Published on January 10, 2008
Enhancing the Mathematical Core Grand Valley State University: Enhancing the Mathematical Core Grand Valley State University Working Groups David Austin Ed Aboufadel Alverna Champion Charlene Beckmann Pam Wells Phyllis Curtiss Will Dickinson John Gabrosek Matt Boelkins Esther Billings David Coffey John Golden Karen Novotny Karen Novotny Clark Wells Steven Schlicker Akalu Tefera Core Math & Stats Courses: Core Math & Stats Courses Faculty Teams: Faculty Teams Each course is being enhanced by a team of faculty made up of 1 or 2 mathematicians or statisticians and a mathematics educator Goals: Goals Maintain a high level of mathematical content for all clients. Motivate, introduce, extend, or apply mathematics content. Make evident to mathematics majors: That the material in undergraduate mathematics courses arises from the mathematics they learned in K-12 and applies to the mathematics prospective teachers will teach. That the level of mathematics required of children (and their teachers) through exemplary K-12 curricula is challenging. Our Process for Each Course: Our Process for Each Course Become more familiar with how students learn the mathematics encompassed by the course. Review the course content and determine the areas most likely to have links to K-12. Review K-12 materials for motivating activities that lead to higher level mathematics. Adapt or write materials to make the course even more interactive while challenging students at a high level. Enhanced Courses: Enhanced Courses 01-02: Course studies and first pilot Communicating in Mathematics (first proof course) Euclidean Geometry Probability and Statistics 02-03: Revisions and further piloting of course materials Beginning Fall 2003: Research of efficacy of courses. Courses to Be Enhanced: Courses to Be Enhanced 02-03: Course studies and first pilot Linear Algebra Modern Algebra Discrete Mathematics 03-04: Revisions and further piloting of course materials Beginning Fall 2004: Research of efficacy of courses. Active Involvement: Active Involvement Building on the Departments of Mathematics and Statistics philosophy of active learning: Activities engage students in a hands-on way. Students investigate and discover mathematics. Students share their discoveries in small groups then with the full class. Assessment: Assessment Authentic and constant Through observations as students work in class, Through students’ written and oral responses to labs and homework, Through oral presentations, Through exams. Enhancing Communicating in Mathematics: Enhancing Communicating in Mathematics Team: Ed Aboufadel Pam Wells Communicating in Mathematics: Communicating in Mathematics A study of proof techniques used in mathematics. Intensive practice in reading mathematics, expository writing in mathematics, and constructing and writing mathematical proofs. Communicating in Mathematics-Content: Communicating in Mathematics-Content Elementary logic Set theory Congruence arithmetic Functions Equivalence relations and classes Communicating in Mathematics: Communicating in Mathematics Activities launch discussion of different proof techniques so that students: See how proof techniques are used to reason informally, both by adults and children. Have concrete references for various proof techniques to help them understand and apply the techniques correctly. Understanding Proof Techniques: Understanding Proof Techniques Several activities were used to help students apply various proof techniques or to assess students’ understanding of the mathematical content of the course. Practice with direct proofs using divisibility Practice on functions and associated terminology Portfolio Proof dealing with twin primes Final Exam question related to twin primes portfolio proof. Proof by Cases: Proof by Cases Sheryl was excited when she arrived home with three small boxes of marbles. She labeled each box with its contents. One box had 2 blue marbles, the second had 2 red marbles, and the third had 1 blue and 1 red marble. Sheryl’s mischievous little sister removed all of the labels and replaced them so that each box was labeled incorrectly. Sheryl wanted to put the correct labels back on without opening all the boxes. She wondered if she could figure out the correct labeling by just seeing the color of 1 marble in one box. Is it possible for Sheryl to pull only one marble from only one box and know the correct labels for each of the boxes? If so, how? If not, why not? Source: Teaching Children Mathematics, December 2000. Twin Primes Theorem: Twin Primes Theorem Theorem #3 in the Portfolio: Twin prime numbers are pairs of prime numbers whose difference is 2. Examples are 5 and 7, 17 and 19, or 41 and 43. Source: Interactive Mathematics Program, Year 3 Theorem: If x and y are twin primes other than 3 and 5, then xy + 1 is a perfect square and is divisible by 36. Twin Primes Theorem (cont’d): Twin Primes Theorem (cont’d) On the Final Exam: a. In 1993, the world record for the largest known twin primes was x = 459 28529 – 1, and y = 459 28529 + 1. Use your knowledge of the proof of Theorem 3 of the Portfolio to explain what the remainder is when 459 28529 is divided by 6. b. Use modulus arithmetic to determine the remainder when 459 28529 is divided by 6. (Show your work.) High-Low Differences: High-Low Differences Follow these steps for any three digit number: Arrange the digits from largest to smallest; Arrange the digits from smallest to largest; Subtract the smaller number from the larger. Example: 743 – 347 = 396, 396 is the high-low difference of 473. Source: Year 1, Interactive Mathematics Program High-Low Differences (cont’d): High-Low Differences (cont’d) Before answering the following questions, create several examples of high-low sequences to develop some intuition about them. For notation, define f(x) to be the high-low difference of x. Decide on sets X and Y so that f: X Y makes sense. Which set is the domain? Which set is the co-domain? Is f a 1-to-1 function? Justify your answer. Is f an onto function? Justify your answer. Writing Assignment: Writing Assignment Imagine you are teaching calculus to seniors in high school. You ask the students the following question: If f is an even function, then must f’ be an odd function? A student gives the response at right: Questions: Questions Suppose this is the response of a Math 210 student. Discuss the strengths and weaknesses of the response using appropriate technology from this course. Now consider the response to be that of a high school student. What questions might you ask to probe the student’s thinking? What explanation would you say or write to the student? Induction: Induction An elementary school student performs the calculations at right to find the sum of the whole numbers 1 to 100. 10(55)+(100+200+300+400+500+600+700+800+900)=5050 Questions: Questions What is the child doing and why does it work? Use terminology from Math 210 in your discussion. How could you use this child’s technique to add the whole numbers from 1-250? From 1-217? Explain. Utilizing the student’s technique, can you find a formula for the sum of the whole numbers from 1 to n, where n>1 and n is a multiple of 10? Do any interesting issues arise? Consecutive Addends: Consecutive Addends The number 15 can be written as a sum of consecutive whole numbers in three different ways: 15 = 7+8 = 1+2+3+4+5 = 4+5+6 The number 9 can be written as a sum of consecutive whole numbers in two ways: 9 = 4+5 = 2+3+4 Adapted from Balanced Assessment Project, Middle Grades Assessment. New Jersey: Dale Seymour Publications, pp. 39-48. Questions: Questions Look at other numbers and find out all you can about writing them as sums of consecutive whole numbers. Begin with numbers from 1 through 36. Decide what kind of numbers can be written as a sum of 2 consecutive whole numbers; 3 consecutive whole numbers; 4 consecutive whole numbers; and so on. What numbers cannot be written as sums of consecutive whole numbers? Write your answers to the above questions in the form of conjectures and prove each conjecture. Enhancing Euclidean Geometry: Enhancing Euclidean Geometry Team: David Austin Charlene Beckmann Will Dickinson How Students Learn Geometry: How Students Learn Geometry Van Hiele Model of Geometric Thought, 5 stages arising from experience (not maturation): Visualization Analysis Informal Deduction Formal Deduction Rigor Outline of Euclidean Geometry: Outline of Euclidean Geometry Triangle Explorations (2 wks) Quadrilateral Explorations (2 wks) Spherical and Hyperbolic Geometries (1 wk) Axiomatic Systems, Neutral Geometry, Euclidean Geometry (6-7 wks) Transformational Geometry (2-3 wks) Triangle Activities: Goals: Triangle Activities: Goals Recall how and why congruence works Work towards precise language See geometry in the real world Surprise and challenge Introduce fundamental concepts in the course Triangle Activities: Triangle Activities Congruence Scavenger Hunt Relationships between angles and sides Similarity What is a straight line Spherical geometry Triangle Congruence: Triangle Congruence Given Angles: Given Side lengths: How many triangles can be made? Explore. Student Responses: Student Responses “There is only one triangle that can be built.” “They are the same.” “There are no differences.” “One can be laid on top of the other.” “They are congruent.” “Congruent means they match up.” Scavenger Hunt: Scavenger Hunt Equilateral triangle Isosceles triangle Scalene triangle Right triangle Equiangular triangle Relationships Between Sides and Angles: Relationships Between Sides and Angles Similarity: Similarity Angle Measures Add to 180: Angle Measures Add to 180 Spherical Geometry: Spherical Geometry Using the Lenart sphere, determine which path is the shortest between two points? Compare “lines” on the sphere and in the plane. Measure the interior angles in a triangle and determine the angle sum. What is the largest angle sum you can find? What is the smallest? Next Time, We Will: Next Time, We Will Trim and refine activities Spread them throughout the term Have more explicit follow up Include more to challenge better students Exploring Quadrilaterals: Exploring Quadrilaterals Starts with intuitive definitions of quadrilaterals and works toward discovering their properties and several equivalent definitions. Quadrilateral Search: Quadrilateral Search Pre-Activity: Search for quadrilaterals around you Square, Rhombus, Kite, Parallelogram Trapezoid, Isosceles Trapezoid, Rectangle and General Quadrilateral Intuitive definitions of these Definition #1 for quadrilaterals Day 1: Constructions with Geometer's Sketchpad Diagonals Activity: Diagonals Activity Day 2: Definition #2 for quadrilaterals Tree diagram of quadrilaterals Diagonal Activity : Diagonal Activity Solution to Diagonal Part: Diagonal Activity: Diagonal Activity Solutions to Tree Diagram: Diagonal Activity: Observations and Student Progress: Diagonal Activity: Observations and Student Progress Overall the activity went well and students were engaged. Some students assumed too many conditions. Students began to see the role of definition in geometry (e.g. If you choose a trapezoid to have exactly one set of parallel sides then you get a different tree diagram). Students moved up in their Van Hiele level to informal deduction and began to progress into formal deduction Later students proved the new definitions equivalent and most began to perform at the formal deduction level. Symmetries with Geometer’s Sketchpad: Day 3: Discover a 3rd definition for the quadrilaterals using the properties grid sheet: Angle, side, diagonal and symmetry properties for each type of quadrilateral Definition #3 for each quadrilateral in terms of symmetries Symmetries with Geometer’s Sketchpad Quadrilaterals on the Sphere and the Hyperbolic Plane: Quadrilaterals on the Sphere and the Hyperbolic Plane Days 4 and 5: Introduction to Spherical and Hyperbolic Geometry Which of our equivalent definitions make sense in the context of each of these geometries? Final Project on Quadrilaterals:: Final Project on Quadrilaterals: Pick any three of the definitions for the same quadrilateral and prove they are equivalent. Point-wise definitions of symmetry make this challenging Next Time, We Will…: Next Time, We Will… Explore the connection between the definitions and the tree diagram more. Some students didn’t see property grid sheet connection Prove definition equivalences earlier Definition #1 Definition #2 Use Geometer's Sketchpad more effectively Make sure everyone has a working quadrilateral construction after Day 1 Define Convex versus Non-Convex figures earlier Axiomatic Systems and Proof: Axiomatic Systems and Proof Develop naturally as students analyze and make conjectures about triangles and quadrilaterals. Undefined terms, axioms and definitions are introduced and discussed as they arise in discussions and through explorations. Proof begins with informal deduction then to formal deduction and finally to rigor. Transformational Geometry: Transformational Geometry Start with informal discussion via patty paper, GeoTools and Miras Matrix / Functional representation and composition EnhancingProbability and Statistics: Enhancing Probability and Statistics Team: Alverna Champion Phyllis Curtiss John Gabrosek Course Syllabus: Course Syllabus Sampling (1 wk) Descriptive Statistics (2 wks) Probability (6 wks) Sampling Distributions (1 wk) Statistical Inference (3 wks) Regression (1 wk) Elementary Grade Activities: Elementary Grade Activities Investigations in Number, Data, and Space (Grade 5 - Probability) Everyday Mathematics (Grade 5 - Law of Large Numbers) Middle School Activities: Middle School Activities MathThematics (Book 1 - Regression) Connected Mathematics Project (Grade 6 - Descriptive Statistics) High School Activities: High School Activities Contemporary Mathematics in Context (Course 3 - Sampling and Confidence Intervals) Math Connections (Volume 3a - Expectation) Activity-Based Statistics (Sampling Distributions) Teaching Statistics (Vol. 15 - Hypothesis Testing) Sampling Distributions: Key Concepts: Sampling Distributions: Key Concepts Statistics vary from sample to sample For a simple random sample (SRS), as the sample size increases the sampling distribution of the sample mean becomes closer to a normal distribution For SRS, as the sample size increases the variability in the sample mean decreases Making Cents Out of Pennies: Making Cents Out of Pennies A population of 100 pennies is given to each group of students. Students draw simple random samples of various sizes from the population. Students investigate the effect of sample size on the distribution of the sample mean of the