The Mathematics Pre-Service Teachers Need to Know R James Milgram Department of Mathematics, Stanford University, Stanford, California, 94305 E-mail address: milgram@math.stanford.edu The author was supported in part by a Grant from the U.S Department of Education Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author and not necessarily reflect the views of the United States Department of Education Copyright 2005, R James Milgram Permission granted to reproduce for educational purposes and distribute for cost of reproduction and distribution For my students at the University of New Mexico and Skip Matthes To my wife Judy, my son Jules, my daughter Jean, and dedicated to the memory of Martha, my sister Contents Preface vii Chapter Introduction Why is math important? The critical role of mathematics in modern society Common misconceptions by pre-service teachers about mathematics The mathematics K - teachers need to know Mathematical problem solving Chapter - the mathematics students need to know Chapter - the core material central to all school mathematics The detailed course discussions, chapters - Chapter 9, experiences teaching pre- and in-service teachers 10 The book The Mathematical Education of Teachers 10 11 13 14 16 21 21 Chapter The Basic Topics in K - Mathematics Introduction Place Value and Basic Number Skills Fractions Ratios, Rates and Percents The Core Processes of Mathematics Functions and Equations Real Measurement and Measurement in Geometry Course Outline for the First Course Course Outline for the Fractions Course 23 23 24 27 33 37 46 51 57 70 Chapter Topics Needing Special Attention in all Four Courses Introduction Precision Making Sense of Mathematics for Students Abstraction Definitions Problem Solving: Overview Well-Posed and Ill-Posed Problems in K-8 Mathematics Problems with Hidden Assumptions Problems where Psychology Affects the Outcome 10 Patterns in School Mathematics iii 73 73 74 79 82 85 92 96 100 101 104 iv CONTENTS 11 12 13 14 Parsing Word Problems Real World Problems Polya’s Four Step Problem Solving Model Working With Problems for Elementary Teachers 107 110 111 114 Chapter Issues in the Basics Course Introduction: Foundational Mathematics in the Early Grades Whole Numbers: First Steps Addition and Subtraction Multiplication and Division Magnitude and Comparison of numbers Place value Decimals Bringing in the Number Line Other systems for writing numbers 10 Algorithms and their Realizations 11 Algorithms: Addition and Subtraction 12 Algorithms: Multiplication 13 Rounding, Approximation, and Estimation 14 Algorithms: Division 15 Factoring, Multiplication and Division 16 Fractions: First Steps 17 Average, Rates, Ratios, Proportions, and Percents 121 121 122 128 146 153 157 168 171 175 177 187 197 207 209 215 218 221 Chapter Fractions, Ratios, Percents, and Proportion Definition of fractions and immediate consequences Negative fractions Arithmetic operations Complex fractions Percent Ratios and Rates Alternative Development of Ratios, Rates, and Percents Finite decimals Infinite decimals 10 False Periods for Repeating Fractions 11 The two-sided number line and the rational numbers 12 The arithmetic operations on rational numbers 13 Ordering rational numbers 14 The Fundamental Assumption of School Mathematics 15 Sample Problems from Other Nations 227 230 240 240 248 250 252 255 263 265 270 271 272 279 280 282 Chapter The Role of Technology in Mathematics Instruction Introduction Introducing the Graphing Calculator and Its Functions Introducing Calculators Into The Classroom Mathematical Activities 289 289 290 292 315 CONTENTS References v 334 Chapter Discussion of Issues in the Geometry Course Introductory Comments Lines, Planes and Figures in Space Length and Perimeter Angles and Arc Length on the Circle Polygons in the plane Measurement, Perimeter, Area, and Volume Congruence and Similarity Grade 8: Scale Factors Coordinate Geometry 10 The Euclidian Group: I 11 Euclidean Group II: Reflections and Applications 12 Optional discussion of relation to optics 13 Similarity and Dilations 14 Geometric Patterns - Symmetry 15 Geometry in Space 16 Length and Euclidian Group in Space 17 A Problem Solving Example in Plane Geometry 337 337 338 342 349 353 354 366 372 374 377 381 386 388 389 392 393 394 Chapter Discussion of Issues in the Algebra Course Introduction Objectives of the Algebra Course and Key Definitions Variables and Constants Decomposing and Setting Up Word Problems Symbolic Manipulation Functions Graphs of Equations Contrasted with Graphs of Functions Symbolic Manipulation and Graphs Linear Functions 10 Polynomials 11 Rational Functions 12 Inductive Reasoning and Mathematical Induction 13 Combinations, Permutations and Pascal’s Triangle 14 Problem Solving Applications of Binomial Coefficients 15 Compound interest 401 