1. ASSESSMENT

2. Bloom’s (revised) taxonomy of cognitive levels

3. «Landscapes of investigation»
Traditional tasks
Landscapes of investigation
References to pure mathematics
(1)
(2)
References to semi-reality
(3)
(4)
Real-life references
(5)
(6)
«Classic mathematics teaching»

4. Implementent tasks that promote reasoning and problem solving
A taxonomy of mathematical tasks based on the kind and level of thinking required to solve them.
Lower-level demands (memorization)
Lower-level demands (procedures without connections)
Higher-level demands (procedures with connections)
Higher-level demands (doing mathematics, mathematical reasoning))

5. Bloom’s (revised) taxonomy of cognitive levels
High-level tasks
Landscapes of investigation
Low-level tasks

6. Assessment How do we assess? Formal Informal During learning process
Formative
With grades
No grades
Formal
Informal
Summative
At the end of learning process

7. ASSESSMENT Informal Formal Formative Concept cartoons Etc
Mid semester test
Mid semester assigment
Summative
Quiz at end of course
Exam

8. «Assessment for learning»
How do we assess?
«Assessment for learning»
Formative
Formal
Informal
Summative

9. «Assessment for learning»
The results of an assessment activity form a feedback to teachers and learners
Feedback may indicate that performance exceeds goal
Feedback may indicate that performance reaches goal
Feedback may indicate that performance falls short of goal

10. «Assessment for learning»
The results of an assessment activity form a feedback to teachers and learners
Feedback may indicate that performance exceeds goal
Feedback may indicate that performance reaches goal
Feedback may indicate that performance falls short of goal
and
Feedback is used by teachers to evaluate teaching practice
if needed, to adjust teaching
Better
learning
and
Feedback is used by learners to evaluate own learning process
if needed, to modify learning processes

11. How is the feedback received by the learners?

12. performance exceeds goals
Table 1
Possible responses to feedback interventions (Kluger & DeNisi, 1996)
Respons type
Feedback indicates
performance exceeds goals
Feedback indicates performance falls short of goal
Change behaviour
Exert less effort
Increase effort
Change goal
Increase aspiration
Reduce aspiration
Abandon goal
Decide goal is too easy
Decide goal is too hard
Reject feedback
Feedback is ignored

13. How can we assess learners’ knowledge?
Dialogue
Tests/quizes
Concept maps
Drawings
Games
Graphic organisers
Concept Cartoons
Experiments
True-false statements
Naylor, S. & Keogh, B. (2012). Concept Cartoons: What have we learnt? Paper presented at the Fibonacci Project European Conference, Inquiry-based science and mathematics education: bridging the gap between education research and practice. Leicester, UK, April 2012

14. Concept map

15. Concept map

16. Graphic organiser 4/5 3/4 Comparison Fill in the open boxes
Smallest? (Y/N)
0.75
2(5 – a)
16/25
a2 ?
4/3
- 0.8
2/ 5

17. COMPARING TWO FUNCTIONS
Graphic organiser
COMPARING TWO FUNCTIONS
y = x and y = -2x + +1
Similarities
Differences
y = x + 1
y = -2x + 1

18. Which is the odd one out in the following and WHY?1 2 49 13 11 7
3 4 x 1 2
0.75 x 0.5
4 3 x 3 32
0.25 x 3 3 4
0,1 10
1/1000 1

20. WHAT DO YOU THINK?
It will become = 6 times smaller
If we halve the length of each of the sides of the box, then the volume will be halved.
I think that the new volume of the box will be 1/3
I think the volume of the smaller box will be 1/8 of the volume of the bigger box

21. WHAT DO YOU THINK? The functions must have the same constant c
Such functions have the same slope x
Question:
What do we know about any two linear functions that cross the y-axis in the same point?
The two functions must have the same y-value when x = 0
They must have opposite directions

22. Yes or no, or maybe? 1 All numbers are fractions 8
The numerator is bigger than the denominator in an improper fraction 2
Every fraction is smaller than 1 9
The numerator is bigger than the denominator in a proper fraction 3
49/21 is in its simplest form 10
1/6 is double 1/3 4
To simplify a fraction, divide by 2 11
0.33 is greater than 1/3 5
Doubling the top and bottom of a fraction makes it twice as big 12
Adding the tops and bottoms of two fractions gives their sum 6
The larger the denominator the bigger the fraction 13
Some fractions are odd numbers 7
Multiplying by a fraction makes a number smaller 14
Fractions are always less than zero
Yes or no, or maybe?

