Major Reflection
Concerns for social justice and for the dignity of all human beings are fundamental principles in mathematics education as numeracy is a basic necessity for operating in the community, and in order to maximise success as an adult. The focus on social justice and equity is necessary to improve outcomes for Indigenous students and those “from low socio-economic backgrounds” (Ministerial Council on Education, Employment, Training and Youth Affairs [MCEETYA], 2008, p. 5), as they are “under-represented among high achievers and over-represented among low achievers” (MCEETYA, 2008, p. 5). Mathematics education is also necessary for success across other learning areas of the curriculum such as History, English and Science (National Curriculum Board, 2009). Peter Gates and Robyn Jorgensen (2009) detail “access to curriculum, access to resources and good teachers, conditions to learn, and feeling valued” (p. 164) as being necessary for social justice in mathematics, but it is essential that these areas are also of high-quality. Additionally, we need to ensure that “socio-economic disadvantage ceases to be a significant determinant of educational outcomes” (MCEETYA, 2008, p. 7).
I regard social justice and the dignity of all human beings a significant consideration for future classroom planning and my expectations of students. According to the Third International Mathematics and Science Study, a large percentage of classrooms were using low complexity problems that “emphasised procedural fluency” (National Curriculum Board, 2009). I will consider the context of the problems I pose in the classroom to ensure they are relevant to students and target relational understanding. Contextual relevance will provide students with an understanding of the importance and power of mathematics as well as “a sense of enjoyment and curiosity about” (Haylock, 2014, p. 17) mathematics. I endeavour to build relationships with students to understand their needs and adapt lesson plans to ensure inclusivity of all students.
Self-esteem is an important indicator of attitude towards maths. Generally, if a person is not confident in their abilities to succeed at something they will avoid it or they could believe it is too hard. This can create stress and impact on their ability to solve problems (Christou, Phillipou, & Menon, 2001). Self-esteem and mathematics have a reciprocal relationship where self-esteem can impact your ability to do mathematics, but experiencing failure in mathematics can then affect self-esteem (Christou, Phillipou, & Menon, 2001). Chris Hurst and Len Sparrow (2010) advise that the more “positive or enjoyable emotive responses” (p. 19) a person experiences while doing mathematics the more positive their beliefs and attitude towards mathematics. I will instil in my students that making mistakes is a positive thing as that is how we learn and with reflection we can improve our understanding of mathematics (Van de Walle, Karp, & Bay-Williams, 2014). Additionally, I will provide students with opportunities to succeed so their self-efficacy improves and they do not become defeated.
References
Christou, C., Phillipou, G., & Menon, M. (2001). Preservice teachers’ self-esteem and mathematics achievement. Contemporary
Educational Psychology, 26(1), 44-60. doi: 10.1006/ceps.1999.1028
Gates, P. & Jorgensen, R. (2009). Foregrounding social justice in mathematics teacher education. Journal of Mathematics Teacher
Education, 12(3), 161-170. doi: 10.1007/s10857-009-9105-4
Haylock, D. (2014). Mathematics explained for primary teachers. London, England: SAGE Publications.
Hurst, C. & Sparrow, L. (2010). Effecting affect: Developing a positive attitude to primary mathematics learning: Len Sparrow and Chris
Hurst remind us of the importance of helping students to develop positive attitudes to mathematics and provide practical
suggestions as to how to engage students in a variety of stimulating activities. Australian Primary Mathematics Classroom, 15(1),
18-24. Retrieved from http://search.informit.com.au.ezproxy2.acu.edu.au/fullText;dn=969014337621185;res=IELHSS
Ministerial Council on Education, Employment, Training and Youth Affairs [MCEETYA]. (2008). Melbourne Declaration on Educational
Goals for Young Australians. Retrieved from http://www.curriculum.edu.au/verve/_resources/National_Declaration_
on_the_Educational_Goals_for_Young_Australians.pdf
National Curriculum Board. (2009). Shape of the Australian curriculum: Mathematics. Barton, Australia: Commonwealth of Australia.
