7.2 The Content of the Elementary Course
7.2.1 Numbers and arithmetical operations
All of the textbooks cover the following subjects:
Counting objects. Names, succession, and notation of numbers from 0 to 1,000,000. Number relations, such as “equal,” “greater than,”
“less than,” and their notation: =, <, >. The decimal numbering system. Classes and digit positions. The positional principle of number notation.
All textbooks, with the exceptions of Alexandrova (2009) and Davydovet al.(2009), are structured concentrically: the students first learn the numbers 1 through 10, then the numbers up to 100, and then up to 1000 and beyond. This corresponds to the child’s experience and to the methodological tradition in Russia. The textbooks present a variety of methods for deriving numbers: counting, addition, and subtraction of 1, measurement, and arithmetical operations with other numbers. In Alexandrova (2009) and Davydov et al. (2009), the main method of deriving numbers (natural as well as rational, etc.) is measurement. By introducing a variety of measuring units, the course
prepares first and second graders for the study of different counting systems and permits them to see the decimal system as one possibility among several.
This approach does not take into account the child’s preschool experience, but it permits students with different levels of preparation to feel confident in discovering new knowledge. One drawback of both Alexandrova (2009) and Davydov et al. (2009) is that the text does not differentiate between notations referring to magnitudes and those referring to sets or figures. This approach may lead to confusion over such concepts as “finite set” and “size of finite set,” “segment,” and
“length of segment; it runs counter to the principle of continuity, since at a later stage the student will be asked to differentiate these concepts through notation (Beltiukovaet al., 2009).
All other textbooks use a concentric structure to teach derivation of numbers, their names and sequencing, the decimal order, positional notation, and various methods of number comparison. Many of the textbooks make use of historical references (Bashmakov and Nefedova, 2009; Demidovaet al., 2009; Ivashova, 2009; Peterson, 2009; Rud- nitskaya and Yudacheva, 2009).
All textbooks make extensive use of various types of modeling. For example, in learning the decimal numbering system, students are asked to use sticks and bundles of sticks or squares for ones, strips for tens, and large squares for hundreds. The great majority of the textbooks make use of the number line; Arginskaya et al.(2009) and Istomina (2009) use the segment of natural numbers, while Alexandrova (2009) and Davydovet al.(2009) discuss various kinds of positional notation.
All of the textbooks cover the following subjects associated with arithmetical operations:
Addition and subtraction, multiplication and division, corresponding terminology. Tables of addition and multiplication. Number rela- tions, such as “greater by … ,” “smaller by … ,” “ … times greater,”
and “ … times less.” Division with remainder. Arithmetical operations with zero. Determining the order of operations in numerical expres- sions. Finding the value of expressions with parentheses and without.
Changing the order of addends and multipliers. Grouping addends and multipliers. Multiplying a sum by a number, and a number by
a sum. Dividing a sum by a number. Oral and written calculations with natural numbers. Using the properties of arithmetical opera- tions in calculations. Finding an unknown component of arithmeti- cal operations. Strategies for checking calculations. Solving word problems (with one or multiple operations) by arithmetical means (using a variety of models: schematic drawings and graphs, tables, and shorthand notations with keywords). Relations of proportional magnitudes (velocity, time, distance traveled; price, quantity, cost, etc.).
As for other subjects, let us note that we do not see the advantages of a detailed study of decimal fractions — i.e. construction, rounding, comparison, performing arithmetical operations, and deriving fraction from number and number from fraction — at the fourth-grade level (Alexandrova, 2009). In moving this material from the fifth- and sixth- grade curriculum into elementary school, the textbook runs counter to the principle of succession and shifts attention from other important topics (for example, at the second- and third-grade levels, this textbook has far too few exercises with geometrical figures).
We are also skeptical about the accessibility for students of propor- tions characterizing work, movement, and buying–selling (Davydov et al., 2009), in order to understand which students need to master such concepts as “process,” “event,” “variable characteristics,” “additional conditions,” “uniform process,” “variable process,” and “speed of a uniform process.” In the corresponding textbooks for the fourth grade, one reads: “The speed of the uniform process K indicates the rate of increase of Y with respect to X. X1 = X2,Y1 > Y2,K1 > K2.” And further on: “The speed of a uniform process is a constant. It indicates how many units of Y correspond to a single unit of X” (Davydovet al., 2009, Book 1, p. 109).
It is interesting to note the use of the calculator (Chekin, 2009;
Istomina, 2009; Moro et al., 2009) not as a substitute for manual calculation, but as a way of verifying results. Here are a few sample exercises:
• Find the value of the expressions37+24−24,52+37−37, and 83−18+18.
In what ways are these expressions similar? What conclusions can you draw? Verify your results with the help of a calculator, using different numbers. (Istomina, 2009, 2nd grade)
• Using a calculator, add 1, 2, 3, and 4 to the number 372. Which digit changes in the number 372? What other numbers could you add to 372 without changing any of the other digits in the number? (Istomina, 2009; 2nd grade)
• Using a calculator, find out whether the greatest three-digit number is a multiple of the greatest six-digit number. (Chekin, 2009; 3rd grade)
• The value of what expression would you be calculating if you pressed the following sequence of buttons on your calculator?
2 3 8 9 7 7 − 2 3 8 9 0 5 ÷ 9
(Chekin, 2009, 3rd grade)
The introduction to algorithms in several of the textbooks includes:
analyzing existing algorithms; constructing new algorithms (including
“everyday life” algorithms — crossing the street, lighting a fire, etc.);
types of algorithm notation — verbal and flowchart; and performing calculations using a flowchart. This addresses the requirements set out in the new standard.
A number of “complexes” pay special attention to estimating value, evaluating results, and verifying results. For example, Ivashovaet al.
(2009) include the following assignment:
Calculate and verify using a different calculation technique, such as:
100÷4=(80+20)÷4= 60÷4 90÷5 100ữ4=(120−40)ữ4= 14ã5 38ã2
Bashamkov and Nefedova (2009) have the following:
Choose the answer out of three given values without performing the calculation to the end. Then evaluate the value and compare it with your choice.
Find the value and compare it with your choice.
(a)173+264+435, (b)236+312+422, (c)329+119+449 m
872 972m 899m 772m 970m 997m 874m 890m 897m
Moroet al.(2009, 4th grade, p. 85) have the following:
Pick out the wrong answers without doing the calculation. Solve and check your answer through multiplication.
7380÷9=82, 3010÷5=62, 56014÷7=8002.
Some of the textbooks include subjects not covered in the standard:
“Common fractions, addition and subtraction of fractions with the same denominator, multiplication and division of fractions,” “Positive and negative integers,” and “Percent.”
As far as calculation techniques are concerned, let us note the following: the majority of the textbooks first teach oral calculations and then written calculations. Davydov et al. (2009) first introduce the digit-position principle of written calculation and only later ask students to compose and memorize a table of addition (and later multiplication) and learn the techniques of oral calculation. It seems advisable to encourage students to calculate orally whenever possible.
Rudnitskaya and Yudacheva (2009) give primacy of place to written calculation, which seems to us a doubtful approach, since in everyday life one is often called upon to calculate “in one’s head.”