#teachingmaths

Mathematical equations can seem quite daunting for children if they are yet to learn the concept or strategies (Singer, Ellerton & Cai, 2013). For example, look at the equation above, it may be straightforward for us as we’ve been learning mathematical concepts for years and have developed a range of strategies and knowledge for answering this concept (Swan, Pead, Doorman & Mooldijk, 2013). Yet, for most children, particularly younger children this problem can look complex as it requires the multiplication of larger numbers. I, personally disliked long multiplication because I could easily get confused with what numbers required multiplication and the step of adding the two numbers.  I believe one of the key reasons why I couldn’t grasp the concept of large number multiplication was because I was not stimulated enough to learn it. I felt defeated and unengaged each time I went through the learning process. Fortunately, I’ve come to learn that this is normal behaviour. Emotions are one of the major issues influencing cognitive learning, the result of mathematical interpretation comes from physiological arousal (McLeod, 1988; Tofade, Elsner & Haines, 2013). The point of teaching is to provide rich learning experiences that capture the attention of children (Tofade, Elsner & Haines, 2013). We want to provide experiences that stimulate the minds and activates mathematical thinking processes (Way, 2011).

Using questions throughout lessons is a teaching technique that assesses students’ knowledge, support their comprehension, and stimulate critical thinking (ACARA, n.d. ACMNA057; Tofade, Elsner & Haines, 2013, Way, 2011). Using appropriate and well-constructed questions leads to rich discussions, new insights, and supports the comprehension of content (ACARA, n.d., ACMNA076; Tofade, Elsner & Haines, 2013; Singer, Ellerton & Cai, 2013). Through inquiry-based experiences such as questioning, children can begin to rationalise the direction of the steps in order to achieve an answer (Singer, Ellerton & Cai, 2013). Reflect back to when you were younger, in order to learn you questioned experiences and ideas and when you received the answer you stored that into your memory for later reference. In reference to the equation above the following questions are used to stimulate children’s thinking process:

Closed questions:

Which is the larger number? (ACARA, n.d., ACMNA052, ACMNA072, Does it make a difference if we calculate the higher number first? (ACARA, n.d., ACMNA060) Can we break this down? (ACARA, n.d., ACMNA057, ACMNA074 Can we make the numbers smaller? (ACARA, n.d., ACMNA073)

Open-ended questions?

How else could we display this so we can count it? (ACARA, n.d., ACMNA081) How are these numbers different? What strategies do we use for normal multiplication? (ACARA, n.d., ACMNA074, ACMNA075) What else could we do rather than draw it out?

References 

Australian Curriculum, Assessment and Reporting Authority. (n.d.) Australian Curriculum: Mathematics, Year 3 to Year 5. Retrieved from: https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/?year=11754&year=11755&year=11756&strand=Number+and+Algebra&capability=ignore&capability=Literacy&capability=Numeracy&capability=Information+and+Communication+Technology+%28ICT%29+Capability&capability=Critical+and+Creative+Thinking&capability=Personal+and+Social+Capability&capability=Ethical+Understanding&capability=Intercultural+Understanding&priority=ignore&priority=Aboriginal+and+Torres+Strait+Islander+Histories+and+Cultures&priority=Asia+and+Australia%E2%80%99s+Engagement+with+Asia&priority=Sustainability&elaborations=true&elaborations=false&scotterms=false&isFirstPageLoad=false

Fessesha, E. & Pyle, A. (2016). Conceptualizing play-based learning from the kindergarten teacher’s perspective. International Journal of Early Years Education. Retrieved from: DOI: 10.1080/09669760.2016.1174105

McLeod, D. B.. (1988). Affective Issues in Mathematical Problem Solving: Some Theoretical Considerations. Journal for Research in Mathematics Education, 19(2), 134–141.

Singer, F.M., Ellerton, N. & Cai, J. (2013). Problem-posing research in mathematics education: new questions and directions. Educational Studies in Mathematics, 83(1). https://doi.org/10.1007/s10649-013-9478-2

Swan, M., Pead, D., Doorman, M., Mooldijk, A. (2013). Designing and using professional development resources for inquiry-based learning. ZDM Mathematics Education, 45(945). https://doi.org/10.1007/s11858-013-0520-8

Tofade, T., Elsner, J., & Haines, S. T. (2013). Best practice strategies for effective use of questions as a teaching tool. American journal of pharmaceutical education, 77(7), 155. doi:10.5688/ajpe777155

Through the study of geometry and spatial awareness, children learn location, structure, shape, and transformation (Horne, 2003; Newcomb & Stieff, 2012). Children use this knowledge make sense of the geometry components including features and properties of shapes and objects (ACARA, n.d.). The study of geometry and spatial thinking is highly valuable to children’s mathematical learning as it can be applied in other areas (Horne, 2003).

