One Mile Wide and One Inch Deep: Giving the Secondary Mathematics Curriculum More Depth with Mathematica 

The author develops a series of strategies for giving students a deeper understanding of the concept of function.



by Kurt Peckman

The scrutiny of mathematics curricula in United States schools has recently intensified with the release of the Third International Mathematics and Science Study which concludes that the mathematical abilities of American students are "average," at best, when compared to those of students in other industrialized countries [1]. One reason often cited for this average status: The secondary mathematics curriculum found within the typical American school is too broad and lacks the depth necessary for mastery of mathematical concepts [2]. Thus, the focus of this paper is to suggest a Mathematica-based approach to secondary instruction that allows teachers to reverse the trend of "too broad, not enough depth" as they work through their curriculum. Of course, such an approach must be pedagogically sound as well as consistent with local, state, and national standards [3]—something that is very easy to accomplish via Mathematica.

Developing the Concept of Function

The Concept of Function as it is Typically Defined

A quick glance at just about any secondary math textbook utilized in the United States today easily demonstrates why students continue to struggle with the concept of function. More often than not, the notion of function is introduced by means of a convoluted definition, with foreign vocabulary, in a context that seems isolated from and irrelevant to many of the other mathematical topics that may be in the student's current cognitive map. For example, consider the following definitions of function taken from various classroom texts.

"A function f from a set X to a set Y is a correspondence that assigns to each element x of X a unique element y of Y. The element y is called the image of x under f and is denoted by f(x). The set X is called the domain of the function. The range of the function consists of all images of elements of X." [author's emphasis, 4]

"A function is a relationship in which there is only one value of the dependent variable for each value of the control variable." [author's emphasis, 5]

Using these definitions and others like them as introductory tools implies a readiness to learn [6] on the part of the student that is very premature. Specifically, the teacher who introduces the concept of function in this manner makes two inaccurate assumptions about his students' mathematical abilities. First, he assumes that students have the ability to immediately concretize key vocabulary words within the definition. For example, a domain must be immediately internalized as a set. Second, he assumes that students have the ability to immediately comprehend the relationship between each of these key words. For example, an image must be immediately internalized as the result of a function having been applied to some element in the domain! From a taxonomic point of view [7], this teacher assumes that students can instantaneously pass through developmental stages like knowledge and comprehension as they wrestle with concepts like "unique" and "correspondence."

At a much broader level, teaching mathematical concepts by first presenting definitions formally, then utilizing these definitions to determine if something meets a certain criteria, runs contrary to many theories currently accepted in educational psychology. From a learning modalities standpoint [8], such an approach assumes that a student can visualize abstract relationships in the mind's eye without the need for the kinesthetic manipulation of concrete objects. From a constructivist standpoint [9], such an approach does not allow the student to build, from the ground up, a personal understanding of the mathematics. However, if teachers abandon the traditional "state the definition, use the definition" approach to instruction and have the tools to expedite such a departure (viz. Mathematica), concepts like function can be introduced and studied in depth and in a pedagogically sound manner.

Allowing Students to Get Their Hands Dirty

At a very basic level, a function is a special relationship between sets of things. Another way to state this is to say that a "function describes how one quantity depends on another" [10] and that this dependence is unusual in some respect. For example, in any given year there would seem to be a relationship between the population of the United States and the tonnage of solid waste that population produces. Similarly, there would seem to be a relationship between the brightness of a light bulb and the wattage of that bulb. What makes such relationships "unusual" or special is that in any given year, say 1980, America produced just one amount of garbage: 151,500,000 tons [11]. That is, in 1980 the people of the U. S. did not produce both 151,500,000 and 200,000,000 tons of waste; it had to have been one or the other but not both amounts.

Using the previous examples and others like them as a vehicle for introducing functions represents a departure from the typical approach to mathematical instruction. For example, this approach causes the most important concepts of function to be pushed immediately to the forefront in the minds of the students; they immediately begin to view a function as a relationship with a uniqueness component. In addition, students have an easier time mentally constructing the concept of function; it is easier to envision the population of the United States collectively setting out their trash than it is to envision each element of some domain being mapped to a unique element in some range. Needless to say, such an approach alters the role of the teacher as, in these classrooms, they present students with interesting relationships and/or sets that lead to the internalization of function rather than static definitions that confuse.

