stephan, M. (2004). Measurement in PreK-2 mathematics. Web.
Colliver, Y. (2017). Fostering young children’s interest in numeracy through demonstration of its value: The Footsteps Study. Mathematics Education Research Journal, 30(4), 407-428. Web.
Kamii, C. (2006). Measurement of length: How can we teach it better? Teaching Children Mathematics, 13(3), 154-158.
Kotsopoulos, D., Makosz, S., & Zambrzycka, J. (2015). Number knowledge and young children’s ability to measure length. Early Education and Development, 28(8), 925-938. Web.
Lee, M., & Francis, D. (2016). 5 Ways to improve children’s understanding of length measurement. Teaching Children Mathematics, 23(4), 218-224. Web.
MacDonald, A., & Carmichael, C. (2017). Early mathematical competencies and later achievement: insights from the Longitudinal Study of Australian Children. Mathematics Education Research Journal, 30(4), 429-444. Web.
McLennan, D. (2018). The Beautiful Tree Project: Exploring measurement in nature. Teaching Children Mathematics, 25(1), 16-23. Web.
Sarama, J., Clements, D., Barrett, J., Van Dine, D., & McDonel, J. (2011). Evaluation of a learning trajectory for length in the early years. ZDM, 43(5), 667-680. Web.
Wright, B., Robertson, S., & Hadfield, L. (2011). Transitivity for height versus speed: To what extent do the under-7s really have a transitive capacity? Thinking & Reasoning, 17(1), 57-81. Web.