Abstract
This study sought to determine the relationship between participation in informal mathematics activities and the formal-to-informal beliefs of university teacher candidates in elementary education. Three classes of preservice teachers participated in the study through their enrollment in a content mathematics course for elementary education majors. Four informal mathematics activities were employed as part of the course requirements. Before and after formal-to-informal beliefs about mathematics and mathematics instruction were measured using a Likert-scale beliefs assessment instrument used by Collier (J Res Math Educ 3(3):155–163, 1972) and Seaman et al. (School Sci Math 105(4):197–210, 2005). Changes in beliefs about mathematics and mathematics instruction were compared to a control group. Student reflection upon personal experience derived from participation in the activities was analyzed for formal and informal belief statements.
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Appendix: informal mathematics activities
Appendix: informal mathematics activities
1.1 Reflective mathematics activity #1: the nets of a cube
Background
In mathematics, a net is a connected two-dimensional figure that can be folded into a three-dimensional object. Nets are particularly powerful teaching tools because they help students to extend their knowledge about two-dimensional objects into notions of three-dimensional objects. The following examples are distinct (i.e., different) nets for the tetrahedron.
It should be noted that these two nets are called distinct because there is no way to transform one into the other by a “rigid motion”, i.e., a rotation (turn), translation (slide), or a reflection (flip). The following examples are not nets for the tetrahedron.
Problem statement
Your task in this reflective mathematics activity is to find all the distinct nets of the cube and then prove that no other nets of the cube exist. At the end of this activity, you will be asked to reflect on your personal experience of coming to understand this mathematical concept and what the experience “teaches you” about learning mathematics. Keep track of your strategies and procedures. Make note of your emotions and feelings, and be prepared to report your findings.
1.2 Reflective mathematics activity #2: inscribed angles of a circle
Background
Choose any three points A, B, and C on a circle with center D. Angle ABC is then an inscribed angle because the points which define it lie on the circle itself. Three examples of inscribed angles are shown below.
Of interest in this investigation is the relationship that exists between the inscribed angle ABC and the central angle ADC that subtends (contains) the same arc. The three examples above are again shown below each with the central angle included.
Problem statement
Your task in this reflective mathematics activity is to make a conjecture about the relationship between an inscribed angle and the central angle, which subtends the same are on any circle, and then prove that conjecture. At the end of this activity, you will be asked to reflect on your personal experience of coming to understand the mathematical concept and what the experience “teaches you” about learning mathematics. Keep track of your strategies and procedures Make note of your emotions and feelings and be prepared to report your findings.
1.3 Reflective mathematics activity #3: the Isis problem
The two fundamental mathematical concepts of two-dimensional figures are area and perimeter. In this investigation, we consider the area and perimeter of rectangles that have side lengths which are integers. Several examples of such rectangles are shown below.
Problem statement
Your tasks in this reflective mathematics activity are to find all rectangles with sides of integral length whose area and perimeter are numerically equal and then prove that there are no others. At the end of this activity, you will be asked to reflect on your personal experience of coming to understand this mathematical concept and what the experience “teaches you” about learning mathematics. Keep track of your strategies and procedures. Make note of your emotions and feelings and be prepared to report your findings.
1.4 Reflective mathematics activity #4: tessellation
A tessellation is an arrangement of two-dimensional figures that cover the entire plane without any overlaps or gaps. Tessellations are commonly found in mosaics, architectural designs, and tile work. An example of a tessellation is given below.
A regular tessellation of the plane is a tessellation that is made up of congruent regular polygons, which meet vertex to vertex such that every vertex arrangement is identical. We will prove in class that only three regular tessellations exist. A semi-regular tessellation of the plane is a tessellation that is made up of two or more congruent regular polygons, which meet vertex to vertex such that every vertex arrangement is identical. Notice that the example given above is a semi-regular tessellation of the plane, which is composed of squares and equilateral triangles.
Problem statement
Your task in this reflective mathematics activity is to find all semi-regular tessellations of the plane and prove that no others exist. At the end of this activity, you will be asked to reflect on your personal experience of coming to understand this mathematical concept and what the experience “teaches you” about learning mathematics. Keep track of your strategies and procedures. Make note of your emotions and feelings and be prepared to report your findings.
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Roscoe, M., Sriraman, B. A quantitative study of the effects of informal mathematics activities on the beliefs of preservice elementary school teachers. ZDM Mathematics Education 43, 601–615 (2011). https://doi.org/10.1007/s11858-011-0332-7
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DOI: https://doi.org/10.1007/s11858-011-0332-7