PART 1

Overview of MCK Test 15%

It is NOT a traditional test in the sense that you have to answer a set of questions but rather a test that you have to devise. This is actually a better assessment of your content knowledge for teaching mathematics.

• Devise a set of assessment tasks comprising four multiple choice questions and a short diagnostic interview to assess children’s understanding of aspects of algebra and/or making effective computational choices.
• Provide a brief rationale for each showing links to the curriculum, likely points of difficulty, and correct answers as appropriate.
• The multiple choice questions should reflect generally accepted conventions for that type of question as used in NAPLAN tests.
• Clearly describe reasons for providing the various choices in terms of identifying misconceptions that may exist.
• The diagnostic interview may be associated with the multiple choice questions and follow the same content theme. Alternatively, it may be a separate entity. Use the Fraction Diagnostic Interview as a guide for the types of tasks to use and questions to ask.

Diagnostic Interview Sample

1. David sells tickets for the school play at $6.00 each. He sells 4 on Monday, 9 on Tuesday, 6 on Wednesday, 5 on Thursday and 11 on Friday.

1. What is the total amount of money he takes for the week?
2. Explain how you calculated the answer?
3. Why did you use particular formulas/operations to calculate your answer? What told you to use these in the question?

1. Can you continue this pattern?

1. Describe what is happening to the pyramids?
2. If you continued this pattern, what would the 7^th, 9^th and 11^th shape look like? How many blocks would you need to make these shapes? How do you know?

MCK Test Sample

A reminder of the readings listed around the notion of Big Ideas in Mathematics.  

Week 2 listed the following:

• • Reys, R., Lindquist, M., Lambdin, D., & Smith, N. (2012). Helping children learn mathematics (10^th edition). New York: John Wiley & Sons. Chapter 3

Other Texts

•  Booker, G., Bond, D., Sparrow, L. & Swan, P. (2014). Teaching primary mathematics (5^th edition). Frenchs Forest, NSW: Prentice Hall. Chapter 11 (pp. 546-560)
•  Van de Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally. (8th edition). Boston: Pearson. Chapters 4 and 5

    Articles and papers

•  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. CHARLES NCSM Big Ideas.pdf
•  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. CLARKE ET AL APMC 2012 Big Ideas.pdf        

•  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. SIEMON ET AL. Big Ideas.pdf

HD Example 1:

MCK TEST

Q1 Sarah wrote a pattern.  She added 4 to each number to get the next number.

Which is Sarah’s pattern?

1. 8, 11, 14, 17 ……. (adding 3 repeatedly)

1. 11, 15, 19, 23 ……. (correct answer)

1. 4, 14, 24, 34 …… (Adding 10 to keep 4 in units)

1. d) 8, 16, 32 64…… (doubling each number)

Curriculum

Investigate number sequences, initially those increasing and decreasing by twos, threes, fives and ten from any starting point, then moving to other sequences (ACMNA026)

Describe, continue, and create number patterns resulting from performing addition or subtraction (ACMNA060)

Rationale

According to Booker, Bond, Sparrow and Swan (2014), using and seeing patterns can form the first introduction to algebraic thinking through additive and multiplicative sequences. This question was designed to reveal the student’s understanding of patterns in number using an additive sequence. Each new number is found by adding 4 to the proceeding number.  The student is required to find the next number in a sequence of 3 numbers. Answer a) reveals a computational error – adding 3 repeatedly, suggesting that the student can identify that an additive sequence is required to solve the pattern.  Answer b) is the correct answer demonstrating identification of the pattern, however the student will still be asked how they got this number to identify possible misconceptions.  Answer c) reveals a growing pattern but 10 is added to each proceeding number to maintain the 4 in the unit place value. Answer d) demonstrates a doubling pattern, where each proceeding number is doubled.

Q2  This pattern of triangles has been made using toothpicks with shared edges.

How many tooth picks are needed for 10 triangles?

1. 20 (multiplied by 2)
2. b) 21  (correct answer)
3. c) 19   (computational error)
4. d) 11 (next odd number in sequence)

Curriculum: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence. (ACMNA133)

Rationale

Patterns and Algebra, within the primary curriculum, demonstrate the importance of early number learning and the development of algebraic thinking (The national education access license for schools (NEALS), 2012; Reys, Rogers, Falle, Bennett, Frid, Lindquist, Lambdin, and Smith, 2012). The emphasis is on number patterns and number relationships with investigation into how one changes in relation to the other. In this question, algebraic thinking is examined using a diagram and table to discover the growing patterns.  Student’s might use an additive approach and increase by 2 toothpicks for each shape by continuing the pattern using the table and still get the answer correct ((b) 21).  However, the student may use a multiplicative approach by multiplying number of triangles ie 10(t) by 3 (base number to make a triangle) then subtract 10 (no of toothpicks spare -1 spare toothpick when triangles are joined) add 1(for the initial triangle) = t x 3-n+1 (10 x 3-10+1).  Or number of toothpicks (t) x 2 +1.  Question (a) reveals identification that the pattern is growing by multiplication of the number of triangles by 2 (two toothpicks are added each time) but fails to add the initial toothpick on. Answer (b) is correct; however, the student will still be asked how they got this number to identify potential misconceptions.  Answer (c) reveals a computational error.  19 is the answer for 9 triangles not 10. This may revel the use of the table to continue the pattern, not generalising and using a formula. Answer (d) reveals that the student is aware that the pattern is growing by selecting the next odd number as the answer, but has disregarded the “gap” in the number of triangles.

Q3) Pip thinks of a number. If she doubles the number and adds 4 she gets 18. What is the number?