Specifically, students using Mathematica can be presented with sets of things (e.g. a set of ordered pairs relating wattage of a bulb to its brightness) and asked a couple of key questions: Does a relationship exist between or amongst the elements of each set, and, if so, is there anything special about the relationship? Of course, words like "relationship" and "special" are, at this point, ambiguous and will be co-defined by the teacher and by the students later in the lesson [12]. But by using Mathematica to introduce the concept, students are forced to concentrate on the relationship and whether or not they deem it "unusual."

To demonstrate how Mathematica can be used to introduce the concept of function in this manner, consider the following lesson. Two sets of data are presented to the student in a Mathematica notebook along with the following question: "Within each set, is there any general rule or statement which relates the first of each ordered pair to the second?"

\(firstSet = {{\(-4\), 16}, {\(-3\), 9}, {\(-2\), 4}, {\(-1\), 1}, {0, 
          0}, {1, 1}, {2, 4}, {3, 9}, {4, 16}}; 


    secondSet = {{16, \(-4\)}, {9, \(-3\)}, {4, \(-2\)}, {1, \(-1\)}, {0, 
          0}, {1, 1}, {4, 2}, {9, 3}, {16, 4}}; \)

Notice that by labeling the collections as firstSet and secondSet, students are forced to consider each as just that—a collection of ordered pairs. When presented with collections, students are instructed to look for a pattern or relationship (if any) which is facilitated by organizing the ordered pairs into other formats.

TableForm[firstSet]

After students describe extant (or approximate) relationships and independent of the terminology and/or notation they use to do so, the teacher offers the following introductory statement: "The relationship found for firstSet can be thought of as a function whereas the relationship found for secondSet is not a function." Then, with the aid of Mathematica, students work to further construct and refine their own definition of function.

To begin a mental construction or refinement of function, the student might plot firstSet and secondSet on a coordinate grid. Doing so reinforces the conclusion that there is indeed a relationship between the elements as, intuition tells the student, such a pattern would never occur by chance. Plotting also helps students form a hypothesis or a justification as to why "firstSet is a function and secondSet is not."

ListPlot[firstSet, PlotStyle -> PointSize[0.02]];

ListPlot[secondSet, PlotStyle -> PointSize[0.02]];

When the points themselves are distracting or do not seem to yield too much information, connecting the dots helps to discern patterns which may or may not exist.

ListPlot[firstSet, PlotJoined -> True];

ListPlot[secondSet, PlotJoined -> True];

Although only two sets of data have been provided here, Mathematica allows teachers to create an infinite number of sets of ordered pairs for presentation, discussion, or student study. Then, as each set is generated, students are asked to: a) Describe the general relationship, if any, between the first and second elements within each set of ordered pairs, b) Visualize each relationship graphically or rewrite collections into other formats, c) Determine if relationships are "special" or not, and d) Explain and justify in words why a given relationship is or is not a function.

In such a Mathematica-based approach, many sound instructional practices are implemented with little discontinuity to the teacher's normal routine. For using this approach, the teacher implements instructional practices as outlined in the NCTM Professional Standards for Teaching Mathematics [13], limits the breadth of the material as outlined in the NCTM Curriculum and Evaluation Standards [14], introduces functions inductively, and generates concrete examples allowing students to approach the concept from an atypical yet effective cognitive vantage point—all with little extra preparation.

Taking such an approach even further, Mathematica's robust programming language can be used to create commands that provide feedback to students as they investigate sets of ordered pairs. For example, the following Mathematica command takes as its argument a set of ordered pairs and, using an alternate definition of function [15], checks to see if the set is indeed a function.

