Understanding Mathematics - Assignment Example

Running head: MCK Test MCK Test Name Course Tutor Date PART 1 Overview of MCK Test 1. calculate in the equation ½ +1/5 a. 2/7 (adds without LCM of denominators) b. 7/10 (correct answer) c. 1/2 (Student never understood concept) d. 1/5 (Student never understood concept) Curriculum Modelling unit fractions and their multiples (ACMNA058) Rationale This question determines a child understands of the unit fractions and their ability to make related computation choices. This is an underpinning element of computation and mathematical reasoning necessary for effective problem-solving relating to fractions, place value, measurement, fractions. Answer (a) indicates that the student does not understand fractions as he adds denominators and numerators without checking calculating least common denominators. Answer (b) is correct; however the student will still be asked how they got this number to identify potential misconceptions. Answer (c) indicates that the student has either disregarded the does not understand the concept of adding fractions. Answer (d) indicates that the student does has understood the question. This may indicate poor number sense as they should have the ability to estimate an approximate number if they understand the question. 2. Choice largest ½, 1/5 1/6, 1/3 a. 1/5 (does not understand question) b. 1/2 (correct answer) c. 1/3 (Student does not understand concept) d. 1/6 (Student does not understand concept) Curriculum Investigate number sequences, initially those increasing and decreasing by twos, threes, five and ten from any starting point, then moving to other sequences Rationale This question determines a child understands of the unit fractions and their ability to make related computation choices. This is an underpinning element of computation and mathematical reasoning necessary for effective problem-solving relating to fractions, place value, measurement, fractions. Answer (a) indicates that the student does not understand fractions. Answer (b) is correct; however the student will still be asked how they got this number to identify potential misconceptions. Answer (c) indicates that the student has either disregarded the does not understand the concept of adding fractions. Answer (d) indicates that the student does has not understood the question. This may indicate poor number sense as they should have the ability to estimate an approximate number if they understand the question. 3. Convert ½ to decimal a. 0.1 (takes 1 as decimal) b. 0.2 (takes 2 as decimal ) c. 0.12 (Student does not understand the concept) d. 0.5 (Correct answer) Curriculum changing fraction to decimals Rationale This question tests a child’s understanding of the conversion of fraction to decimal and their ability to make related computation choices. This is an underpinning element of computation and mathematical reasoning necessary for effective problem-solving relating to number, fractions and decimals. Answer (a) indicates that the student does not understand the term convert to decimals. Answer (b) indicates that the student has either disregarded or does not understand the concept of decimals. Answer (c) indicates that the student has not understood the question. Answer (d) is correct; however the student will still be asked how they got this number to identify potential misconceptions. 4. Add ¼ +1/4 a. 1/8 b. 1/2 (correct answer) c. 1/4 d. 1/16 Curriculum Modelling unit fractions and their multiples (ACMNA058) Rationale This question determines a child understands of the unit fractions and their ability to make related computation choices. This is an underpinning element of computation and mathematical reasoning necessary for effective problem-solving relating to fractions, place value, measurement, fractions. Answer (a) indicates that the student does not understand fractions as he adds denominators and numerators without checking calculating least common denominators. Answer (b) is correct; however the student will still be asked how they got this number to identify potential misconceptions. Answer (c) indicates that the student has either disregarded the does not understand the concept of adding fractions. Answer (d) indicates that the student does has understood the question. This may indicate poor number sense as they should have the ability to estimate an approximate number if they understand the question. Diagnostic Interview Diagnostic Interview The Notion of Equality, balance and Relational Thinking One of the difficulties with algebraic thinking is getting students to understand that the equal sign represents a relationship on each side of the = sign, not just a signal to perform an operation (Darr, 2003; Falkner, K., Levi, & Carpenter, 1999; Sheffield, Chapin, & Gavin, 2010;Victorian department of education and training, 2015). For example, in the number sentence 2a + 4 = 20, the answer is not 20. This research reveals that students need assistance to construct meaning for equality and they should be taught to think of equivalence and balance when using an = sign. This relational thinking is crucial in the development of computation skills and algebraic thinking (Darr, 2003). Falkner et al. (1999), suggest that its understanding the equivalence between terms and operations each side of the equal sign enables children to think relationally. A key with relational thinking is seeing relationships within number sentences and the next step is to analyze the change in context to each other. The relationship may be expressed using natural language, graphs and tables or algebraic notation (or a combination of the three). Equality is important as it signifies a relationship between two mathematical expressions which hold the same value. This is vital for