This is an informal content review of the Mathematik Plus Gymnasium curriculum. Please see the Education Page of Bas Braams - Links, Articles, Essays, and Opinions on K-12 Education - for related matter.

In the Summer of 2001 I was in Greifswald, Germany (Mecklenburg - Vorpommern). My host has two kids who were just entering 5th Grade, which is the first year of Gymnasium in that State, and as an interested observer of K-12 mathematics and science curriculum in the United States and a frequent visitor to Germany I was curious to see their textbooks. For mathematics this was the series Mathematik Plus for Gymnasium from Volk und Wissen / Kamp Verlag, Berlin.

I was impressed by the quality of content of the 5th Grade textbook of that series, and obtained the complete series for Grades 5-9. (Meanwhile also a Grade 10 text is available.) In October and November of 2001 I contributed content summaries of the Grades 5-8 textbooks to an education email list by the name of Kto15 that has a special interest in mathematics and science curriculum. The list has its origins in the Mathematically Correct group.

The present contribution, done September 2002, brings these content summaries to the web-reading public and adds some commentary on what I see as positive and negative content qualities of the series. To summarize my assessment: the 5th Grade textbook is indeed of high quality, but the quality declines in further years and the 8th and 9th Grade textbooks are rather disappointing given that they are intended for a select group of students.

I would be delighted to know of other reviews of German K-12 mathematics textbooks, especially comparative reviews. My contact information is at the bottom of the page.

First some information for U.S. readers. Germany has a Federal system and responsibility for education belongs to the States. Standards and systems vary. In most States Grades K-4 are comprehensive and in Grade 5 the school paths diverge between Gymnasium, Realschule and Hauptschule. About 35% of the kids enter Gymnasium, which is the more academically oriented school type. Kids are generally six years of age or nearly six when they enter 1st Grade, so I think the correspondence between age and grade is the same in Germany as it is in the United States.

Using New York City points of reference we should think of Gymnasium as attracting a student population similar to what is found in our run-of-the-mill magnet schools: better than average, but certainly not rarefied. Gymnasium is selective, but it is far, far less selective than our Stuyvesant or Bronx Science High Schools.

The textbooks discussed here are the series "Mathematik Plus" for Gymnasium, Grades 5 through 9. The responsible authors are StD Dietrich Pohlmann and Prof. Werner Stoye. The publisher is Volk und Wissen / Kamp Verlag. Volk und Wissen has its roots in the German Democratic Republic, in which it was the near monopoly publisher of school textbooks. Presently the choice of textbook is left to the individual schools and Volk und Wissen (now joined with the West-German publisher Kamp and part of the Cornelsen publishing group) is only one of several suppliers, still focussed mainly on the eastern states. I suspect, but do not know for sure, that this series would be considered a quite traditional and solid one within Germany generally.

The five books are each 200-240 pages in length. I paid $65 for the lot, including postage (surface mail) to the United States (and received a second set courtesy of the publisher). The ones I obtained are for Bundesland Sachsen, but I don't think there is much difference between the various State-specific instances of this series.

My content summaries for the Grades 5-8 texts are linked here below. On this page I only reproduce the tables of contents.

Grade 5 content summary. ToC: The Natural Numbers # Fractions # Units and Dimensions # Geometric Figures # Symmetry and Reflection # Measurement, Representation and Computation.

Grade 6 content summary. ToC: Divisibility, Factorization # Fractions # Reflection, Translation, Rotation # Triangles and Congruence # Proportionality # Geometric Bodies, an Overview # Circumference and Area.

Grade 7 content summary. ToC: Computations with Percentages # The Rational Numbers # Equations and Inequalities # Triangles, Quadrilaterals and Polygons # Prisms # Elements of Stochastics # Miscellaneous Exercises.

Grade 8 content summary. ToC: Working with Variables # Linear Equations and Functions # The Circle # The Pythagorean Theorems # The Circular Cylinder # Random Events # Miscellaneous Exercises and Applications.

Grade 9 ToC: Real Numbers # Linear Equation Systems # Quadratic Functions and Equations # Powers, Roots, Logarithms # Similarity # Representations and Computations of Bodies # Elements of Stochastics # Miscellaneous Exercises and Applications.

In reviewing the quality of content of this series we need to keep a few things in mind. First of all, the series is intended for pupils that are better than average, although by no means exceptional. Approximately the top 1/3 of all students will be in Gymnasium. Second, at the start of Gymnasium, in 5th Grade, we have no reason to expect a base level that is particularly strong by international standards.

[Peripherally, and admitting that this is very impressionistic, I'll amplify that second statement. My impression is that the German educational style is for the younger years quite hostile to anything intellectual or academic. School days in K-1 are short, and kids are hardly expected to learn anything. This impression is based on interaction with colleagues over a prolonged stay and many visits, mainly in Munich, Bavaria, where the educational system is supposedly better than in most States.]

In any case, a first glance at the 5th Grade Mathematik Plus text gives a very favorable impression of the level of expectations to which the students will be held. (See the content summaries linked earlier.) The presentation is attractive throughout, and there appears to be a good balance between routine and challenging problems, and a good balance between basic symbolic manipulation and problems in context.