401 402 405 408 416 417 422 424 425 430 439 450 454 466 473 Chapter Experiences in Teaching Math to Pre-Service and In-Service Teachers Comments on the need for mathematician involvement in pre-service teacher training Some points to consider in teaching pre-service elementary teachers A Mathematician’s Thoughts on Teacher In-service Learning Comments on the Issues of Pre-Service Teachers 475 475 478 482 484 vi CONTENTS Mathematics for elementary teachers: “Explaining why” in ways that travel into the school classroom 488 Teaching Math for elementary Ed majors 490 The Geometry of Surprise 495 Vermont Mathematics Initiative (VMI) 497 Appendix A Singapore Grade Level Standards Arranged by Topic Multiplication and division Decimals Standard multiplication algorithm Rounding, approximation, and estimation Standard division algorithm Fractions Rates, ratios, proportion, and percent Lines, planes, space Length and perimeter 10 Angles and arc-length on the circle 11 Length, perimeter, area, volume 12 Congruence and similarity 499 499 500 501 501 502 502 504 505 505 506 509 510 Appendix B 511 Algorithms from the Education Perspective Appendix C The Foundations of Geometry A Model for Geometry on the Line A Model for the Plane and Lines in the Plane Distance in the Plane and Some Consequences Further Properties of Lines in the Plane Rays and Angles in the Plane Euclid’s Axioms in the model for Plane Geometry 515 516 520 523 526 528 531 Appendix D The Sixth Grade Treatment of Geometry in the Russian Program 533 Appendix E The Sixth Grade Treatment of Algebra in the Russian Program 543 Preface It has long been felt that the mathematical preparation of pre-service teachers throughout the country has been far too variable, and often too skimpy to support the kind of outcomes that the United States currently needs Too few of our K - 12 graduates are able to work in technical areas or obtain college degrees in technical subjects This impacts society in many and increasingly harmful ways, and it is our failure in K - mathematics instruction that is at the heart of the problem This is especially true when we compare outcomes in the United States with outcomes in countries that a better job of teaching mathematics, countries such as Poland, Hungary, Bulgaria, Romania, Singapore, China, and Japan, to name a few It has also been increasingly recognized that if we are to improve our performance in K - mathematics instruction, pre-service teachers should take focused, carefully designed courses directly from the mathematics departments, and not, as is often the case, just a single math methods course taught in the Education School A focused two year sequence in the basic mathematics teachers have to know is the minimal mathematics sequence that pre-service teachers need in order to to successfully teach students in K - The United States Department of Education under the guidance of Secretary Paige awarded an FIE (Fund for the Improvement of Education) grant to Doug Carnine, Tom Loveless and R James Milgram in 2002 to analyze the reasons for the success of these foreign programs and produce a book, designed for the use of mathematics departments in constructing a two year sequence sequence of courses that will achieve this goal A critical part of the project was an advisory committee comprised of many of the top people in this country concerned with the issues of K - 12 mathematics education and outcomes Their advice has been critical in the development of this book The members of the advisory committee: Prof Richard Askey, Department of Mathematics, University of Wisconsin (emeritus) Prof Deborah Ball, School of Education, University of Michigan Prof Hyman Bass, Department of Mathematics and School of Education, University of Michigan Prof Sybilla Beckmann, Department of Mathematics, University of Georgia vii viii PREFACE Dr Tom Fortmann, Mass Insight Education, Boston, Massachusetts Prof Sol Friedberg, Department of Mathematics, Boston College Prof Karen Fuson, School of Education, Northwestern University (emerita) Prof Ken Gross, Department of Mathematics, University of Vermont Prof Roger Howe, Department of Mathematics, Yale University Kathi King, Messalonskee High School, Oakland, Maine Prof Jim Lewis, Department of Mathematics, University of Nebraska Prof David Klein, Department of Mathematics, California State University, Northridge Prof Stan Metzenberg, Department of Biology, California State University, Northridge Prof Ira Papick, Department of Mathematics, University of Missouri Prof Tom Parker, Department of Mathematics, Michigan State University Prof Paul Sally, Department of Mathematics, University of Chicago Prof Uri Treisman, Department of Mathematics, University of Texas at Austin Prof Kristin Umland, Department of Mathematics, University of New Mexico Prof H.