Geometry and spatial awareness are one of the mathematical areas that can be utilised and benefit from visual imagery (Spelke, Ah Lee & Izard, 2010). During a practical experience last year, I observed year three children engaging in a geometry lesson, the properties of shape. The teacher has used the smartboard to show a 3D image of shapes. She used an app that allowed her to use her cursor to move the shape into different angles. She then extended the activity by allowing children to use the cursor themselves and take notes of the different orientation, turns, and properties of the shapes. I really enjoyed viewing this experience as I witness the classroom become extremely engaged, they were able to comprehend the content through the support of technological media.

For my stimulus in this module, I wanted to use the context of visual imagery in relation to geometry and mapping. I have decided to use a google maps image of a school, in the context of this module I have used a random google image school, but for the real-life experience, I would use the actual school of attendance. I used this image because it allows children to conceptualise an understanding of length, shape, size, scales, true size and comparison (MacDonald & Lowrie, 2011; Spelke, Ah Lee & Izard, 2010). Throughout this experience, I would use open-ended questions that allowed children to make sense of the image in relation to geometry.

Note. This experience is designed for a year two or year three classroom.

What does this image tell us?

This allows children to make connections to their everyday environment by locating key features relating to geometric shapes. Children will build awareness towards angles and measurement within their everyday environment (ACMMG064), identify and locate symmetry within the image (ACMMG066) and locate shapes, structures, and measurement (MacDonald & Lowrie, 2011).

What can we compare this scale too?

This will allow children to consider ratio in terms of geometry reasoning. Using a physical item for comparison, place the item next to the image and ask this question (Copley, 2010). It allows children to compare the size, is this realistic? (ACARA, n.d., ACMMG064). How do they know it’s not true to size? What can we do to compare? Then extend the learning by inviting children to think what we use to measure this large area surface (ACARA, n.d., ACMNG061)

If we are guiding a new student around the school, how can we direct them to the library (or any given location)?

Children will have the opportunity to observe for location, positions, and pathways (ACARA, n.d., ACMMG05). This question will allow children to make comparisons of measurement with their peers. It will allow a rich discussion of transitivity and conservation (MacDonald & Lowrie, 2011)

References

Australian Curriculum, Assessment, and Reporting Authority. (n.d.) Australian Curriculum: Mathematics.

Copley, J .V. (2010). Geometry and spatial sense in the early childhood curriculum. In J. V. Copley (Ed.), The young child and mathematics (2nd ed.), (pp. 99-117). Reston, VA: National Council of Teachers of Mathematics

Horne, M. (2003). Properties of shape. Australian Primary Mathematics Classroom, 8(2), 8. http://ezproxy.acu.edu.au/login?url=http://search.informit.com.au/documentSummary;dn=408130323462644;res=IELHSS

MacDonald, A., & Lowrie, T. (2011). Developing measurement concepts within context: Children’s representations of length. Mathematics Education Research Journal, 23(1), 27-42. http://ezproxy.acu.edu.au/login?url=http://dx.doi.org/10.1007/s13394-011-0002-7

I found the research completed by Bobis (2007) to be so intriguing. I was almost memorised to reflect upon where I had fallen short within my years of learning maths. A key paragraph that captured my attention the most:

"The worry with an early emphasis on standard algorithms is that students will shift their focus to executing convenient procedures rather than on understanding the mathematics." - Bobis (2007)

Thinking back to when I would be completing maths tasks, I felt so lost and confused which resulted in me writing and using an algorithm, even if wasn't related to that mathematical area. This piece of computational knowledge would have served as a strength throughout the development of my mathematical knowledge (Grover & Pea, 2013; Bobis, 2007). Fortunately, these reflections of my own experiences allow me to realise where I struggled and what key information I missed that lead me to struggle. It allows me to consider how am I providing the right information in the most engaging and mind stimulating way.

Number Line Strategies to Support Mental Computation:

The introduction of strategies such as number lines promotes children's awareness of multiple relationships among numbers, understanding of mathematical procedures, the examination of assumptions and opportunity for visual references for mental computations (Kaminski, 2002; Cheeseman, 2010). Number line strategies include:

Patterns Placement:

Research explains that the development of number sense supports the ability to build upon the mental computation of children (Yadav, Mayfield, Zhou, Hambrusch & Korb, 2014) Through pattern placement, children learn number sense by adding and subtracting numbers through numerical value (Bobis, 2007).

Empty Number Lines:

The procedure of an empty number lines allows for children to access and use conceptual thinking processes, such as logical reasoning, decomposition, a recreation of patterns and abstract thinking (Berry & Csizmadia, 2016). By using an empty number line, children are able to map out visually techniques to build upon sophisticated strategic thinking by using different methods of sequencing by number groups (Cheeseman, 2010).

