Judah L Schwartz, Ronan Downes, Mar 14, 2020

This GeoGebra book is the principal [i][b]online resource[/b][/i] for the second course in a two semester sequence on posing K-12 mathematics problems. The courses are designed for middle and secondary school teachers in the "Mathematics for Teaching" graduate program at the Harvard Extension School. The primary goal of this course is to give teachers a more systematic and theoretical framework for formulating the mathematics problems they pose to their students. Many (although not all) of the mathematics problems that we pose to K-12 students in order to stimulate invention and creativity can be classified along several several pairs of perspectives. These pairs of perspectives seem to overlap in some cases and in some cases to contrast with one another. In this course we explore some of these perspectives and the degree to which a sensitivity to them can help teachers pose problems both in class and for homework. The aim is to help teachers help their students to develop deeper insight into, and appreciation for, mathematics. In the course teachers work on problems that stimulate their inventiveness and creativity and then analyze the approaches both they and their colleagues have taken. The course relies heavily on [i][b]interactive images[/b][/i], computer applets that use multiple graphical representations of the mathematical problems. One perspective can be described as follows: - the mathematical situation described by the problem presents a phenomenon that must be[i] [b]decomposed[/b][/i] into component representations and analyzed to be understood - the mathematical situation described by the problem presents phenomena that must be combined into an [i][b]aggregated[/b][/i] representation to be understood Another perspective is one that characterizes problems on the basis of whether the resolution of the problem calls for [i][b]analysis[/b][/i] (deduction) or [i][b]synthesis[/b][/i] (induction). [i][b]Classifying[/b][/i] and[i] [b]exemplifying[/b][/i] form the basis of yet another perspective. This perspective is well-known by teachers although probably not as widely used as it should be. It is the contrast between - Given a specific mathematical object and asking “What is this mathematical object a case of?” and, - Given a set of attributes and asking to generate (one, many, all) the mathematical object(s) that possess this set of attributes. Another perspective is one of [b][i]Universal[/i][/b] and [b][i]Existential[/i][/b] quantification. This rather formal perspective is increasingly present as students progress through their mathematical education. In some respects it echoes elements common to all of these perspectives. But these issues are also present in informal settings. The words we use in common parlance in building models of spatially distributed phenomena ([b][i]All over, Somewhere, Nowhere[/i][/b]) or temporally evolving events ([b][i]All the time, Sometimes, Never[/i][/b]) reflect the essence of Universal and Existential quantification. Finally, Polya in his seminal book on mathematics education “Mathematical Discovery” distinguishes two categories of problems – [i][b]problems to prove[/b][/i] and [i][b]problems to find[/b][/i]. [This distinction is fully discussed in Chapter V of Volume 1]. Students will be asked to explore the degree to which Polya’s distinctions map onto the other perspectives explored in the course (and vice versa). [b][color=#c51414]Case studies drawn from introductory geometry, algebra and arithmetic and their application will be analyzed in detail. Suggested background readings include the following essays on the mathMINDhabits website [/color][/b] [url]https://sites.google.com/site/mathmindhabits/-[/url] [color=#c51414]"What Makes A Good Challenge ?" "On Problem Posing & Making Conjectures"" "Formulating Measures" "Chicken Is To Data As Model Is To Egg" [/color] [color=#0a971e]Some comments on the organization of these materials [i][b]The interactive images[/b], or applets in this collection(1) are organized in “matrix” form – there are three “ROWS” (algebra, arithmetic, geometry} and three “COLUMNS” {Decomposition & Aggregation, Induction & Deduction, Classification & Exemplification} [b] Decomposing / aggregating Deductive / inductive approaches Classifying / exemplifying Algebra 1 4 7 Arithmetic 2 5 8 Geometry 3 6 9[/b] Users of the materials are urged to examine each [i][b]interactive image[/b][/i] or applet not only by its particular COLUMN coordinate but by the appropriateness of each of the other columns as a perspective for understanding the applet and its usefulness in helping to pose problems. Users are also urged to explore how the links among the algebra, arithmetic and geometry ROWS might permit them to pose problems that span {algebra & arithmetic}, {arithmetic & geometry}, and {geometry & algebra}.[/color][/i] (1) The applets in this collection are drawn from the "mathMINDhabits" archive [url]https://sites.google.com/site/mathmindhabits/[/url]