-H Wu, Department of Mathematics, University of California, Berkeley We have also benefitted from the advice of Barry Garelick and Karen Jones-Budd Prof Klein played a critical role in the writing of most of the chapters - Prof H.-H Wu also deserves special thanks for help beyond the call, as Prof Beckmann, Prof Fuson Prof Parker, and Prof Umland A second component of the FIE grant was to study the issues needed to construct successful in-service mathematics training Both Prof Sally and Prof Gross have been running long-term in-service training and the grant has helped them collect data on their outcomes, though, at this time, the data is still being analyzed We would like to thank Susan Sclafani, Assistant Secretary of Education, and above all Pat Ross of the U.S Department of Education for their help and support We would also like to thank Tom Kelly at Cappelli Miles [Spring] for assistance with design and layout, as well as the people at Direction Service who managed the grant, particularly Aimee Taylor and Marshall Peter CHAPTER Introduction It is well known that for many years mathematics outcomes for K - 12 students in this country have lagged far behind what they should be This is clearly illustrated by the results of the TIMSS tests, which show our students about average internationally in grade 4, significantly below average in grade 8, and near the bottom by grade 12 It is also illustrated by the very low numbers of United States students who graduate from college with degrees in technical areas The level and quality of the highest mathematics courses that students successfully take in K - 12 is the greatest single predictor of degree completion in college, and the data clearly show that Algebra II is the college “gatekeeper.”1 Mathematics is the key component of success in any technical area If we are to prepare our students to maximize their opportunities to succeed in today’s society, then improving their backgrounds in mathematics is the key Moreover, there is only so much that can be done to improve outcomes by improving the quality of the texts they use and focusing instruction on the most critical topics In California, in 1997 - 1998, for the first time in many years, mathematicians were asked to write the state mathematics standards Clifford Adelman, Answers in the Tool Box, Academic Intensity, Attendance Patterns, and Bachelor’s Degree Attainment, U.S Dept of Education, 1999, p 17 D THE SIXTH GRADE TREATMENT OF GEOMETRY IN THE RUSSIAN PROGRAM 541 (1) How many triangles can be constructed congruent to a given scalene triangle ABC such that they share a given segment as a side? (2) Prove that any two such congruent triangles are symmetrical with respect to some axis (39) Pg 87, #290 According to tradition, the ancient Greek mathematician Thales was the first to solve the problem of calculating the distance from shore to a ship To this he measured the distance |AB| and angle ABC (fig 139) and then, performing several constructions and measurements on land, he calculated the distance |AC| What constructions and measurements might Thales have performed in solving the problem? What was his solution based on? (40) Pg 93, #309 Sides AB and CD and also angles A and D of quadrilateral ABCD are congruent Prove the congruence of angles B and C (41) Pg 93, #310 Given two pointe A and B What is the figure formed by the set of points X such that (1) |AX = |BX|; (2) |AX| ≥ |BX|; (3) |AX| < |BX|? (42) Pg 93, #311 Towns A and B are located on the same side of a railway line (fig 153) At what point on this line should a platform C be built so that (1) the distances |AC| and |BC| are equal; (2) the sum of the distance |AC| and |BC| is the smallest possible; (3) the difference between |AC| and |BC| is the greatest possible? (43) Pg 96, #324 How can you use a hinged mechanism with sections of equal length (fig 159) to construct (1) the bisector of a given 542 D THE SIXTH GRADE TREATMENT OF GEOMETRY IN THE RUSSIAN PROGRAM angle; (2) the midpoint of a given segment; (3) the center of a given circle? (44) Pg 100, #332 Given a point M within triangle ABC, prove that the sum of the distances from this point to the vertices of the triangle is greater than the sum of the distances from this point to the sides of the triangle Circles again: (45) Pg 107, #356 A circle is divided by two points into two arcs What is the measure of each of these arcs if: (1) the measure of one of them is 30◦ greater than the measure of the other; (2) the measure of these arcs is proportional to the numbers and 3? (46) Pg 107, #357 (1) Construct and equilateral triangle whose vertices lie on a given circle (2) Use a ruler and a protractor to construct a regular five-pointed star (47) Pg 110, #366 Construct a circle of a given radius r tangent to a given line a at a given point M on the