What is the most powerful mathematical result

Learning is the most powerful mechanism of cognitive development: the acquisition of mathematical skills

Research report 2003 - Max Planck Institute for Human Development

Educational Science and Education Systems (Baumert) (Prof. Dr. Jürgen Baumert)
MPI for Educational Research, Berlin

In mathematics, too, a lack of knowledge cannot be compensated for by intelligence. Even before school starts, some basic conditions necessary for the acquisition of mathematical competencies are defined on an individual level, but school support is essential for the transition from intuitive to cultural mathematics. Everyone - including the less gifted children - has to learn at an early age that numbers can not only be used for counting and that the benefits of addition and subtraction are not limited to describing changes in quantity. Good mathematical skills at the end of school are the result of an intelligent practice process with intellectually stimulating tasks that starts at an early stage.
Intelligence cannot make up for lack of knowledge: this is also true for school mathematics. Part of the mental framework that is essential for acquiring mathematical competencies is determined, in the individual, even before school starts, but passing from intuitive mathematics to cultural mathematics requires the support of school settings. All children - even the less gifted - should learn early on that there is more to numbers than mere counting and that describing changing sets is not the only use addition and subtraction may be put to. Good mathematical competencies at the end of schooling are the result of an early start and an intelligent exercise procedure engaging pupils in intellectually stimulating tasks.

"Regardless of the different abilities and talents of the students, everything has to be learned what is later known and mastered. Learning is the most powerful mechanism of cognitive development. This applies without restriction to both gifted children and less gifted students Didactic support is necessary and effective. No matter how well intended motivational psychological or socio-educational measures are, they cannot be a substitute for the actual learning act, but only an often very effective prerequisite. " [1]

Franz E. Weinert recognized earlier than most other teaching-learning researchers that a high level of intelligence is only an advantage if it has been converted into area-specific knowledge beforehand. In the much-cited work by Schneider, Körkel and Weinert [2] it was shown that a lack of intelligence can be compensated for by knowledge, while a lack of knowledge cannot be compensated for by a high level of intelligence. However, some intelligence researchers questioned the generalizability of the results to more intelligence-related content areas. This article presents results from the longitudinal studies LOGIK and SCHOLASTIK, headed by Franz E. Weinert, which demonstrate the importance of prior knowledge for a subject area that is associated with intelligence like no other: mathematics.

Developing Mathematical Skills: From Intuitive to Cultural Mathematics

The ease with which children learn to count in the smaller range of numbers and to model the change in quantities contradicts the results that demonstrate the immense difficulties that mathematics can cause as a school subject. The results of infant research now impressively show the modularized fundamentals of mathematical competencies [3]. The universally available, intuitive mathematical knowledge facilitates the transfer of mathematical language to situations in the perceptible world. A set of objects or events can be recorded by counting, and the increase or decrease in quantities can be traced with the aid of addition and subtraction. On the basis of simple mathematical symbols, especially in connection with graphic-visual illustrations, complex mathematical concepts have emerged in the course of cultural development, which advanced science and technology. Despite the overwhelming importance of mathematics in modern industrial societies, people who can easily acquire mathematical skills are in the minority. Even with many people with a high school diploma, their mathematical knowledge does not go beyond the percentage calculation. Deficits in mathematical understanding become particularly evident when solving complex word problems for which no finished solution can be obtained.

Kintsch and Greeno [4] presented fourteen simple word problems with equations like 8 − 3 = 5 or 5 + 3 = 8 could be solved. Nevertheless, there were clear differences in difficulty between the tasks. The resolution rate for exchange tasks (Maria had eight marbles. Then she gave Hans three marbles. How many marbles does Maria have now?) was ninety percent, while the solution rate for comparison problems (Maria has eight marbles. She has three more marbles than Hans. How many marbles does Hans have?) was twenty percent. For tasks to combine quantities (Maria and Hans have eight marbles together. Maria has three marbles. How many marbles does Hans have?) resolution rates of fifty percent have been reported. The discrepancy in the solution rate between tasks with the same mathematical structure is particularly evident in the following task: The task Five birds are hungry. You find three worms. How many birds don't get a worm? is solved by eighty percent of preschoolers. If the task ends with the question How many more birds than worms are there?, the resolution rate is less than thirty percent, even with third graders.

An inadequate understanding of language and a lack of conceptual mathematical knowledge cause children to fail in certain tasks [4]. In order to solve comparison problems, one needs an advanced understanding of numbers that goes beyond the counting function of numbers [5]. The in sentence Hans has five more marbles than Peter given information does not designate a concrete, existing set, but describes the relation between two sets. You have to develop a mental model - that is, a mental idea abstracted from the concrete things - of the situation described in the text problem. Who, for example, with the number 5 only connects five objects, he will not understand the sentence. Who, however 5 understood as a section on the number line that marks the relation between two other numbers - for example between 2 and 7 or between 4 and 9 - who can understand comparison problems. Quantity comparison word problems are a good indicator of advanced math understanding in elementary school age.