Number Line Addition and Subtraction:

Number lines can be used to exercise children left-to-right spatial-numerical representations of numbers (Aulet & Lourenco, 2018; Kaminski, 2002). Number lines allow children to map out numbers and the spaces between them when adding or subtracting (Aulet & Lourenco, 2018). Children are able to make a relationship between algorithms and spatial organisation (Aulet & Lourenco, 2018; Kaminski, 2002).

References:

Aulet, L., & Lourenco, S. (2018). The Developing Mental Number Line: Does Its Directionality Relate to 5- to 7-Year-Old Children's Mathematical Abilities? Frontiers in Psychology, 9, 1142.

Bobis, J. (2006). From here to there: The path to computational fluency with multi-digit multiplication. Australian Primary Mathematics Classroom, 12(4), 22- 27.  http://ezproxy.acu.edu.au/login?url=http://go.galegroup.com/ps/i.do?&id=GALE|A170817120&v=2.1&u=acuni&it=r&p=AONE&sw=w&authCount=1

Cheeseman, J. (2010). Empty number lines: How can we help children to use them? In J. Mousley, L. Bragg, & C. Campbell (Eds.), Mathematics-Celebrating achievement 100 (Proceedings of the 42nd annual conference of the Mathematical Association of Victoria pp. 49-58) Melbourne, Vic: MAV. http://ezproxy.acu.edu.au/login?url=http://search.informit.com.au/documentSummary;dn=998640761921722;res=IELHSS

Grover, S., & Pea, R. (2013). Computational Thinking in K-12: A Review of the State of the Field. Educational Researcher, 42(1), 38-43.

Kaminski, E. (2002). Promoting mathematical understanding: Number sense in action. Mathematics Education Research Journal, 14(2), 133-149.

In the child-initiated dramatic play of Jabil, Elle, and Joey, they show to be using discussion and analysing to comprehend mathematical concepts (Touhill, 2013). The mathematical concepts which they are trying to understand are division, addition, subtraction and shape, and size. To further define in line with the Mathematical for Kindergarten, these children are investigating the concepts of subtilising small collections (ACARA, n.d., ACMNA003), sorting and describing shapes (Queensland Studies Authority, 2006; ACARA, n.d., ACMMG010), data representation and interpretation (ACARA, n.d., ACMSP011). The Australian Curriculum suggests that as Jabil, Elle, and Joey are in Kindergarten and they should be learning to understand the concepts of quantities and numerals using problem-solving skills using materials and using the discussion to identify a reasonable answer (ACARA, n.d.). 

Through my recent teaching experiences in both an early childhood and primary setting, I've come to believe that children learn best through inquiry learning and play-based experiences. I strongly believe that it allows children to hypothesise, experiment, question and conclude an understanding. This mindset is supported through the consideration of what kind of learner, I am. I am not a theoretical person but rather a practical learner. I need to observe and use my senses to make a theory in which I can put into practice. This learning strategy is highly common for many people, especially children. Our attention spans can be easily distracted and bored if we are not engaged, this is also common for children with Attention Deficit Disorder (ADD). What I believe and what is supported by theory is that, teaching of the curriculum should be a sum of engaging, interest-based and rich content activities (Zeki, 2017; Ginsberg & Amit, 2008). Therefore, my teaching approaches with Jabil, Elle and Joey would be: 

• A deep discussion of the possible outcomes, allowing children to use counting skills and subtraction strategies 

• Experimenting by using a practical use; playdoh (one for each child) • Allowing children to create and cut shapes then discussing why they chose that particular shape (ACARA, n.d., ACMMG010)

 • Discuss with children how many times that shape could be cut inside of the pizza to make an equal amount for all three people (ACARA, n.d., ACMMG010, ACMSP011) 

• Discussing with the group to find out what shape works best and how many pieces they could provide each person 

• Capturing the thinking process and outcomes through photographs that can be displayed through the classroom 

The domain of subtraction features several growth points that are required to extend on children’s learning (Gervasoni, 2011). These available growth points allow educators to identify children’s skills and especially children who may be vulnerable in particular areas. The Early Numeracy Interview (ENI) was a strategy used to identify children’s vulnerability within mathematical domains. Below, I have provided a sample of growth points required for subtraction as stated by the Victorian Government Education and Training and the research of Gervasoni (2011): 

1. Early stages: Unable to combine and count collections 

2. Counting all through physical representation – using perceptual strategies to subtract with numbers from ten 

3. Counting on from one number to find a total of two collections 

4. Using appropriate strategies to count down or count back 

5. Using new (basic) strategies: doubles, adding 10, ten facts

6. Derived strategies: near doubles, build to next ten, family facts, intuitive – more sophisticated strategies and using them to solve larger number problems. 

7. Extending and applying subtraction strategies using basic, derived and intuitive strategies – Solving mentally while using appropriate strategies and understanding key concepts.