10/19/2011

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What Mathematics is

important for (future) Work?

How to achieve a balanced curriculum

Henk van der Kooij

Freudenthal Institute, Utrecht University

h.vanderkooij@uu.nl

Goals for math education

1. To prepare for (critical) citizenship

2. To prepare for future work

3. To prepare for further learning

In most countries, ‘further learning’ still seems the most (or only?) important drive

2

No balanced curriculum at all!!

3

100?

?

?

Abstract version of a problem from a Dutch textbook (1986)

5347

10

100x

z

y

x + y = 100

x + z = 63

y + z = 57

5

100?

?

?

Strategy 1: 63+57+100 = 220 and 220:2 = 110

110 – 57 = 53 110 – 100 = 10 110 – 63 = 47

6

Strategy 2: L + R = 100 and L = R + 6

100?

?

?L

so L = 50 + 3 and R = 50 – 3

and U = 10

R

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7

?

?Start here with 0

So this number has to be -6

Then this number is 63

But the sum should be 100, so I am 106 away. Therefore start with half the difference: 53

Strategy 3:

100?

?

?

8

?

Start here with x

So this number has to be 57 – (63 – x) = x – 6

Then this number is 63-x

x + (x – 6) = 100, so 2x = 106

And thus: x = 53

Strategy book:

100?

?

?

Non-professional ‘mathematicians’ solve problems different from professional mathematicians

When you are not a professional mathematician, you try to find an answer to the stated problem by making use of the context in which the problem is stated

Lesson learned:Pilots at work

10

Research among practitioners in England, carried out in the 90-sCelia Hoyles and Richard Noss

They observed, interviewed and “worked” with bank staff, nurses and pilots

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150 km/h

N

S

90°

N

12

N

S

130°

75 km/h

wind

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13

A

B

C

c=150 a = 75

γ = 40°

a

sinα=

b

sinβ=

c

sinγ

40°

?

?

75

sinα=

bsin β

=150

sin400

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Problems like this were offered to pilots in a school-like setting : sitting in a classroom with pencil and paper

� some pilots were not able to do it outsidethe cockpit

� others were not able to do it without theinstruments at hand

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A

q

pr

B

2 m/s

1 m/s

r

a

-q

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A number of pilots could not solve this problem!

The numbers were so much smaller than the ones they were used to, that the problem did not make sense to them.

Situated abstraction

Practitioners at work do use situated abstraction inwhich local mathematical models and ideas are usedthat are only partly valid in a different context becausethey are connected to anchors within the context of theproblem itself

T = 2πl

g y = 2 x

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F m a= ⋅ V I R= ⋅and

� The (work) context is of influence on the way in

which mathematical activities take place

� Transfer of mathematical knowledge is not selfevident

Research findings onwork-related math

(Noss & Hoyles, Steen, …)

• reading and interpreting tables, charts and graphs

• use of ICT (like spreadsheets)

• dealing with numbers, often not precise andcoming with units of measurement

• proportional reasoning

• statistical process control activities

• representing and analyzing data

• multi-step problem solving

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Focus on ‘math for work’

My interest in this area started in 1996:

Curriculum reform project in vocational education (engineering) with the aim to:

Make Math and Physics really supportive for vocational courses

1920

20

Dutch school system

age

12

14

16

18

Primary Education ( age 4 - 12)

Secondary Education

VWO

pre university

generalHAVO

middle level

general

VMBO

lower level

pre-vocational

university higher

vocational

educationSenior high

vocational

education

reading and interpreting tables, charts and graphs

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f =c ⋅ F ⋅ l 3

E ⋅ w ⋅ h 3⋅ 9.81

proportional reasoning

Flexion of a beam

23 24

Find the relationship between b, f and the entries d of the table

26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

12 467 502 538 574 610 646 682 718 754 790 826 861 897 933 969

14 400 431 461 492 523 554 585 615 646 677 699 739 769 800 830

16 350 376 403 431 458 484 511 538 565 593 619 646 673 700 727

18 311 335 359 383 407 431 455 479 502 526 550 574 598 622 646

20 280 301 323 345 366 388 409 431 452 474 495 517 538 560 581

22 254 274 294 313 333 353 372 392 411 431 451 470 490 509 528

24 233 251 269 287 305 323 333 359 377 395 413 431 449 467 484

26 215 231 249 265 282 298 315 331 348 365 381 398 414 431 447

28 199 215 230 246 262 277 292 308 323 338 354 369 384 400 415

f

b

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Geometry: cut/copy and pasteSchool Mathematics: Ideal, abstract world

Mathematics at Work: Real, complex world

Elementary use of sophisticated mathematics

Sophisticated use of elementary mathematics

Lynn Arthur Steen (USA)

Models and continuous functions

Discrete data and measurements: Data analysis

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exists in this world as a real object2

in this world is an operation not yet finished;given the tolerance it will be 1.4…2

If a ‘balanced curriculum’ is what we want to have

(citizenship, work and further learning) …

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… how can that be organized?

Difficult when viewed from the existing content perspective

PISA Framework (OECD, 2003)Modern view on mathematics emphasizes‘the study of pattern’ and ‘dealing with data’

To be used to define a balanced curriculum?

Ove

rarc

hin

g th

em

es

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• (Patterns in) Space and Shape

• (Dealing with) Uncertainty

• (Patterns in) Change and Relationships

• (Patterns in) Quantity

Policies and Politics (Economic factor) 1

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Policies and Politics (Economic factor) 2

We believe, after examining the findings of cognitive science, that themost effective way of learning skills is "in context," placing learningobjectives within a real environment rather than insisting that students firstlearn in the abstract what they will be expected to apply.(SCANS, 1991, USA)

Mathematical competence is the ability to develop and applymathematical thinking in order to solve a range of problems in everydaysituations. Building on a sound mastery of numeracy, the emphasis is onprocess and activity, as well as knowledge. Mathematical competenceinvolves, to different degrees, the ability and willingness to usemathematical modes of thought (logical and spatial thinking) andpresentation (formulas, models, constructs, graphs, charts).(Key competence 3, 2007, EU)

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Why do curricula worldwide not (or only poorly) reflect these visions?

Probably because of the deeply rooted belief that math has to be learned through a linear sequence:from arithmetic, via algebra to functions and calculus (with some probability/statistics and geometry alongside)

This is a safe environment, in particular seen from the perspective of teaching and assessment.“It worked for us, so it will work for them”

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What ‘work-math’ aspects should be included in a balanced curriculum?

• interpreting and analyzing tables, graphs

• use (and design) of spreadsheets

• proportional reasoning

• some discrete math

• dealing with data, uncertainty

• multi-step problem solving activities

• modeling activities

• logical and critical thinking

• making decisions34

Time for a balanced curriculum?!

35source: Jan de Lange, Freudenthal Institute

quantitative literacy

(pre-) vocational(62%)

pre-university(22%)

work-related math

Informal math formal math

Work

Learning

Citizenship

pre-higher voc(16%)

Consequences for curricula in secondary education

� including workplace related topics needs time, soreconsider the value of traditional topics

?? skip quadratics (polynomials) as a main topic??

� reconsider the curriculum from the ‘PISA- perspective’

� create time for project days or – weeks to challengestudents with modeling activities

� stop thinking about linear sequencing (difficult!) of acurriculum

� add ‘workplace elements’ in current classroom practice