Following the social democratization of the 1990s, alternative edu- cational programs gained official recognition alongside that of Moro et al.; these included both Zankov and Davydov, as well as N. Ya.
Vilenkin, L. G. Peterson, and N. B. Istomina (Programmy, 1998). The appearance of competing programs in general education and preschool training led to the development of an Educational Standard and the Conception of the Content of Continuous Education.
The Educational Standard for Russian Schools (Uchebnye stan- darty, 1998) acknowledges the changing role of mathematics in general culture and education. The Conception of the Content of Continuous Education sets out the following objectives for elementary mathematical education:
• Development of the basic forms of intuitive and logical thinking and mathematical language; development of intellectual operations (analysis, synthesis, comparison, classification, etc.); ability to oper- ate with symbolic systems;
• Command of a specific system of mathematical notions and com- mon operations;
• A basic grasp of the leading mathematical method for understand- ing the physical reality — mathematical modeling (Kontseptsiya, 2000, pp. 16–17).
The Federal Educational Standards (second generation), set to take effect in 2010, lay out performance requirements, structural guidelines, and conditions (staffing, financial, technical, material, etc.). Standards consider subject-specific performance alongside metadisciplinary and personal accomplishments. Personal accomplishments include readi- ness and capacity for self-development, motivation for study and acquisition of knowledge, system of values, and foundations of civic identity.Metadisciplinaryaccomplishments include universal learning operations (which form the basis of learning capability) and interdis- ciplinary notions.Subject-specificperformance includes acquisition and application of new subject-specific skills, as well as a system of elements of scientific knowledge at the basis of the contemporary scientific understanding of the world.
Subject-specific performance requirementsin Mathematics and Infor- matics include the following (Ministry, 2009, pp. 12–13):
• Using mathematical skills to describe and explain objects, processes, and phenomena, and to evaluate their quantitative and spatial characteristics;
• Receiving the foundations of logical and analytical thinking, spatial imagination and mathematical speech, measurement, enumeration, estimation and assessment, visual representation of data and pro- cesses, notation and performance of algorithms;
• Basic experience in using mathematical skills to solve theoretical and practical problems;
• Ability to perform arithmetical operations (oral and written) with numbers and numeral expressions; solve word problems; follow an algorithm and construct basic algorithms; examine, identify, and reproduce geometrical figures; work with tables, diagrams, graphs, sequences, and populations; visualize, analyze, and interpret data;
• Basic notions of computer literacy.
The main objectives of the study of Mathematics and Informatics, according to the same Standards, are as follows: development of mathematical speech, logical and algorithmic thinking, imagination, and preliminary notions of computer literacy (p. 22).
At this time, there are 15 curriculum “series” or “complexes,”14 as they are called, in Russia in mathematics for the elementary school, evaluated and included in the federal register of textbooks recommended by the RF Ministry of Education and Sciences for use in Russian schools (see http://www.edu.ru/db-mon/mo/Data/d_09/
m822.html).
Different methodological ideas underlie the various “complexes;”
however, all of them give primacy of place to the developmental aims of education. Ivashova et al. (2009) stress the equal importance of developmental and discipline-specific aims.
All of the “complexes” break down the material according to the basic components of learning activity (positing an objective, proposing ways of attaining the objective, planning, following the plan, self- monitoring and self-evaluation, reflection). Bashmakov and Nefedova (2009) and Ivashovaet al. (2009) include an overview at the start of the textbook (section titled “What will we learn?”), quarterly review sections, and reference materials.
Several textbooks make use of creating so-called “problem sit- uations” in the material: for example, in Ivashova et al. (2009) and Istomina et al. (2009), students are asked to evaluate solving strategies, explain underlying reasoning, choose the best option, and find and correct errors. In Ivashovaet al.(2009), correction of errors presupposes variability, as in the following exercise:
Check the calculations. Correct one of the terms or the final value.
14Typically, such a “complex” includes not only textbooks, but teachers’ manuals, problem books, and other supplemental materials.
Arginskaya et al. (2009), Davydov et al. (2009), and Istomina et al. (2009), among others, encourage students to find a solution strategy independently and draw their own conclusions. Exercises involving “problem situations,” composition, or transformation of existing problems, numbers, expressions, investigative exercises, and so on promote creative thinking in students. For example:
What is the rule governing the transformations of the original expression in each column?
7ã4+18−9ã3 28+18−9ã3 28+18−27 46−27
86−7ã3−49ữ7 86−21−49÷7 86−21−7 65−7
Use the same rule to construct a new column beginning with the expres- sion9ã5−6ã4ữ8(Istominaet al., 2009, 3rd grade).
