Unconscious assumptions in mathematical examples
Collaborative didactical project.
The aim of the project is raising awareness concerning common practices in teaching high-school mathematics.
In mathematics we are supposed to make "general" examples: the example should better not have additional properties which in general do not hold. For example if we draw a "generic" parallelogram, we wouldn't draw a rectangle. (However, it is also important to present the rectangle as a special example of parallelogram.)
Of course one tries to pick nice examples for simplicity, for example if one considers a quadratic equation with real numbers as coefficients, then one opts for some nice-looking small integers rather than for horribly looking irrational numbers... What I suggest in this case is however that the teacher makes also one ugly example to give an idea of how the generic case truly looks like.
I am making a list of things which are so common that they are almost taken for granted. I am not saying that this is bad practice, but the pupils should see at least one occurrence in which the additional assumption is not satisfied, or an exception to the common practice. To broaden the mind, I would say. Also because the pupils may feel uncomfortable outside the common setting (which has become their comfort zone)...
- A triangle usually has a horizontal side and lies in the upper half-plane with respect to this side. People often take the horizontal side to be the "base", so that the "height" is measured vertically: this limits the effectiveness of the popular area formula, A=bh/2.
For example, place a 3-4-5 right triangle with a horizontal hypotenuse. What's its area? If you say the base is 5, then you have a little puzzle ahead as you try to figure out the height h relative to that base. But if you are comfortable taking another point of view, you might say 3 is the base, so the height is 4 and the area is 6. This is actually one of the easiest ways to compute that mystery height from before: since 5h/2=6, the unknown height is 12/5. Consider this applet to explore triangles. Entry by Brad Ballinger and Antonella Perucca
- Triangles are never skinny (who has seen a triangle in which the basis is 100 times the altitude, although it would be possible to display it with a computer (by scrolling)?).
There is a similar issue for other figures, for example parallelograms (and this causes problems because if one draws a parallelogram which looks like a rhombus one is led to think that the diagonals are angle bisectors, which is evidently false for a non-skinny parallelogram). Speaking of parallelograms, they usually have an horizontal side and lean towards the right. Entry by Antonella Perucca
- Another example of unconscious assumptions: at the lowest van Hiele level, a student might identify a square with vertical and horizontal sides as a square; but if you rotate it by 45 degrees, it no longer looks like a square, so it isn't identified as one (more likely as a "diamond"). They probably learn this by looking at examples drawn by people who know, but choose not to express, that squares come in different orientations. In a similar fashion, some people would have a hard time working with a rhombus which is displayed "as a parallelogram" so that one side is horizontal (in the common presentation "as a diamond" the diagonals are vertical and horizontal). Entry by Brad Ballinger and Antonella Perucca
- There is also the issue of not recognizing special cases as belonging to the definition. For example, we never refer to a square as a rectangle. What if, when we draw a rectangle, sometimes it's a square? (It happens that children are judged not to identify shapes correctly if, when asked to pick out all the rectangles, they pick the squares too.) Entry by Debra Seidell
- Wanted: right triangles with irrational values for sides. Easy examples are those with sides a=p^(1/4)-1, b=2p^(1/8), and c=p^(1/4)+1.
Entry by Jane Gillette and José Hernández
Analytic Geometry, Area/Perimeter
- With coordinate plane geometry, one works in the first quadrant. Entry by Antonella Perucca
- There are math problems that start out by saying something like "the area of a square equals its perimeter". This is impossible, since area and perimeter are measured in different units. For example, a square with a side length of 4 meters has a perimeter of 16 meters and an area of 16 square meters. I guess that means the area and perimeter are equal? But the same square measured in different units has a perimeter of 1600 cm and an area of 160000 square cm. Perhaps the unconscious assumption is that units have been chosen in such a way that the area (in the chosen area unit) equals the perimeter (in the chosen perimeter unit). Entry by Brad Ballinger
Polynomials, Equations and Functions
- Missing: polynomials with irrational coefficients; polynomials with complex and non-real coefficients (warning: some textbook defines polynomials as having only rational coefficients). Entry by Jane Gillette
- Two of my big ones in Algebra 1 are linear equations and systems of equations with integer solutions. Some of my students get so used to problems with integer solutions that they think that if a solution comes out as a fraction either a) there must be something wrong, or b) it has no solutions. Entry by David Srebnick
- Missing: polynomials in quadratic form that are other than 4th, 6th, and 8th degree. For example, x+ 5*sqrt(x) +6. Entry by Jane Gillette
- We often teach using the variables x and y, where y represents a function of x. We "know" that the horizontal axis is called x (and the vertical, y). Sometimes we use other variables, but we almost never see x be the variable that depends on y, except perhaps in a discussion about inverse functions; nor do we see the dependent variable graphed on the horizontal axis. We can make sense of supply/demand curves by thinking that they showed the independent variable p (price) on the vertical axis, and dependent variables for supply or demand on the horizontal.
(Note: we wouldn't describe the positions of x, y, z axes in 3D graphs as unconscious assumptions, because often these are discussed explicitly.) Entry by Brad Ballinger
- Brad Ballinger, Professor at Humboldt State University, California, USA.
- Jane Gillette, Self-Employed Math Tutor, Teacher-Author TeachersPayTeachers, www.JaneGillette.com, Massachusetts, USA.
- José Hernández
- Antonella Perucca, Professor at the University of Luxembourg, Luxembourg.
- Debra Seidell, Malden, Massachusetts, USA.
- David Srebnick, Math Teacher, Eighth Grade Head Advisor, Acton, Massachusetts, USA.
This is a collaborative didactical project with various contributors, started in March 2021 by Antonella Perucca and currently ongoing. This web-page is regularly updated, for questions or comments please email email@example.com.