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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 [", ButtonBox["2", ButtonData:>"footnote2", ButtonStyle->"Hyperlink"], "]. Thus, the focus of this paper is to suggest a ", StyleBox["Mathematica-", FontSlant->"Italic"], "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 [", ButtonBox["3", ButtonData:>"footnote3", ButtonStyle->"Hyperlink"], "]\[LongDash]something that is very easy to accomplish via ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell["Developing the Concept of Function", "Subsection"], Cell[CellGroupData[{ Cell["The Concept of Function as it is Typically Defined", "Subsubsection"], Cell["\<\ 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. \ \>", "Text"], Cell[TextData[{ "\"A ", StyleBox["function", FontWeight->"Bold"], " ", StyleBox["f", FontSlant->"Italic"], " ", "from a set ", StyleBox["X", FontSlant->"Italic"], " to a set ", StyleBox["Y", FontSlant->"Italic"], " is a correspondence that assigns to each element ", StyleBox["x ", FontSlant->"Italic"], "of ", StyleBox["X", FontSlant->"Italic"], " a unique element ", StyleBox["y", FontSlant->"Italic"], " of ", StyleBox["Y", FontSlant->"Italic"], ". The element ", StyleBox["y", FontSlant->"Italic"], " is called the ", StyleBox["image", FontWeight->"Bold"], " of ", StyleBox["x", FontSlant->"Italic"], " under ", StyleBox["f", FontSlant->"Italic"], " and is denoted by ", StyleBox["f", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "). The set ", StyleBox["X", FontSlant->"Italic"], " is called the ", StyleBox["domain", FontWeight->"Bold"], " of the function. The ", StyleBox["range", FontWeight->"Bold"], " of the function consists of all images of elements of ", StyleBox["X", FontSlant->"Italic"], ".\" [author's emphasis, ", ButtonBox["4", ButtonData:>"footnote4", ButtonStyle->"Hyperlink"], "]" }], "Text", CellTags->"def of function"], Cell[TextData[{ "\"A ", StyleBox["function", FontWeight->"Bold"], " is a relationship in which there is ", StyleBox["only one", FontSlant->"Italic"], " value of the dependent variable for each value of the control variable.\" \ [author's emphasis, ", ButtonBox["5", ButtonData:>"footnote5", ButtonStyle->"Hyperlink"], "]" }], "Text", CellTags->"def of function"], Cell[TextData[{ "Using these definitions and others like them as introductory tools implies \ a readiness to learn [", ButtonBox["6", ButtonData:>"footnote6", ButtonStyle->"Hyperlink"], "] 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 [", ButtonBox["7", ButtonData:>"footnote7", ButtonStyle->"Hyperlink"], "], this teacher assumes that students can instantaneously pass through \ developmental stages like knowledge and comprehension as they wrestle with \ concepts like \"unique\" and \"correspondence.\"" }], "Text"], Cell[TextData[{ "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 [", ButtonBox["8", ButtonData:>"footnote8", ButtonStyle->"Hyperlink"], "], 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 [", ButtonBox["9", ButtonData:>"footnote9", ButtonStyle->"Hyperlink"], "], 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 (", StyleBox["viz", FontSlant->"Italic"], ". ", StyleBox["Mathematica", FontSlant->"Italic"], "), concepts like function can be introduced and studied in depth and in a \ pedagogically sound manner. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Allowing Students to Get Their Hands Dirty", "Subsubsection"], Cell[TextData[{ "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\" [", ButtonBox["10", ButtonData:>"footnote10", ButtonStyle->"Hyperlink"], "] 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", StyleBox[" ", FontWeight->"Bold", FontSlant->"Italic"], "just one amount of garbage: 151,500,000 tons [", ButtonBox["11", ButtonData:>"footnote11", ButtonStyle->"Hyperlink"], "]. 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." }], "Text"], Cell["\<\ 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.\ \>", "Text"], Cell[TextData[{ "Specifically, students using", StyleBox[" Mathematica", FontSlant->"Italic"], " 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 [", ButtonBox["12", ButtonData:>"footnote12", ButtonStyle->"Hyperlink"], "]. But by using ", StyleBox["Mathematica ", FontSlant->"Italic"], "to introduce the concept, students are forced to concentrate on the \ relationship and whether or not they deem it \"unusual.