Assessment Opportunities: Enhancing Learning and Communication

1. Assessment 2:00 pm to 3.30 pm

2. Opportunity to… • learn • show • know what to show

3. In communication during day-to-day classroom work In communication during entire class sessions at the end of teaching units Marking (in Sweden, secondary and upper secondary school only Assessment Implicit and explicit In connection with diagnostic and other test In connection with teacher/student/parent meetings Summary in assessment forms/matrices

4. Lisa Björklund Boistrup Discourses in mathematics classrooms: a multimodal social semiotic study su.diva-portal.org/smash/record.jsf?pid=diva2:355024

5. The consequences of assessment (Pettersson, 2005)

6. Classroom assessment • Summative assessment Test on a local or national level • Formative assessment “all those activities undertaken by teachers, and/or by their students, which provide information to be used as feedback to modify the teaching and learning activities in which they are engaged” (Black and Wiliam, 1998)

7. Feedback Information provided by an agent (e.g. teacher, peer, book, parent, or one’s own experience) about aspects of one’s performance or understanding” (Hattie, 2009)

8. Feedback • Feed back what has happened ← The teacher evaluates the student´s demonstrated performances in terms of ”good” or ”bad”. Rewarding. Punishing, approval, disapproval • Feed forward the future → The teacher use the student´s demonstrated (lack of knowing) as a basis for continued learning • Feed up Students´demonstrated knowing ↕ explicit goals/criteria that should/could have been reached and/or to be reached in the future

9. In Sweden Presence of Assessment acts in Classroom visited • Feed back 58 • Feed forward 72 • Feed up 2

10. Analysis of knowledge Therefore the analysis ought to focus in what extent the pupil has access to his/her knowledge in different situations. Higher quality can e.g. be that the pupil shows his/her understanding of a concept in different ways and in different situations.

11. 1/4 of a string 1/4 of water in a glass 1/4 of a lump of play dough 1/4 of 8 stones 1/4 of a paper Length Volume Weight Quantity Area 1/4

12. Weakness in understanding are often pinpointed by weakness in representations. Deep understanding of a mathematical item or topic is displayed by the ability to transfer the idea between different representations: pictorial, verbal, symbolic and the real world. (Alistair McIntosh)

13. Four kinds of representations Event The fishing huts Number Picture # 1 1 + 1 · 4 = 5 # 2 1 + 2 · 4 = 9 # 3 1 + 3 · 4 = 13 # 4 1 + 4 · 4 = 17 etc Word Symbols First I built a hut with 5 sticks, then I built a second hut with 4 sticks. After that I continued … 1 stick + 4 sticks for each new hut 1 st + 4 st · number of huts 1 st + 4 st · n 1 + 4 · n 1 + 4n 4n + 1

14. Documentation which highlights the mathematics

15. Documentation which highlights the mathematics

16. Assessment concerning … • Problem solving capability • Comprehension and method • Accomplishment • Communication capability • Mathematical language and/or representation • Clarity of presentation

17. Three levels

19. The area of a triangle on small pieces of papers

20. Documentation which shows development of knowledge What I know about … What I have learnt about …

21. Documentation which shows development of knowledge What have you learnt since …?

22. Documentation which highlights the mathematics

23. Documentation which shows development of knowledge The ellipse model

24. Triangle – Rectangle Similarities · both consist of straight lines · both have angles · you calculate the perimeter in the same way: you add the sides · both have a base and height · both have an area Differences · three sides · three angles · have not parallel sides · the three angles are not always the same · maximum one angle is a right angle · four sides · four angles · have parallel sides · the four angles are always the same · all angles are right angles Name:______________________ Class:____ Similarities and Differences

25. Addition • I think it is when you are doing different sports • I think it is computor • I don´t know, but I take a chance: it´s different jobs • When you are outdoors walking • Something with a pocket calculator • I think it is what ever … or theatre • It means plus, because it sounds like that • Something on the car • It is plus • It is like 3+3=6 • Addition is when you get warm when you are running • Additon is a counting sign. The signs name is addition and is plus. You use plus when you are plussing something like 4+4=8 • Additon is when someone is alone and another one is coming.

26. Oral analyses • The pupils discuss in pairs or in small groups. A respondent is speaking on behalfs of others.

27. Oral analyses • The pupils discuss in pairs or in small groups. A respondent is speaking on behalfs of others. • Ask all to write down an answer and then reading out a selected one

28. What happens …? • Make an equilateral triangle. • What happens with the shape if you add another side to the triangle? • Describe the shape! • Check! Triangle (3 sides) Quadrilateral (4)

29. What happens …? • What happens if you add another side … • What´s the name of the new shape? Pentagon (5) Hexagon (6) Heptagon (7) Octagon (8)

30. Oral analyses • The pupils discuss in pairs or in small groups. A respondent is speaking on behalfs of others. • Ask all to write down an answer and then reading out a selected one • Give pupils a choice between different possible answers and ask them to vote on the options.

31. Small cubes in a big cube Daniela has got cubes of equal size. She has put some of them into a box shaped like a cube in the way you see on the picture. What maximum number of cubes can she put into the box? a/ 9 b/ 13 c/ 17 d/ 21 e/ 27 Ecolier 2007

32. Area and perimeter A parallellogram is divided in two parts P and Q as shown in the figure What statement is surely true? a/ P has a bigger perimeter than Q b/ P has a smaller perimeter than Q c/ P has a smaller area than Q d/ Q and P have the same area e/ Q and P have the same perimeter Benjamin 2007

33. Four routes Between two points four routes are drawn. Which route is the longest? A B C D E: All are equal Eccolier 2006

34. Folding cubes We are going to build a cube like this Which of the following foldings are impossible? a/ 1 and 3 b/1 and 5 c/ 3 and 4 d/3 and 5 e/ 2 and 4 Ecolier 2008

35. National tests in mathematics The side in an equilateral triangle is 5 dm.How large area has the triangle?One choice is right. Circle your answer. 6,3 dm2 10,8 dm2 12,5 dm2 15 dm2 25 dm2 Solution frequency Girls: 0,10 Boys: 0,13

36. Area with sticks area < 0,5 a.u. area ≈ 0,43 a.u.

37. Pedagogical material Geoboards

38. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · National tests in mathematics • The area of the rectangle is 18 cm2 and the perimeter is 22 cm. • Draw another rectangle with the same area (18 cm2) but with a shorter perimeter. Each of the four vertices of the rectangle must be located at a dot. • Find the perimeter of your rectangle. Show your calculations here: