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What Important Mathematics do Elementary School Teachers “Need to Know”? (and)

What Important Mathematics do Elementary School Teachers “Need to Know”? (and). How Should Elementary School Teachers “Come to Know” Mathematics? Jim Lewis University of Nebraska-Lincoln. Math Matters. A Mathematics – Mathematics Education Partnership at the

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What Important Mathematics do Elementary School Teachers “Need to Know”? (and)

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  1. What Important Mathematics do Elementary School Teachers “Need to Know”? (and)

  2. How Should Elementary School Teachers “Come to Know” Mathematics? Jim Lewis University of Nebraska-Lincoln

  3. Math Matters A Mathematics – Mathematics Education Partnership at the University of Nebraska-Lincoln Ruth Heaton and Jim Lewis

  4. What’s the big deal? Can’t any good elementary school teacher teach children elementary school mathematics?

  5. What is so difficult about the preparation of mathematics teachers? • Our universities do not adequately prepare mathematics teachers for their mathematical needs in the school classroom. Most teachers cannot bridge the gap between what we teach them in the undergraduate curriculum and what they teach in schools. • We have not done nearly enough to help teachers understand the essential characteristics of mathematics: its precision, the ubiquity of logical reasoning, and its coherence as a discipline. • The goal is not to help future teachers learn mathematics but to make them better teachers. H. Wu

  6. What is so difficult ….? • For most future elementary school teachers the level of need is so basic, that what a mathematician might envision as an appropriate course is likely to be hopelessly over the heads of most of the students. • Forget what seems like a good thing to do in some idealized world, and get real: adapt the courses to the actual level and needs of the students. • The mathematics taught should be connected as directly as possible to the classroom. This is more important, the more abstract and powerful the principles are. Teachers cannot be expected to make the links on their own.

  7. Get teacher candidates to believe, that mathematics is something you think about - that validity comes from inner conviction that things make sense, that mathematical situations can be reasoned about on the basis of a few basic principles. • The goal is to have them develop some flexibility in their thinking, to be able to reason about elementary mathematics. Roger Howe

  8. Adding it Up argues that mathematical proficiency has five strands: • Conceptual understanding • Comprehension of mathematical concepts, operations, and relations • Procedural fluency • Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

  9. mathematical proficiency … • Strategic competence • Ability to formulate, represent, and solve mathematical problems • Adaptive reasoning • Capacity for logical thought, reflection, explanation, and justification • Productive disposition • Habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one’s on efficacy.

  10. Educating Teachers of Science, Mathematics, and Technology New Practices for the New Millennium A report of the National Research Council’s Committee on Science and Mathematics Teacher Preparation

  11. Educating Teachers … argues: 1) Many teachers are not adequately prepared to teach science and mathematics, in ways that bolster student learning and achievement. 2) The preparation of teachers does not meet the needs of the modern classroom. 3) Professional development for teachers may do little to enhance teachers’ content knowledge or the techniques and skills they need to teach science and mathematics effectively. and recommends: “a new partnership between K-12 schools and the higher education community designed to ensure high-quality teacher education and professional development for teachers.”

  12. The Mathematical Education of Teachers Is guided by two general themes: • the intellectual substance in school mathematics; and • the special nature of the mathematical knowledge needed for teaching.

  13. The MET recommends: • Prospective teachers need mathematics courses that develop a deep understanding of the mathematics they will teach. • Prospective elementary grade teachers (K-4) should take at least 9 semester-hours on fundamental ideas of elementary school mathematics.

  14. Mathematics courses should • focus on a thorough development of basic mathematical ideas. • develop careful reasoning and mathematical ‘common sense’ in analyzing conceptual relationships and in solving problems. • develop the habits of mind of a mathematical thinker and demonstrate flexible, interactive styles of teaching.

  15. The mathematical education of teachers should be based on • partnerships between mathematics and mathematics education faculty. • collaboration between mathematics faculty and school mathematics teachers.

  16. The Math Matters 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

  17. MATH MATTERS Timeline • Project began 01/01/2000 with NSF support. • AY 2000/2001, first Math Matters cohort (16 students) participated in a year-long, 18-hour block of courses. Each semester had a math class, a pedagogy class, and a field experience. • Significant progress in learning mathematics and in developing a positive attitude towards teaching math • Significant growth in potential to be an outstanding teacher • Strong bonding between students and faculty

  18. MATH MATTERS Timeline • Math Matters was repeated with a second cohort (16 students) during AY2001/2002 • Findings similar to first cohort • AY 2002/2003 – Two experiments with a one-semester, 12 hour block of courses (28 students, 19 students) • Fall 2003 – The Mathematics Semester begins • Current program is a one-semester, 10-hour, math, pedagogy, & field experience program.