\(\(isIt[dataSet_List] := 
	
      With[{
			
          temp1 = Length[
              Union[Table[
                  Take[dataSet[\([i]\)], 1], {i, 1, Length[dataSet]}]]], 
          temp2 = Length[
              Union[Table[
                  Take[dataSet[\([i]\)], 2], {i, 1, Length[dataSet]}]]]}, 

        		Which[
			temp1 >= temp2, "Yes, it is a function.", 

          			temp1 < temp2, "No, it is not a function."]]; \)\)

It is interesting to note that even the syntax used to define isIt is valuable from an instructional standpoint. The use of mathematical words like Union and programming words like Length serves as the basis for an extremely valuable discussion in any math class (e.g. "How does this particular command determine what set is a function?"). However, if teachers do not or cannot devote time to the study of syntax, it is easy for this command to become a button effectively shielding students from Mathematica syntax as well as from the need to type the command name over and over again.

isIt[\[SelectionPlaceholder]]

isIt[firstSet]

isIt[secondSet]

newSet = {{1, 1}, {2, 2}, {3, 3}, {4, 4}};

isIt[newSet]

isIt[{{1, 1}, {1, 2}, {1, 3}}]

Of course, the above definition of isIt is not as elegant as one can construct with the Mathematica programming language. Rather, isIt has been constructed in such a way as to provide an intuitive mini-programming sampler to teachers who may be unfamiliar with Mathematica syntax. A more elegant definition of isIt, one that puts to use Mathematica's functional programming style, might look like isIt2 which has been defined below.

\(isIt2[dataSet_List] := 
	
      If[Apply[GreaterEqual, Map[Length[Union[#]]&, Transpose[dataSet]]], 
        "Yep, it's a function.", "Nope, not this one."]\)

isIt2[firstSet]

isIt2[{{-131, E}, {-132, π}, {-131, 
          ϕ}}]

In fact, all of the Mathematica functions used in this introductory lesson can be put into palette form, saved as a floating palette, placed in the palettes folder [16], and used with many classes and at many occasions.

({{isIt[\[SelectionPlaceholder]]}, {isIt2[\[SelectionPlaceholder]]}, {ListPlot[\[SelectionPlaceholder]]}})

In summary, Mathematica allows students to create an infinite number of sets of ordered pairs, to visualize these creations graphically, to determine perceived or extant relationships, to immediately differentiate functions from non-functions, and to document findings in a notebook for submission to an instructor or to classmates for discussion. Such an approach is consistent with Standards-based teaching, is exciting for the student, reduces the breadth of material presented while increasing the depth, facilitates comprehension of the concept by de-emphasizing the mechanics and vocabulary involved, and mimics the mathematical exploration necessary to skillfully attack future mathematical concepts and problems.

Introducing Functional Notation

Teachers who avoid rigorous definitions in general often introduce the concept of function as rules. An example of a rule might be to "take any number, multiply it by 2 then add 3." These rules are immediately put into symbols and students are asked to practice using them by creating output, or sets of input and output, for the purpose of graphing the sets on a coordinate plane. Consider the following example which represents a typical introduction to functions using this approach.

"Let f(x) = 2  x + 3 where f(2) means 2*2 + 3. Find f(-2) and f(4) . "

However, proceeding in such a manner asks students to focus not on the concept of function but rather on the mechanics of arithmetic. No thought has been given to the construction of the function nor to the mapping that is taking place; the majority of the students' attention is on performing the necessary calculations and on getting the correct output. If fact, as testimony to the trivial, mechanical nature of such an approach, the aforementioned questions can be answered using just 3 simple lines of Mathematica code.

f[x_] := 2  x + 3myTable = Table[{x, f[x]}, {x, -8, 8, 2}]ListPlot[myTable, PlotStyle -> PointSize[0.03], 
      PlotRange -> {{-10, 10}, {-20, 20}}];

Although such questions are rather low-level and not very thought-provoking, using Mathematica to answer them causes some very interesting things to occur from a cognitive standpoint. For example, the manner in which one typically defines a function within Mathematica forces contemplation of the meaning of f(x). That is to say, because functions that are set delayed [17] in Mathematica require the use of x_ and := within their definition, students are forced to carefully consider the role of f and that of x and the relationship between them [18]. Consequently, an interesting side effect to having students become more familiar with Mathematica syntax is to have comments like "I thought f(x) meant f times x" subside. Furthermore, by allowing Mathematica to perform the arithmetic, students are free to focus on the fact that f takes 1 to 5, thatf : 3⟶9, that f turns 10 into 23, and so on.

Ultimately using Mathematica to answer many low-level questions does not really address the Standards, does not utilize the potential of Mathematica, and is not very relevant to the real world. Teachers taking a slightly different approach can introduce students to functions (by means of functional notation) using intriguing, complex examples that force them to focus on the relationship between input (x) and output (y). Such an approach asks Mathematica to worry about arithmetic.

f[x_] := x - ((x^3)/6) + ((x^5)/120) - ((x^7)/5040) + ((x^9)/362880)

f[1]

N[f[1]]

f[2]

N[f[2]]

As noted earlier, the responsibility of the teacher shifts when introducing functions by means of Mathematica. No longer is the teacher's role to evaluate accuracy of output; rather the teacher's role is to emphasize the relationship between 1 and roughly 0.8 or to make statements like "2 is being mapped to (2578/2835)." The responsibility of Mathematica is to provide accurate output and to plot this output allowing students to focus on the relationship and thus construct the concept.

manyPoints = Table[{x, f[x]}, {x, 0, 7, 0.25}]

ListPlot[manyPoints, PlotRange -> {-1, 1}, 
      PlotStyle -> PointSize[0.02]];

By asking Mathematica to tend to the mechanical concerns, teachers and students are free to ask higher-level questions as well as to make higher level observations. For example, in looking at the graph of f(x) a legitimate question to pose would be: For what values of c does f(c) = 0? That is, what values get mapped to 0? An examination of the construction of manyPoints indicates a value of c somewhere between x = 3 and x = 3.25 .

Table[{x, f[x]}, {x, 3, 3.25, 0.05}]

Continuing this process subliminally exposes the students to the concept of continuity.

Table[{x, f[x]}, {x, 3.1, 3.15, 0.005}]

Because the Mathematica environment encourages exploration, it is conceivable that students stumble upon more sophisticated or easier ways to find such c's.

NSolve[f[c] == 0, c]

Having students work with functions and functional notation in this manner is very consistent with the NCTM Standards as it encourages active participation, true investigation, and thoughtful reflection from the student. For example, students who have studied trigonometry may recognize the graph of manyPoints as an approximation to a trigonometric function. This revelation could and should lead to other questions like "can rules be constructed that do a reasonable job approximating any function...like Log or Exp?" [19] or "Can one alter f(x) such that its tail comes back down?" Regardless of the questions that develop, students walk away from this type of introduction with a mathematical rather than an arithmetic understanding of function.

Seeing Cause and Effect

After students have been introduced to functions by means of formal definitions or functional notation, they are often given some special functions with which they are asked to translate, rotate, or scale. This is problematic as students quickly lose sight of the fact that they are working with functions and not just graphs or plots. Furthermore, such lack of contrast between functions and non-functions is reinforced throughout the career of secondary students as they are rarely asked to plot anything but functions. However, if students are introduced to functions in a manner to that which was outlined in previous sections of this paper, then Mathematica becomes the ideal environment for investigating the translation and scaling of special functions. Consider the palette below which allows students to focus on the relationship between the translation of a function and its "name."

({{Define g, move up 1 unit, move right 1 unit, compress 1 unit}, {Clear All, move down 1 unit, move left 1 unit, expand 1 unit}})

g[x_]:=x^2

Of course, a number of these plots could easily be strung together to create effective classroom animations or individual, self-paced tutorials.

Working with Composites

Composite functions are rich in both mathematical content and structure yet are often presented to secondary students at only a superficial level. One reason for this may be attributed to the fact that the study of composite functions requires, at the outset, a more-than-adequate understanding of function itself. Another reason may be that, within a traditional approach, the prerequisite dexterity with mechanical, arithmetic, and algebraic processes has not yet been met. Whatever the reason, the traditional approach is to construct composite functions using contrived, trivial functions thereby emphasizing procedural and mechanical concerns over conceptual ones.

Specifically, students are typically given very simple functions f(x) and g(x) and are asked to find f(g(z)) for some z ∈ ℤ. In this context, "simple" is defined as selecting f,  g,  and z such that they are relatively easy for the typical secondary student to manipulate from both an algebraic and an arithmetic standpoint. Consider the following which typifies a student's introduction to, and interaction with, composite functions.

"Given f(x) = x + 1 and g(x) = (x^2), calculate f(g(1))  and  g(f(-3))."

Such an approach diverts the students' attention away from the crucial idea of composite functions. Namely, that elements from a given set will be experiencing back-to-back mappings or rules, which, if one wishes, may be written as a single mapping in its own right. Therefore, instead of asking students to concentrate on mechanical concerns by providing them with linear and quadratic functions, studying composite functions within Mathematica allows students to work with more interesting functions that, in a sense, force them to ignore the mechanics and arithmetic and focus on the back-to-back nature of the mapping.

Needs["Graphics`Colors`"];

\(f[x_] := x\^3 - 3  x\^2 - 11  x + 4; 
g[x_] := x\^2 - 2  x + 3; 

    Plot[{f[x], g[x]}, {x, \(-4\), 6}, PlotStyle -> {Green, Blue}]; \)

Specifically, students using Mathematica can easily create and work with sophisticated functions that might otherwise be inaccessible. In this instance, the functions created are composite functions and the purpose for doing so includes learning how composites look in comparison to the original functions as well as identifying the properties of the originals that are retained. For example, using f(x) and g(x) from the previous example, let h(x) = (f + g) (x) and let m(x)  = (f∘g) (x).

Clear[h]\(h[x_] := g[x] + f[x]; 
h[x]\)

Once h(x) has been defined [20], it extremely instructive to plot all three on the same set of coordinate axes and to notice, for example, that h(-4) = f(-4) + g(-4),  
      h(0) = f(0) + g(0),  and that, in general, h(x) = f(x) + g(x) for any x chosen.

Plot[{f[x], g[x], h[x]}, {x, -4, 6}, 
      PlotStyle -> {Green, Blue, Red}];

Using the same approach (that of plotting all three functions on the same set of coordinate axes) with m(x) = f∘g(x) requires a little more effort on the part of the teacher. Here, the teacher must explicitly demonstrate, by pointing to the plot, that g(0) = 3,  that f(3) = -29, and that h(0) = f(g(0)) = f(3) = -29.

m[x_] := f[g[x]]

m[x]

Simplify[m[x]]

Plot[{f[x], g[x], m[x]}, {x, -4, 6}, 
      PlotRange -> {{-4, 6}, {-40, 20}}, 
      PlotStyle -> {Green, Blue, Red}, 
      Epilog -> {{AbsolutePointSize[8], Point[{0, g[0]}]}, {
            AbsolutePointSize[8], Point[{3, f[3]}]}, {AbsolutePointSize[8], 
            Point[{0, m[0]}]}}];

A major instructional goal of approaching the study of composite functions in such a way is to demonstrate to students that mastery of this concept will require more than the simple recall of arithmetic procedures. Specifically, mastery of this concept will require the ability to perform mechanical procedures as well as the ability to discuss underlying mathematical structure. Students should be held accountable for understanding that creating certain functions and grouping them as a "collection" causes some collections to exhibit behavior similar to that of other seemingly unrelated collections with which they are already familiar. While this may seem out of reach for the typical secondary math student, this discussion takes place effortlessly within Mathematica.

