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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