\"" }], "Text"], Cell[TextData[{ "To demonstrate how ", StyleBox["Mathematica", FontSlant->"Italic"], " 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 ", StyleBox["Mathematica", FontSlant->"Italic"], " 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?\" " }], "Text"], Cell[BoxData[ \(firstSet = {{\(-4\), 16}, {\(-3\), 9}, {\(-2\), 4}, {\(-1\), 1}, {0, 0}, {1, 1}, {2, 4}, {3, 9}, {4, 16}}; \n\n secondSet = {{16, \(-4\)}, {9, \(-3\)}, {4, \(-2\)}, {1, \(-1\)}, {0, 0}, {1, 1}, {4, 2}, {9, 3}, {16, 4}}; \)], "Input", InitializationCell->True], Cell[TextData[{ "Notice that by labeling the collections as ", StyleBox["firstSet", "Input"], " and ", StyleBox["secondSet", "Input"], ", students are forced to consider each as just that\[LongDash]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." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TableForm[firstSet]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {\(-4\), "16"}, {\(-3\), "9"}, {\(-2\), "4"}, {\(-1\), "1"}, {"0", "0"}, {"1", "1"}, {"2", "4"}, {"3", "9"}, {"4", "16"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "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:", StyleBox[" ", FontWeight->"Bold"], "\"The relationship found for ", StyleBox["firstSet", "Input"], " can be thought of as a function whereas the relationship found for ", StyleBox["secondSet", "Input"], " is not a function.\" Then, with the aid of ", StyleBox["Mathematica, ", FontSlant->"Italic"], "students work to further construct and refine their own definition of \ function." }], "Text"], Cell[TextData[{ "To begin a mental construction or refinement of function, the student \ might plot ", StyleBox["firstSet", "Input"], " and ", StyleBox["secondSet", "Input"], " 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. 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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." }], "Text"], Cell[TextData[{ "In such a ", StyleBox["Mathematica-", FontSlant->"Italic"], "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 ", StyleBox["Professional Standards for Teaching Mathematics ", FontSlant->"Italic"], "[", ButtonBox["13", ButtonData:>"footnote13", ButtonStyle->"Hyperlink"], "], limits the breadth of the material as outlined in the NCTM ", StyleBox["Curriculum and Evaluation Standards ", FontSlant->"Italic"], "[", ButtonBox["14", ButtonData:>"footnote14", ButtonStyle->"Hyperlink"], "], introduces functions inductively, and generates concrete examples \ allowing students to approach the concept from an atypical yet effective \ cognitive vantage point\[LongDash]all with little extra preparation." }], "Text"], Cell[TextData[{ "Taking such an approach even further, ", StyleBox["Mathematica'", FontSlant->"Italic"], "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 ", StyleBox["Mathematica", FontSlant->"Italic"], " command takes as its argument a set of ordered pairs and, using an \ alternate definition of function [", ButtonBox["15", ButtonData:>"footnote15", ButtonStyle->"Hyperlink"], "], checks to see if the set is indeed a function. " }], "Text"], Cell[BoxData[ \(\(isIt[dataSet_List] := \n\t With[{\n\t\t\t temp1 = Length[ Union[Table[ Take[dataSet[\([i]\)], 1], {i, 1, Length[dataSet]}]]], temp2 = Length[ Union[Table[ Take[dataSet[\([i]\)], 2], {i, 1, Length[dataSet]}]]]}, \n \t\tWhich[\n\t\t\ttemp1 >= temp2, "\", \n \t\t\ttemp1 < temp2, "\"]]; \)\)], "Input", InitializationCell->True], Cell[TextData[{ "It is interesting to note that even the syntax used to define ", StyleBox["isIt", "Input"], " is valuable from an instructional standpoint. The use of mathematical \ words like ", StyleBox["Union", "Input"], " and programming words like ", StyleBox["Length", "Input"], " 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 ", StyleBox["Mathematica", FontSlant->"Italic"], " syntax as well as from the need to type the command name over and over \ again." }], "Text"], Cell[BoxData[ ButtonBox[\(isIt[\[SelectionPlaceholder]]\), ButtonStyle->"Paste"]], "Input", Active->True], Cell[CellGroupData[{ Cell[BoxData[ \(isIt[firstSet]\)], "Input"], Cell[BoxData[ \("Yes, it is a function."