  19. Barriers to a Successful Partnership • Math expectations seem to overwhelm students in Elementary Education • Student evaluations critical of math faculty Type of Course Faculty GPA #Students • Honors class 3.20 1,367 • All faculty courses 3.04 16,693 • Large Lectures 2.88 6,060 • Education Majors 2.48 726

  20. Comments from a math class for elementary school teachers: (the course GPA was 2.93) • This wasn't a course where we learn to teach math. Why do we have to explain our answers. • I did not like getting a 0 on problems that I attempted. I could have just of left the problem blank then. • tests are invalid. They ask questions we have never seen before. It would help if we knew more about the questions on the exams - If examples in class were used on the exams.

  21. Comments from a math class for elementary school teachers: (the course GPA was 2.93) • Her way of assessing her class aren't fair. • there was never partial credit. When 20 people drop a class ... there is an obvious problem. [Note: Only 3 of 33 students dropped the class.] • Test materials were not consistent or reliable with the material covered in class. Grading was very biased.

  22. Comments from a Contemporary Math class • (She) does a good job making the subject matter interesting. She always seems very enthusiastic about the class and and actual work. More teachers should be like her. • (She) is a great teacher with a love for her subject that becomes addictive. It has really been my lucky pick to have gotten her as an instructor. • (She) made the class exciting. It is obvious she enjoys math and teaching. She was always clear in her expectations and directions.

  23. Comments from a Contemporary Math class • This was a very useful class. I also think that (she) is a great teacher. • (She) was one of the best teachers I have had here at UNL. She was always available for questions! • This was a very good class. I failed the class last semester with a different teacher but (she) did a much better job and I am doing great in the class!

  24. Barriers to a Successful Partnership • Students took math courses before admission to Elementary Education Program • Math for Elementary Education was often taught by graduate students or part-time lecturers • Cultural differences in how instruction delivered and students assessed • Fall 2000 Undergraduate GPA by Dept. • Math 2.53 (UNL’s lowest) • Curr & Inst 3.64 (among highest)

  25. Beliefs of Math and El Ed Faculty1 2 3 4Strongly Disagree Disagree Agree Strongly Agree El Ed Math Question • 1.71 2.12 Stated from Traditional Viewpoint ------------------------------------------------------------------------------ • 2.00 2.92 Algorithms are best learned through repeated drill and practice. • 1.57 2.55 An advantage of teaching math is that there is one correct answer. • 2.00 3.22 Frequent drills on the basic facts are essential in order for children to learn them. • 1.83 2.70 Time should be spent practicing computational procedures before children are expected to understand the procedures. • 2.83 2.14 The use of key words is an effective way for children to solve word problems.

  26. Beliefs of Math and El Ed Faculty1 2 3 4Strongly Disagree Disagree Agree Strongly Agree • 3.27 3.07 Stated from Reform Viewpoint ------------------------------------------------------------------------ • 3.29 2.25 Teachers should let children work from their own assumptions when solving problems. • 3.86 2.78 Mathematics assessment should occur every day. • 2.71 3.40 Leading a class discussion is one of the most important skills for a math teacher.

  27. What is the typical ability of a future elementary school teacher at UNL? How would our students do on two famous problems? 1) How do you solve the problem 1 ¾ / ½ ? Imagine you are teaching division with fractions. Teachers need to write story-problems to show the application of some particular piece of content. What would be a good problem for 1¾/½ ?

  28. In Knowing and Teaching Elementary Mathematics this question was asked of 23 US teachers. Liping Ma reports: • 9 of 23 US teachers could perform the calculation using a correct algorithm and provide a complete answer. • Only 1 of 23 US teachers could create a valid story problem for the computation. It was pedagogically problematic in that the answer involved 3 ½ children.

  29. Surely our teachers and future teachers can do better? • In the summer of 2002, Ruth interviewed a group of Lincoln Public School teachers who were participating in a summer workshop Jim had organized. • 14 of 17 (elem and middle level) LPS teachers tested could perform the division of fractions algorithm correctly. • Only 4 out of 17 could create a valid word problem. • 3 of these 4 were middle school teachers • We also asked our students to work this problem. Algorithm Word Problem • Fall 2003 Students 23 of 25 8 of 25 • Year 2 (early in sem) 10 of 16 6 of 16 • Year 1 (mid year) 12 of 16 10 of 16

  30. Sample solutions from our class last Fall: Among the better answers we received were these: • Bobby only ran ½ the distance that he hoped to run in practice. If Bobby ran 1 ¾ miles and that was only ½ of the amount he wanted to run, what was his original goal? • Bill is running a race that is 1 ¾ miles. The director of the race wants to put markers every ½ mile. How many markers does he need?

  31. Sample solutions from our class this Fall: Other “solutions” included the following: • Suzie has one and ¾ pies. Ed is hungry and decides he is going to eat ½ of what Suzie has. When Ed is finished, how many halves of the pie does Suzie have left? • Jane has 7 quarters. She wants to put the quarters into groups of 2 quarters each. How many groups can she make? • You have 1 ¾ of a pie! That is not enough. You want ½ more of what they started with. How much more will you need? • Mom baked 2 pies and divided the pies in quarters. After giving away one slice she’s left with 1 ¾ pies. She then divides each left over piece into halves. How many pieces does she have now?

  32. 2) A student comes to class excited. She tells you she has figured out a theory you never told the class. She says she has discovered that as the perimeter of a closed figure increases, the area also increases. She shows you a picture to prove what she is doing. The square is 4 by 4 Perimeter = 16 Area = 16 The rectangle is 4 by 8 Perimeter = 24 Area = 32 Perimeter and Area

  33. The first day of our Fall 2002 class, we asked our (geometry) students to respond to the area – perimeter question. • 24 of 32 future elementary teachers believed the child’s theory was correct and indicated that they would congratulate the child. • Eight of 32 future elementary teachers questioned the child's theory about area and perimeter. • Only six of the eight explained that the theory was definitely not true.

  34. Mental Math Quiz(Work all problems in your head.) 1) 48 + 39 = 2) 113 – 98 = 3) 14 x 5 x 7 = 4) is closest to what integer? 5) 4 x 249 = 6) 6(37 + 63) + 18 = 7) .25 x 9 = 8) 12.03 + .4 + 2.36 = 9) 1/2 + 1/3 = 10) 90% of 160 = 11) The sum of the first ten odd positive integers (1+3+5+…+17+19) is equal to what integer? 12) If you buy items (tax included) at $1.99, $2.99 and $3.98, the change from a $10 bill would be? 13) To the nearest dollar, the sale price of a dress listed at $49.35 and sold at 25% off is _____?

  35. Mental Math Quiz(Work all problems in your head.) 14) The area of a square of perimeter 20 is ___? • The ratio of the area of a circle of radius one to that of a circumscribed square region is closest to? a) .5, b) .6, c) .7, d) .8, e) .9 16) The average (arithmetic mean) of 89, 94, 85, 90, and 97 is _____ ? 17) If 4/6 = 16/x, then x = _____ ? 18) If 2x + 3 = 25, then x = _____ ? 19) The square root of 75 is closest to what integer? 20) To the nearest dollar, a 15% tip on a restaurant bill of $79.87 is _____ ?

  36. How do UNL’s future elementary school teacher do on the Mental Math Quiz? • Group Number Median Average Correct Correct • MM Fall 2000 16 15.5 14.8 • GW Fall 2003 21 13 13.1 • LOK Fall 2003* 19 12 12.7 • SW Fall 2003* 23 12 12.4 • MM Fall 2001 16 12 12.3 • BH Fall 2000 32 12 11.9 • JL Fall 2003 24 12 11.8 • MM Fall 2002* 28 12 11.6 • MG Fall 2001 35 12 11.5 • WH Fall 2000 24 11 11.0 • WH Fall 2001 32 10 10.9 • TM Fall 2003* 20 10.5 10.4 • Total 290 12 11.9 * LPS Sum Wks. 22 13.5 13.7 * Have already taken the “Arithmetic” course.

