This essay will analyze two children’s mathematical thinking respectively in the context of mathematics assessment interview (foundation detour) and the context of free play. The analysis will review children’s current thinking level by the Australian Curriculum Mathematics Foundation Level learning content and achievement standards. After the analysis, critique will be given on the extent to which children’s mathematical thinking were enhanced in the event. In the end of the both sections, suggestions will be provided for the children’s further development on mathematical thinking.
1. Observation of the Foundation Detour
The observation was conducted in an outstanding early learning center which exceeds five of the National Quality Standards and meets the rest two (Australian Children’s Education & Care Quality Authority, Starting Blocks, 2017). An estimating number of 20 children were present at the 4-5 kinder room at the time of the interview.
With the help of my partner, who is an experienced in-service early year teacher, children quickly found their ease sitting with us. They became more open and talkative with us and gradually showed great interest in the interview activities.
The boy I interviewed, T, uses English as his first language and has been going to this early learning center at a daily basis for more than a year. He is very confident in making friends and solving daily mathematics-related problems. His educator reflected that in this room they planned multiple ways to blend mathematical learning subtly in various play like construction play.
1.2 Analysis – Linking the child’s thinking to Australian Curriculum
We decided that Nassem would interview first while I assisted and T watched. So at the point that T had his turn to play with the manipulatives, he was confident and informed about the activities he was going to engage in.
T can now “connect number names, numerals and quantities, including zero, initially up to 10” (Australian Curriculum, Assessment and Reporting Authority[ACARA], Australian Curriculum[AC], ACMNA002, n.d), which is evident when he made a set of 5 teddies and answered “How many” questions. In another activity, T again showed that he was working towards this level by matching numeral cards with dot cards by 9/10 tests.
T has also established “understanding of the language and processes of counting by naming numbers in sequences” (ACARA, AC, ACMNA001, n.d), which is observed when he was checking the answers for the “(5+3) teddies” question by counting from 1 to 8. His ability to successfully put numeral cards 1-9 in correct order suggests that he could connect number with numerals and he has mastered the sequence of number then numerals. Meanwhile, T could subitise dot cards from 0 to 9, which outstands the content standards of “subitising a small group of objects” (ACARA, AC, ACMNA003, n.d).
As for geometry, T conveyed good understanding on the position words “besides”, “behind” and “in front of” when he put the teddies in the correct position following the directional language instruction. This is a sign that T is approaching the standards of “describing position and movement” (ACARA, AC, ACMMG010, n.d)
Besides, T is working towards the level to “compare, order and make correspondence between collections, initially to 20, and explain reasoning” (ACARA, AC, ACMNA289, n.d). He can now compare a small group of objects, 4 is the outmost number that was tested. Although he understood that one group had teddies than another small group and he could explain why, he had not yet understood the exact difference between sequential numbers. So he could not identify a number that was “one more than four”, or a number that is “one more than three”. As for ordinal numbers, T missed the correct answers when asked to identify the color in the teddies pattern. Hence, no evidence from the interview indicates that T could understand and use ordinal numbers.
Furthermore, T successfully adopted three ways to sum the number six using his fingers. Together with the success in the straw-sharing tasks, T demonstrated his skill of “using a range of practical strategies for adding small groups of numbers, such as visual displays or concrete materials” (ACARA, AC, ACMNA004, n.d).
Lastly, T’s developing thinking on ordering and comparing lengths is strongly supported by his success in ordering 4 candles from the shortest to the longest, underlying the ability to use “suitable language associated with measurement attributes” (ACARA, AC, ACMMG006, n.d)
1.3 Critique of the Foundation Detour.
From the child’s point of view, the interview was a sequence of tasks that they could handle using their existing mathematical knowledge and skills. Mathematical tasks are beneficial for problem-solving applying children’s mathematical knowledge, as well as their adaptive reasoning and strategic competence (Sullivan, 2011). For example, in the counting “5+3” teddies tasks, T was able to count on from 5 when a set of 3 additional teddies were unscreened. When all the teddies were screened, he went back to his counting-all strategy. Such switch implies how the challenging task affected his choices of strategy in different problem solving contexts. In addition, the tasks also place a context for children to make meaning of concept which has not yet been mastered or understood (Linder, Powes-Costello & Stegelin, 2013). So T’s thinking about some strange mathematical concept like ordinal number was concretized when he followed Naseem’s instruction counting the teddies in order.
