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row ~ - ~ 10~ Berm ~1 Art The sessions on content and learning mathematics in the middle grades focused on the questions: . What developmental considerations are important in thinking about middle school students as learners? As learners of mathematics? Are these compatible? · What do we know about middle school students' capacity for learning? For learn- ing mathematics? · What are important ideas in mathematics for the middle grades and how are these related to developmental learning considerations? REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS Nancy Doda, Professor of Education, National-Louis University. MATHEMATICS CONTENT AND LEARNING ISSUES IN MIDDLE GRADES MATHEMATICS Kathleen Hart, Chair of Mathematics Education (retired), University of Nottingham, United Kingdom. SUMMARY OF SMALL GROUP DISCUSSION ON CONTENT AND LEARNING ISSUES IN MIDDLE GRADES MATHEMATICS
Hi ',,r~.r~--~ Nancy Doda Professor of Education, National-Louis University, Washington, DC OVERVIEW Young adolescence is a remarkable and challenging time in human life not always appropriately appreciate(1 nor well understood. In 1971, loan Lipsitz published a review of research on the middle grades child and learner in which her chosen book title, Growing up Forgotten, was essentially her most provocative conclusion that young adolescents were American education's most neglected and least well under- stoo(1 age group (Lipsitz, 19711. Since this seminal work's debut, there has been without question a steady ca- cophony of hearts and minds that have joined to recognize, celebrate, and better understand and serve this unique age group. Indeed, several decades of reform initiatives now stand before us, yielding wisdom to guide future efforts to improve schooling for young a(loles- cents (Lipsitz, 1981; George and Shewey, 19941. While we are fortunate to have this rich history of mi(l(lle gra(les reform, with its now well-documented dividends (e.g., FeIner et al., 1997), it is nonethe- less clear that the (livi(len(ls of greater student learning and achievement are still more illusive than we might have hoped. There still remain enormously stubborn achievement gaps between white children and children of color; between poor children and financially advantaged children, between girls and boys. If asked the question, "How are we doing in the U.S. with regards to student achievement?", the truthful answer would be, 'That depends on which children we are discussing." And in the case of our focus here, perhaps it also depends on which subject we are · ~ examining. We face a crossroads in middle school reform. One that calls upon us to reexamine not only the fundamental philosophy of the middle school con- cept, but the beliefs and practices that
remain as potential deterrents to our gravest challenge that of ensuring academic success for all our children. There is an urgent need to (lig (1eeply to understand what is needed in the reform conundrum, particularly with student learning in mathematics. Stu- dent success in middle school math- ematics remains the great equalizer or divider, as it were, and our students' individual and our collective futures depend upon it. CHANGING THE IMAGE- CHANGING THE CURRICULUM While the contemporary dialogue about the nature of young adolescents has often been delivered with affection- ate humor, such humor often highlights the least dignifying portrait of the young teen. ~ have often quoted Linda Reiff's comments to evoke an affirming chuckle from most middle grades educators or parents. She wrote: Working with teenagers is not easy. It takes patience, humor and love. Yes, love of kids who burp and fart their way through eighth grade. VVho tell you "Life sucks!" and everything they do is "Boring!" VVho literally roll to the floor in hysterical laughter when you separate the prefix and suffix from the word "predic- tion" and ask them for the root and what it means. VVho wear short, skin-tight skirts and leg-laced sandals, but carry CONTENT AND LEARNING ISSUES teddy bears in their arms. VVho use a paper clip to tattoo km Morrison's picture on their arm during quiet study, while defending the merits of Tigger's personal- ity in Winnie-the -Pooh. VVho send obscene notes that would make a football player blush, written in pink magic marker, blasting each other for stealing or not stealing a boyfriend, and sign the note, "Love, back." (Reiff, 1992, p. 90-91) P.S. Please write In their light-hearted intent, such comments can spur teacher camarade- rie, but they also can serve to remind us of how easily this unique stage of development might be misconstrued. In fact, caution is in order since for a variety of reasons beyond such lan- guage, the intellectual character and energy of young adolescents have often been un(lerrate(l. Most certainly we have nearly eliminate(1 reductive images portraying young a(lolescents as "hor- mones with feet," and yet the achieve- ment question ought to call upon all of us to make a more (leliberate effort to acknowle (lge an (1 celebrate the incre (l- ibly powerful intellectual character of this age group. As ~ travel around the nation, ~ have ha(1 the (delightful opportunity to (lia- logue with many young adolescents, and I'(1 like to share with you that our young people, from all ethnic anti cultural affiliations, from all levels of income anti from all levels of school competence, repeate(lly (remonstrate that they are