line (48) Pg 111, #371 Construct a circle tangent to all the sides of a given triangle (49) Pg 115, #376 Construct a tangent to a given circle passing through a given point APPENDIX E The Sixth Grade Treatment of Algebra in the Russian Program UCSMP translated the sixth grade Russian textbook Algebra, Fifth Edition, (1981) by Yu N Makarychev, N.G Mindyuk, and K.S Muravin, which is currently only available for private research purposes We give here a brief survey of the program, along with a number of sample problems to give a feel for what this part of the Russian sixth grade mathematics curriculum is like: The table of contents for this volume is as follows: 543 544 E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM The beginning discussion of expressions is quite a bit more precise and careful than is typically the case with U.S programs For example, here is the last paragraph: 38 The expression (48/6)−8 has no numerical value since not all of the indicated operations can be performed (you cannot divide by zero!) Such expressions are said to be meaningless Which of the following expressions are meaningless: 7.845 a) 4.18−2.09·2 ; b) ·9−20 ; c) −3 ? 0.8−1.2· Without doing the calculations, compare the values of the following expressions: 7 a) 640 · 16 and 640/ 16 ; b) 243 · − 10 and 243/ − 10 27 27 (1) 41 Give some pair (x, y) of values for the variables x and y which turns the following propositions into true statements: a) city x is farther north than city y; b) word x is part of speech y 47 Let n be a natural number Give: a) the number following n in the sequence of natural numbers; b) the number preceding n (n > 1) in the sequence of natural numbers; c) the product of two sequential natural numbers, the lessor of which is n; d) the product of three sequential natural numbers, the greatest of which is n (n > 2) 64 One reservoir contains 380m3 of water and another contains 1,500m3 of water 80m3 of water enters the first reservoir every hour, and E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM 545 60m3 flows out of the second every hour In how many hours will there be an equal amount of water in the reservoirs? 86 a) One side of a triangle is twice as long as the second side, and 3cm shorter than the third side Find the lengths of the sides of the triangle, given that the perimeter of the triangle is 33cm b) A rectangular sports field is enclosed by a fence 320m long Find the area of the field, given that it is 40m longer than it is wide 126 One shelf contained twice as many books as a second shelf After 14 books were removed from the first shelf and books were removed from the second shelf, there were again twice as many books on the first shelf as on the second How many books did each shelf contain originally? Paragraph of Chapter II starts out as follows: If the domain of definition of a function is a finite set, the number of elements of which is not very great, then the function can be given by a listing of all pairs of corresponding elements Tush, the function f , given by the description “each two-digit number less than 16 has a corresponding remainder from the division of the number by 4,” can be given by listing all pairs of corresponding elements: (10, 2), (11, 3), (12, 0), (13, 1), (14, 2), (15, 3) Arrows can be used to represent pairs of corresponding elements The same set of pairs can be written in a table, where x is the variable whose value forms the domain of definition of function f and y is the variable whose value comprises the range of values of the function: (Note the care taken to make sure that the domain and range in these cases are small finite sets The implication that such methods can be used to give a function with infinite domains and ranges is a persistent error in U.S texts.) 546 E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM This is further clarified in Paragraph where methods for giving a function via formulas are discussed 176 The number corresponds to each two-digit prime number, and the number corresponds to each two-digit non-prime number What number corresponds to 12, to 17, to 29, to 99? Does this description express a function? 