**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

*Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations: Proceedings of a National Convocation and Action Conferences*. Washington, DC: The National Academies Press. doi: 10.17226/9764.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

*Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations: Proceedings of a National Convocation and Action Conferences*. Washington, DC: The National Academies Press. doi: 10.17226/9764.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

*Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations: Proceedings of a National Convocation and Action Conferences*. Washington, DC: The National Academies Press. doi: 10.17226/9764.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

**Suggested Citation:**"Teaching and Learning Mathematics at the Middle Grades: Setting the Stage." National Research Council. 2000.

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

~r.e Err The emphasis on "teaching the whole child and the whole curriculum" advocated by some can be seen as opposed to the emphasis on "teaching mathematics content area" as advocated by others. MATHEMATICS IN THE MIDDLE: BUILDING A FIRM FOUNDATION OF UNDERSTANDING FOR THE FUTURE Glenda Lappan, University Distinguished Professor, Department of Mathematics, Michigan State University. THE MIDDLE SCHOOL LEARNER: CONTEXTS, CONCEPTS, AND THE TEACHING CONNECTION Thomas Dickinson, Professor of Curriculum and Instruction at Indiana State University.

Glenda Lappan University Distinguished Professor, Department of Mathematics, Michigan State University What makes mathematics in the middle grades so important? There are factors that have to do with adolescent growth and development. Others have to do with the subject matter itself, its increasing complexity, and its increas- ing importance as a foundation on which science, mathematics, and tech- nology literacy is to be built. Each of these factors that influence mathemat- ics' importance in the middle years will be elaborated individually and as they interact with each other. Students in the middle grades go from pre-adolescence to adolescence at different rates. At the beginning of the year in grade six, students may all look much as they did the year before in grade five. By the end of this year or the next, most students, both girls and boys,willgrowseveralinches. Stu 1 , ~, ~ . . . dents reactions to these rapid changes in their bodies are, of course, quite varied. But for all students these changes are accompanied by emotional challenges. Students are faced with getting to know a new self, including a new body, with new emotions. Strong ,' ,. . - . among these new emotions is a need to be like others, to belong. Parents are no longer the center of the world. Peers are now the focus for what is "cool" and ~ . . . tor emotional support or crushing · . · ~ . .. rejection. At the same time, a student's intellectual capacity to reason is expand- ing rapidly. Students in middle grades are grow- ing in their ability to reason abstractly. They become capable of generalization, abstraction, and argument in math- ematics. This signals the need for programs that give students the oppor- tunity to expand their experiences with "doing" mathematics, with controlling variables and examining the conse- quences, with experimenting, making conjectures, and developing convincing arguments to support or disconfirm a conjecture. Taken together, these changing intellectual capacities and the

fragile emotional state of these vuIner able middle grade students call for very carefully crafted mathematical experi ences that allow students to satisfy their social needs as well as their intellectual needs. The following section looks at the mathematics of the middle grades, its challenges and its fit with these emerging adolescents. MATHEMATICS OF THE MIDDLE GRADES As students encounter mathematics in the upper elementary gra(les, the emphasis changes from a focus on the additive structure of numbers and relationships to the multiplicative structure of numbers and relationships. This means that the students are faced with new kinds of numbers, fractions operations and decimals that rely on multiplication for their underlying structure. These numbers numbers are useful in making new kinds of comparisons that rely on two measures (or more) of a phenomena. For example, which is best, 5 cans of tomatoes for $6 or ~ cans for $9? A simple subtraction will not resolve the issue. A comparison that takes into account both the quantity of cans and the prices is called for. These new mathematical ideas contain intellectual challenges for the students that are as conceptually difficult as any thing anywhere else in the K-12 math . T E A C H I N G A N D L E A R N I N G MAT H E MAT I C S emetics curriculum. Many ofthese mathematical ideas win not reach their fuD maturity in the mi(l(lle years, but it is in the mi(l(lle gra(les that the firm foundation for understanding is laid. It is here that students have time to experiment, to ponder, to play with mathematical ideas, to seek relationships among ideas and concepts, and to experience the power of mathematics to tackle problem situations that can be mathematized or modeled. It is also here in the middle years that the serious development of the language of mathematics begins. It is easy to make a laundry list of topics in mathematics important in the middle years. Here is such a list: · Number theory, factors, multiples, division, products, relationships among numbers and among · Rational numbers, integers, irrational · Fractions, decimals and percents · Ratios, rates, proportions, quantitative · , ~ reasoning aria comparison · Variables, variables changing in relation to each other, rates of change among related variables · Representations of related variables in tables, graphs, and symbolic form · Slope, linearity, non-linear relation- ships, families of functions · How things grow in both an algebraic and a geometric sense Maximum and minimum