students to understand for two reasons. Firstly, student’s need to understand the relationships evident in number sentences and then use this to represent and communicate these ideas. For example, using the number sentence; 6 + 7 = 6 + 6 + 1. The student does not have to know what 6 + 7 = (equals), but can use notions of equality to solve this computation problem. If this is clearly understood, the student may be able to solve increasingly difficult understandings, for example; 35-16, by expressing 35 - 16 = 35 - 20 + 4 (using a compensating strategy). A second reason for the need for understanding equality as a relationship is that it is a common stumbling block as students move from arithmetic to algebra (Kieran, & Matz, (1982), in Falkner et al., (1999). To develop the notion of equivalence, balance and relational thinking, concrete materials and balancing devices are used initially to support the understandings. To develop the understandings further involves the use of symbols, pictures and finally algebraic representations (more formal notations). This diagnostic interview is intended to explore equality, balance and relational thinking in primary students and reveal any misconceptions that are present. Area for a vegetable garden (variables – length x width = 64) Haylee and Sean decided to build a rectangular vegetable garden in their backyard with an area of 64m2. They need to work out how much fencing to purchase. TASK possible rectangles that they could build with a total area of 64m2 Rectangles 32 8 16 Length 2 2 4 Width 1 4 1 Area 64 64 64 Suppose we already had 32m of fencing. What are the dimensions of the pen and what would the area be? What is the biggest area you could make? Fencing is the perimeter which is 32m and dimensions are 32/4 = 8m The area is 8 x8m =64m2 What is the biggest area you could make? Rectangles 32 8 16 Length 2 2 4 Width 1 4 1 Area 64 64 64 Shopping choices (relational thinking) Haylee and Sean went shopping for a garden supplies to plant their new garden. They can only fit 25kg into the car. 1 bag of soil, 1 bag of bulbs and 1 box of mushrooms weigh 17 kg. 2 boxes of mushrooms and 2 bags of soil weigh 14kg. 2 bags of bulbs and 1 bag of soil weigh 22 kg. TASK Let 1 bag of soil be x , 1 bag of bulbs be y and 1 box of mushrooms be k the weight of each item x+y+k=17. 2k+2x=14 2y+x=22. x=22-2y 2k+2x=14 2k=14-2(22-2y) k=7-(22-2y) 22-2y+y+7-22+2y=17. 7+y =17 y=10kg 2(10)+x=22. x=2 2+10+k=17 k=5kg 1 bag of soil is 2kg , 1 bag of bulbs is 10kg and 1 box of mushrooms be 5kg The most you can fit in the car if you have to have a least 1 of each item 1 1 bag of bulbs, 5 bags of soil and 1 box of mushrooms The combinations could you have that add up to at least 20 kg but not over 25 kg 1 bag of bulbs and 1 box of mushrooms and 5 bags of soils, The combination would you choose to purchase is a bag of bulbs and a box of mushrooms and 5 bags of soils because it will cover the space available References Australian Curriculum, Assessment and Reporting Authority [ACARA]. (2016). Australian Curriculum v 8.2: Mathematics – Foundation to Year 10 Curriculum. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1#cdcode=ACMNA175&level=7 Booker, G., Bond, D., Sparrow, L., & Swan, P. (2014). Teaching primary mathematics (5th ed.). Melbourne, Australia: Pearson Australia.  Booker, G., Bond, D., Sparrow, L. & Swan, P. (2014). Teaching primary Mathematics. Frenchs Forest, NSW: Prentice Hall. Chapter 11 (pp. 546-560) Darr, C., (2003). The meaning of equals.   Research information for teachers. Set 2.4-7. Retrieved from http://www.nzcer.org.nz/system/files/set2003_2_04.pdf Department of Education. (2010). First Steps in Mathematics: Number 1. Retrieved from http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps mathematics/games. Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(1). Ormond, C. (2012). Developing algebraic thinking: Two key ways to establish some early algebraic ideas in primary classrooms. Australian Primary Mathematics Classroom, 17 (4), 13-21. Reys, R., Rogers, A., Falle, J., Bennett, S., Frid, S., Lindquist, M., Lambdin, D., & Smith, N. (2012). Counting and number sense in early childhood and primary years. Helping children learn mathematics (1st Australian ed.). Milton, Queensland: John Wiley & Sons. Sheffield, L., Chapin, S., & Gavin, K. (2010).  A Balancing Act: Focusing on Equality, Algebraic Expressions and Equations.  National council for teaching mathematics, 20(8) Sparrow, L. (n.d.) Adding variety with game playing in mathematics teaching. Unpublished conference notes. State of Victoria [Victorian department of education and training]. (2013). Common Misunderstandings - Level 6 Generalising. Retrieved from http://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/lvl6general.aspx  Charles, R. (2005). Big ideas and understandings as the foundation for early and middle school mathematics. NCSM Journal of Educational Leadership, 8(1), 9-24.   Clarke, D.M., Clarke, D.J., & Sullivan, P. (2012). Important ideas in mathematics: What are they and where do you get them? Australian Primary Mathematics Classroom, 17(3), 13-19.   Siemon, D., Bleckly, J., & Neal, D. (2012). Working with the big ideas in number and the Australian Curriculum: Mathematics. In B. Atweh, M. Goos, R. Jorgensen & D. Siemon, (Eds.). Engaging the Australian National Curriculum: Mathematics - Perspectives from the Field. Online publication: Mathematics Education Research Group of Australasia, pp. 19-45.  Reys, R., Lindquist, M., Lambdin, D., & Smith, N. (2012). Helping children learn mathematics (10th edition). New York: John Wiley & Sons. Chapter 3 Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally. (8th edition). Boston: Pearson. 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