Consider the following example in which mathematical structure is emphasized by generating and studying a collection of related functions. For this discussion, composite is interpreted as "multiplication" and the additive and multiplicative inverses for f(x) = (x^3) - 3  (x^2) - 11  x + 4 are created and grouped together. The teacher then demonstrates that this collection (if supplemented by repeating this procedure using other polynomial functions) begins to look and act just like the set of rational numbers—a set whose every element has an additive and a multiplicative inverse [21].

f[x_] := (x^3) - 3  (x^2) - 11  x + 4

Solve[f[x] + y == 0, y]

negativef[x_] := -(x^3) + 3 (x^2) + 11 x - 4

f[x] + negativef[x]

Plot[{f[x], negativef[x]}, {x, -5, 5}, 
      PlotStyle -> {Green, Red}];

Needs["Miscellaneous`RealOnly`"];

Solve[f[y] == x, y]

oneOverf[x_] := 
      1 + ((14 ((((2/3)))^(1/3)))/(((
              81 + 9 x + (Sqrt[3]) (Sqrt[-8789 + 486 x + 27 (x^2)])))^(1/3
              ))) + ((((81 + 9 x + (Sqrt[3]) (Sqrt[-8789 + 486 x + 27 (x^2)]))
              )^(1/3))/((2^(1/3)) (3^(2/3))))

Simplify[f[oneOverf[x]]]

A better motivation and context for finding the inverse of a function appears in the next section.

Inverses of Functions

Within the typical chapter or unit on functions, students are usually given an overview of how to find the inverse of a given function. However, textbooks and teachers usually ask students to only find the inverses of linear functions. This is due to the fact that the process and mechanics involved in finding those of higher degree polynomials can be intimidating and out of reach for the typical secondary student. Consider the following which represents a typical textbook question.

"Given that f(x) = 2  x - 7,  
      find  g(x) = (f^(-1))(x) and verify your results."

To answer questions such as these, students are usually directed to switch the role of the independent and dependent variables (in the aforementioned example, x and y respectively) and then to isolate, or solve, for the "new" independent variable.

\(Clear[f, g]; 
f[x_] := 2  x - 7\)Solve[f[y] == x, y]

g[x_] := ((7 + x)/2)

f[g[x]]

g[f[x]]

Even though students are able to calculate the inverse and verify their findings by demonstrating that f(g(x)) = g(f(x)) = x, many cannot fully explain why this is the case. That is, many know "how" but cannot explain "why." However, as alluded to in previous sections of this paper, Mathematica can help students answer the question of why by providing a forum for investigation and exploration devoid of mechanical concerns.

f[x]

g[x]

elems = Table[i,  {i, -5, 5}]

f[elems]

g[elems]

f[g[elems]]

f[g[elems]] == elems

By generating multiple examples, students begin to truly understand why f(g(x)) = g(f(x)) = x. In the example provided, the student sees that f : 5⟶3, g : 3⟶5, and f∘g : 5⟶5. This is no coincidence as f : 2⟶-3,  
      g : -3⟶2,  and f∘g : 2⟶2. Thus, students are able to take a given function, immediately find its inverse, study the relationship between the two using values or sets of values, and explain it to others—all without concern for output accuracy [22]. This, in turn, helps students to understand and explain why the plots of f(x) = 2  x - 7 and  g(x) = (f^(-1))(x) look the way they do.

Plot[{f[x], g[x], f[g[x]]}, {x, -5, 10}, 
      PlotStyle -> {Green,  Red, Black}, AspectRatio -> Automatic];

For those comfortable with the fact that \(TraditionalForm
      \`f(\(f\^\(-1\)\)(x)) = \(\(f\^\(-1\)\)(f(x)) = x\)\) , Mathematica allows these students to apply a traditional line of questioning to more interesting, realistic functions. For example: Given h(t) = -16  (t^2) + 10  t + 5,  
      find (h^(-1))(t), verify the results, and explain what you found.