Practice early: the long-term stability of inter-individual differences in math performance

In the LOGIK study as well as in the SCHOLASTIK study, word tasks were given for the additive and multiplicative comparison of quantities throughout primary school. Since such tasks are very rare in German primary school lessons, the students have to work out solution strategies on their own. In a follow-up study, the LOGIK Follow-Up Study, which was carried out when the subjects were on average seventeen years old, mathematical competence was also recorded by assigning tasks from the Third International Mathematics and Science Study (TIMSS) under time pressure were. Although this test mainly relates to the school material of the eighth grade, the solutions are not easy to solve even for mathematically educated people, as the following example shows: Which x-value satisfies the equation x2 −14x + 49 = 0 ?: A) 7 and 0, B) 7, C) −14, D) 7 and −7, E) 14 and 0. In order to achieve a wide spread of the results, so many tasks of this type were given that it is impossible even for an expert in mathematics to solve all the tasks in the limited time.

Intelligence and mathematics performance in elementary school age as well as performance in the eleventh grade are related to one another as follows: There is a close correlation between solving mathematical word problems in the second grade and mathematics performance in the eleventh grade. On the other hand, there is no correlation between the intelligence of the second grade and the mathematics performance of the eleventh grade. According to this, the intelligence of the eleventh grade does not correlate as highly with the mathematics performance of the eleventh grade as the performance in solving word problems in the second grade. This correlation is not based on outliers. At the same time, it also shows that no participant in the LOGIK study who did not already show above-average performance in solving word problems in the second grade achieved good to very good values ​​in the eleventh grade. On the other hand, there were a number of students who performed above average in the second grade, but later fell back into the average or even below average range. The data show that an early understanding of mathematics, which is expressed in solving demanding word problems, is a necessary, but by no means sufficient, requirement for later mathematical skills.

In order to separate the influence of specific knowledge and current intelligence on solving mathematical problems, a so-called commonality analysis was carried out, the results of which showed that the proportion of "pure" intelligence in the inter-individual differences in mathematical problem solving is only very small. The influence of intelligence is mainly shown in the "confounded variance". This states that children with a higher intelligence acquire more mathematical knowledge in the long run and therefore perform better. However, the improvement in performance due to intelligence is significantly less than that due to greater mathematical knowledge. Deficits in intelligence can obviously be compensated by prior knowledge, but deficits in mathematical prior knowledge cannot.

The results of the LOGIK follow-up study show that, even in an area close to intelligence such as mathematics, good performance depends crucially on previous knowledge. As early as the second grade, knowing that numbers can be used not only to illustrate the power and the change in sets, but also to depict relationships between sets, seems to be a necessary, if not sufficient, condition for high mathematical performance in the to be late secondary school. The LOGIK data even suggest that early failures in supporting a demanding mathematical understanding can no longer be compensated for later. This result could lead to fatalistic attitudes. One might think that if the "critical period" for access to cultural mathematics was missed, "the train has left". Or one could equate an early understanding of cultural mathematics with a genetically determined mathematical talent. Both interpretations are, however, premature at present. Different studies on mathematics teaching in primary schools show that the potential of children is not being used adequately here. It has been criticized several times that especially demanding word problems to compare quantities and later to the Cartesian product almost never occur [5]. Possibilities for the use and visualization of word problems have so far been little anchored in teacher training [6]. In the following, the results of the SCHOLASTIK study show that teachers have an indirect influence on solving demanding word problems despite these unfavorable boundary conditions.

The teacher makes the difference: the influence of the pedagogical content knowledge of the teacher on the learning progress of the students

In countries with relatively homogeneous school conditions, in which the training of teachers and their assignment to schools is centrally regulated by the state, teacher characteristics contribute comparatively little to clarifying differences in performance between individuals. Minimum standards are generally guaranteed, and fixed specifications in the curriculum often leave the teachers little leeway to try out independently planned teaching methods. However, certain characteristics and behaviors of teachers can be critical to the performance of some students.

"Although the individual differences in ability, learning and performance remain relatively stable over time, the (individually variable) learning and performance progress is a function of the quantity and quality of learning and is more or less strongly influenced by the effectiveness of the teaching. School achievements are therefore always achievements of the students that are favored or made more difficult by the school. " [1]