immersed in one of the most intellectu- ally pivotal times in human develop ment. ~ ask young people to share with me the questions and concerns they have about the world. Their musings should remind us at how phenomenal our educational opportunity is during the middle school years. Let me share a few of their recurring questions and concerns: · Will there ever be world peace? · Can we clean up the environment? . Will men and women ever be equal? · Will there ever be a black President? . What happens when you die? · Why do people hate each other? · Are millions of people in the world really starving? · Why is there so much hatred and violence? · Why does anyone have to be poor? How can we cure AIDS and cancer? · Is there life in outer space? · Why are we here? . Young adolescents are philosophical, investigative, renective, hypothetical, and skeptical. They love to debate, query, conjecture, moralize, judge, and predict. They are filled with the joy of self-discovery and the inevitable disillu- sionment of world discovery. They are paradoxical. While plagued with self- doubt, they are armed with a heroic invincibility (Elkind, 19841. In sum, REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS these young people are developmentally ripe for intellectual growth. But when they enter our middle schools every day, not many find in their classrooms a match for the intellectual intensity their questions renect. For too many, curriculum is not seen as excit- ing, useful, meaningful, or helpful. In numerous and lengthy focus group interviews with students, ~ have found their responses quite telling. ~ asked one energetic sixth grade student what he was learning and why. He responded dutifully, "Latin America." ~ then asked, ' Why are you studying this? What's really important about this topic?" He was equally honest and said, "I have no clue, but ~ think it's in the curriculum." His peer offered, "I think we need to learn it cause we might need it later on." Another added, "No, ~ don't really think so because my father is very smart and successful, and ~ know he never uses this stuff." Something huge is missing in how students are experiencing curriculum. We shouldn't presume that it is only those struggling students who raise serious question with our curriculum. ~ asked a group of honor roll students in a middle school to tell me about what they had learned from the fall until January. They couldn't recall much. In consider- ing their plight of failed memory, one eager student perked up with some sense of enlightenment, "I think ~ know
why we're slumped. Wewerein the accelerated program and we went so fast, we don't remember much." In the very least, his comments affirmed my often nagging notion that much of what we define in schools as accelerated is merely more content taught faster, more home- work done hastily, and not much more learned deeply. Clearly, the TIMSS findings (Silver, 1998) illuminate this "mile wide and inch deep" curriculum problem that challenges our pedagogy. That curriculum in all fields of knowI- edge in middle school must be more meaningful, challenging, anti engaging is unquestionable. When observing students and teachers at work, ~ dis- cover that many of our students are often not engaged in challenging learn- ing experiences. ~ watched extremely bored students sit in a well managed classroom listening to a litany from peers who were to each report to the class on the content of an individually rea(1 current event. This was an entire class period of high disengagement and notable disenchantment. This is where an urgency for reform should be felt. This still happens far too often. THE REFORM PICTURE While the middle school movement's thirty year history has secured a place on the reform map and has contributed CONTENT AND LEARNING ISSUES greatly to the overall improvement of many middle level schools, the movement's reform recommendations and efforts have, ~ believe been more successful in altering the climate and structure of mi(l(lle level schools than the curriculum and instruction our young people have experienced (FeIner, etal.,19971. Organizing smaller,more personalize(1 learning communities, commonly called teams, creating teacher scaffol(ling anti support for all students, emphasizing interdisciplinary planning anti teaching, anti creating more flexible sche(lules liberate(1 from tracking, have without question, raised teacher efficacy, encouraged profes- sional (lialogue, re(luce(1 school ano- nymity, improve(1 school climate, anti even in some pockets, raised school achievement. They have not always resulte(1 in the (1ramatic shift in teaching and learning that was ~ believe a bold hope of the mi(l(lle school movement's many advocates anti champions. Perhaps the simplest explanation that draws nods from many is that the mi(l(lle school movement has (levote(1 too much of its energy and attention to reforming the organizational character of our mi(l(lle level schools. James Beane (199Sa) woul(1 suggest that such a situation with the state of reform in mi(l(lle school teaching anti learning was inevitable since we never fully achieve(1 consensus on the goals anti
purposes of the middle school curricu- lum. Likewise, some speculate that three decades of work devoted to the creation of more humane schools has resulted in soft attention to the intellec- tual development of our young people. Others argue that achievement, particu- larly in mathematics, has been short- changed by our advocacy and imple- mentation of thematic teaching which often highlights social studies and language arts or (lisparagingly, reduces mathematics to labeling correct mea- surements on an interdisciplinary exhibit. Still others ~ might say, quite legitimately, have argued that mis- guided interpretations of progressive instructional methods have yielded sloppy attention to intellectual develop- ment and authentic and substantive student learning. Regardless of the complex puzzle of causes that we are now facing, the TIMSS results in mathematics are not surprising. In the last three decades of classroom practice, approaches to mathematics instruction in middle schools have not changed as consis- tently and dramatically as some of us might have hoped following the publica- tion of the NCTM standards. The islands of excellence are simply too few. None of the recommendations for middle school structural reform are void of underlying theory about their rela- tionshipw~thstudentlearning. While REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS not always clear to the public nor consistently conveyed in professional development, we should not be too quick to blame the current achievement conundrum in mathematics or any field on the middle school concept. Interdisciplinary teaming, for in stance, remains at the core of middle school reform, in large part because of its research credibility to raise teacher sense of efficacy a key element in high performing classrooms. Team organization has also been associated with reduced school anonymity and teacher collegiality, additional features in safe and productive schools. More- over, attempts at increased personaliza- tion, in the form of Teacher Advisory programs and similar middle school initiatives, were fundamentally grounded in the belief that the quality of teacher-stu(lent relationships greatly impacts student motivation anti perfor- mance. That learning is a social en- deavor, embedded in relationships is not an assumption unique to middle school philosophy (Glasser, 19921. That curriculum was inten(le(1 to be exploratory was not meant to suggest that it could not also be (leman(ling. That mi(l(lle schools were inten(le(1 to be humane anti caring places was not inten(le(1 to be antithetical to serious mathematics education. In(lee(l, there is a tremendous need to pursue and refine these elements of the mi(l(lle
