2:00 pm to 3.30 pm
AssessmentAssessment
Opportunity to…
· learn· show· know what to show
Marking(in Sweden, secondary and upper secondary school only
In communication during day-to-day classroom work In communication
during entire class sessions at the end of teaching units
In connection with diagnostic and other test
In connection withteacher/student/parent meetings
Summary in assessment forms/matrices
AssessmentImplicit and explicit
Lisa Björklund BoistrupDiscourses in mathematics classrooms: a multimodal social semiotic study
su.diva-portal.org/smash/record.jsf?pid=diva2:355024
(Pettersson, 2005)
The consequences of assessment
Classroom assessment
• Summative assessmentTest 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)
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)
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 forwardthe future →
The teacher use the student´s demonstrated (lack of knowing) as a basis for continued learning
• Feed upStudents´demonstrated knowing
↕
explicit goals/criteria that should/could have been
reached and/or to be reached in the future
In Sweden
Presence of Assessment acts in Classroom visited
• Feed back 58• Feed forward 72• Feed up 2
Analysis of knowledgeHigher quality can e.g. be that the pupil shows his/her understanding of a concept in different ways and in different situations.
Therefore the analysis ought to focus in what extent the pupil has access to his/her knowledge in different situations.
1/4
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
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.
Weakness in understanding are often pinpointed by weakness in representations.
(Alistair McIntosh)
Four kinds of representations
Event
Picture Number
Word Symbols
The fishing huts
# 1 1 + 1 · 4 = 5
# 2 1 + 2 · 4 = 9
# 3 1 + 3 · 4 = 13
# 4 1 + 4 · 4 = 17
etc
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
Documentation which highlights the mathematicsDocumentation which highlights the mathematics
Documentation which highlights the mathematicsDocumentation which highlights the mathematics
Assessment concerning …
· Problem solving capability · Comprehension and method · Accomplishment
· Communication capability · Mathematical language and/or representation · Clarity of presentation
Three levels
Qualitative levels
Mathematical language and/or representation
Poor and occasionally wrong
Acceptable but with some shortages
Correct and appropriate
The area of a triangle The area of a triangle on small pieces of paperson small pieces of papers
Documentation which shows development of knowledge
What I know about …
What I have learnt about …
Documentation which shows development of knowledge
What have you learnt since …?
Documentation which highlights the mathematics
Documentation which shows development of knowledge
The ellipse model
Similarities
Differences
Name:______________________ Class:____
Triangle – Rectangle
· 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
· 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
Similarities Similarities andandDifferencesDifferences
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.
Oral analyses
• The pupils discuss in pairs or in small groups. A respondent is speaking on behalfs of others.
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
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)
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)
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.
Ecolier 2007
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/ 9b/ 13c/ 17d/ 21e/ 27
Benjamin 2007
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 Qb/ P has a smaller perimeter than Qc/ P has a smaller area than Qd/ Q and P have the same areae/ Q and P have the same perimeter
Eccolier 2006
Four routes
Between two points four routes are drawn.Which route is the longest?
A B C D E: All are equal
Ecolier 2008
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
Folding cubes
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 frequencyGirls: 0,10 Boys: 0,13
Area with sticks
area < 0,5 a.u.
area ≈ 0,43 a.u.
Pedagogical materialGeoboardsGeoboards
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:
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