ASSESSMENT

Bloom’s (revised) taxonomy of cognitive levels

«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»

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))

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

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

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

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

«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

«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

How is the feedback received by the learners?

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

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

Concept map

Concept map

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

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

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 49 4 625 3 8 3 27

WHAT DO YOU THINK? It will become 2 + 2+ 2 = 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

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

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?

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»

Relational Understanding and Instrumental Understanding Richard R. Skemp First published in Mathematics Teaching, 77, 20-26 (1976)

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

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

2 3 of 4 5 ? 2 3 × 4 5 = 2 × 4 3 × 5 = 8 15

2 3 of 4 5 ? 2 3 × 4 5 = 2 × 4 3 × 5 = 8 15 2 3 of 4 5 Do the learners understand why is the same as ? 2 3 × 4 5

3 5 × 10 13 = 3 ×10 5 ×13 = 30 65 = 6 13 2 3 5 × 10 13 = 6 13

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

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 ….

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 ….?

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!

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

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»

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»

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!

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

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

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

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

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

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

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

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