school concept as we engineer a new plan towards higher performing middle schools with greater learning for all. MIDDLE SCHOOL MATHEMATICS- SEARCHING FOR SOLUTIONS The standards movement is accompa- nied by a wave of achievement panic that threatens to diminish the focus on learning in the middle grades. In this panic context, the TIMMS data suggest to some that classrooms have failed to conduct sufficient skill and drill work. Others cast the blame on thematic teaching or detracking. Fortunately, there are several themes that are recurring in current conversations among middle school advocates and those interested in mathematics reform which, when united, bring clarity and perspective to some of the more emo- tional attacks. Both groups call for a curriculum that is challenging and engaging for young people, offers connections across disciplines, chal- lenges students to apply knowledge, putting mathematics to use, emphasizes problem-centered learning, provides opportunities for collaboration, and seeks to en(1 inequitable practices like tracking (Beane, 199Sb). The earnest call to engage more young adolescents in meaningful math- ematics has led exemplary districts and CONTENT AND LEARNING ISSUES schools to pilot new programs and test their own results. In Corpus Christi, Texas, where standards-based reform is ongoing, all but one middle school has extended the invitation to take algebra to all eligible 6, 7, ~ gra(lers. Eligibility is still the sticky issue since aigebrafor all (toes not mean all students are guaranteed exposure to algebra con- cepts by the Sth gra(le. In fact, while it opens the door to early maturing, frequently advantaged young people, it still fails to embrace the very young people we have missed all along. In that same district one lone middle school has employe(1 Connecte(1 Math- ematics, a program developed out of Michigan State University, anti the student engagement anti learning success they are observing are inspir- ing. The manipulative and collaborative nature of this curriculum approach finds a place for varying levels of readiness in a way ~ have not observed with tradi- tional pre-algebra anti algebra ap- proaches. Their story and the story of other schools engage(1 in renective practice and study will continue to offer promise to our steady search for solu- tions. ~ am frequently asked, "Should all middle school students take algebra before moving on to the high school?" might begin by posing a clarifying question, "As it is most often taught?" If the response is "yes," then I am com
pelted to answer, "no." We recognize that many of our young adolescents are not formal operational thinkers with a strong logical-mathematical intelligence and that algebra has traditionally been taught to them as if they were. ~ am not sure, however, that that is the right question to ask; rather, shouldn't we be asking, what kind of mathematics should all young adolescents be learn- ing to enhance their understanding of the world and the role mathematics plays in it? Algebra as a course is problematic. Middle school algebra for high school credit is even more problematic. The presence of algebra as a select course with eligibility and teaching certification requirements faithfully diminishes a middle school's chances at academic equity. When students are grouped for mathematics instruction, they are divided as well by race, economics, and learning orientation. As middle schools organize in small learning communities to ensure the noted benefits of teaming, students are grouped by mathematics levels in ways that can result in tracking and the reduction of mathematics learning for non-aIgebra students. These students deemed less ready or able, travel apart from algebra students, and may spend an entire year relearning mathematics concepts many already know, while they wait to enter "the algebra course." REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS For young adolescents, meaning is everything. In fact, human learning involves meaning making, does it not? Should we ted our young teens to endure our current version of algebra because they will increase their chances of going to college or because then we can outper- form our international partners? If we are eager to embrace more learners in mathematics education, these suggestions will hardly inspire the tentative. The same students with involved parents or from advantaged homes will be at our college doors while those who wonder if there is life after middle school or hope in life at all will remain out of reach. Even among sup- ported students we still must go further as not one of my son's Sth gra(le hien(ls, all in Sth grade algebra, can explain to me why or how algebra is or even could be useful in the world. Perhaps it is time for a bold step to move towards creating in middle schools, mathematics for life far more challenging and meaningful than what many currently experience in algebra? Perhaps what all middle school students should experience is the kind of foundation algebra that few of us re- ceived the kind that would make it possible today for you to Lent the many ways in which algebra is at work in the world. Perhaps we might even be able to recognize when we use it? Few of us who (lo not teach mathematics can (lo this well.