\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(isIt[secondSet]\)], "Input"], Cell[BoxData[ \("No, it is not a function."\)], "Output"] }, Open ]], Cell[BoxData[ \(\(newSet = {{1, 1}, {2, 2}, {3, 3}, {4, 4}}; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(isIt[newSet]\)], "Input"], Cell[BoxData[ \("Yes, it is a function."\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(isIt[{{1, 1}, {1, 2}, {1, 3}}]\)], "Input"], Cell[BoxData[ \("No, it is not a function."\)], "Output"] }, Open ]], Cell[TextData[{ "Of course, the above definition of ", StyleBox["isIt", "Input"], " is not as elegant as one can construct with the ", StyleBox["Mathematica ", FontSlant->"Italic"], "programming language. Rather, ", StyleBox["isIt", "Input"], " has been constructed in such a way as to provide an intuitive \ mini-programming sampler to teachers who may be unfamiliar with ", StyleBox["Mathematica", FontSlant->"Italic"], " syntax. A more elegant definition of ", StyleBox["isIt", "Input"], ", one that puts to use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s functional programming style, might look like ", StyleBox["isIt2", "Input"], " which has been defined below." }], "Text"], Cell[BoxData[ \(isIt2[dataSet_List] := \n\t If[Apply[GreaterEqual, Map[Length[Union[#]]&, Transpose[dataSet]]], "\", "\"]\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(isIt2[firstSet]\)], "Input"], Cell[BoxData[ \("Yep, it's a function."\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(isIt2[{{\(-131\), \[ExponentialE]}, {\(-132\), \[Pi]}, {\(-131\), \[Phi]}}]\)], "Input"], Cell[BoxData[ \("Nope, not this one."\)], "Output"] }, Open ]], Cell[TextData[{ "In fact, all of the ", StyleBox["Mathematica", FontSlant->"Italic"], " functions used in this introductory lesson can be put into palette form, \ saved as a floating palette, placed in the palettes folder [", ButtonBox["16", ButtonData:>"footnote16", ButtonStyle->"Hyperlink"], "], and used with many classes and at many occasions." }], "Text"], Cell[BoxData[GridBox[{ { ButtonBox[\(isIt[\[SelectionPlaceholder]]\)]}, { ButtonBox[\(isIt2[\[SelectionPlaceholder]]\)]}, { ButtonBox[\(ListPlot[\[SelectionPlaceholder]]\)]} }, RowSpacings->0, ColumnSpacings->0, GridDefaultElement:>ButtonBox[ "\\[Placeholder]"]]], "Input", Active->True], Cell[TextData[{ "In summary, ", StyleBox["Mathematica", FontSlant->"Italic"], " 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 ", StyleBox["Standards-", FontSlant->"Italic"], "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." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Introducing Functional Notation", "Subsubsection"], Cell[TextData[{ "Teachers who avoid ", ButtonBox["rigorous definitions", ButtonData:>"def of function", ButtonStyle->"Hyperlink"], " 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." }], "Text", TextAlignment->Left, TextJustification->0], Cell[TextData[{ "\"Let ", Cell[BoxData[ \(TraditionalForm\`f(x) = 2 x + 3\)]], Cell[BoxData[ \(TraditionalForm\`\(\ where\ \(f(2)\)\ means\ 2*2 + 3\)\)]], ". Find ", Cell[BoxData[ \(TraditionalForm\`\(f(\(-2\))\)\ and\ \(f(4) . \)\)]], "\" " }], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "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 ", StyleBox["Mathematica", FontSlant->"Italic"], " code. 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For example, students who have studied \ trigonometry may recognize the graph of ", StyleBox["manyPoints", "Input"], " 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 ", StyleBox["Log", "Input"], " or ", StyleBox["Exp", "Input"], "?\" [", ButtonBox["19", ButtonData:>"footnote18", ButtonStyle->"Hyperlink"], "] or \"Can one alter ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " such that its tail comes back down?\" Regardless of the questions that \ develop, students walk away from this type of introduction with a ", StyleBox["mathematical", FontVariations->{"Underline"->True}], " rather than an arithmetic understanding of function." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Seeing Cause and Effect", "Subsubsection"], Cell[TextData[{ "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 ", StyleBox["functions", FontSlant->"Italic"], " 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 ", StyleBox["Mathematica", FontSlant->"Italic"], " becomes the ideal environment for investigating the translation and \ scaling of special functions. 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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.\ \>", "Text"], Cell[TextData[{ "Specifically, students are typically given very simple functions ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`g(x)\)]], " and are asked to find ", Cell[BoxData[ \(TraditionalForm\`f(g(z))\)]], " for some ", Cell[BoxData[ \(TraditionalForm\`z \[Element] \[DoubleStruckCapitalZ]\)]], ". In this context, \"simple\" is defined as selecting ", Cell[BoxData[ \(TraditionalForm\`f, \ g, \)]], " and ", Cell[BoxData[ \(TraditionalForm\`z\)]], " 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. " }], "Text"], Cell[TextData[{ "\"Given ", Cell[BoxData[ \(TraditionalForm\`f(x) = x + 1\)]], " and ", Cell[BoxData[ \(TraditionalForm\`g(x) = x\^2\)]], ", calculate ", Cell[BoxData[ \(TraditionalForm\`\(f(g(1))\)\ \ and\ \ \(g(f(\(-3\)))\)\)]], ".\"" }], "Text", TextAlignment->Center], Cell[TextData[{ "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. 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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 ", Cell[BoxData[ \(TraditionalForm\`\(f(x)\)\ and\ \(g(x)\)\)]], " from the previous example, let ", Cell[BoxData[ \(TraditionalForm\`h(x) = \((f + g)\) \((x)\)\)]], " and let ", Cell[BoxData[ \(TraditionalForm\`m(x)\ = \((f\[SmallCircle]g)\) \((x)\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[h]\), \(h[x_] := g[x] + f[x]; \nh[x]\)}], "Input"], Cell[BoxData[ \(7 - 13\ x - 2\ x\^2 + x\^3\)], "Output"] }, Open ]], Cell[TextData[{ "Once ", Cell[BoxData[ \(TraditionalForm\`h(x)\)]], " has been defined [", ButtonBox["20", ButtonData:>"footnote 20", ButtonStyle->"Hyperlink"], "], it extremely instructive to plot all three on the same set of \ coordinate axes and to notice, for example, that ", Cell[BoxData[ \(TraditionalForm\`h(\(-4\)) = f(\(-4\)) + g(\(-4\)), \ h(0) = f(0) + g(0), \)]], " and that, in general, ", Cell[BoxData[ \(TraditionalForm\`h(x) = f(x) + g(x)\)]], " for any ", Cell[BoxData[ \(TraditionalForm\`x\)]], " chosen." }], "Text"], 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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 ", StyleBox["Mathematica.", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ "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 ", Cell[BoxData[ \(TraditionalForm\`f(x) = x\^3 - 3 x\^2 - 11 x + 4\)]], " are created and grouped together. 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finding the inverse of a function appears \ in the next section.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Inverses of Functions", "Subsubsection"], Cell["\<\ 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. \ \>", "Text"], Cell[TextData[{ "\"Given that ", Cell[BoxData[ \(TraditionalForm\`f(x) = 2 x - 7, \ find\ \ \(g(x)\) = \(f\^\(-1\)\)(x)\)]], " and verify your results.\"" }], "Text", TextAlignment->Center], Cell[TextData[{ "To answer questions such as these, students are usually directed to switch \ the role of the independent and dependent variables (in the aforementioned \ example, ", Cell[BoxData[ \(TraditionalForm\`x\ and\ y\)]], " respectively) and then to isolate, or solve, for the \"new\" independent \ variable. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Clear[f, g]; \nf[x_] := 2 x - 7\), \(Solve[f[y] == x, y]\)}], "Input"], Cell[BoxData[ \({{y \[Rule] \(7 + x\)\/2}}\)], "Output"] }, Open ]], Cell[BoxData[ \(g[x_] := \(7 + x\)\/2\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(f[g[x]]\)], "Input"], Cell[BoxData[ \(x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[f[x]]\)], "Input"], Cell[BoxData[ \(x\)], "Output"] }, Open ]], Cell[TextData[{ "Even though students are able to calculate the inverse and verify their \ findings by demonstrating that ", Cell[BoxData[ \(TraditionalForm\`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, ", StyleBox["Mathematica ", FontSlant->"Italic"], "can help students answer the question of why by providing a forum for \ investigation and exploration devoid of mechanical concerns." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[x]\)], "Input"], Cell[BoxData[ \(\(-7\) + 2\ x\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[x]\)], "Input"], Cell[BoxData[ \(\(7 + x\)\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(elems = Table[i, \ {i, \(-5\), 5}]\)], "Input"], Cell[BoxData[ \({\(-5\), \(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 4, 5}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f[elems]\)], "Input"], Cell[BoxData[ \({\(-17\), \(-15\), \(-13\), \(-11\), \(-9\), \(-7\), \(-5\), \(-3\), \(-1\), 1, 3}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[elems]\)], "Input"], Cell[BoxData[ \({1, 3\/2, 2, 5\/2, 3, 7\/2, 4, 9\/2, 5, 11\/2, 6}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f[g[elems]]\)], "Input"], Cell[BoxData[ \({\(-5\), \(-4\), \(-3\), \(-2\), \(-1\), 0, 1, 2, 3, 4, 5}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f[g[elems]] == elems\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "By generating multiple examples, students begin to truly understand why ", Cell[BoxData[ \(TraditionalForm\`f(g(x)) = \(g(f(x)) = x\)\)]], ". In the example provided, the student sees that ", Cell[BoxData[ \(TraditionalForm\`f : 5\[LongRightArrow]3\)]], ", ", Cell[BoxData[ \(TraditionalForm\`g : 3\[LongRightArrow]5\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`f\[SmallCircle]\(g : 5\[LongRightArrow]5\)\)]], ". This is no coincidence as ", Cell[BoxData[ \(TraditionalForm\`f : 2\[LongRightArrow]\(-3\), \ g : \(-3\[LongRightArrow]2\), \)]], " and ", Cell[BoxData[ \(TraditionalForm\`f\[SmallCircle]\(g : 2\[LongRightArrow]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\[LongDash]all without concern for output \ accuracy [", ButtonBox["22", ButtonData:>"footnote 22", ButtonStyle->"Hyperlink"], "]. 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The typical math student in a traditional math curriculum is \ expected to learn, understand, and remember the following:\n\n\t\ -Definition(s) of function, \n\t-Function vocabulary (such as \"function\", \ \"domain\", \"range\"),\n\t-How to plot functions,\n\t-Many types of \ functional notation (such as ", Cell[BoxData[ \(TraditionalForm\`\(f(x), \ f\^\(-1\), \ f : x \[Rule] f(x))\)\)]], ",\n\t-How to find the inverses of functions (and verify them),\n\t-How to \ find composite functions (such as ", Cell[BoxData[ \(TraditionalForm \`\(f\ \[EmptySmallCircle]\ g, \ f(g(x)), \(f\^\(-1\)\)(f(x)))\)\)]], ",\n\t-How to work with and identify special functions (such as ", Cell[BoxData[ \(TraditionalForm\`f(x) = \[LeftCeiling]x\[RightCeiling]\)]], "), and\n\t-How changing various parameters of functions affect their \ plots, values, and the like.\n\t\nIronically, 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?" }], "Text"], Cell[TextData[{ "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 ", StyleBox["Mathematica", FontSlant->"Italic"], " environment. To implement such an approach requires just four things: a \ copy of ", StyleBox["Mathematica", FontSlant->"Italic"], ", a copy of the NCTM ", StyleBox["Standards,", FontSlant->"Italic"], " 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." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References & Footnotes", "Subsection"], Cell[TextData[{ "1. U. S. Department of Education. National Center for Education \ Statistics, ", StyleBox[ "Pursuing Excellence: A Study of U. S. Twelfth-Grade Mathematics and \ Science Achievement in International Context", FontSlant->"Italic"], ", NCES 98-049. Washington, DC: U. S Government Printing Office, 1998. \n\n\ 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 ", ButtonBox["www.nces.ed.gov", ButtonData:>{ URL[ "http://www.nces.ed.gov"], None}, ButtonStyle->"Hyperlink"], "." }], "Text", CellTags->"footnote1"], Cell["\<\ 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.