Unpacking foreshadowing in mathematics teachers’ planned practices

Abstract

This paper provides an empirical exploration of mathematics teachers’ planned practices. Specifically, it explores the practice of foreshadowing, which was one of Wasserman’s (2015) four mathematical teaching practices. The study analyzed n = 16 lessons that were planned by pairs of highly qualified and experienced secondary mathematics teachers, as well as the dialogue that transpired, to identify the considerations the teachers made during this planning process. The paper provides empirical evidence that teachers engage in foreshadowing as they plan lessons, and it exemplifies four ways teachers engaged in this practice: foreshadowing concepts, foreshadowing techniques, foregrounding concepts, and foregrounding techniques. Implications for mathematics teacher education are discussed.

Appendix. Study task prompts

1.1 Prompt 1: Distance

You are teaching a middle/high school class and are about to move into a unit on coordinate geometry. The aim of your lesson today is for students to be able to understand and apply the distance formula, $d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$, to find the distance between two points, $({x}_{1}, {y}_{1})$ and $({x}_{2}, {y}_{2}).$ Your students have never seen the distance formula, but they do know how to plot points $(x, y)$ on a coordinate plane and they have learned the Pythagorean theorem, including using square roots. Design a task—or sequence of shorter tasks—for students to perform to help them learn about the distance formula.

1.2 Prompt 2: Quadratic

You are teaching a high school class and are working with students on graphing quadratic functions—in particular, graphing functions from their factored form. The aim of your lesson today is for students to connect the zeros of the parabola with the function’s factored form. At this point, students already know how to factor quadratic equations (of the form $a{x}^{2}+bx+c$) and how to solve linear equations ($ax+b=0$). They have also recently graphed quadratic functions of the form $f\left(x\right)=a{x}^{2}+bx+c$ and learned about how the parameter $a$ effects the graph (skinnier/wider) and how the parameter $c$ effects the graph ($y$-intercept/vertical shift). Design a task—or sequence of shorter tasks—for students to perform to help them learn about graphs, zeros/roots, and different forms of quadratic functions.

1.3 Prompt 3: Area

You are teaching a middle/high school class and are working with students on justifying area formulas. The aim of your lesson today is for students to understand and justify the formula for the area of a parallelogram, $A=bh$. At this point, students know the area formula for a rectangle—and, for the most part, they understand that it quantifies the number of square units. They have also been briefly introduced to finding the area of a parallelogram, but have not seen many different cases and cannot always justify their answer. Design a task—or sequence of shorter tasks—for students to perform to help them better understand the area formula for parallelograms and be able to justify it more completely.

1.4 Prompt 4: Volume

You are teaching a middle/high school class and are working with students on determining the volume of a prism. (A prism has two end faces of similar and equal shape whose sides are parallelograms.) The aim of your lesson today is for students to understand the formula for the volume of a prism, $V=bh$. At this point, students know the basic vocabulary of polygons and are familiar with calculating the areas of all types of quadrilaterals, but they have not yet been exposed to calculating volumes for three-dimensional figures. Design a task—or sequence of shorter tasks—for students to perform to help them understand the formula for the volume of a prism.

1.5 Prompt 5: Trig

You are teaching a high school pre-calculus class and are working with students on solving trigonometric equations. The aim of your lesson today is for students to understand how to solve equations of the form, $A\;\sin(Bx)=C$ (and also for cosine or tangent functions). At this point, students are fluidly solving equations with any of the arithmetic operations (e.g., multiplication), they know the unit circle and the graphs of the trigonometric functions, and may recollect some of the transformations from polynomial functions. Students have recently been introduced to the inverse trigonometric functions (e.g., arcsin, arccos, arctan). Design a task—or sequence of shorter tasks—for students to perform to help them understand how to solve trigonometric equations of the form $A\mathrm{sin}(Bx)=C$.

1.6 Prompt 6: Exponent

You are teaching a high school class and are preparing to begin a unit on exponents. The aim of your lesson today is for students to learn about the three main exponent properties—${a}^{b}\cdot {a}^{c}={a}^{b+c}$, ${\left({a}^{b}\right)}^{c}={a}^{bc}$, and $\frac{{a}^{b}}{{a}^{c}}={a}^{b-c}$. At this point, students understand that exponents are repeated multiplication (i.e., $3\cdot 3\cdot 3\cdot 3={3}^{4}$) but are unfamiliar with how to handle a zero exponent (${3}^{0}$), a negative exponent (${3}^{-5}$), or a fractional exponent (${3}^{1/2})$. Design a task—or sequence of shorter tasks—for students to perform to help them learn about exponents, including the three main exponent properties.

1.7 Prompt 7: Circumference

You are teaching a middle/high school class and are working with students on determining the circumference of a circle. The aim of your lesson today is for students to understand the formula for the circumference of a circle, $C=\pi d$ and/or $C=2\pi r$. At this point, students know the basic vocabulary of circles (e.g., diameter), but they do not have a sense of how to determine the circumference. They know how to determine the perimeter of most polygons, including squares ($4s$), rectangles ($2l+2w$), and non-regular polygons (by summing the side lengths). Although all have heard of $\pi$, and some know its approximate value, they do not know much else. Design a task—or sequence of shorter tasks—for students to perform to help them understand the formula for the circumference of a circle.

1.8 Prompt 8: Rational

You are teaching a high school class, and are working with students on simplifying rational expressions. The aim of your lesson today is for students to understand how to simplify various rational expressions (e.g.,  $\frac{{x}^{2}-9}{2x+6}$, $\frac{4x}{{x}^{2}-3x}$ ÷ $\frac{{x}^{2}}{{x}^{2}-9}$) and what the simplified expression represents. At this point, students are fluent in factoring algebraic expressions and understanding the operations—they know that division is equivalent to multiplying by the reciprocal and that a common denominator is necessary for addition or subtraction. Design a task—or sequence of shorter tasks—for students to perform to help them understand simplifying rational expressions.

1.9 Prompt 9: Rate

You are teaching a middle/high school class, and are working with students about real-world applications that have to do with rates of change—particularly constant rates of change. The aim of your lesson today is to engage students with a real-world application to help them understand how constant rates of change are evident within a table of values, and to make connections to attributes in other representations of a function (e.g., graphs, equations). At this point, students are familiar with slope, graphing linear equations, and different representations of functions (e.g., graphs, equations, tables). Design a task—or sequence of shorter tasks—for students to do to help them understand rates of change within tables, and their relationship to the other representations.