177 Function r is given by the description “for every natural number there is a corresponding remainder when the number is divide by 4.” What number is the image of 13, 120, 162, 999? What is the range of values of function r? Direct proportion is also handled very carefully After a thorough discussion students are expected to solve problems like these: 227 Divide the number 468 into parts proportional to the numbers 3, 4, 228 Find the angles of a triangle, given that they are proportional to the numbers: a) 2, 3, 10; b) 1, 3, 231 An alloy consists of copper, zinc, and nickel, the masses of which are proportional to the numbers 13, 4, What is the mass of the alloy if it is known to contain 2.4kg more copper than nickel? item[302.] Given the sets A = {−3, −1, 0, 7} and B = {−2, 3, 5} The relation between sets A and B is given by the proposition “a + b > 0,” where a ∈ A and b ∈ B Give this relation by means of arrows 303 The relation between sets A = {3, 8, 11, 15} and B = {5, 7, 10} is given by the proposition “The difference a − b is a positive integer,” where a ∈ A and b ∈ B Give this relation by means of a list of pairs 371 The domain of definition of the function y = f (x) is the set of all non-zero numbers If x ∈ (−∞, 0), then the function is given by the formula y = − 12 , and if x ∈ (0, +∞), then by the formula y = 12 x x Graph the function f How can the function f be given by a single formula? E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM 547 The handling of inverse proportion is worth noting: 253 Are the following assertions true: a) The time it takes a train to travel from A to B is inversely proportional to its speed, on the condition that the train moves at a constant speed 548 E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM 256 261 363 364 b) The time it takes a typist to type a manuscript is inversely proportional to the number of pages she can type in one hour c) The number of pages in a book that have been read is inversely proportional to the number of pages which remain to be read A bar of aluminum and a bar of iron have identical masses Which bare has the greater volume, and how much greater, if it is given that the density of aluminum is 27g/cm3 and the density of iron is 7.8g/cm3 ? It took 18 hours for a ship to travel downstream How long will the return trip take if the ship’s speed is 26km/hr and the river current is 2km/hr? The length of a rectangular parallelepiped was doubled, and its width was tripled Hom must the height be changed for the volume of the parallelepiped to remain the same? In hours a freight train covers the distance that a passenger train covers in hours The trains left two towns at the same time, heading towards one another When they met the passenger train had traveled 180km How far had the freight train traveled? What was the distance between the towns? Here is the way the program handles division of powers: E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM 460 Prove that for any natural number n, the value of the fraction is a natural number 549 10n +2 550 E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM Here is the way the program begins the general discussion of polynomials: 566 A student found that the values of two expressions in a single variable were different for a certain value of the variable Is this enough to assert that these expressions are not identically equal on the set of all numbers? E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM 551 The discussion continues as follows after a complete discussion of monomials: The problems that follow are some of the more interesting problems from the rest of the book except for the final chapter on systems of equations: 594 Some algebra students were given the problem: “Find the value of the expression (7a3 − ga2 b + 5ab2 ) + (5a3 + 7a2 b + 3ab2 ) − (10a3 + a2 b + 8ab2 ) for a = 0.25, b = −0.347.” One of the students stated that there was superfluous data in the problem Was he correct? 599 Prove: a) that the sum of any two-digit numbers of the form ab and ba is a multiple of 11; 552 E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM 605 618 620 645 669 670 671 696 720 750 772 b) that the difference of any two-digit numbers of the form ab and ba is a multiple of Given that x = 5a2 + 6ab − b2 ; y = −4a2 + 2ab + 3b2 ; z = 9a2 + 4ab Substitute these polynomials for x, y, and z in each expression and simplify it a) x + y + z; b) x − y − z A hiker traveled distance AB, equal to 110km, in three days On the second day of the hike he traveled 5km less than on the first day, and on the third day he hiked 3/7 of the distance covered on the first two days How many kilometers did the hiker travel each day? A freight train left station A for station B at a speed of 66km/hr and 20 minutes later a passenger train left B in the direction of A at 90km/hr How long did the passenger train travel before meeting the freight train, if the distance between stations A and B is 256km? A hiker figures that if he walked to the train station at a speed of 4km/hr, he would be a half hour late for the train, and if he walked at 5km/hr, then he would arrive at the station minutes before the train leaves What distance does the hiker have to walk? Find four consecutive natural numbers, given that the difference between the product of the two greatest numbers and the product of the two remaining numbers equals 58 Find three consecutive even numbers, given that the difference between the product of the two largest