\(Clear[d, h, t, time, x]; 
h[t_] := \(-16\)\ *t\^2 + 10  t + 5\)

Solve[h[t] == d, t]

time[d_] := (1/16) (5 + (Sqrt[105 - 16 d]))

Simplify[h[time[ x]]]

Needs["Graphics`ImplicitPlot`"];

ImplicitPlot[{h[t] == d, h[d] == t, t == d}, {t, 0, 6.6}, 
      PlotStyle -> {Green, Red, Black}, PlotRange -> {{0, 8}, {0, 8}}, 
      AspectRatio -> Automatic];

Of course, defining inverse functions requires some attention to detail and harkens back to an initial discussion as outlined in an earlier section of this paper.

Concluding comments

If secondary math teachers step back to objectively consider what typical students are asked to learn while studying functions, the realizations are quite sobering. The typical math student in a traditional math curriculum is expected to learn, understand, and remember the following:

	-Definition(s) of function, 
	-Function vocabulary (such as "function", "domain", "range"),
	-How to plot functions,
	-Many types of functional notation (such as f(x),  (f^(-1)),  f : x -> f(x)),
	-How to find the inverses of functions (and verify them),
	-How to find composite functions (such as \(TraditionalForm
      \`\(f\ ◦\ g, \ f(g(x)), \(f\^\(-1\)\)(f(x)))\)\),
	-How to work with and identify special functions (such as f(x) = ⌈x⌉), and
	-How changing various parameters of functions affect their plots, values, and the like.
	
Ironically, such a unit is usually slated to be completed in 10-15 days! Such realizations lead to the following (loaded) question: should math teachers be surprised when students leave math classrooms with little-to-no understanding of functions?

What has been proposed in this paper is an instructional approach to functions that strives to improve retention and ability without sacrificing content. This is accomplished primarily by de-emphasizing the procedural and emphasizing the structural within the Mathematica environment. To implement such an approach requires just four things: a copy of Mathematica, a copy of the NCTM Standards, the sophistication to realize that the current mathematics curriculum is "one mile wide and one inch deep", and the willingness to do something about it.

References & Footnotes

1. U. S. Department of Education. National Center for Education Statistics, Pursuing Excellence: A Study of U. S. Twelfth-Grade Mathematics and Science Achievement in International Context, NCES 98-049. Washington, DC: U. S Government Printing Office, 1998. 

Copies of this study as well as a wealth of information about it can be downloaded (in .pdf format) from the National Center for Educational Statistics website at www.nces.ed.gov.

2. The phrase "one mile wide, one inch deep" has been used by many to describe the survey approach taken by many secondary mathematics curricula in the United States. However, the origin of this phrase has is not very well documented.

3. For example, many school districts in Illinois must create and administer district-level, standardized tests in addition to the mandatory administration of the Illinois Goal and Placement (IGAP) exams. The former is typically an assessment of district/local goals whereas the latter is an assessment of the Illinois State Goals which are loosely based upon national policy statements like the NCTM Standards.

4. Swokowski, Earl W., Fundamentals of College Algebra, Fourth Edition, Boston: Prindle, Weber & Schmidt, Inc., 1979, p. 114.

5. Rubenstein, Rheta N., Timothy V. Craine, and Thomas R. Butts, Integrated Mathematics, Boston: Houghton Mifflin Company, 1995, p. 220.

6. "Readiness to learn" is an important component to the theory of learning put forth by Jerome Bruner. The concept, and others like it, has been used by some to evaluate instructional technique. See, for example, Bruner's The Process of Education published in 1960 by Harvard University Press.

7. Gage, N. L. and David C. Berliner, Educational Psychology, Boston: Houghton Mifflin Company, 1988, pp. 42-46.

Bloom's Taxonomy is a classification system that allows teachers to create and evaluate both content and classroom interaction from a cognitive standpoint. The levels of this taxonomy–knowledge, comprehension, application, analysis, synthesis, evaluation–begin with a student's ability to recall facts and information and work towards having the student use, integrate, and evaluate facts and information and information in the creation of original work.

8. Gage, N. L. and David C. Berliner, Educational Psychology, Boston: Houghton Mifflin Company, 1988, p. 76.

Many researchers like Howard Gardner (and his theory of multiple intelligences) assert that the channel through which information is processed plays an extremely important role in learning. Thus, mathematics classrooms conducted in tradtional lecture format give visual and auditory learners considerable advantage over kinesthetic learners or those with a high kinesthetic intelligence.