23. False or true? Informal assessment is without grades
Formative assessment is also called «Learning for assessment»
Summative assessment takes place after the learning process
Bloom’s taxonomy has been revised
The lowest cognitive level in Bloom’s taxonomy is Knowledge/Remembering
The verb explain is used in questions/tasks when assessing learners’ understanding
7. Analyzing is a lower cognitive level than evaluating 8. Analyzing is a lower level than applying 9. Exam is an example of formal summative assessment 10. The verbs define and list are used in questions/tasks when assessing learners’ understanding 11. Assessment can only take place after the end of a course 12. All informal formative assessment is «Assessment for learning»

24. Relational Understanding and Instrumental Understanding
Richard R. Skemp
First published in Mathematics Teaching, 77, (1976)

25. AN EXAMPLE
A teacher reminds the class that the area of a rectangle is given by
A = L x B
A pupil says she does not understand.
The teacher gives her an explaination:
«The formula tells you that to get the area of a rectangle, you multiply the length by the breadth»
The pupil responds: «Oh, yes, I understand». B L

26. Does she really understand?
She knows how to get the area of the rectangle, but does she understand?

27. 2 3 of ?
2 3 ×
= 2 × 4 3 × 5 =

28. 2 3 of ?
2 3 ×
= 2 × 4 3 × 5 =
2 3 of
Do the learners understand why
is the same as ?
2 3 ×

29. 3 5 ×
= 3 ×10 5 ×13
= = 2
3 5 × =

30. Instrumental approach vs Relational approach
Two kinds of mathematical mismatches which can occur:
Pupils whose goal is to understand instrumentally, taught by a teacher who wants them to understand relationally.
The other way around

31. Two kinds of mathematical mismatches which can occur:
Pupils whose goal is to understand instrumentally, taught by a teacher who wants them to understand relationally.
All the pupils want is som kind of rule for getting the answer. As soon as this is reached, they are fine and ignore the rest ….

32. Two kinds of mathematical mismatches which can occur:
Pupils whose goal is to understand instrumentally, taught by a teacher who wants them to understand relationally.
All the pupils want is som kind of rule for getting the answer. As soon as this is reached, they are fine and ignore the rest ….
If the teacher asks a question that does not quite fit the rule ….?

33. This is a more damaging mis-match!
Two kinds of mathematical mismatches which can occur:
Pupils whose goal is to understand instrumentally, taught by a teacher who wants them to understand relationally.
The other way around
This is a more damaging mis-match!

34. Instrumental understanding
Another mis-match is that which may occur between teacher and text.
Teacher Text
Instrumental understanding
Relational approach

35. WHY SHOULD TEACHERS TEACH INSTRUMENTAL MATHEMATICS?
1. Usually easier to learn
Examples:
(-2) x (-3) = 6
«Minus times minus equals plus»
3 : ½ = 3 x 2 = 6
«To divide by a fraction you turn it upside down and multiply»

36. WHY SHOULD TEACHERS TEACH INSTRUMENTAL MATHEMATICS?
2. Because less knowledge is involved, one can often get the right answer more quickly and reliable by instrumental thinking than relational
Examples:
(-2) x (-3) = 6
«Minus times minus equals plus»
3 : ½ = 3 x 2 = 6
«To divide by a fraction you turn it upside down and multiply»

37. WHY SHOULD TEACHERS TEACH INSTRUMENTAL MATHEMATICS?
2. Because less knowledge is involved, one can often get the right answer more quickly and reliable by instrumental thinking than relational
Even relational mathematicians often use instrumental thinking!

38. WHY SHOULD TEACHERS TEACH MORE RELATIONAL MATHEMATICS?
1. It is more adaptable to new tasks
Tasks that deviate from the «standard tasks»
2. There is less to remember
Fewer rules and formula to remember
3. There is more to learn, but less to remember And the result, once learnt, is more lasting

39. WHY SHOULD TEACHERS TEACH MORE RELATIONAL MATHEMATICS?
4. Relational knowledge can be effective as a goal in it self
»Deeper» Self-confidence. Motivation
5. Relational schemas are organic in quality
Leads to exploring

40. 3 2 4 5 6 7 6 4 8 10 12 14

41. 3 2 4 5 6 7 7 5 9 11 13 15

42. 2 3 3 4 5 6 7 5 7 9 11 13

43. What is inside the red box, the blue box, the green box?
What do this boxes do to the numbers we enter
LONG before we introduce x and y
and
y = mx + c

44. First Conceptual understanding, then Procedural Fluency
First: x + 3 = 5
Subtract 3 on both sides of the «equals symbol»
x + 3 − 3 = 5 – 3
x = 2
Then: x + 3 = 5
Move 3 to the other side and change from pluss to minus
x = 5 – 3

45. Relational understanding
First Conceptual understanding, then Procedural Fluency
Instrumental understanding
First: x + 3 = 5
Subtract 3 on both sides of the «equals symbol»
x + 3 − 3 = 5 – 3
x = 2
Then: x + 3 = 5
Move 3 to the other side and change from pluss to minus
x = 5 – 3