\ \>", "Text", CellTags->"footnote2"], Cell[TextData[{ "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 ", ButtonBox["Illinois State Goals", ButtonData:>{ URL[ "http://www.isbe.state.il.us/oldisbe/standards/math.html"], None}, ButtonStyle->"Hyperlink"], " which are loosely based upon national policy statements like the NCTM ", StyleBox["Standards.", FontSlant->"Italic"] }], "Text", CellTags->"footnote3"], Cell[TextData[{ "4. Swokowski, Earl W., ", StyleBox["Fundamentals of College Algebra, Fourth Edition", FontSlant->"Italic"], ", Boston: Prindle, Weber & Schmidt, Inc., 1979, p. 114. " }], "Text", CellTags->"footnote4"], Cell[TextData[{ "5. Rubenstein, Rheta N., Timothy V. Craine, and Thomas R. Butts, ", StyleBox["Integrated Mathematics", FontSlant->"Italic"], ", Boston: Houghton Mifflin Company, 1995, p. 220." }], "Text", CellTags->"footnote5"], Cell[TextData[{ "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 ", StyleBox["The Process of Education", FontSlant->"Italic"], " published in 1960 by Harvard University Press." }], "Text", CellTags->"footnote6"], Cell[TextData[{ "7. Gage, N. L. and David C. Berliner, ", StyleBox["Educational Psychology", FontSlant->"Italic"], ", Boston: Houghton Mifflin Company, 1988, pp. 42-46.\n\nBloom'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\[Dash]knowledge, comprehension, \ application, analysis, synthesis, evaluation\[Dash]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." }], "Text", CellTags->"footnote7"], Cell[TextData[{ "8. Gage, N. L. and David C. Berliner, ", StyleBox["Educational Psychology", FontSlant->"Italic"], ", Boston: Houghton Mifflin Company, 1988, p. 76.\n\nMany 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." }], "Text", CellTags->"footnote8"], Cell["\<\ 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.\ \>", "Text", CellTags->"footnote9"], Cell[TextData[{ "10. Best, George, David Penner, and Robert Schneider, ", StyleBox[ "Exploring Algebra, Precalculus, and Statistics with the TI-83 Graphing \ Calculator", FontSlant->"Italic"], ", Andover, Mass.: Venture Publishing, 1997. " }], "Text", CellTags->"footnote10"], Cell[TextData[{ "11. U. S. Bureau of the Census, ", StyleBox["Statistical Abstract of the United States: 1995", FontSlant->"Italic"], " (115th Edition) Washington DC, 1995." }], "Text", CellTags->"footnote11"], Cell["\<\ 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.\ \>", "Text", CellTags->"footnote12"], Cell[TextData[{ "13. National Council of Teachers of Mathematics, ", StyleBox["Professional Standards for Teaching Mathematics", FontSlant->"Italic"], ", Reston, Virginia: National Council of Teachers of Mathematics, 1991.", " " }], "Text", CellTags->"footnote13"], Cell[TextData[{ "14. National Council of Teachers of Mathematics, ", StyleBox["Curriculum and Evaluation Standards for School Mathematics", FontSlant->"Italic"], ", Reston, Virginia: National Council of Teachers of Mathematics, 1989. " }], "Text", CellTags->"footnote14"], Cell[TextData[{ "15. For example: \"A function with domain ", Cell[BoxData[ \(TraditionalForm\`X\)]], " is a set ", Cell[BoxData[ \(TraditionalForm\`W\)]], " of ordered pairs such that for each ", Cell[BoxData[ \(TraditionalForm\`x\ in\ X\)]], ", there is exactly one ordered pair ", Cell[BoxData[ \(TraditionalForm\`\((x, y)\)\ in\ W\)]], " having ", Cell[BoxData[ \(TraditionalForm\`x\)]], " in the first position.\" See ", ButtonBox["footnote 4", ButtonData:>"footnote4", ButtonStyle->"Hyperlink"], " above." }], "Text", CellTags->"footnote15"], Cell[TextData[{ "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: ", StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["\[Rule]3.x\[Rule]SystemFiles\[Rule]FrontEnd\[Rule]Palettes", FontWeight->"Bold"], ". 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." }], "Text", CellTags->"footnote16"], Cell[TextData[{ "17. As one reviewer pointed out, ", StyleBox["Mathematica", FontSlant->"Italic"], "'s pattern-matching ability makes it possible to define functions in more \ than one manner. Specifically, ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], " can be defined as ", Cell[BoxData[ \(f[x_] = x\^2\)], "Input"], ". 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(*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)