Retrieved from http://www.acara.edu.au/verve/_resources/australian_curriculum_-_maths.pdf
Van De Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Week 1
I found the reading about relational and instructional understanding enlightening. Skemp (1978) discussed how he felt that instructional learning, ‘learning without rules’ (Skemp, 1978, p. 9), was not learning at all. Prior to reading the text I had not considered different types of mathematical understanding. In Primary School I was taught using the instructional understanding approach which I had not considered to be an issue until now. I now understand how much easier it would be to adapt mathematical knowledge to new tasks if I understood why I needed to do something, and not just the rules. Using hands on manipulatives during activities has provided me a visual aspect that has assisted with grasping concepts. I feel relational learning is important. In future, I endeavour to provide learning opportunities and activities that are related to the real world in order to build on students’ existing knowledge. Additionally, I will provide multiple representations of concepts to improve understanding; and I will provide tools and manipulatives that are hands on to explore concepts and relationships (Van de Walle et al., 2014). Based on the research, the relational understanding approach will is less time efficient but I feel the long term benefits will compensate for the time challenge (Skemp, 1978).
References
Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15. Retrieved from
http://www.jstor.org.ezproxy1.acu.edu.au/stable/41187667?seq=1#page_scan_tab_contents
Van De Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Concerns for social justice and for the dignity of all human beings are fundamental principles in mathematics education as numeracy is a basic necessity for operating in the community, and in order to maximise success as an adult. The focus on social justice and equity is necessary to improve outcomes for Indigenous students and those “from low socio-economic backgrounds” (Ministerial Council on Education, Employment, Training and Youth Affairs [MCEETYA], 2008, p. 5), as they are “under-represented among high achievers and over-represented among low achievers” (MCEETYA, 2008, p. 5). Mathematics education is also necessary for success across other learning areas of the curriculum such as History, English and Science (National Curriculum Board, 2009). Peter Gates and Robyn Jorgensen (2009) detail “access to curriculum, access to resources and good teachers, conditions to learn, and feeling valued” (p. 164) as being necessary for social justice in mathematics, but it is essential that these areas are also of high-quality. Additionally, we need to ensure that “socio-economic disadvantage ceases to be a significant determinant of educational outcomes” (MCEETYA, 2008, p. 7).
I regard social justice and the dignity of all human beings a significant consideration for future classroom planning and my expectations of students. According to the Third International Mathematics and Science Study, a large percentage of classrooms were using low complexity problems that “emphasised procedural fluency” (National Curriculum Board, 2009). I will consider the context of the problems I pose in the classroom to ensure they are relevant to students and target relational understanding. Contextual relevance will provide students with an understanding of the importance and power of mathematics as well as “a sense of enjoyment and curiosity about” (Haylock, 2014, p. 17) mathematics. I endeavour to build relationships with students to understand their needs and adapt lesson plans to ensure inclusivity of all students.
Self-esteem is an important indicator of attitude towards maths. Generally, if a person is not confident in their abilities to succeed at something they will avoid it or they could believe it is too hard. This can create stress and impact on their ability to solve problems (Christou, Phillipou, & Menon, 2001). Self-esteem and mathematics have a reciprocal relationship where self-esteem can impact your ability to do mathematics, but experiencing failure in mathematics can then affect self-esteem (Christou, Phillipou, & Menon, 2001). Chris Hurst and Len Sparrow (2010) advise that the more “positive or enjoyable emotive responses” (p. 19) a person experiences while doing mathematics the more positive their beliefs and attitude towards mathematics. I will instil in my students that making mistakes is a positive thing as that is how we learn and with reflection we can improve our understanding of mathematics (Van de Walle, Karp, & Bay-Williams, 2014). Additionally, I will provide students with opportunities to succeed so their self-efficacy improves and they do not become defeated.