The following strategies reflect the movement toward personalized education:
• Students are asked to characterize exercises as easy or difficult, interesting or boring, to choose the most comfortable solution strategy (Alexandrova, 2009; Davydov et al., 2009; Ivashova et al., 2009), explain the solution process (Alexandrova, 2009;
Arginskayaet al., 2009; Ivashovaet al., 2009), compose an original exercise and teach it to others (Alexandrova, 2009), and compose problems based on personal observations (Davydovet al., 2009).
• Exercises are worded in a personalized manner: “Do you know … ?”
“How much would you have to spend if you wanted to buy … ?”
“Draw up a plan of action and tell it to others” (Ivashova et al., 2009; Rudnitskaya and Yudacheva, 2009).
• Emphasis is placed on alternative solving strategies (Alexandrova, 2009; Istomina, 2009; Ivashovaet al., 2009), and on choosing the most appealing exercises (Rudnitskaya and Yudacheva, 2009) and solving strategies (Ivashovaet al., 2009).
• Exercises of varying difficulty include advanced-level (Ivashova et al., 2009), required, and supplemental exercises (Bashmakov and Nefedova, 2009; Moroet al., 2009; Rudnitskaya and Yudacheva, 2009).
One of the peculiarities of Ivashova et al. (2009) is psychological differentiation. Exercises intended for students with different styles of perception and information processing are marked accordingly:
sign stands for kinesthetic perception (exercises dealing with movements and notions about movement), sign stands for visual perception (exercises involving images and diagrams), aural perception (listening), and sign stands for verbal representation. Here are some exercises for the derivation of the number 5 (first grade):
Lay out four circles. Add one more. How many circles are there?
Examine the drawing and explain how the number 5 was derived.
Name four girls, then name another one. Now say the five names all together.
Typically, textbooks break down the material into discrete lessons;
the exceptions are Alexandrova (2009), Istomina (2009), and Rud- nitskaya and Yudacheva (2009), where the material is presented thematically. Rudnitskaya and Yudacheva (2009) include a review section after each theme, while Davydovet al.(2009) and Alexandrova (2009) gather all the review materials into a single section at the end of the textbook, titled “Check your skills and knowledge” or “Check yourself!”. In providing review sections, textbooks encourage self- monitoring by students.
Several sections are aimed at broadening or deepening the students’
mathematical skills, e.g. “This is interesting!” (Alexandrova, 2009;
Rudnitskaya and Yudacheva, 2009), “Problems for those who like to work hard” (Alexandrova, 2009), “For the math enthusiast” (Demi- dovaet al., 2009), “From the history of mathematics” (Bashmakov and Nefedova, 2009; Ivashova et al., 2009; Rudnitskaya and Yudacheva, 2009), “Let’s play with the kangaroo” (Bashmakov and Nefedova, 2009), and so on. Here is a sample exercise from the third-grade textbook of Bashmakov and Nefedova (2009):
Which number matches the following description? It is even, none of its digits are the same, and the digit in the third position is double that in the first position. (A) 1236, (B) 3478, (C) 4683, (D) 4874, (E) 8462.
We should note that for the enrichment of their courses, teachers of elementary mathematics frequently make use of the publication by Kalininaet al.(2005), which essentially doubles as an encyclopedia for the elementary school, written in accessible language.
Many textbooks include group exercises, aimed at developing communication skills in children. Working in dialog, the students acquire new skills and knowledge and learn to accept another’s point of view. For example, Istomina (2009), Ivashova et al. (2009), and Rudnitskaya and Yudacheva (2009) make use of recurring characters with competing viewpoints, which are sometimes correct and some- times incorrect.
A number of texts include reference materials (such as average speeds of various types of transportation and animals, or weights of various types of objects and materials), which train the students’ ability to work with data and promote interest in mathematics and creativity in composing one’s own exercises.
Many of the “complexes” place special emphasis on the cultural aspect of mathematics through word problems (including problems with interdisciplinary content) and calculation exercises that require students to decipher certain names, terms, etc., contextualize numerical data, and identify geometrical figures in their immediate surroundings or in architectural structures. In certain textbooks, entire lessons are structured around a narrative. For example, the review sections in Bashmakov and Nefedova (2009) for the second and third grades have unifying themes: “Little Boy and Karlsson” (recalling Astrid Lindgren’s story), “A Flight to the Moon,” and “The Golden Fleece.”
For example, a calculation exercise in “A Flight to the Moon” asks the student to decipher the name of the first astronaut to step on the surface of the moon, which requires a series of calculations to determine the correspondence between numbers and letters.
Overall, many of the “complexes” in elementary mathematics may be characterized as “next generation.” Their content is primarily scientific, personalized, and aimed at general development, follows the
“active” approach to elementary education, and generally conforms to current standards (Uchebnye standarty, 1998) and forthcoming edu- cational standards (http://standart.edu.ru/catalog.aspx?CatalogId=
531).