What really stands in the way of true reform in middle school mathematics has less to do with education, however, and more to do with politics. Math- ematics is a subject area with social status. To suggest that algebra not be taught as a course reserved for "ca- pable learners," is to invite a public relations disaster of epic proportions (Beane, 199Sa). In fact, in James Beane's work with mathematics teach- ers who have had success in teaching mathematics in the context of curricu- lum integration, many have begun to schedule their mathematics as one of the separate subjects in their pro- grams, not because they think it is sound educational practice, but be- cause they would lose the rest of their programs if they did not. JOINING FORCES We do have many important ques- tions to consider: what is the purpose of the middle school curriculum? Is it to prepare students to be academic schol- ars? To understand themselves and their world? To make our students score higher than students Tom other countries anti with what assurance that it translates into benefits in life and work? To decide early who wait take what path in schooling? CONTENT AND LEARNING ISSUES These questions also renect the major points of deliberation such as whether international test scores should shape curriculum goals, whether all math- ematics must be taught as a separate subject, and to what extent higher expectations ought to involve vertical acceleration through mathematics areas or application of knowledge to increas- ingly sophisticated problems. ~ believe we have answers to what constitutes best mathematics education. Perhaps the really critical question we nee(1 to a(l(lress is how can we make the rhetoric of best practice a reality for more of our young people. In answering this question, we in fact push ourselves towards a vision of mathematics e(luca- tion that offers great hope for equity and academic excellence. REFERENCES Beane, J. (1998a). The middle school under siege. Paper presented at the National Middle School Association's annual conference, Denver, CO. Beane, J. (1998b). Paper prepared for Middle Grades Mathematics Convocation, September, 24-25,1998. Washington, DC. Elkind, D. (1984~. All grown asp & no place to go. Reading, MA. Addison-Wesley. Felner, R. D. Jackson. A.W., Kasak, D., Mulhall, P., Brand, S., Flowers, N. (1997~. The impact of school reform for the middle years. Phi Delta Kappan, 78~7), 528-532, 541-550. George, P.S., & Shewey, K. (1994~. New evidence for the middle school. Columbus, OH: National Middle School Association.
Glasser, W. (1992~. The quality school: Managing sta~dentswitho?`t coercion. New York: Harper Perennial. Lipsitz, J. (1971~. Growing asp forgotten: A review of research and programs concerning early adolescence. New Brunswick, NI: Transaction Books. Lipsitz, J. (19843. Successful school foryoa`ng adolescents. New Brunswick, NI: Transaction Books. REFLECTIONS ON MIDDLE SCHOOL MATHEMATICS Reiff, L. (1992~. Seeking diversity: Language arts with adolescents. Portsmouth, NH: Heinemann. Silver, E.A (1998~. Improving mathematics in middle school: Lessons from TIMSS and related research. Report prepared for the U.S. Department of Education, Office of Educa- tional Research and Improvement.
