The paper reflects on mathematics education courses for K-12 teacher candidates through the lens of asymmetry enhanced by using various digital tools. It demonstrates both explicit and hidden presence of asymmetry in the topics of the courses and shows what knowledge of asymmetry as antithesis of symmetry brings to the study of mathematics for teaching. The topics include pizza sharing, equations and inequalities with parameters, generalized Golden Ratios, and Fibonacci-like polynomials.
Young children can be given symmetrical shapes (e.g., isosceles triangles and trapezoids, rectangles, rhombuses) to put them together and see whether the combination of two (or more) symmetrical figures (with at least one side length being equal among the shapes) retains the attribute of symmetry or not ( i.e. , becomes asymmetrical). They can trace a symmetrical figure on a piece of paper and then use paper folding, holes punching, or scissors to demonstrate the relationship between the two parts. Through these activities, one can learn that in the case of a single symmetrical figure, its partitioning in two figures through paper folding results in asymmetrical parts more often than in symmetrical ones.
Using the context of pizza sharing one can create an environment to deal with fractions visually without using any mathematics other than counting and measuring. In such an environment, dividing fairly m identical circular pizzas among n people, n > m , with the minimal number of slices using a pizza wheel can result in either mutually asymmetrical or symmetrical slices. Visualization allows one to see the results of division of pizzas through the lens of symmetry and asymmetry.
High school algebra provides classroom opportunities to talk about symmetry and asymmetry in the context of digital fabrication of the images of loci of inequalities with parameter. Connecting algebra to geometry allows one to learn how visual is controlled by symbolic and vice versa, how certain changes in geometric shapes create conditions for analytic description of their boundaries, and to appreciate the sensitivity of the interplay between symmetry and asymmetry in terms of analytics.
Many problems of science and engineering require knowledge of roots of one-variable polynomials and their location in the coordinate plane. Investigating quadratic equations with parameters enriches secondary mathematics teacher education courses with explorations redolent of real research experience in STEM (science, technology, engineering, mathematics) fields that teacher candidates need at least to know about. An example about the location of real roots of the trinomial x 2 +bx+c about an interval on the number line can demonstrate that in the ( c , b )-plane of parameters to the right of the vertical axis, a symmetrical outcome when the interval includes the smaller root only is very small in comparison with asymmetrical outcomes when the interval does not include the roots. One can observe bifurcation of symmetry into asymmetry (and vice versa) when the parabola y= x 2 +bx+c is symmetrical about the y -axis.
The entries of the classic Pascal’s triangle are symmetrical within each row. The top right-bottom left diagonals of the triangle consist, respectively, of ones, natural numbers, triangular numbers, triangular pyramidal numbers, pentatope numbers, and so on. An asymmetrical rearrangement of those numbers when the diagonal with ones forms the first column, the diagonal with natural numbers forms the second column shifted down about the first one by two rows, the diagonal with triangular numbers becomes the third column shifted down about the second one by two rows, the diagonal with triangular pyramidal numbers forms the fourth column shifted down about the third one by two rows, and so on, is consequential. The numbers appearing in each row of the rearranged Pascal’s triangle can serve as coefficients of one-variable polynomials (called Fibonacci-like polynomials) having all their roots located within the interval ( 4, 0). Furthermore, out of two polynomials of the same degree, one polynomial has symmetrical location of roots, and another polynomial has asymmetrical location of roots within the interval. For example, the numbers 1, 6, 10, 4 (with the sum equal to the 8 th Fibonacci number 21), form the polynomial f x = x 3 +6 x 2 +10x+4 the graph of which intersects the interval (−4, 0) three times in such a way that the smaller and the larger roots are symmetrical about the third root. At the same time, the numbers 1, 5, 6, 1 (with the sum equal to the 7 th Fibonacci number 13), form the polynomial f x = x 3 +5 x 2 +6x+1 the graph of which intersects the interval (−4, 0) three times as well, yet the points of intersection are not mutually symmetrical. These observations would not be possible without the use of digital technology.
The joint use of Wolfram Alpha, Maple, and a spreadsheet in the context of Fibonacci-like polynomials is critical for demonstrating asymmetry and symmetry of the roots of the polynomials and the introduction of generalized Golden Ratios as cycles in the form of strings of numbers. These findings although discussed in the context of mathematics education turned out to be new for the very discipline of mathematics. This points at the importance of using the modern-day computational tools in revealing to the learners of mathematics, including teachers of the subject matter as the major custodians of knowledge in general, the hidden presence of asymmetry inside the variety of mathematical structures.
This paper “From symmetry to asymmetry with digital tools in mathematics teacher education” was published in Asymmetry .
Abramovich, S. From symmetry to asymmetry with digital tools in mathematics teacher education. Asymmetry 2025(1):0002, https://doi.org/10.55092/asymmetry20250002.
Asymmetry
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From symmetry to asymmetry with digital tools in mathematics teacher education
19-Feb-2025