References
Christou, C., Phillipou, G., & Menon, M. (2001). Preservice teachers’ self-esteem and mathematics achievement. Contemporary
Educational Psychology, 26(1), 44-60. doi: 10.1006/ceps.1999.1028
Gates, P. & Jorgensen, R. (2009). Foregrounding social justice in mathematics teacher education. Journal of Mathematics Teacher
Education, 12(3), 161-170. doi: 10.1007/s10857-009-9105-4
Haylock, D. (2014). Mathematics explained for primary teachers. London, England: SAGE Publications.
Hurst, C. & Sparrow, L. (2010). Effecting affect: Developing a positive attitude to primary mathematics learning: Len Sparrow and Chris
Hurst remind us of the importance of helping students to develop positive attitudes to mathematics and provide practical
suggestions as to how to engage students in a variety of stimulating activities. Australian Primary Mathematics Classroom, 15(1),
18-24. Retrieved from http://search.informit.com.au.ezproxy2.acu.edu.au/fullText;dn=969014337621185;res=IELHSS
Ministerial Council on Education, Employment, Training and Youth Affairs [MCEETYA]. (2008). Melbourne Declaration on Educational
Goals for Young Australians. Retrieved from http://www.curriculum.edu.au/verve/_resources/National_Declaration_
on_the_Educational_Goals_for_Young_Australians.pdf
National Curriculum Board. (2009). Shape of the Australian curriculum: Mathematics. Barton, Australia: Commonwealth of Australia.
Retrieved from http://www.acara.edu.au/verve/_resources/australian_curriculum_-_maths.pdf
Van De Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Week 1
I found the reading about relational and instructional understanding enlightening. Skemp (1978) discussed how he felt that instructional learning, ‘learning without rules’ (Skemp, 1978, p. 9), was not learning at all. Prior to reading the text I had not considered different types of mathematical understanding. In Primary School I was taught using the instructional understanding approach which I had not considered to be an issue until now. I now understand how much easier it would be to adapt mathematical knowledge to new tasks if I understood why I needed to do something, and not just the rules. Using hands on manipulatives during activities has provided me a visual aspect that has assisted with grasping concepts. I feel relational learning is important. In future, I endeavour to provide learning opportunities and activities that are related to the real world in order to build on students’ existing knowledge. Additionally, I will provide multiple representations of concepts to improve understanding; and I will provide tools and manipulatives that are hands on to explore concepts and relationships (Van de Walle et al., 2014). Based on the research, the relational understanding approach will is less time efficient but I feel the long term benefits will compensate for the time challenge (Skemp, 1978).
References
Skemp, R. R. (1978). Relational understanding and instrumental understanding. The Arithmetic Teacher, 26(3), 9-15. Retrieved from
http://www.jstor.org.ezproxy1.acu.edu.au/stable/41187667?seq=1#page_scan_tab_contents
Van De Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Week 2
This week I have learnt how important place-value education is for mathematical success. Experts suggest a strong understanding of place-value is necessary in order for students to experience ongoing mathematical success (McGuire, 2013). It is therefore crucial that a relational approach is used to teach place-value concepts. Place-value is understood first through groupings of ten so it is important for students in the early years to create groups of tens using ones (Van der Walle, Karp, Bay-Williams, 2014). This can then be extended to counting groups of ten. Multiple concrete representations are valuable for assisting students to grasp these concepts. I will exercise vigilance in the classroom to ensure that students’ misconceptions are remedied through activities utilising hands on manipulatives. These include place-value flip charts, place-value pipes and place-value mats, along with proportional tools such as MABs and non-proportional tools that include coloured counters or money. Language is also important to understanding place-value. Different language can be used for grouping such as twenty-four being 2 tens and four ones or 24 ones (Van der Walle et al., 2014). Using a variety of language will provide more opportunities for students to understand.