9. A widely accepted theory of learning is Piaget's Constructivism. In general terms, this theory states that students do not and can not learn mathematical truths and rules by simple memorization. Rather, they must construct a personal copy of the truths and rules based upon their experience and interaction with them.

10. Best, George, David Penner, and Robert Schneider, Exploring Algebra, Precalculus, and Statistics with the TI-83 Graphing Calculator, Andover, Mass.: Venture Publishing, 1997.

11. U. S. Bureau of the Census, Statistical Abstract of the United States: 1995 (115th Edition) Washington DC, 1995.

12. Of course, the teacher's mission here is to steer the students towards a semi-rigorous definition; one in which the students themselves have had a hand in creating.

13. National Council of Teachers of Mathematics, Professional Standards for Teaching Mathematics, Reston, Virginia: National Council of Teachers of Mathematics, 1991.

14. National Council of Teachers of Mathematics, Curriculum and Evaluation Standards for School Mathematics, Reston, Virginia: National Council of Teachers of Mathematics, 1989.

15. For example: "A function with domain X is a set W of ordered pairs such that for each x in X, there is exactly one ordered pair (x, y) in W having x in the first position." See footnote 4 above.

16. Access to customized palettes via the pull-down menus can be achieved if active palettes are placed within the "Palettes" folder using the following path: Mathematica->3.x->SystemFiles->FrontEnd->Palettes. Of course, if one is going to use a customized palette, care must be taken in constructing and evaluating the cells that initialize any user-created commands.

17. As one reviewer pointed out, Mathematica's pattern-matching ability makes it possible to define functions in more than one manner. Specifically, f(x) can be defined as f[x_] = (x^2). However, there is a very subtle distinction between a function that has been defined using SetDelayed to that which has been defined using only Set (f[x_] := (x^2) and f[x_] = (x^2), respectively); SetDelayed does not evaluate the right hand side of a definition.

For example, in the first definition below x has been "set" to π and f(x) has been "set" to (x^2). Thus, the pattern "f(anything)" will be assigned the value of (π^2). Because the equal sign was used in defining f, Mathematica immediately evaluates the right hand side of f using what it already knows about x. In the second definition below, x has been "set" to π as before but the definition of f has been delayed. That is, Mathematica defers the evaluation of the right hand side of f until it runs across the pattern "f(anything)" at which time it takes the argument "anything," and squares it.

\(Clear[a, f, x]; 
x = π; 
f[x_] = x\^2\)

f[5]

f[a]

\(Clear[a, f, x]; 
x = π; 
f[x_] := x\^2; \)

f[5]

f[a]

18. Because it is much more precise, a recent Wolfram Research Visiting Scholar admitted to using Mathematica syntax (e.g. f[x]) when writing mathematics on the chalkboard.

19. The fact that a function with a finite number of terms can be constructed, with arbitrary precision, to "look like" a transcendental function is a truly powerful concept. However, this powerful and remarkable fact often takes a backseat to the mechanics of such calculations.

20. In this example, Mathematica has expressed h(x) using its default output style StandardForm where \(TraditionalForm
      \`h(x) = a\_0 + \(a\_1\) x\^1 + \(a\_2\) x\^2 + \(a\_3\) x\^3\). Math teachers who claim that Mathematica has written h(x) in the "wrong order" can easily convert the expression to TraditionalForm using the pull down menu item Cell->Convert To->TraditionalForm. By using this menu item, Mathematica expresses h(x) in "math class form" where \(TraditionalForm
      \`h(x) = \(a\_3\) x\^3 + a\_\(2\ \)\ x\^2 + a\_1\ x\^1 + a\_0\).

21. Of course, one must be careful with such generalities as there has been no mention of discontinuous functions or the need to remain on some fixed interval like 𝕀 = (0, 1].

22. Of course, accuracy of output is important in some instances, just not in this one.