  37. The Mathematics Semester(For all Elementary Education majors starting Fall 2003) MATH • Math 300 – Number and Number Sense (3 cr) PEDAGOGY • CURR 308 – Math Methods (3 cr) • CURR 351 – The Learner Centered Classroom (2 cr) FIELD EXPERIENCE • CURR 297b – Professional Practicum Exper. (2 cr) (at Roper Elementary School) • Students are in Roper Elementary School on Mondays and Wednesdays (four hours/day) • Math 300 & CURR 308 are taught as a 3-hour block on Tuesday and Thursday • CURR 351 meets at Roper on Wednesdays

  38. A look inside Math Matters(and The Mathematics Semester) • Early math assignments establish our expectations • Professional/Reflective Writings • Curriculum Materials • Number and Number Sense Items • Working together as a team • The Curriculum Project • Some Geometry Assignments • Activities at Roper • Teaching a Math Lesson • Child Study • Learning and Teaching Project

  39. A look inside Math Matters(and The Mathematics Semester) • Early math assignments establish our expectations • Professional/Reflective Writings • Curriculum Materials • Number and Number Sense Items • Working together as a team • The Curriculum Project • Some Geometry Assignments • Activities at Roper • Teaching a Math Lesson • Child Study • Learning and Teaching Project

  40. Fall 2001 – The Rice Problem • (This assignment builds on a N&NS problem.) Recall our discussion about the game of chess and how a humble servant for a generous king invented it. The king became fascinated by the game and offered the servant gold or jewels in payment, but the servant replied that he only wanted rice—one grain for the first square of the chess board, two on the second, four on the third, and so on with each square receiving twice as much as the previous square. In class we discussed how the total amount of rice was 264 grains of rice. (To be completely precise, it is this number minus one grains of rice.) Suppose it was your job to pick up the rice. What might you use to collect the rice, a grocery sack, a wheelbarrow, or perhaps a Mac truck? Where might you store the rice?

  41. Fall 2001 – A letter from 1st & 2nd graders 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 2 and then a 1 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 do you call a number if it has more than 100 zeros? Is there a name for it?

  42. 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 do you put the commas in? It doesn’t divide evenly into groups of 3 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

  43. clipping from Kid City magazine Apple Of My Eye The tiny female apple aphid is a champ as an egg-layer. This insect can lay as many as 21000000000000 00000000000000000000000 00000000000000000000000 00000000000000000000000 00000000000000000000000 0000 eggs in 10 months.

  44. Fall 2003 – 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?

  45. A look inside Math Matters(and The Mathematics Semester) • Early math assignments establish our expectations • Professional/Reflective Writings • Curriculum Materials • Number and Number Sense Items • Working together as a team • The Curriculum Project • Some Geometry Assignments • Activities at Roper • Teaching a Math Lesson • Child Study • Learning and Teaching Project

  46. Professional/Reflective Writings Purpose: To make connections within and across mathematical, pedagogical, and field experiences through writing.

  47. Professional/Reflective Writings Read “What do Math Teachers Need to Be?” The author is Herb Clemens, a mathematics professor at The University of Utah, and the article was published in 1991 in Teaching academic subjects to diverse learners (pp. 84-96). In this article, Herb Clemens lists what he thinks teachers of mathematics need to be. After reading his article and his meaning and use of these words, where does your own practice of teaching mathematics stand in relationship to what Clemens says mathematics teachers need to be:  unafraid, reverent, humble, opportunistic, versatile, and in control of their math. On p. 92, Clemens lists four fundamental questions about mathematics teaching that matter to him. If he came to your practicum classroom and watched you teach a math lesson tomorrow, how would he answer his own last question about your practice: Can this teacher teach it [math] with conviction, and with some feeling for its essence? Explain.

  48. Professional/Reflective Writings Some educators argue that there is real value in teaching children mathematics in diverse, heterogeneous classrooms. Some teachers may counter this position, contending that it is best for children if students are homogeneously grouped for mathematics instruction. Pick a position in this argument and articulate it in writing. State your position and explain why you believe what you do. Your reasons for believing what you do may come from past teaching and learning experiences (your own and others’), and things that you’ve read, or learned in other courses.

  49. Professional/Reflective Writings Read "Teaching While Leading a Whole-Class Discussion," Chapter 7 from Lampert’s book. In this chapter Lampert examines problems of practice that arise while addressing a whole group of students or choosing students to answer questions. As you read the chapter find places in the chapter where you can relate Lampert's writing to your own experiences in the practicum setting while teaching math and maybe even other subjects. Use quotes from the text that connect to your experiences. Explain how and why they relate.

  50. A look inside Math Matters(and The Mathematics Semester) • Early math assignments establish our expectations • Professional/Reflective Writings • Curriculum Materials • Number and Number Sense Items • Working together as a team • The Curriculum Project • Some Geometry Assignments • Activities at Roper • Teaching a Math Lesson • Child Study • Learning and Teaching Project

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