To some extent, the tasks also built confidence into T by creating a safe environment where his mathematical opinions were shared, valued and not judged, which will further empowers him being a creative learner (Smith Ⅲ, 1996). This is evident when T completed all the tasks with great confidence, and he was able to create a color pattern when putting straws into the cups. However, the fact that most of the tasks were designed as closed ended does not necessarily prepare T to solve the mathematical problems that he might encounter in real life situations (Boaler, 1998).
1.4 Suggestion and advice to support further mathematical learning
Research finding has suggested a robust relation between frequency of children’s involvement in indirect numeracy activities at home and mathematical fluency they will achieve in school settings (LeFevre, Skwarchuk, Smith-Chant, Fast, Kamawar, & Bisanz, 2009). Out of all the commonly used approaches to facilitate children’s mathematical learning at home, asking questions, playing, instructing and pointing to examples are effective modes for children and parents to become co-learners and co-educators in mathematics (Eloff & Miller, 2006). In fact, a daily activity like cooking and be a great learning experience for children to be exposed to numeracy with parents’ guidance and questions and encouraging different explorations such as measuring and counting.(Vandermaas-Peeler, Boomgarden, Finn & Pittard, 2012).
To develop T’s understanding of number relationships, educators can utilize all kinds of everyday settings that T can practice the comparison of numbers and to consolidate his understand of the cardinal change in the number sequence. A stair can be a great tool for children to understand the cardinal change of counting numbers (Montague-Smith, Cotton, Hansen & Price, 2018). Many similar scenarios and setting, for example stepping onwards a staircase, beading, helps him understand.
Real life can also provide T with numerous opportunities to observe patterns. For example, T could observe how repeated patterns extend and how patterns grow on the exterior or interior decorations of buildings. (Montague-Smith et.al, 2018). He can even create his own drawing to copy the patterns he observed or create a new pattern by adding new elements to the original one. A recent research argues that children understand the concept of pattern more easily by creating their own pattern not copying a pattern that others created (Rodrigues, 2015). This approach will also evolve their awareness of the unit as a “code” for the patterns. Hence educators and parents might incorporate the learning of patterns in coloring or pasting for T to create his own pattern, read the pattern then transfer the pattern to another media (for example, present the pattern using body movements).
As T is now using a “counting all” strategy for addition, practicing developing counting on and counting back will allow him to count from any number in the sequence so as to solve addition or subtraction problem with bigger numbers (Montague-Smith et.al, 2018). Research has found the positive effect that children will gain from play or game experiences in the classroom (Edo, Planas & Badillo, 2015) and at home (Nicholas & Schneider, 2013) as educators and parents provide support in addition or subtraction tasks in games. Parents and educators can implement dice play or card games in a daily basis to assist T’s skill of counting. To support T’s learning, it is essential to model the counting skill when it is time to add or subtract points/dots. After the modeling, educators can encourage T to practice the strategy by himself.
Furthermore, in order to understand ordinal numbers, it is critical to connect the ordinal number words to their meaning for children and explicitly discuss correspondence between these two (Clements, Sarama & Ebook Corporation, 2014). Lining up is a great everyday activity for T to visualize ordinal numbers and observe how the number extend when the teacher counts “who is first, who is second, who is third….”. Similarly, parents can reinforce the learning of ordinal numbers when children naturally line up or build up a group of toys.
2. Observation of two children’s free play
The observation was conducted in the same room for the Foundation Detour interview. At that time the 4-5 year-olds are all engaging in various forms of free play. In a room that was distributed into four areas (a construction area, a mini park/lawn, a reading area and a play table area), the two boys were playing at the construction corner, with other three groups of children playing on other toys at that corner. An educator was sitting by the children and watching what the children were doing. Very few conversations were noted because the room was considerably noisy.