mint date. Source: Activity-Based Statistics Given a frequency table of the mint dates, students construct a histogram and describe its center, spread, and shape.: Given a frequency table of the mint dates, students construct a histogram and describe its center, spread, and shape. Each student draws a SRS of size 3 and finds the mean mint date.: Each student draws a SRS of size 3 and finds the mean mint date. Example: Dates: 1986, 1994, 1990 Students create a frequency table and histogram using the class sample means from SRS of size 3. : Students create a frequency table and histogram using the class sample means from SRS of size 3. Each student draws a SRS of size 10 and finds the mean mint date.: Each student draws a SRS of size 10 and finds the mean mint date. Example: Dates: 1998, 1994, 1981, 1999, 1997, 1996, 1998, 1970, 1998, 2000 Students create a frequency table and histogram using the class sample means from SRS of size 10.: Students create a frequency table and histogram using the class sample means from SRS of size 10. Students compare histograms and answer questions including:: Students compare histograms and answer questions including: Look at the three histograms. What can you say about the shape of the distribution of the sample means as the sample size increases? What can you say about the mean of the sample means as the sample size increases? What can you say about the standard deviation of the sample means as the sample size increases? An “A” Student Response: An “A” Student Response As the sample size increases, the values bunch closer to the mean of the population. As the sample size increases, the mean gets closer to the mean of the population, and the standard deviation gets smaller. Gets closer to the normal curve. A “B” Student Response: A “B” Student Response As the sample size increases the graph becomes more symmetrical. As the sample size increases the sample means seem to stay relatively close. As the sample size increases the standard deviation decreases. A “C” Student Response: A “C” Student Response SD = smaller as size increases. Mean is about the same. Becomes more symmetric. Enhancing Linear Algebra: Enhancing Linear Algebra Team: Matt Boelkins John Golden Clark Wells Our Process : Our Process Identify mathematical goals of the class Identify relevant parts of the standards Find relevant research about student learning of linear algebra Find and adapt activities to meet our goals Course Outlinethe Big Topics: Course Outline the Big Topics Systems of Linear Equations Matrix Algebra (Matrices as Quantities) Matrices as and in Functions Linear Dependence and Independence Invertible Matrix Theorem (and consequences) Vector Spaces Eigenvalues and Eigenvectors Relations to NCTM Content Standards: Relations to NCTM Content Standards One of the perceived problems with preservice teachers and linear algebra is the idea that the material is irrelevant, or mostly so, to them. But the NCTM standards make it clear that the content is directly relevant. Number and operation: develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases Number and operation The high school number and operation standards have explicit linear algebra content such as: develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices Algebra: but it was also clear how strongly all four strands of the algebra standards for high school and middle school applied. Some specific examples: High School: understand vectors and matrices as systems that have some of the properties of the real-number system; develop an understanding of properties of, and representations for, the addition and multiplication of vectors and matrices; develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases. Algebra Algebra, cont’d: Algebra, cont’d model and solve contextualized problems using various representations, such as graphs, tables, and equations. Even though middle school students do not use matrix algebra (typically) to address these, it’s clear that a linear algebra course could deepen the preservice teacher’s understanding. Middle School: represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules; explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope; How Students Learn Linear Algebra: How Students Learn Linear Algebra The best information we could find has already been collected in a single text: Resources for Teaching Linear Algebra, published by the MAA and edited by Carlson, Johnson, lay, Porter, Watkins and Watkins. The most useful materials (on curriculum planning) are probably those from the Linear Algebra Curriculum Study Group. Ed Dubinsky’s response to the LACSG has the most information about student learning of linear algebra. LA Curriculum Study Group Recommendations: LA Curriculum Study Group Recommendations The LACSG made five recommendations (paraphrased here), as well as making a pretty detailed suggested syllabus. The course