Decomposing and Aggregating can be used as a way of understanding different and seemingly contrasting approaches to mathematical situations - the mathematical situation described by the problem presents a phenomenon that must be decomposed into component representations and analyzed to be understood - the mathematical situation described by the problem presents phenomena that must be combined into an aggregated representation to be understood

• 1. The Parallelogram Factory
• 2. Linear functions - parameter plane
• 3. QUADRATIC functions - parameter Plane
• 4. Build & Fit Linear Functions
• 5. Build & Fit Absolute Value F'ns - I
• 6. Build & Fit Absolute Value function - II
• 7. Rotating Squares - an algebraic perspective

Decomposing and Aggregating can be used as a way of understanding different and seemingly contrasting approaches to mathematical situations - the mathematical situation described by the problem presents a phenomenon that must be decomposed into component representations and analyzed to be understood - the mathematical situation described by the problem presents phenomena that must be combined into an aggregated representation to be understood

• 1. 'between-ness' in addition/subtraction
• 2. 'between-ness' in multiplication/division
• 3. Build a million numbers
• 4. Place Value - Decomposing for addition & subtraction
• 5. Place Value - Decomposing Multiplication

Decomposing and Aggregating can be used as a way of understanding different and seemingly contrasting approaches to mathematical situations - the mathematical situation described by the problem presents a phenomenon that must be decomposed into component representations and analyzed to be understood - the mathematical situation described by the problem presents phenomena that must be combined into an aggregated representation to be understood

• 1. kite space
• 2. Polygonal pancakes
• 3. ISOSCELES TRIANGLES
• 4. Polya's Triangle Space
• 5. angle parameter space for triangles
• 6. Rectangles in the {Perimeter, Area} plane
• 7. Rotating Squares - geomteric perspective
• 8. Two rectangles in a square II

In dealing with a problematic situation a person thinking [b]deductively[/b] will have some theory about how s/he might resolve the problem. A hypothesis is formulated and the observations made may either confirm or confute the hypothesis. This is the essence of [i][b]analysis[/b][/i]. Dealing with a problematic situation, a person thinking [b]inductively[/b] will collect observations about the situation, attempt to discern patterns, and use these patterns to formulate hypotheses to be tested. This is the essence of [i][b]synthesis[/b][/i]. Here are some thoughts adapted from an essay on analysis and synthesis...[http://shell.cas.usf.edu/~mccolm/pedagogy/HWanalysissynthesis.html] The word analysis comes from a Greek word meaning "to break up." It is often helpful to break a problem or a phenomenon into small pieces: if one studied each piece, independently of the other pieces, one might have a better chance of understanding the pieces. From that, one might better understand the whole. The word synthesis comes from a Greek word meaning "to put together." Knowledge is created, according to Locke in his Essay Concerning Human Understanding, by combining perceptions, ideas, and other bits of knowledge. Kant, in his Critique of Pure Reason, pointed to two modes of understanding - Analysis. One understands something by taking it apart and looking at the pieces. Synthesis. One understands something by combining it or comparing it with other things, or by looking at interrelations between its constituent parts. An argument in support of the importance of synthesis: The human body, very nearly in its entirety, consists of oxygen, carbon, hydrogen, nitrogen, calcium, and phosphorus atoms - it's the way all the pieces are put together. An argument in support of the importance of analysis: Eugene Wigner, in his “The unreasonable effectiveness of Mathematics in the Natural Sciences”, wrote that progress in physics would have been impossible if it wasn't possible to take phenomena apart and then study the parts in isolation. This is the essence of analysis. Despite the fact that many "reductionists" who claim to rely entirely on analysis, synthesis appears to be necessary for creativity and invention. There are two faces of synthesis. First, creative acts often consist of combining notions that one usually doesn't imagine having much to do with each other… the construction of entirely new things from old parts. Second, many things cannot be understood in isolation - for example the separated gears and spring of a mechanical clock offer no clue as to its function when assembled. Sometimes mathematics problems are like this, too. Consider the problem: find all solutions to: 3x + y = 2 -2y - z = 2 x = 1 This can be solved by substitution: x = 1, so y = 2 - 3 = -1, so z = 2 - 2 = 0, This is a reductionist approach. But what about: x - y + 2z = 2 3x + 2y + z = 1 x + y + z = 0 Substitutions doesn't work so well here: we need to use a method that deals with the entire system at once: Cramer's rule, or the Gauss-Jordan method.