References
McGuire, P. (2013). Analysis of place palue instruction and development in pre-kindergarten mathematics. Early Childhood Education
Journal, 41(5), 355-364. doi: 10.1007/s10643-013-0580-y
Van de Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
This week I have learnt how important place-value education is for mathematical success. Experts suggest a strong understanding of place-value is necessary in order for students to experience ongoing mathematical success (McGuire, 2013). It is therefore crucial that a relational approach is used to teach place-value concepts. Place-value is understood first through groupings of ten so it is important for students in the early years to create groups of tens using ones (Van der Walle, Karp, Bay-Williams, 2014). This can then be extended to counting groups of ten. Multiple concrete representations are valuable for assisting students to grasp these concepts. I will exercise vigilance in the classroom to ensure that students’ misconceptions are remedied through activities utilising hands on manipulatives. These include place-value flip charts, place-value pipes and place-value mats, along with proportional tools such as MABs and non-proportional tools that include coloured counters or money. Language is also important to understanding place-value. Different language can be used for grouping such as twenty-four being 2 tens and four ones or 24 ones (Van der Walle et al., 2014). Using a variety of language will provide more opportunities for students to understand.
References
McGuire, P. (2013). Analysis of place palue instruction and development in pre-kindergarten mathematics. Early Childhood Education
Journal, 41(5), 355-364. doi: 10.1007/s10643-013-0580-y
Van de Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Week 3
Addition and subtraction are a critical resource for students. This a tool that they will use frequently for the rest of their lives, and just as importantly to develop their mathematical thinking (Erdogan, 2010). Even though “addition and subtraction are complimentary operations” (Baroody, 1999, p. 138), research shows that students do not always understand the relationship between these operations (Baroody, 1999). For that reason, I will need to consider the structure of lessons and activities so that we explore both addition and subtraction together. Additionally, I will incorporate a variety of problem types such as “change problems, compare problems and part-part-whole problems” (Van de Walle, Karp, & Bay-Williams, 2014, p. 159) to ensure my students have a broad understanding of these concepts. I think it is important that teaching addition and subtraction is in a context that is relevant to students’ lives and that they have many opportunities to participate in hands-on activities. I endeavour to create contextual problems that link to other classroom activities, excursions or personal experiences in order for students to accommodate new concepts more easily (Van de Walle et al., 2014). Then the use of multiple concrete representations will deepen their understanding of the concepts (Van de Walle et al., 2014). I would also like to use the think board in my classroom to provide students the opportunity to explore problem solving through multiple representations.
References
Baroody, A. (2009). Children's Relational Knowledge of Addition and Subtraction. Cognition and Instruction, 17(2), 137-175. Retrieved
from http://www.jstor.org.ezproxy1.acu.edu.au/stable/3233824?seq=1#page_scan_tab_contents
Erdogan, E. (2010). A comparison of curricula related to the teaching of addition and subtraction concepts. Procedia: Social and
Behavioral Sciences, 2(2), 5247-5250. Retrieved from http://ac.els-cdn.com/S1877042810008943/1-s2.0-S1877042810008943- main.pdf?_tid=d5be7876-c9f4-11e4-a1cb-00000aab0f27&acdnat=1426301592_a533036532ef4ee3c62318a4b19a01de
Van de Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Addition and subtraction are a critical resource for students. This a tool that they will use frequently for the rest of their lives, and just as importantly to develop their mathematical thinking (Erdogan, 2010). Even though “addition and subtraction are complimentary operations” (Baroody, 1999, p. 138), research shows that students do not always understand the relationship between these operations (Baroody, 1999). For that reason, I will need to consider the structure of lessons and activities so that we explore both addition and subtraction together. Additionally, I will incorporate a variety of problem types such as “change problems, compare problems and part-part-whole problems” (Van de Walle, Karp, & Bay-Williams, 2014, p. 159) to ensure my students have a broad understanding of these concepts. I think it is important that teaching addition and subtraction is in a context that is relevant to students’ lives and that they have many opportunities to participate in hands-on activities. I endeavour to create contextual problems that link to other classroom activities, excursions or personal experiences in order for students to accommodate new concepts more easily (Van de Walle et al., 2014). Then the use of multiple concrete representations will deepen their understanding of the concepts (Van de Walle et al., 2014). I would also like to use the think board in my classroom to provide students the opportunity to explore problem solving through multiple representations.
