Place value problem solving and written assessment

¶ … Value: Problem Solving and Written Assessment by: Sharon R. Ross

This paper provides an analysis of an article by Ross (2002), which describes the learning characteristics exhibited by young children in relation to a particular topic in mathematics, known as place value. Place value refers to the value assigned to each digit within a multiple-digit numeral, within the familiar decimal number system that is the basis for contemporary mathematics in academia and commerce. The premise of the article is that the concept of place value is inherently difficult for elementary grade students to grasp, and correspondingly difficult for instructors to teach. The reason for this difficulty is attributed to the complicated symbology inherent in the decimal system of numeration, which may be summarized as being the product of four mathematical properties, namely the additive, positional, base-ten and multiplicative elements inherent in each and every multiple-digit numeral. Ross (2002) describes a practical classroom study in which researchers analyzed the results of five problem-solving assignments administered to different groups of elementary school students between grades three and five. These assignments were designed to exercise the cognitive skills and aspects needed to deal with the four mathematical properties related to place value. Results of the experiment were compared between individual and collaborative group efforts to solve the assigned problems.

To recap the core concept of place value in the decimal system, the value of each place in a numeral is ten times the value of the place to the right. As such, the rightmost digit in a numeral preceding the decimal point represents "ones," the next digit to the left represents "tens," the third digit to the left represents "hundreds," and so on. Children in the Ross (2002) study were assigned five problem solving tasks in which they were called upon to use intuitive, pictorial and group discussion methods to decide upon the relationship between parts of a given numeral and the actual count of objects in a given collection. The mathematical content involved in each of these five exercises will now be examined.

1. 144 Squares problem

In this exercise, students were made to collaborate in groups of four to assess whether several paper shapes with slight modifications had the same area as a 12 cm x 12 cm square, whose area was 144 cm2. The shapes to be compared included two rectangles and a square with clipped corners. Students were asked to identify the significance of each of the three digits comprising the number 144 relative to the area of the shape.

2. 124 Cubes problem

Groups of students in this exercise were asked to exercise their three-dimensional visualization capabilities to identify all possible ways to construct 124 cubes, using a fixed number of blocks with base-ten dimensions. Students were provided ten long blocks and one-hundred flattened blocks, with instructions to approach the problem in two different ways. Initially students were made to find all three-dimensional combinations resulting in 124 cubes, and subsequently asked to formulate combinations of blocks corresponding to each of the three digits in the numeral 124, thus representing the total volume.

3. 26 Wheels problem

Students in this exercise were asked to compute the total number of wheels associated with six vehicles (cars), added to the remaining number 2. They were then asked to pictorially associate on paper the significance of each digit in the numeral 26 to the actual number of wheels in the set. Researchers observed that a majority of the students in the study were unable to associate the digit 2 to the quantity 20 in this exercise. Instead, some students believed that the digit 2 represented the count of twenty wheels in five cars, with six remaining, while others believed that the digit 2 represented the wheels left over when six cars had twenty-four wheels assigned. Clearly, neither of these interpretations were correct, corroborating the author's initial assertion of the general difficulty of place value as a mathematical learning subject.

4. 62 Wheels problem

In this problem, students exercised visualization skills to address the inverse problem to the 26 Wheels challenge. Given the premise that one car used four wheels, students were asked to derive arithmetically and pictorially the number of whole cars that could be created using 62 wheels, and the number of wheels remaining unassigned. They were then asked to associate the digits 2 and 6 pictorially with the actual total count of 62 wheels.

5. 35 Beans problem

In this final exercise, students were given a two-dimensional matrix consisting of five rows of seven beans, for a total of 35 beans. Without ever identifying the digits 3 and 5, the researcher pictorially split the beans into different groupings and asked students to identify the correct correspondence of numbered groups to the appropriate places in the numeral 35. Researchers discovered that over eighty per cent of student subjects were unable to derive the correct association that the digit 3 represented the quantity 30.