1. a) 5 (incorrect order of operations)

1. b) 7 (Correct answer)

1. c) 40 (incorrect operation – doubling rather than dividing by 2)

1. d) 8 (computational error with division)

Curriculum

Describe, continue, and create number patterns resulting from performing addition or subtraction. (ACMNA060)

Explore the use of brackets and order of operations to write number sentences (ACMNA134)

Rationale

Answer (a) reveals that student may have used a number sentence using inverse operations, but has incorrectly ordered the operations therefore making a transformation error – the student has taken 18 and divided it by 2 and then taken 4 rather than subtract 4 first from 18 which gives 18, then divide 14 by two (the reverse of doubling). Answer (b) is correct; however, the student will still be asked how they got this number to identify potential misconceptions.  Answer (c) reveals that the student may have used the incorrect operation, not using the inverse operation by doubling the initial number rather than dividing it– 18 to get 36 and then adding 4. This demonstrates a common misconception when working with Algebra (Victorian Department of Education and Training, 2015). The student may have a limited understanding of the properties of numbers and operations (for example, multiplication only understood in terms of groups of, division not seen as the inverse of multiplication). Answer (d)reveals a computational error.  The correct procedure for working out the number sentence – 18 subtract 4 is 14 (correct), however a computational error has occurred when 14 divided by 2 = 8 to get the answer (should be 7).

Question 4

Elli bought 2 chocolate bars and a drink for $6.20. If the drink cost $1.60, how much did she pay for one chocolate bar? Calculators can be used.

1. $2.30  (correct answer)
2. $1.50  (transformation error – division of total amount
3. $2.50 (calculating error)
4. $3.10 (not dividing by 2 to find the cost of one chocolate bar)

Curriculum

Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence (ACMNA133)

Rationale

This question can reveal whether the student is able to apply generalizing using a problem solving approach to algebraic thinking (Booker et al., 2015).  When solutions are obtained, generalisations can be formed based on the relationships within problems using equivalence and equations.  According to the Victorian Department of Education and training (2015), a major source of difficulties in algebra is the failure to recognise equivalence between different forms of the same relationship, or between the left and right sides of an equation. To complete this question correctly the student would demonstrate some understanding of equivalence, underlying properties and arithmetic approach to finding unknowns (Victorian Department of Education and training, 2015). Answer (a) reveals the correct answer, as the answer requires; $6.20 subtract $1.60 (drink) which leaves $4.60.  Divide this by two to give the cost of one chocolate bar which equals $2.30.  If the student selected answer (b) they would demonstrate a transformation error.  That is, they have divided the total amount ($6.20) by 2 to get $3.10.  They have then taken the cost of the drink ($1.60) to get a total of $1.50.  The answer (c) demonstrates a calculation error when subtracting $1.60 from $6.20 to get $5.00 and then divide this by two to get $2.50. If the student selects (d), they would reveal that they hadn’t divided the answer by two to get the cost of one chocolate bar.

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.

1. Weighing Problem (Balance and equivalence)

A set of balance scales are needed to investigate the notion of balance.

TASK

Interviewer shows student a selection of stones with each colour weighing equivalent amounts.  Blue stones(lightest), black rocks (equivalent to 2 blue rocks) and pink stones (equivalent to 1 blue and 1 black stone)

• Interviewer places 1 blue stones on one side and asks student to make the scales balance (student should place 1 blue stone onto the other side of scale).
• Interviewer places 2 blue stones on one side and asks student to make the scales balance (student should place 2 blue stones onto the other side of scale).

• Interviewer places 1 black on one side and asks student to make the scales balance (student should place 2 blue stones onto the other side of scale).
• Interviewer asks student to help make rule which explains the relationship.
• Interviewer asks student to notate the relationship using general language and algebraic notation.
• Interviewer repeats process for other combinations of stones in an increasingly complex nature to access the depth of the student’s understanding.  Interviewer records findings and rules and generalisations that the student makes.

1. 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

• Investigate all possible rectangles that they could build with a total area of 64m2
• Record each of the dimensions into a table with the width, length and total perimeter.
• 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?
• Record this information in a table.

1.  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

• What is the weight of each item?
• What is the most you can fit in the car if you have to have a least 1 of each item?
• What combinations could you have that add up to at least 20 kg but not over 25 kg?
• What combination would you choose to purchase and why?

Example 2 of diagnostic interview:

Diagnostic Interview

Year 3 – Fractions 

Model and represent unit fractions including 1/2, 1/4, 1/3, 1/5 and their multiples to a complete whole (ACMNA058)

Question 1

FSiM Key Understanding 1: When we split something into two equal-sized parts, we say we have halved it and that each part is half the original thing

Classify the following shapes into half and non-half group by drawing an X on shapes that do not show half.

• What did you do to identify them as halves and non-halves shapes.
• Why did you classify them this way?
• Can you define what halving is?
• What do you need to consider before splitting any item?
• Color a half of each shape shown below.

• How did you identify the half of each shape?
• Explain your answer.

Question 2

FSiM Key Understanding 2: We can partition objects and collections into two or more equal-sized parts and the partitioning can be done in different ways.

Draw lines to partition the shapes below into two equal-sized parts.

• How did you know where to draw the lines to make each shape into a half?
• Draw more lines in each shape to show four equal-sized parts.
• Why did you draw the lines that way?
• Color one equal part in each shape blue.
• What fraction does the blue section show?
• Explain your answer.
• How can you partition the shapes in different ways to show the same fraction?

USEFULL

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 (5^th ed.).  Melbourne, Australia: Pearson Australia.

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

The national education access license for schools [NEALS], (2012).  Naplan 2012 teaching strategies.  Retrieved from http://www.schools.nsw.edu.au/learning/7-12assessments/naplan/teachstrategies/yr2012/index.php?id=numeracy/nn_paal/nn_paal_s2a_12