• 1. Solving Linear Equations Graphically & Symbolically
• 2. Linear Inequalities solved w/ graphs & symbols
• 3. Solving Quadratic Equations Graphically & Symbolically
• 4. Quadratic Inequalities solved w/ Graphs & Symbols
• 5. UNsolving Linear Equations & Inequalities
• 6. UNsolving Quadratic Equations & Inequalities
• 7. Inequality & Equation Explorer

In dealing with a problematic situation a person thinking [b]deductively[/b] will have some theory about how s/he might resolve the problem. A hypothesis is formulated and the observations made may either confirm or confute the hypothesis. This is the essence of [i][b]analysis[/b][/i]. Dealing with a problematic situation, a person thinking [b]inductively[/b] will collect observations about the situation, attempt to discern patterns, and use these patterns to formulate hypotheses to be tested. This is the essence of [i][b]synthesis[/b][/i]. Here are some thoughts adapted from an essay on analysis and synthesis...[http://shell.cas.usf.edu/~mccolm/pedagogy/HWanalysissynthesis.html] The word analysis comes from a Greek word meaning "to break up." It is often helpful to break a problem or a phenomenon into small pieces: if one studied each piece, independently of the other pieces, one might have a better chance of understanding the pieces. From that, one might better understand the whole. The word synthesis comes from a Greek word meaning "to put together." Knowledge is created, according to Locke in his Essay Concerning Human Understanding, by combining perceptions, ideas, and other bits of knowledge. Kant, in his Critique of Pure Reason, pointed to two modes of understanding - Analysis. One understands something by taking it apart and looking at the pieces. Synthesis. One understands something by combining it or comparing it with other things, or by looking at interrelations between its constituent parts. An argument in support of the importance of synthesis: The human body, very nearly in its entirety, consists of oxygen, carbon, hydrogen, nitrogen, calcium, and phosphorus atoms - it's the way all the pieces are put together. An argument in support of the importance of analysis: Eugene Wigner, in his “The unreasonable effectiveness of Mathematics in the Natural Sciences”, wrote that progress in physics would have been impossible if it wasn't possible to take phenomena apart and then study the parts in isolation. This is the essence of analysis. Despite the fact that many "reductionists" who claim to rely entirely on analysis, synthesis appears to be necessary for creativity and invention. There are two faces of synthesis. First, creative acts often consist of combining notions that one usually doesn't imagine having much to do with each other… the construction of entirely new things from old parts. Second, many things cannot be understood in isolation - for example the separated gears and spring of a mechanical clock offer no clue as to its function when assembled. Some arithmetic examples of the roles of analysis and synthesis can be seen in the DIY estimation applets in this chapter.

• 1. Common Denominator Finder
• 2. Common Factor Finder
• 3. DIY estimation - addition
• 4. DIY estimation - multiplication
• 5. power to the base!
• 6. Inverse Prime Factorization

• 1. Polygon Synthesizer
• 2. Parametric polygon synthesizer
• 3. Polygon Explorer
• 4. Parametric polygon explorer
• 5. Circles & ellipses in the {radius, eccentricity} plane
• 6. Three mutually tangent circles - synthesis
• 7. Three mutually tangent circles - analysis
• 8. measuring the Unit Circle

Classifying and exemplifying form the basis of yet another perspective. It is the contrast between - You are classifying if when given a specific mathematical object you answer the question “What is this mathematical object a case of?” and, - You are exemplifying if when given a set of attributes you generate (one, many, all) the mathematical object(s) that possess this set of attributes A specific example - An instance of a Classifying task - given a collection of quadratics, sort them into classes whose real roots differ by 1, 2, 3,... An instance of an Exemplifying task - The equation [math]x^2-3x+2=0[/math] has a specific solution set i.e., x=-2 and x=-1. The solution set {-1, -2} belongs to an entire equivalence class of equations - any one of which is an example of a quadratic whose roots differ by one. This latter kind of task is the essence of the UNsolving applets to be found on the mathMINDhabits website

