The Mathematical Education of K-8 Teachers at the University of Nebraska-Lincoln, a Mathematics – Mathematics Education Partnership Jim Lewis and Cheryl Olsen University of Nebraska-Lincoln The Mathematics Semester for future elementary school teachers and Math in the Middle Institute Partnership a professional development program for middle level teachers The Mathematics Semester Math Matters, an NSF-funded CCLI grant began in 2000, and became The Mathematics Semester in Fall 2003 Vision • Create a mathematician – mathematics educator partnership with the goal of improving the mathematics education of future elementary school teachers • Link field experiences, pedagogy and mathematics instruction • Create math classes that are both accessible and useful for future elementary school teachers The Mathematics Semester (For all Elementary Education majors starting Fall 2003) • • • • MATH Math 300 – Number and Number Sense (3 cr) PEDAGOGY TEAC 308 – Math Methods (3 cr) TEAC 351 – The Learner Centered Classroom (2 cr) FIELD EXPERIENCE TEAC 297b – Professional Practicum Exper (2 cr) (a field experience in a local elementary school) – Students are in an elementary school on Mondays and Wednesdays (four hours/day) – Math 300 & TEAC 308 are taught as a 3-hour block on Tuesday and Thursday – TEAC 351 is taught on site at a participating elementary school A look inside The Mathematics Semester • Curriculum materials • Homework to develop mathematical habits of mind • Professional writings • The curriculum project • Learning and Teaching Project • Activities at the elementary school – Teaching a Math Lesson – Child Study Curriculum Materials • Sowder, J et al (2007) Reasoning about Numbers and Quantities W H Freeman (prepublication copy) • Schifter, D., Bastable, V., & Russell, S.J (1999) Number and operations, part I: Building a system of tens Parsippany, NJ: Dale Seymour • Reys, Lindquist, Lambdin, Smith, & Suydam (2007) Helping Children Learn Mathematics John Wiley & Sons • Lampert, M (2001) Teaching problems and the problems of teaching New Haven, CT: Yale University A problem to get started Making change What is the fewest number of coins that it will take to make 43 cents if you have available pennies, nickels, dimes, and quarters? After you have solved this problem, provide an explanation that proves that your answer is correct? How does the answer (and the justification) change if you only have pennies, dimes, and quarters available? Note: We first encountered this problem in a conversation with Deborah Ball A Typical Weekly Homework All Shook Up Five couples met one evening at a local restaurant for dinner Alicia and her husband Samuel arrived first As the others came in some shook hands and some did not No one shook hands with his or her own spouse At the end Alicia noted that each of the other people had shaken the hands of a different number of people That is, one shook no one's hand, one shook one, one shook two, etc., all the way to one who shook hands with of the people How many people did Samuel shake hands with? Note: This problem is a slightly modified version of Exercise #14, page 32, in The Heart of Mathematics, by Burger and Starbird, Key Curriculum Publishing, 2005 Professional Writings Dear Math Professors, We are 1st and 2nd graders in Wheeler Central Public School in Erickson, Nebraska We love to work with big numbers and have been doing it all year! Every time we read something with a big number in it we try to write it Then our teacher explains how to write it We are getting pretty good at writing millions and billions! We have a problem that we need your help with We were reading amazing ‘Super Mom’ facts in a Kid City magazine It told how many eggs some animals could lay We came across a number that we don’t know It had a and then a followed by 105 zeros!! We wrote the number out and it stretches clear across our classroom! We know about a googol We looked it up in the dictionary A googol has 100 zeros Then what you call a number if it has more than 100 zeros? Is there a name for it? Another problem is that we learned about using commas in large numbers In the magazine article they used no commas when writing this large number That confused us Also, if you write a ‘googol’ with 100 zeros, how you put the commas in? It doesn’t divide evenly into groups of zeros There will be one left over We appreciate any help you can give us solving this “big” problem Thank you for your time Sincerely, Mrs Thompson’s 1st & 2nd graders Megan Kansier, Mark Rogers Marcus Witt, Ashley Johnson Math in the Middle Institute Partnership M2 courses focus on these objectives: • enhancing mathematical knowledge • enabling teachers to transfer mathematics they have learned into their classrooms • leadership development and • action research Math in the Middle Institute Design Summer Fall Spring Yr Wk1 Wk2&3 M800T Teac800 & M802T M804T M805T Yr M806T Teac801 & Stat892 Teac888 M807T Yr M808T Teac889/M809T and the Masters Exam - A 25-month, 36-hour graduate program M2 Summer Institute • • • • • Combination of week and week classes Teachers are in class from 8:00 a.m - 5:00 p.m 32-35 teachers – instructors in class at one time Substantial homework each night Substantial End-of-Course problem set – Purpose – long term retention of knowledge gained – Presentation of solutions/celebration of success at start of next class M2 Academic Year Courses • Two-day (8:00 – 5:00) on-campus class session • Course completed as an on-line, distance education course using Blackboard and Breeze – – – – – Major problem sets Professional Writings Learning and Teaching Projects End-of-Course problem set Substantial support available for teachers M2 Institute