Landspróf miðskóla sem inntökupróf í menntaskóla og kennaraskóla og síðar fleiri skóla
var haldið á árunum 1946–1976. Prófað var í átta námsgreinum þar sem íslenska vó tvöfalt.
Landsprófið var upphaflega grundvallað á reglugerð nr. 3/1937 um námsefni til prófs upp
úr öðrum bekk Menntaskólans í Reykjavík. Landsprófið í stærðfræði hélst að miklu leyti
óbreytt á árunum 1947–1965. Prófað var í lesnum dæmum sem nemendur höfðu lært áður
og í ólesnum dæmum. Frá árinu 1966 breyttist prófgerðin: lesin dæmi voru felld niður,
prófið var stytt og tekið var að prófa úr svokallaðri nýrri stærðfræði. Þar var lögð aukin
áhersla á tölur og eiginleika þeirra en einnig á mengi ásamt tilheyrandi rithætti og aðgerðum samkvæmt Drögum að námsskrá í landsprófsdeildum miðskóla frá árinu 1968. Markmiðið með innleiðingu nýju stærðfræðinnar var að auka skilning nemenda í stærðfræði.
Fjögur landsprófsverkefni í stærðfræði frá árunum 1953, 1966, 1971 og 1975 voru greind
með tilliti til inntaks og færnikrafna samkvæmt greiningarlykli TIMSS. Greiningin sýnir
að inntak prófverkefnanna breyttist í samræmi við Drög að námsskrá en um leið styttust
dæmin, orðadæmum fækkaði og jafnframt fjölgaði innbyrðis óskyldum prófatriðum. Færnikröfur færðust frá lausnaleit yfir í aukna beitingu rútínuaðferða en dæmum með flóknum
samsettum aðferðum fækkaði.
Gögn sýna að meðaleinkunn í stærðfræði, sem áður hafði verið lægri en meðaleinkunn
allra átta námsgreinanna á landsprófi, færðist nær heildarmeðaleinkunn og nemendum
gekk hlutfallslega betur en áður. Vonir höfðu staðið til þess að innleiðing mengjafræði yki
skýrleik og skilning. Vart var þó hægt að búast við því. Nemendur þurftu á sama tíma að
kynnast venjulegri algebru sem mörgum hefur reynst torskilin og námsefnið studdi samhengið milli mengjafræði og algebru aðeins að litlu leyti. Eftir sitja spurningar um hvort
glíma við sundurlaus atriði stuðli fremur að vélrænum skilningi en lengri samsett dæmi, og
hvort lesin dæmi eigi rétt á sér.
The goal of new education legislation in Iceland in 1946 was to create a uniform
education system, with eight-year compulsory education and equal access to college
preparatory education. Previously, the two schools, which may be considered parallel to British grammar schools, had selected their students under strict admission
control and their own entrance examinations. As a compromise, regulations for the
Reykjavík Grammar School, dating from 1937, were chosen as a basis for a national entrance examination in eight school subjects, to be run in all larger towns and rural
boarding schools. The examination was intended to ensure a certain and standardized minimum knowledge in a number of subjects; the selection of the fittest with
respect to certain attributes considered necessary for studies in a grammar school
and a university or other establishments of higher education; and to offer all students
and their relatives a certain assurance of an assessment of the examination papers
by impartial persons.
In the national examination’s first year it became clear that examining in Euclidian
geometry did not work as teachers at lower levels had no such training. The mathematics examination was divided into two parts with equal weight, seen problems
and unseen problems, tested two days in a row. The content of the unseen examination became typically 6–8 problems; 4–6 story problems on area, volume and proportions, some to be solved by setting up equations; and two rather complicated fractions with algebraic denominators. The story problems either described situations in
contemporary daily life, or were versions of old problems.
By the mid-1960s, the examination, originally intended to provide equal opportunities
to education, became considered as a hindrance on young people’s path to preparation for life. Initially, a constant rate of 20% of the cohort attempted the examination
and 13-14% reached grammar school admission. By 1969 the rate had risen to 34%
vs. 21%. The seen problems were dropped, the examination was shortened considerably and the number of problems rose to 50 small and often unrelated items to ease
grading but also to help the less able students to show basic competences.
In the midst of this demand for education for all, New Math was implemented for the
purpose of facilitating understanding. The paper contains an analysis and comparison of typical examination papers before and after the implementation of New Math.
The analysis was made according to TIMSS Monograph no. 1: Curriculum framework
for mathematics and science by Robitaille et al. (1993).
The results of an analysis of examination papers from 1953, 1966, 1971 and 1975 indicate that the content swayed towards using set-theoretical notation and number notation to various number bases and back again. The proportion of simplifying algebraic
expressions and solving equations stayed around 50% with a slight increase, while
new topics were introduced to statistics and probability. Performance expectations
became less oriented towards independent development of notation, vocabulary,
and algorithms. The expectations developed away from using complex procedures
towards routine procedures, and investigating and problem solving was reduced. In
general, the examination developed into a host of incoherent details of diffused focus. However, data show that the average mathematics grade, which had been about
0.5 lower on the 0–10 scale than the average grade in all eight subjects during the
1950s and 1960s, had reached other subjects in 1972 during this process of increased
number of simpler problems.
Some questions remain as to what kinds of syllabi and examinations enhance mathematical thinking and understanding. Richard Skemp (1978) expressed concerns that
the backwash effect of examinations and overburdened syllabi promoted superficial
instrumental understanding at the cost of deeper relational understanding. George
Polya (1973) suggested that students think of a familiar problem in devising a plan for
solving problems. Could the seen problems enhance understanding on that ground?
Could multi-step word problems provide opportunities for teachers, together with
their students, to delve into mathematical processes and thus create lattices of acts
of understanding, as proposed by Anna Sierpinska (1994)?