When looking at the development of the various mathematical strands over the course of the five years, Grades 5-9, this favorable impression is not entirely maintained.

The series seems to do a fine job overall on 5th and 6th Grade arithmetic, centered around the use of fractions and decimal numbers. The students are clearly expected to know their basic arithmetic well at the start of 5th Grade. They are started off working with very large and very small numbers, and then there is a substantial treatment of fractions, which is completed in 6th Grade.

Although the treatment of arithmetic seems fine overall there are some curious anomalies. Pupils work systematically with decimals throughout 5th and 6th Grade, but the notion of percents (and per-milles, more familiar in Europe than in the US) is deferred until 7th Grade. The use of percents strikes me as a good basic application of decimal arithmetic, and the concept of percents strikes me as a natural part of a fifth grader's language skills even outside the school environment. Although I can't see serious harm in deferring the introduction of percents to 7th Grade when one is anyway teaching decimals in 5th Grade, I don't see much point in this delay either.

A stranger anomaly, to me, is that the systematic working with negative numbers is also deferred until 7th Grade, where it is then introduced at once for integers and rational numbers. Even the integer number line does not make its appearance before 7th Grade. Of course the pupils know subtraction as the inverse of addition (they knew that before 5th Grade), but they don't work systematically with negative numbers before 7th Grade.

Another item of some wonderment is that there is hardly any review of the basic arithmetic skills of paper and pencil multiplication and division. I may trust that the students will know their multiplication algorithms without review, but I wonder if there really would not be a need for occasional practice with the division algorithm.

Algebraic notions are introduced quite tentatively already in 5th Grade, where pupils see occasionally an expression involving a single unknown variable, and where also the commutative, associative, and distributive properties of arithmetic are stated algebraically. It is only a start, and nothing substantial is added to the algebra base in 6th Grade.

Systematic algebra starts in 7th Grade, where pupils learn to formulate and solve simple linear equations and also inequalities. In 8th Grade also simple polynomial expressions are met.

The concept of a function is introduced rather late and slowly, in my estimation. It starts properly only in 8th Grade and in that year it is restricted to linear funtions. In 9th Grade quadratic functions and equations are covered, and that year also brings power functions with integer powers and a first brief look at logarithms.

All this would be perfectly fine, and perhaps ambitious, when addressed to the general student population. But Gymnasium is selective, and I was very surprised to see how the ambition that looks so promising in the 5th Grade text is lost in 8th and 9th Grade. Integer power functions and polynomials generally should really be 8th Grade material for this community, and in fact in this series one does not find a systematic treatment of polynomials beyond the quadratic ones even in 9th Grade.

In 5th Grade, as might be expected, geometry is only pictorial. In 6th Grade there is quite a bit of pictorial geometry as well, focussing on the recognition of symmetries and congruences, but there is a start of more formal geometric reasoning. Pupils see equal angles when a line intersects a pair of parallel lines, and there is a careful treatment of angles in a triangle adding up to 180 deg. I consider this a fine and promising start of geometry for 6th Grade Gymnasium.

Not much is added to geometry in 7th grade, but the 8th grade geometry is quite substantial. A high point is the very good treatment of the theorem of Pythagoras, the theorem being proved there with due care in four different ways. Also in 8th grade there is a good treatment of the circumscribed and the inscribed circle of a triangle, and thus of the properties that the perpendicular bisectors and the medians each intersect in a point, and of conditions under which the vertices of a quadrilateral lie on one circle.

Through Grade 9, solid geometry is developed to the level that students can recognize and name the various common figures (they know what is a rhomboid), and they can compute surface areas and volumes. In Grade 9 the formula for the volume of a pyramid is derived geometrically and then Cavalieri's rule is used to obtain the volume of a sphere.

Probability and data analysis is covered systematically starting in 7th Grade. I would gladly see this delayed in favor of a better treatment of algebra and functions, but in any case, statistics and data analysis only take up a small chapter of the text in each of 7th through 9th Grade, and the choice of topics is reasonable.

The Mathematik Plus series looks fine for 5th and 6th Grade, where I have only relatively minor quibbles with the texts. The momentum is lost, however, in the higher grades. In my estimation, in spite of the good start in Grades 5 and 6, at the end of Grade 9 the students are about a year behind a proper target level.

My review is focussed on the mathematical content of the texts and not so much on the pedagogical qualities, which I can only assess impressionistically. Impressionistically, then, the pedagogy appears basically sound with one qualification. The series offers a good mix of exercises of different kinds. There are occasional calls for group work, but not to an extent that it would detract from learning.

My qualification with regard to the pedagogy is that I don't see much review in the series. There is occasionally a boxed piece of text reviewing earlier material, but a systematic pattern of review exercises does not seem to be programmed into the text. I don't know if teachers can be trusted to arrange their own review cycles into the material, and I don't even know if this would be expected of them.

I remind the reader of the links to the Grade 5, Grade 6, Grade 7, and Grade 8 content summaries.

Bas Braams

--

Bastiaan J. Braams
- braams@math.nyu.edu

Courant Institute, New York University

251 Mercer Street, New York, NY 10012-1185