Courses • Eight new mathematics and statistics courses designed for middle level teachers (Grades – 8) including: – – – – Mathematics as a Second Language Experimentation, Conjecture and Reasoning Number Theory and Cryptology for Middle Level Teachers Using Mathematics to Understand our World • Special sections of three pedagogical courses: – Inquiry into Teaching and Learning – Curriculum Inquiry – Teacher as Scholarly Practitioner • An integrated capstone course: – Masters Seminar/Integrating the Learning and Teaching of Mathematics Math 800T - Mathematics as a Second Language • The “text” was written by Kenneth and Herbert Gross of the Vermont Mathematics Initiative • Ken helped us “kick off” our first weekend • Innovations (i.e additions) – Habits of Mind problems – Learning and Teaching Project M2 Innovations “Habits of Mind” Problems A person with the habits of mind of a mathematical thinker can use their knowledge to make conjectures, to reason, and to solve problems Their use of mathematics is marked by great flexibility of thinking together with the strong belief that precise definitions are important They use both direct and indirect arguments and make connections between the problem being considered and their mathematical knowledge When presented with a problem to solve, they will assess the problem, collect appropriate information, find pathways to the answer, and be able to explain that answer clearly to others While an effective mathematical toolbox certainly includes algorithms, a person with well developed habits of mind knows both why algorithms work and under what circumstances an algorithm will be most effective M2 Innovations “Habits of Mind” Problems Mathematical habits of mind are also marked by ease of calculation and estimation as well as persistence in pursuing solutions to problems A person with well developed habits of mind has a disposition to analyze situations as well as the selfefficacy to believe that he or she can make progress toward a solution This definition was built with help from Mark Driscoll’s book, Fostering Algebraic Thinking: A guide for teachers grades 6-10 The Triangle Game A “Habits of Mind” Problem E D F A B C (Paul Sally, U Chicago) Consider an equilateral triangle with points located at each vertex and at each midpoint of a side The problem uses the set of numbers {1, 2, 3, 4, 5, 6} Find a way to put one of the numbers on each point so that the sum of the numbers along any side is equal to the sum of the numbers along each of the two other sides (Call this a Side Sum.) – Is it possible to have two different Side Sums? – What Side Sums are possible? – How can you generalize this game? M2 Innovations Learning & Teaching Projects Select a challenging problem or topic that you have studied in MSL and use it as the basis for a mathematics lesson that you will videotape yourself teaching to your students How can you present this task to the students you teach? How can you set the stage for your students to understand the problem? How far can your students go in exploring this problem? You want your students to discover as much as possible on their own, but there may be a critical point where you need to guide them over an intellectual “bump.” Produce a report analyzing the mathematics and your teaching experience Action Research Each teacher takes TEAC 888, Teacher as Scholarly Practitioner, an action research course Each teacher then conducts their own research project and writes a report about their findings – “Action research is research done by teachers for themselves; it is not imposed on them by someone else” (Mills, 2003, p 5, italics in original) – In conducting action research, drawing conclusions isn’t about making generalizations for others but about deciding on a course of action for one’s own teaching – In 2006, 31 teachers had 29 different research projects involving 29 IRB documents – Each teacher posed research questions, used forms of data collection and used at least from their literature review M2 Masters Degrees • Two options for the Masters Degree – MAT (Specialization in the teaching of middle level mathematics (Mathematics Department) – MA (Teaching, Learning and Teacher Ed.) • Masters exam in mathematics – – – – Take home exam (two math questions) Write 5-7 page report on action research project Write an 8-10 page expository paper Give an oral presentation about the paper A Sample Masters Exam Question A math class with “n” students sits in a circle to play mathematical chairs The students choose an elimination number “d” and then count off in order, 1, 2, 3, … When the count gets to d, that student is eliminated from the game The next student starts the count over and the students count 1, 2, 3, … Again, when the count gets to d, that student is eliminated Continue in this manner until only one student is left That student wins the game Where should you sit in order to win the game? Hint: Solve the problem first for elimination number or and then try to solve it for elimination number d Note: This is a version of The Ring of Josephus problem M2 Research Questions • What are the capacities of teachers to translate the mathematical knowledge and habits of mind acquired through the professional development opportunities of M2 into measurable changes in teaching practices? • To what extent observable changes in mathematics teaching practice translate into measurable improvement in student performance? What are we learning? • Integrate content and pedagogy courses • Keep expectations of students or teachers high • Emphasize learning how to learn and offer continued opportunities • Build on existing relationships • Commitment to the partnership need to be long term