• 1. Roots & Asymptotes of Polynomials
• 2. Rate of change - constant or accelerating
• 3. Even & Odd Functions
• 4. A swirl of spirals
• 5. spirals
• 6. The Two Faces of Z
• 7. Powers of Complex Numbers
• 8. The Roots of Unity
• 9. A swirl of spirals

Classifying and exemplifying form the basis of yet another perspective. It is the contrast between - Given a specific mathematical object and asking “What is this mathematical object a case of?” and, - Given a set of attributes and asking to generate (one, many, all) the mathematical object(s) that possess this set of attributes A specific example - Adding the two fractions 9/16 and 3/32 is a way of representing the specific rational number 21/32. On the other hand a/b + c/d = 3/5 is a way of describing the equivalence class of pairs of fractions a/b and c/d that sum to 3/5

• 1. Assessing Understanding of Fraction Ordering
• 2. Exploring the Ordering of Fractions
• 3. Mystery Fraction Search
• 4. The Fraction Workbench - working w/ Uncommon Denominators

Classifying and exemplifying form the basis of yet another perspective. It is the contrast between - Given a specific mathematical object and asking “What is this mathematical object a case of?” and, - Given a set of attributes and asking to generate (one, many, all) the mathematical object(s) that possess this set of attributes A specific example - Consider a {width,height} plane [width and height non-negative]. A point in this plane defines a rectangle with a given area and perimeter. 'Exemplifying' means given the magnitude of an area and the magnitude of a perimeter [i.e., length] produce an example of a rectangle that has these properties. 'Classifying' means given a collection of rectangles group them by area, or group them by perimeter. Is it certain that a point in a {perimeter,area} plane defines only ONE rectangle? Another example - Given three lines that are known to be medians {MED}, or altitudes {ALT}, or angle bisectors {ANG} can you produce an example of a triangle that has these MEDs, or ALTs, or ANGs ? This is an 'Exemplifying' task. An interesting 'Classifying' task - given a collection of triangles as specified by their MEDs, or ALTs or ANGs, what is true of all the isosceles triangles?

• 1. Build a triangle from medians?
• 2. Build a Triangle from Medians - II
• 3. Build a triangle from Angle Bisectors?
• 4. Build a triangle from Altitudes?
• 5. Quadrilateral factory ?
• 6. Equilateral & Equiangular N-gons
• 7. Van Aubel's theorem

Suggested background readings include the following essays on the mathMINDhabits [url]website https://sites.google.com/site/mathmindhabits/[/url]- "What Makes A Good Challenge ?" "On Problem Posing & Making Conjectures" "Formulating Measures" "Chicken Is To Data As Model Is To Egg"

• 1. Hens & Rabbits [for teachers]
• 2. two lap average [for teachers]
• 3. There are planes & there are planes
• 4. TWO MOTIONS - one path in space
• 5. A Leaky Bathtub - competing rates & equilibrium
• 6. Retirement & leaky bathtubs: 2 contexts -1 model

Suggested background readings include the following essays on the mathMINDhabits [url]website https://sites.google.com/site/mathmindhabits/[/url]- "What Makes A Good Challenge ?" "On Problem Posing & Making Conjectures" "Formulating Measures" "Chicken Is To Data As Model Is To Egg"

• 1. adding & subtracting uncertain measurements
• 2. multiplying & dividing uncertain measurements
• 3. SeeSaw - multiplying signed numbers

Suggested background readings include the following essays on the mathMINDhabits [url]website https://sites.google.com/site/mathmindhabits/[/url]- "What Makes A Good Challenge ?" "On Problem Posing & Making Conjectures" "Formulating Measures" "Chicken Is To Data As Model Is To Egg"

• 1. a road up a mountain
• 2. The most efficient can of vegetables
• 3. Rain Gutter
• 4. scaling the volume of a cone