**2. Basic competencies in elementary mathematics: understanding place value**

As pointed out in the introduction, mathematical literacy represents a central goal for mathematics education for all grades and all levels. In this context, foundations on the primary level are of major importance. Van de Walle et al. identify as central topics for the primary level the development of


Place value represents as well one of the fundamental ideas of mathematics, in particular of arithmetic [[4], see also [5]]. A firm understanding of the place value concept is necessary for understanding our decimal number system in general, for coping with bigger numbers and number sense (e.g., estimating). In detail, the importance becomes clear for developing effective computation strategies (e.g., to replace one-by-one finger counting), for understanding the written algorithms, or for extending integers to fractions. Moreover, place value relates to measurement with decimal and non-decimal structures. Place value understanding has a deep impact on further arithmetical development, and the relevance for different fields of school mathematics can be stated [see also [3]].

Place value understanding has been identified as a good predictor of mathematics performances as well as of mathematical difficulties [for example, see [6]]. Research shows that especially low achievers, even in higher grades, have great difficulties with this mathematical topic [7, 8].

Students with difficulties in mathematics often use time-consuming, inefficient, or error-prone strategies to solve simple calculations. In contrast, average-achieving

#### *Fostering Basic Mathematical Competencies: Concepts and Materials for Teachers and Students… DOI: http://dx.doi.org/10.5772/intechopen.113257*

students recall basic elements quickly and accurately [9], and this might be ascribed to missing basic competencies like understanding place value.

What are the key principles and specific challenges for understanding place value [for example, see [10]]? A key to understanding place value is the principle of *bundling*: 10 ones can be exchanged for 1 ten, 10 tens for 1 hundred, 10 hundreds for 1 thousand, …, and children have to become familiar with these base-ten equivalents [11]. Moreover, the principle of *position,* that the position encodes relevant information, is of major importance: "Children who do not understand that position encodes information may disregard digit order (e.g., sometimes read 71 as "seventeen") or overlook the number of digits (e.g., read 1047 as "one hundred forty-seven"). Because decade terms are written with a 0, some children may assume that a 0 should be written whenever they hear a decade term (e.g., write 'forty-two' as 402)" ([11], p. 209).

In this context, Fuson highlights the irregular structure of number names in different languages, especially compared to many Asian languages [12]. This might complicate understanding of place value. Moreover, linguistic similarities and confusions with numbers like "fourteen" and "forty" or in combination with decimals "hundreds" and "hundredths" may cause further difficulties. Comparable linguistic similarities also exist in the German language [for example, see [8]]. Moreover, the German language—as well as others—show an irregular building of number words with respect to tens and ones: "43" is read as "three and forty" that might result in inversion errors [13].

Problems also occur when it comes to non-routine tasks [8]: For composing a number out of hundreds, tens, and ones, standard tasks like 300 + 50 + 4 might not cause any problems. In contrast, vacant place values as well as the unfamiliar order of place values can play an important role: In a study with low achievers, for example, composing a number out of the term 70 + 200 + 3 led to the number 723 [8].

These research findings emphasize that for teaching place value, the design of classroom activities is of major importance. Although for mathematics education, active acquisition of knowledge aiming at conceptual understanding represents a central principle for primary mathematics [for example, see [4]], this might not be true when looking into inclusive classrooms and students with special needs [8].

It is important to choose appropriate manipulatives, models, and representations and give learners enough time and suitable experiences to build up mental images of numbers and operations. Besides the adequate material, the concrete processes of using effective structures and reflection of number relations are of major importance.

For insightful learning, connecting several models and representations (e.g., concrete/enactive, iconic/pictorial, and symbolic level, see also [14]) or making use of virtual representations [15] is essential. For understanding place value, adequate activities, even in higher grades with bigger numbers, should facilitate these transfer processes.