The scientific potential of the SCHOLASTIK study results in particular from the mandatory change of teachers from the second to the third grade in Bavarian elementary schools. This makes it possible to attribute the differences found between the classes in the increase in performance from the second to the third grade to the influence of the teacher teaching in the third grade. Even if German elementary school teachers have little freedom in choosing the content of their mathematics lessons, quite subtle factors can possibly become significant. One characteristic that is becoming increasingly important are the subject-specific pedagogical attitudes of the teachers. This is understood to mean the merging of content and pedagogy to an understanding of how certain topics, problems or questions should be structured, presented, adapted to the interests and abilities of the learner and prepared for teaching. A good teacher knows how students learn certain content. From incomplete solutions and mistakes, he can tell whether children, even if they do not yet meet the performance criterion, are on the right track. The mental activity of understanding is crucial for math lessons. Even if the cognitive sciences and teaching-learning research are still far from being able to explain the phenomenon of understanding, there are still some generally accepted basic assumptions. This means that understanding is the result of an active construction process on the part of the learner. They have to try things out, go astray and be able to recognize them before they can really understand an object. So understanding is not the result of the transfer of knowledge from the teacher to the student. This view is summarized under the concept of constructivist learning. For understanding and solving word problems, an active construction of the underlying situation model and its conversion into a mathematical equation is crucial.

With the help of questionnaires the teachers' attitudes towards the active role of students in solving word problems can be recorded [6]. A constructivist attitude is reflected, for example, in the following statements in the questionnaire:

Students should be given word problems before they have a good command of arithmetic procedures. - Teachers should encourage students to find their own way of solving math problems, even if they are inefficient. - Mathematics should be taught in school in such a way that the students can discover connections for themselves.

In contrast, a receptive attitude towards understanding word problems is expressed in the following statements:

Teachers should provide detailed procedures for solving word problems. - To learn math, it is important that students can listen well. - Effective teachers demonstrate the correct way in which to solve a word problem.

At the suggestion of Fritz Staub, the translated questionnaire was sent to the participating teachers two years after the end of the SCHOLASTICS study. There was an astonishingly close connection between a constructivist attitude expressed in the questionnaire and the average learning progress of the class in solving word problems [6]. Prior knowledge, i.e. the math performance measured at the end of the second grade, is a better predictor of the math performance at the end of the third grade than the intelligence, although it must be taken into account that the "confounded variance" of intelligence and math performance also determines the prior knowledge. When it comes to solving word problems, it can be seen that the teacher convictions explain almost as much variance as the "pure" intelligence. Although in German elementary school mathematics there is hardly any use of the opportunity to expand mathematical understanding with the help of word problems, indirect effects of the teachers' convictions on the mathematical problem-solving skills of the students can be demonstrated. Teachers who are aware of the importance of active, problem-oriented learning of mathematics also provide indirect support for solving word problems. In fact, it was found that teachers with a constructivist attitude were more likely to present conceptually stimulating math problems. The results show that a mathematics lesson geared towards understanding does not neglect "learning to do arithmetic".Classes with teachers who took a constructivist attitude did not show any worse performance in addition and subtraction tasks than classes with receptive teachers. In the case of multiplication and division problems, there was even a positive trend. There was also no evidence to support the objection that demanding, comprehension-oriented mathematics lessons are at the expense of weaker students [6]. In addition to the basic attitude of the teachers, no other influencing characteristics such as class size or average intelligence and performance level of the class could be identified.

Even if the proportion of the variance explained at the individual level through teacher beliefs is not excessively large, it should be noted that at least twenty-five percent of the variance observed between the classes in learning growth for word problems for addition and subtraction can be traced back to the teacher’s beliefs. The importance of these effects should not be underestimated for at least three reasons. Firstly, given the current shortage of qualified mathematicians and natural scientists, it could be seen as a success if one to two percent more students were to orient themselves in this direction through stimulating lessons. Second, it must be remembered that in the analysis of Staub and Stern [6] only the effect of a single school year was taken into account. Adding up over the years, considerable effects can be seen at the end of schooling. Thirdly, it must be taken into account that the heavily regulated curriculum specifications leave little room for maneuver even for teachers with a constructivist attitude to perform demanding word problems. It can be expected that a constructivist attitude on the part of teachers would have a greater impact on the increase in performance in solving demanding word problems if these were included in the primary school curriculum.


[1] F.E. Weinert: School performance - performance of the school or the pupils? In: Performance Measurements in Schools. (Ed.) F.E. Weeps. Beltz, Weinheim 2001, p. 85.

[2] W. Schneider, J. Körkel and F. Weinert: Domain-specific knowledge and memory performance. Journal of Educational Psychology, 81, 306-312 (1989).

[3] K. Wynn: Addition and subtraction by human infants. Nature, 358, 749-750 (1992).

[4] W. Kintsch and J.G. Greeno: Understanding and solving word arithmetic problems. In: Psychological Review, 92, 109-129 (1985).

[5] E. Stern: The development of mathematical understanding in childhood. Pabst, Lengerich 1998.

[6] F.C. Staub and E. Stern: The nature of teacher’s pedagogical content beliefs matters for students ’achievement gains: Quasi-experimental evidence from elementary mathematics. Journal of Educational Psychology, 93, 144-155 (2002).