Kathleen Hart Chair of Mathematics Education (retired, University of Nottingham, United Kingdom In the United Kingdom compulsory schooling starts at the age of five and continues until 16 years of age. The provision of free education continues for another two years. The structure of the school system varies and children can proceed through infant school (age 5-7), junior school (~-~) and secondary school (~-16/~) or through a system in which they change schools at 9 and 13 years of age. So in England a child in the middle gra(les is probably changing (or has just changed) schools. In the first schools, the teacher is a generalist and probably teaches all the subjects the child meets during a week. For the pupil, being promoted to a secondary school (or even a middle school) means that there are many teachers to face in any one day. These teachers tend to be interested in only one curriculum subject. In the primary school, it is likely that the teacher has tried to present the curriculum through project work, which might mean that the intention was to exploit a topic (e.g., The Vikings) for its possibilities to illustrate English, geography, religion, art, science, and mathematics. In the secondary school, these subjects are allotted separate lime slots, anti the teacher only teaches that subject. The institutional life in school changes to a greater focus on formal learning, and in mathematics the con- centration is on competencies and skills and their application. Often assump- tions are ma(le concerning the reper- toire of skills the child already pos- sesses, and she may lose confidence when it is shown her repertoire is limite(l. A(l(1 to this the changes in chil(lren's physical makeup anti the new interests which occupy them, anti it is a wonder that they learn anything. FORMALISATION MANIPULATIVE LINK The influence of Piaget on e(lucational theory has meant that much of the
child's primary education has been imbedded in the idea that prefix years of age the child is operating at the concrete level. Mathematics educators for many years have interpreted this stage as one which requires concrete embodiments or manipulatives to promote what are essentially abstract mathematical ideas. The chasm be- tween the manipulatives and the ab- straction has not been addressed very thoroughly. The research project "Children's Mathematical Frameworks" (CMF) sought to investigate the transi- tion made by children when moving from "concrete" experiences to a for- mula or mathematical generalisation. Teachers who were pursuing a masters' degree in mathematics education enrolled for a module which required them to: a. Identify a topic which they would normally introduce with the use of manipulatives, which experience was to lead to a formula, algorithm or other mathematical generalisation. b. Prepare a series of lessons and teach them to a target class. c. Allow the CMF research team to interview six children, before the teaching started, just before the formalisation took place, just after it, and three months later. d. Alert the researchers to when the "formalisation" lesson or acceptance of MIDDLE GRADES MATHEMATICS the rule would take place and allow the lesson (s) to be observed and tape recorded. e. Interview two other chil(lren in the class and report on the responses. Additionally, an analysis of the tape recorded lesson would be written (the transcript of the recording being sup- plied by the researchers). Topics which were included in the study, fulfilling the description of con- crete embodiments leading to formalisation, were area of a rectangle, volume of a cuboid, subtraction of two and three digit numbers With decompo- sition, the rule for fractions to be equiva- lent, the circumference of a circle, and enlargement of a figure. The advice given to teachers in teach- ing manuals etc., often describes the experiences the children should have and then implies (or even states) that "the children will come to realise" the formula. In practice, it seems that a few children in a class might come to the realization, and the rest be encouraged to accept the findings of their fellows. The teacher feels that time is short, and the class must move on. Part of the three month follow-up interviews was to ask the pupils for the connection between the two experiences, concrete and formal. Only one of the interviewees (out of 150) remembered that one experience led to the other and provided a base for it.
Most of the replies are summed up by the girl who said "Sums is sums and bricks is bricks." The forgetting would be unimportant if the concrete experi- ence (which is often arduous and time consuming) had resulted in a successful use of the formalisation, but it had not. The observations of lessons and analysis of the transcripts of what the teacher said brought to light how very disparate were the views of teacher and pupil. The teacher knew the mathematics, knew the formula or rule, and had devised a set of "manipulative" moves to convince the child of the truth of the rule. The child did not know where the manipulations were leading and to him/her a red brick made from two centimetres of wood was exactly that. The teacher might refer to it as "x, 2, 4" and could even say "let us pretend it is 17." Figure 1. Terence's Diagram for Equivalence CONTENT AND LEARNING ISSUES JOINING THE GROWN-UPS From the observations in CMF (an(1 some subsequent research), it was plain that teachers and children embarked on a voyage of (liscovery to a place well known to the teacher. None of the teachers observed (some 20 experi- ence(1 practitioners) explaine(1 why the pupils woul(1 want to abandon bricks, naive methods, anti even invented chil(l- methods in favour of the formalisation. Nobody explained the power of the new knowledge. The nearest statement to a reason for its adoption was "you do not want to carry around bricks for the rest of your life." The teacher's attitude was one of frien(lly guidance, more in the sense that the lessons were a review of something we already knew rather than an introduction to the complete un- known. Consi(ler how few teachers (lraw on the boar(1 accurate subdivisions when partitioning a circular (lisc into equal fractional segments. The illustra- tion is produced free-hand and quickly split into sections. Little wonder that Terence pro(luce(1 this set of (1iagrams when he was trying to convince the interviewer that 9/27 = 3/9 (Figure 11. Andrew was in a group learning subtraction when the teacher pro(luce a three (ligit subtraction which resulte in zero in the hundreds place. The ensuing conversation with the class of eight pupils was as follows:
(Note: T = Teacher; P = Pupil): P: And that wouIc3 be one huncirec3 take away one huncirec3 is nothing. T: Is nothing, so c30 I put that there? P: No. T: Shall I put that there? Who thinks I shouic3 put that there? Who doesn't think I shouIc3 put that there? Well, I mean, you can, but if I was to ask you to write clown 99 . . . in your books, just write clown 99, you wouIcin't write clown 099 wouIc3 you? P: No. T: You wouic3 just write the 99, wouicin't you? So we don't really need to put that there. 1 take away 1 leaves you with an empty space, so we might as wed leave an empty space, okay? Andrew's attempt at 304-178 gave the answer 2 and his argument went as follows: A: Say if you said, four take away 8, it's 4. You've got 4 and you can't take 8 from 4, so there's nothing there and . . . nothing take 7, you can't c30 that and 3 take away 1 gives you 2. T: I see. So take 4 away 8 I can't c30, right. So c30 I write anything underneath there or c30 I not bother? A: Not bother. MIDDLE GRADES MATHEMATICS He had absorbed the "don't bother" but not when to use it. There were other instances in the interviews of children selecting a specific part of a teacher's statement and generalising incorrectly or of remembering the one erroneous statement the teacher had made. FRACTIONS AND DECIMALS The middle grades are the years when the elements of arithmetic cease to be exclusively whole numbers, and much energy and time is spent on the study of fractions and decimals. There has been a lot of research on children's understanding of these "new" numbers. Generally most eight-nine year olds can recognise and name a region as 1/2, 1/4, 1/3, 1/~; fewer recognise that a region split into twelfths can also be labeled in sixths. Far fewer chil(lren can success- fully carry out operations on fractions. The mode} for introduction currently in most of our textbooks is that of regions (square, circle, line). This enables us to talk of shares, but the result is a tangible amount (slice of pizza, cube of choco- late) which (toes not neatly fit within the operations of a(l(lition, subtraction, multiplication anti (livision. How can you multiply two pieces of pizza? The other meanings of a/b are often not a(l(lresse(1 separately in school text- books, anti the chil(1 is expecte(1 to infer
division or ratio from the same region model. The result is confusion and a heavy reliance on rote-learned rules. Many children reject the whole idea of noncounting numbers and attempt to deal with secondary school mathemat- ics without them. The research project "Concepts in Secondary Mathematics and Science" (CSMS) obtained data by both interviews and tests. A representa- tive sample of 11-16 year olds was asked to give the answers to various division questions, and they were told that if the question was "impossible" they should say so. Table ~ shows some results. A parallel test had required pupils to write a story to illustrate ~ - 4 and 128 - 8. Very nearly all the responses involved the sharing of sweets among hiends. This interpretation of the divi- sion sign is in connict with 16 - 20 as there are obviously not enough sweets for the Fiends to share. Do teachers redefine the operations to accommodate fractions and decimals? The middle grades is when children are trying to graft new concepts on hopelessly inadequate foundations put in place for counting numbers. Algebra is likely to be intro- duced during these years and viewing it as generalised arithmetic seems fraught with difficulties. In algebra, we need to sped out all connections among Me numbers now represented by letters, and in ari~medc Me aim is to carry out Me operations anti to obtain a result as quickly as possible. "x +y" stays as such and cannot be processed to become xy whereas we find it unwieldy to work wad 5 + 3 and replace by ~ as soon as possible. Collis (1975) (lescnbe(1 a level of algebraic un(lerstan(ling as "Allowing Lack of Closure" (ALC). When a chil(1 can accept anti even work wig (x +y), a significant step has been taken. Table 1. CSMS Results to Division Questions Large Survey Results Divicleby20 1.21.0 1.A 1 rem A Impossible (i) 2A (n=170) 11-12yr 9%7% 8% 12% 15% (n=2~0) 1~-15yr 3~%1% 15% 3% 6% 0.80.0 0.16 0 rem 16 Impossible (ii) 16 11-12yr 7%2% A% - 51% 1 A-1 5 yr 36%- 6% - 23% CONTENT AND LEARNING ISSUES
MATCHING THE MATHEMATICS TO THE CHILD It has long been known that from any class lesson, the child participants take away very different pieces of knowI- e(lge. In CMF a group of eight pupils, all thought by the teacher to be "ready" for subtraction and all taught in the same way, were found to have very different paths to success. A child's success depends on what was known before Now many of the pre-requisite skills are in placed; how much of the current content matter is understood; his attention span (was he even in school, absences matter) and the confidence with which the mathematics is approached (does the child fee} in control of the mathematics content or is it magical. A child cannot be confident if all the mathematics exercises he does are marked incorrect. By the middle gra(les it is likely that any group con- tains a number of "low attainers" who, without positive action, are unlikely to become even "average attainers." The curriculum development project "Nuffield Secondary Mathematics" was designed to provide suitable material for all attainment levels in the secondary school (ages 11-16 years). The books were: a) Topic books; short "content" orientated material in four sets- Number, Space, Probability and Statis- tics, and Measurement; b) Core books; MIDDLE GRADES MATHEMATICS books of problems for an entire year to allow groups of mixed attainment to work together applying their mathemat- ics; anti c) Teachers gui(les; very full information for teachers. To find where to start the Number strand, children in primary school (some 100 pupils), including some who were identified as (lisplaying "special needs" were teste with items that researchers had previ- ously used with six year olds. Follow-up interviews disclosed that there were pupils about to enter secondary school, who had very limited number skills. A list of pre-requisite skills was drawn up, and we stated that to start on the Num her books, children had to demonstrate that they could do the following: · Arrange car(ls showing configura- tions of (lots for ~ to 6 in order. · Give the number before anti after a written two-digit number. · Count on (rather than count from 1) when given two strips of stamps. · Write correctly two-(ligit numbers when they are rea(1 out (oral). · Put written numbers, less than 25, in order (written). · Interpret the wor(ls "more" anti "less" when given two sets of (lots · Count a pile of coins Hess than one pound) accurately, taking account of the (lifferent face values. · Choose the correct single coin for a purchase of 45p.