2.2 Analysis – Linking to Australian Curriculum
In this first half of the playing episode, the boys built a “parking lot” and tried to park all the toy cars they had. They put the cars in lines and columns to make sure that all the cars could fit in.
Then, the boy who proposed to build the car park suggested that this time that they were going to build a “tall” house. They took part of the box away and added a second story to their “parking lot”. When they finished building, they found out that even the houses they built were different in size and shapes, they could both have all the cars fit in. This indirection comparison of the area referencing toy cars as units underlines the learning content of using “direct and indirect comparisons to decide which is longer, heavier or holds more, and explain reasoning in everyday language” (ACMMG006, 2016).
At the end of this learning episode, the boys tried to create the “triangular” style of roof, which is the gable roof they often spotted in real life. Spontaneously, a boy connected two square magnets at one side and formed an angle, suggesting that they were developing senses of the attributes of a triangular prism. When the other boy was not sure about what to do, he was helped by his friend telling him to put exactly “two squares together” at a time. The activity to create 3D shapes, vary the shapes and name 2D shapes falls to the learning to “Sort, describe and name familiar two-dimensional shapes and three-dimensional objects in the environment” (ACMMG008, 2016).
2.3 Critique of the free play activity
The Victorian Early Years Learning and Developing Framework (Department of Education and Training Victoria [DET], 2016) proposed a play-based curriculum because in play children relate to real life situation and develop conceptual understanding. The room where the observation was conducted was a vastly resourceful room that provides an ideal physical environment for children to explore.
During the entire observation, the educator did not join any of the play. Instead, she was watching the children and making sure that they got access to the toys they wanted and that the children were keeping things in order. However, research has found that educators who can join the play and tailor themselves to address children’s need in play will achieve greater development in children’s mathematical thinking (Tramwick-Smith, Swaminathan & Liu, 2016). Whitin and Whitin (2003) also argue that it is necessary that children talk about their mathematical ideas and explain their understanding. Teachers should be able to prompt and guide these conversations to probe students’ thinking on mathematical concepts. It would’ve been a great opportunity for the educator to invest some intentional learning about the number and shapes of faces of a polyhedron, measurements using the cars and the differences and similarities of triangles and squares.
2.4 Suggestion and advice to support further mathematical learning
Measurement is closely connected to life and is common in a range of settings. Educators can now engage the boys in more measuring activities that they can indirectly compare length or area of two objects by referencing a third object as their “ruler”, which helps them develop the concept of transitivity which is also important for future learning of measurement (Clements & Stephan, 2004). Educators and parents can also probe verbal discussion comparing objects that they encounter in life, such as fruits, toys, vehicles, etc. The use of math-related vocabularies such as “tall or taller”, “short or shorter”, “heavy and heavier” is critical. Research suggests that children who were exposed to richer maternal math talks are equipped with more sophisticated mathematical knowledge (Susperreguy & Davis-Kean, 2016).
To engage in the boys’ learning interest of construction and 3D shapes composition, educators and parents can involve them in wood crafting or the recycling of paper/plastic boxes where they can observe freely about corners, arches, crosses and enclosures and how their movements can change these structures. In the process of taking the parts of furniture or utilities apart, children will learn how different pieces of shapes are composed. Meanwhile when assembling parts altogether, they will learn about how space is created and how different properties of shapes, including faces, lines, angles, etc. Also, when the children are manipulating the paper boxes, they particularly get to decompose a 3D objects to 2D shapes, which further expand their thinking about hollowness and solidity of 3D objects (Montague-Smith et.al, 2018).
Educators can ask “what if” questions to expand their explorations of possibilities. Asking why their attempts succeed or fail will encourage them to think about properties of shapes and how different movements vary the properties to alter the shape of objects (Montague-Smith, et.al, 2018).
This essay provides a brief glimpse on the active and growing mathematical thinking of 4-5 year old children. This little research has fostered the author’s thinking about assessing children’s current development and plan for research-based activities to support further mathematical learning from the perspectives of Australian Curriculum and sets a start point for the author’s mathematics teaching practice.