must respond to needs of client disciplines. Math dept.s should make the first linear algebra course matrix oriented. Students’ needs and interests as learners should be considered. Technology should be utilized. A second course in linear algebra should be a high priority for every math curriculum. Searching the curricula: Searching the curricula We found relevant problems and examples in all the middle school and high school curricula in which we looked, and eventually drew examples from: Math in Context (8/9): the Toy Factory (Linear combinations) Contemporary Math in Context (Core+) (9): Why I is like 1 (matrices as number-like objects) Core + (10)/David Lay’s Text:Owls and Rats (eigenvalues, matrix valued functions) SIMMS (10): Family Snack (Matrix Multiplication) Interactive Mathematics Project (12): Merry Go Round (vectors) What is missing?: What is missing? We were impressed at the amount of linear algebra content in the curricula. But much of the more abstract content in the course, while it informs the applications, had no direct corollary in the grade 6-12 curricula. There is also so much linear algebra content in the curricula that not all of it is covered in our classes, the most notable and most common being a lot of linear programming. We estimate that we would need at least two weeks in the college course to address it in any depth, and there is not room in the class. One activity – a contextfor matrix multiplication: One activity – a context for matrix multiplication In general we felt the MS and HS curricula provided excellent concrete applications and contexts. One of the areas we thought needed more concrete examples was the idea of matrices as a quantity. The students spend a lot of time performing algebraic operations on matrices, but the course was missing an element where the matrices represent real life quantities. The following is from the activity on matrix multiplication. The context: Family Snack: The context: Family Snack A company called Family Snack is in the business of selling nuts, beef jerky and jam. They sell these items individually and in various combinations. Their Snack Pack consists of two boxes of nuts and 6 beef jerkies. The Gift Pack consists of two snack packs and three jars of jam. The Family Pack consists of 3 gift packs and three jars of jam. In the snack industry business the nuts, jerky and jam are referred to as simple components, and the packs as composite products. A Question: A Question One way of displaying this information is in a matrix, and there are several different ways that could be done. 1Explain how each matrix displays the information. 2 4 12 b) 1 0 0 2 4 12 6 12 36 0 1 0 6 12 36 0 3 12 0 0 1 0 3 12 0 0 0 1 2 6 0 0 0 0 1 3 0 0 0 0 0 1 A Question of Interpretation: A Question of Interpretation Once students are representing such quantities in matrices, questions can be asked both about interpretation and use of the more matrices in compatible ways. Such as: Family Snack has received an order for 20 boxes of nuts, 60 pieces of jerky, 48 jars of jam, 24 Snack Packs, 12 Gift Packs and 2 Family Packs. How many boxes of nuts, pieces of jerky and jars of jam are required to fill this order? Use one of the matrices above, and describe how you would use it to solve this problem. Further Questions in Context: Further Questions in Context Family Snack employs three sales people, Keyes, Zhang and Troy. In the first two weeks of September, Keyes sold 16 Snack Packs, 28 Gift Packs, and 8 Family Packs. In the same week, Zhang sold 12 Snack Packs, 36 Gift Packs, and 4 Family Packs and Troy sold 18 Snack Packs, 12 Gift Packs, and 11 Family Packs. Calculate (and show your calculations) the total value of the packs they sold. Sales people at Family Snack are paid in part by commission: $2.25, $4.20 and $7.20 for the different size packs. Write this information in a matrix, and use matrix multiplication to figure out the total commission earned by each salesperson in the first two weeks of September. In Conclusion: In Conclusion Student Benefits: Student Benefits Students acquire deeper and more connected understanding of mathematics and statistics. Prospective teachers (and parents) experience the level of mathematics embedded in K-12 curricula. Students are motivated to develop profound understanding of the mathematics they will teach (as teachers or parents). Faculty Benefits: Faculty Benefits Growth in understanding of national standards for school mathematics and content of K-12 mathematics curricula Deeper appreciation for the level of mathematics present in K-12 materials Stronger collegial and collaborative environment Large collection of activities to teach mathematics and statistics with understanding As We Continue...: As We Continue... We will benefit from your assistance. Questions? Comments? Suggestions? Thank You!: Thank You! Your participation today is greatly appreciated!