References
Baroody, A. (2009). Children's Relational Knowledge of Addition and Subtraction. Cognition and Instruction, 17(2), 137-175. Retrieved
from http://www.jstor.org.ezproxy1.acu.edu.au/stable/3233824?seq=1#page_scan_tab_contents
Erdogan, E. (2010). A comparison of curricula related to the teaching of addition and subtraction concepts. Procedia: Social and
Behavioral Sciences, 2(2), 5247-5250. Retrieved from http://ac.els-cdn.com/S1877042810008943/1-s2.0-S1877042810008943- main.pdf?_tid=d5be7876-c9f4-11e4-a1cb-00000aab0f27&acdnat=1426301592_a533036532ef4ee3c62318a4b19a01de
Van de Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
Week 4
I had not realised how many alternative strategies are available for working through addition and subtraction, most of these assisting with mental computations. “A study found that adults used mental computation methods for 85 percent of the calculations they did in their daily lives” (Northcote & McIntosh, as cited in Van de Walle et al., 2014, p. 231). This signifies the importance of mastering the skill. It is essential to provide students with ample opportunities to explore their own strategies as well as to share strategies with peers so they can find the strategies that “play to their strengths” (Van de Walle et al., 2014, p. 232). I would like to create posters as students build on their repertoire of alternative strategies so they are able to refer back to them, and so they have support if they decide to try strategies they have not yet mastered. Since I was at school there has been a large shift from memorisation to understanding. Having a meaningful context for learning basic facts assists students to gather a variety of strategies to solve problems (Choi-Koh, 1999). For this reason I will ensure my lessons include manipulatives that will help develop a deeper understanding. Research has shown that if children move from alternative strategies to standard algorithms too quickly they become reliant on standard algorithms so it is important to allow enough time to immerse into alternative strategies (Van de Walle et al., 2014).
References
Choi-Koh, S. (1999). Developing meaning for addition and subtraction. Australian Primary Mathematics Classroom, 4(4), 19-25. Retrieved
from http://search.informit.com.au.ezproxy1.acu.edu.au/fullText;dn=460936164008382;res=IELHSS
Van de Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.
I had not realised how many alternative strategies are available for working through addition and subtraction, most of these assisting with mental computations. “A study found that adults used mental computation methods for 85 percent of the calculations they did in their daily lives” (Northcote & McIntosh, as cited in Van de Walle et al., 2014, p. 231). This signifies the importance of mastering the skill. It is essential to provide students with ample opportunities to explore their own strategies as well as to share strategies with peers so they can find the strategies that “play to their strengths” (Van de Walle et al., 2014, p. 232). I would like to create posters as students build on their repertoire of alternative strategies so they are able to refer back to them, and so they have support if they decide to try strategies they have not yet mastered. Since I was at school there has been a large shift from memorisation to understanding. Having a meaningful context for learning basic facts assists students to gather a variety of strategies to solve problems (Choi-Koh, 1999). For this reason I will ensure my lessons include manipulatives that will help develop a deeper understanding. Research has shown that if children move from alternative strategies to standard algorithms too quickly they become reliant on standard algorithms so it is important to allow enough time to immerse into alternative strategies (Van de Walle et al., 2014).
References
Choi-Koh, S. (1999). Developing meaning for addition and subtraction. Australian Primary Mathematics Classroom, 4(4), 19-25. Retrieved
from http://search.informit.com.au.ezproxy1.acu.edu.au/fullText;dn=460936164008382;res=IELHSS
Van de Walle, J., Karp, K., & Bay-Williams, J. (2014). Elementary and middle school mathematics teaching developmentally (8th ed.). Essex,
England: Pearson Education.