Practising secondary school teachers during in-service courses were shown the contents of Level One Number and asked if they had pupils who could only "do this." Usually they assured us that none of their pupils knew so little, but later we were bombarded with requests for the material at this level. In the trial schools there were usually about ten 11-12 year olds who needed it. We only found one pupil in a normal school for whom the work was too difficult. The material was put through several, wed regulated trials and rewritten if it proved too difficult for the pupils using it. The intention was that the child should experience success and so become confident in mathematics, no matter how limited. An early result was that al- though the first book took perhaps three months to complete, the second took much less time. As confidence built, so did the speed with which the child worked. The classes using this early material were usually small, and the pupils worked in pairs or groups of three with a lot of teacher help. No child progressed to the next book until he/she had demonstrated that the content of the previous book had been absorbed by passing a test at the 80% (or better) level. No test was given until the teacher was sure that the child would pass because all sufficient. the book had been understood. Failing a test helps neither teacher nor chil(l. This concept was very difficult for some CONTENT AND LEARNING ISSUES teachers because they assumed there would always be failures. We were closely involved in all the trials and often marked the tests. In one school, the teacher agreed that the reason six children hall not reache(1 "mastery" level in the test was because they had not covered the entire book but only part of it. This must perpetuate a situation which is bound to be deficient half learned mathematics grafted onto holes in knowledge. The teacher explaine(1 that she could not wait for these pupils. CONSTRAINTS AND BELIEFS Running parallel to any new curricu- lum effort, and having an unseen but powerful influence on it, are the con- straints and beliefs of the general populace, politicians, headmasters, publishers, and even classroom teach- ers. Some of these are listed here: 1. There are certain topics which every chil(1 should be taught. 2. There are specific topics that every chil(1 should have learne(1 before a certain time in his/her life. 3. A certain amount of time spent in school on mathematics lessons is 4. A certain amount of material ([books, worksheets or scheme) is enough. 5. Mathematics is (difficult.
We should be wary of these beliefs because they are very strongly held and usually backed up by appeals to "raising standards." In the UK during the summer of 199S, the Minister of Education boasted that the number of pupils passing the school-leavers' examination in mathematics had fallen, in order to claim that standards had been maintained. The expecta- tions the community has of pupils vary from country to country and what may seem obvious to a Japanese writer, is not obvious to an Italian observer. Howson (1991) published a list of ages at which specific math- ematics content was presented to pupils. An excerpt is shown in Table 2. There is obviously no "obvious" age for the introduction of a topic. CONCLUSION Mathematics in the middle grades is still "Mathematics for All," although the move is towards a formalisation of the subject. Failure to understand destroys the child's confidence so any introduc- tion of new concepts, such as numbers which cannot be used for counting objects, must be built up carefully and with few assumptions on the part of the teacher. Learning mathematics is a series of leaps, so it is good to know that the ground from which you take off is soli(l. Table 2. Age of Introduction of Content (aclaptecl from Howson, ~ 991~ Belgium France Italy Japan England Decimals9 9- 11 8- 11 8 9 Negative numbers8 11-12 11-1A 12 9 Operations on these1 2 1 2-1 3 1 1 -1 ~1 2 1 3 Fractions7 9- 11 8- 11 8 11 Use oflelters12-13 11-12 11-1A 10 13 REFERENCES Cockcroft Committee of Inquiry Into The Teaching of Mathematics in Schools. (1982~. Mathematics counts. London: HMSO. Collis, K (1975~. Cognitive development and mathematics learning Chelsea College, P.M.E.W. MIDDLE GRADES MATHEMATICS Hart, K (Eddy. (1981~. Children's understanding Mathematics: 11-16. London: John Murray. Howson, G. (1991~. National cavrric?`la in mathematics. Leicester, UK: The Mathematical Association. Johnson, D.C. (Ed). (1989~. Children's math- ematicalframeworks 8-13: A study of classroom teaching Windsor, UK: NFERNelson.