numbers and the square of the third equals 188 The perimeter of a rectangle equals 60cm If the length of the rectangle is increased by 10cm, and the width is decreased by 6cm, the area of the rectangle is reduced by 32cm2 Find the area of the rectangle Compare the area of a square and a rectangle, given that the base of the rectangle is 10cm greater and its height 10cm less than the length of a side of the square Two trains set out at the same time from points A and B, 1,020km apart, heading towards one another The speed of one train was 10km/hr greater than that of the other hours later the trains had already passed on another and were now 340km apart Find the speed of each train Prove that the following expressions cannot take on negative values: a) b2 + 6b + 9; b) a2 − 12a + 36; c) x2 + 10x + 25; d) y + 8y + 16 If the number is written immediately to the left of a three-digit number, and 619 is added to the resulting four-digit number, then E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM 553 this sum is 40 times greater than the original three-digit number Find the original three-digit number 781 A motorboat traveling downstream for hours covers the same distance that it can travel upstream in hours, 15 minutes Find the speed of the boat in still water, if the speed of the river current is 2.4km/hr 836 One number leaves a remainder of after being divided by 5, and another leaves a remainder of Is the sum of the squares of these numbers divisible by 5? 903 Do the following systems of equations have solutions:  3x + 5y = 34 a) 4x − 5y = −13  2x − y = 1; 6x − 5y = −15, b) 13x + 3y = −86,  3x + y = −18? Here is the discussion of setting up and solving problems using systems of equations: 554 E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM Sample problems from this section: 919 In the soccer championships the Dynamo team did not lose a single game of the eleven games they planed, and they gained 17 points How many games did Dynamo win and how many did it tie (a team gets points for a win and for a tie)? 922 A tank is filled by means of two pipes If water flows from the first pipe for 20 minutes and from the second for 10 minutes, there will be 120 hectoliters of water in the tank But if the first pipe is open 15 minutes and the second minutes, then 88.5 hectoliters of water will flow into the tank How many hectoliters of water per minute flows into the tank through each pipe? 923 I am thinking of two numbers If you add half of the second number to the first number, then you get 65; but if you subtract one third of the first number from the second, then you get the first number What numbers am I thinking of? 925 It is necessary to make up 36 kilograms of a mixture of two types of dried fruit (their prices are ruble 20 kopecks and ruble 50 kopecks per kg) which will cost ruble 30 kopecks per kg How many kilograms of each type of dried fruit are needed? 988 Two tourists, with only one bicycle, have to travel a distance of 12km in an hour and a half Each of them can attain a speed of 20km/hr on the bicycle and 5km/hr on foot Can the tourists E THE SIXTH GRADE TREATMENT OF ALGEBRA IN THE RUSSIAN PROGRAM 997 1002 1003 1004 1013 1017 1020 555 cover the distance without being late? (Two people cannot ride the bicycle at the same time) Prove that the value of the expression 116 + 146 − 133 is a multiple of 10 If you take a certain two-digit number and split up its digits by inserting the same two-digit number in between them, then the four-digit number you get will be 77 times greater than the original number Find this number Find a three-digit number which is equal to the square of a two-digit number and the cube of a one-digit number When the polynomial 2x3 − 5x2 + 7x − was multiplied by the polynomial ax2 + bx + 11, the product was a polynomial which did not contain either and x4 term or an x3 term Find the coefficients a and b and find the polynomial which was the product Can the difference of two three-digit numbers, the second of which has the same digits as the first, but in reverse order ,be the square of a natural number? Prove that p2 − 1, where p is a prime number greater than is divisible by 24 10 1o11 Which is greater 1011 +1 or 1012 +1 10 +1 +1 ... math important? The critical role of mathematics in modern society Common misconceptions by pre-service teachers about mathematics The mathematics K - teachers need to know Mathematical problem... discussion reflects the mathematics that students need to know, and consequently the core mathematics that pre-service K - teachers must know, though it goes without saying that they must know more than... numbers for the variables 32 THE BASIC TOPICS IN K - MATHEMATICS placed places (corresponding to the in 105 ) to the left of the last (right) digit of the numerator Thus, by the same token, 1200