~] Working through a problem in the role of student can serve as a springboard for a discussion of the issues around content and learning mathematics. Contrasting student solutions with adult solutions (Appendix 4J furthergrounds the conver- sation in a situation reading student responses that is in fact, part of the practice of teaching Marcy's Dots ~ ~_~ (Figure ~J from the 1992 National Assessment of Educational Progress grade test provoked a variety of responses rangingirom concern over the clarity of directions to surprise at the many and diverse ways students found their solu- tions. Common themes that emerged from the group discussions are described below. Figure ~ . Marcy's Dots Problem from the ~ 992 National Assessment of Eclucational Progress A pattern of clots is shown below. At each step, more clots are aclclecl to the pattern. The number of clots aclclecl at each step is more than the number aclclecl in the previous step. The pattern continues infinitely. ( 1 St step) (2n~ step) (try step) · · 2 Dots 6 Dots 1 2 Dots Marcy has to determine the number of clots in the 20th step, but she does not want to clraw all 20 pictures and then count them. Explain or show how she could clo this and give the answer that Marcy should get for the number of clots. Note: See Appendix 4 for sample student solutions and the guiding questions for the discussion.
STUDENT WORK "Student work helps teachers think about how students learn mathematics and about the depth of their own under- standing." (Participant comment) Studying how students come to learn mathematics by using a "site of prac- tice," an activity that is something teachers do as part of teaching, led some discussants to conclude that the richness of student thinking about a problem accompanied by student solutions can be buried in the raw statistics reporting student achieve- ment. The variety of strategies used by the students in their solutions charts, recursive rules, formulas, listing all of the cases, drawing diagrams paralleled the strategies used by the adults. Discussing how students used their understanding of the mathematics to select an approach led to a discussion about the reasoning used by the adults, and in fact, led to some stimulating mathematical discussions. In some cases, there was concern over the lack of consistency between the work stu- dents did and their description of what they did. This was attributed to a lack of communication skills on the part of the students, although learning to commu- nicate is an important middle grades topic. An effective strategy to promote student learning could be to have SUMMARY OF SMALL GROUP DISCUSSION students themselves learn by using other student work. The question of quality answers vs. quality thinking became an issue, however. How do teachers reward and reinforce correct thinking even though the desired solution is not presented? Groups identified non-mathematical causes for student errors, answering the wrong question, not reading carefully, or jumping to conclusions, as well as mathematical reasons such as identify- ing the wrong pattern. A si(le effect of the analysis of the (liversity in the student work was a reminder to the teachers not to impose their solution method on their students. THE MATHEMATICS The question of teacher knowledge and capacity to deal with problems such as Marcy's Dots is a serious one. There was a strong feeling that many middle grade teachers do not have the neces- sary background to deal with some of the broad mathematical and algebraic concepts involved in the problem: variables and an introduction to sym- bols, functional relationships, linearity, sequences and series, recursion. The problem links algebra and geometry, is multi-step involving logical thinking, and leads to making generalizations. The perception that teachers are not pre
pared to teach this kind of mathematics was reinforced by studying the algebra portion of the draft version of the revised National Council of Teachers of Mathematics standards: Principles and Standards for School Mathematics: Discussion Draft. The issue of certifica- tion for teaching mathematics in the middle grades, the nature of preservice programs, and professional develop- ment were repeatedly identified as critical in helping teachers move beyond their comfort with number to other mathematical strands that should be part of the middle grades curriculum. There was also strong agreement that content knowledge is not enough; teachers must learn how to help stu- dents bridge from the concrete to the abstract. ALGEBRA IN THE MIDDLE GRADES The issue of whether ad eighth grad- ers are (levelopmentally really for aIge- bra and how to position algebra in the learning environment of the child raised more questions. There was a consistent belief that the study of patterns was important, but there was tension over how to move from the specific to a CONTENT AND LEARNING ISSUES generalization with ah students. The nature of the problem allowed students with different abilities and understanding to find a solution, an important feature for good problems. It is important for students to see each others work and then come to consensus on an effective way to solve the problem at the most abstract level that the students are developmentally capable of understand- ing. Common beliefs were that the use of a problem without a context lacks motivation for students and that prob- lems should be relevant and real to engage students. Students should be able to see where a problem is going and how it connects to other areas they are studying. They need to understand why formulas and generalizations are impor- tant, as well as how to think about and use them appropriately. ORGANIZATIONAL ISSUES Two organizational issues surfaced as barriers against practicing teachers approaching any such problem in a thoughtful and analytic way: the lack of time during their school life to engage in this kind of thinking and the current emphasis on testing, where most of their energy is concentrated on what is being tested or the need for accountability.