· Shape and shape relations, similarity, symmetry, transformations · The Pythagorean Theorem Chance and reasoning about uncertainty Prediction using probability and statistics · Measurement systems and measure- ment of attributes such as length, area, volume, weight, mass, angle, time, distance, speed · Visualization and location of objects in . space Representation of three-dimensional objects in two-dimensional drawings and vice versa The real challenge is to help students see these ideas in their relationship to each other. This requires a different way of bringing mathematics and students together. Rather than see these ideas as a series of events to be covered, good middle school curricula are finding ways to help students engage in making sense of these ideas in their complex relation- ship to each other. Symbolic representa- tions of patterns in algebra are seen as related to symbolic representations of transformations, slides, nips and turns, in the plane. In other words, we need to think about how we set students' goals for mathematics. Rather than expound- ing a list of individual ideas to be "cov ered," goals need to help a teacher and a student see what is to be learned, how the ideas can be used, and to what the ideas are connected. What follows is an argument orga- nized around strands of mathematics and important related i(leas within those strands for thinking about what is important for students to learn in mathematics in the middle years. In each of these sections an argument for inclusion is based on the universality of the i(leas in the strand no matter what the future ambitions of the students will be. Whether students will enter post- secondary educational institutions or the world of work directly after high school, the following ideas are key. They are mathematics that is central to being able to manage affairs as an adult and to be a good citizen who makes good decisions based on evidence rasher then persuasive rhetoric. They are also mathematical ideas that have their roots in the middle years. RATIONAL NUMBERS AND PROPORTIONAL REASONING One area of mathematics that is fundamental to the middle grades is rational numbers and proportional iThis strand argument is taken from a paper I presented at the Fourth International Mathematics Education Conference at the University of Chicago in August 1998 entitled Preparing Students for College and the Work Place: Can We Do Both? MATH EMATI C S I N TH E Ml D DLE

reasoning. This includes fractions, decimals, percents, ratios, rates, propor- tions, and linearity, as well as geometric situations such as scales and scaling, similarity, scale factors, scale modeling, map reading, etc. Another important aspect of this mathematics is that it gives a powerful way of making com- parisons. Having two measures, rather than one, on attributes that we are trying to compare leads us into the world of derived measures that are often per quantities or rates. This is a core of ideas that relate to quantitative reason- ing or literacy and connects directly to science. Almost everything ofinterest to scientists is a quantity, a number with a label. Study of mathematics involving quantitative reasoning invariably means reasoning about mathematics in con- texts. Part of what makes mathematics so powerful is its science of abstraction from real contexts. To quote Lynn Steen (p. xxiii, 19971: The role of context in mathematics poses a dilemma, which is both philosophical and pedagogical. In mathematics itself.... context obscures structure, yet when mathematics connects with the world, context provides meaning. Even though mathematics embeciclec3 in context often loses the very characteristics of abstrac- tion and clecluction that make it useful, when taught without relevant context it is ad but unintelligible to most students. Even the best students have difficulty applying context-free mathematics to T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S problems arising in realistic situations, or applying what they have learned in another context to a new situation. This is an argument that works for students hea(le(1 for the workplace as well as college. Clearly applications, but even more so, mathematics skills, pro- cedures, concepts, and ways of thinking and reasoning taught through contextu- al~zed problem situations, help students of all kinds and ambitions learn math- ematics. The modern high-performance workplace involves problems that require sophisticate(1 reasoning anti yet often only the mathematics of a good middle school education. The high level of reasoning that empowers a student to use these ideas in new and creative ways is not complete by the end of middle school but requires continue(1 experi- ences in the high school curriculum. DATA ANALYS15, REPRESENTATION, AND INTERPRETATION Another aspect of mathematics that pervades our modern life is (lata, or as statistician David Moore cabs it, numbers with a context. In the workplace, in jobs at all levels, employees are (leafing with the problem of either reducing or un(ler- stan(ling the reduction of large quantities of complex (lata to a few numbers or graphical representation. These num

hers or statistics are expressed in frac- tions, decimals and percents. Thus, knowing something about data analysis and interpretation and rational numbers is important to ad students. The society in which we live has become quantitative. We are surrounded by data. To han(lle our money, to make trans- actions, to run our businesses, as well as to do almost any job well, we need mathematics that is underpinned with a high level of reasoning and understand- ing. As Iddo Gal puts it, "There are no "word problems" in real life. Adults face quantitative tasks in multiple situations whose contexts require seamless integration of numeracy and literacy skills. Such integration is rarely dealt with in school curricula" (Gal, 1997, p. 361. But, it should be! Using con- texts that show the world of work help make mathematics more authentic to students regardless of their future career goals. Dataanalysis,representa- tion, and interpretation are an important and rich area for such examples. GEOMETRIC SHAPES, LOCATION, AND SPATIAL VISUALIZATION Whether we are talking about an intending calculus student or an employee at a too} and die shop, geometry, location, and spatial visualization are important. Just as a calculus student nee(ls to visual MATH EMATI C S I N TH E Ml D DLE Size a curve moving to form the boundary or surface of a solid in space, a too} and die employee uses such skis to set a machine to locate and cut a huge piece of steel to specifications that truly exceed our imagination. Here again is an area of mathematics that has application in the workplace and should be a part of the education of all students. This is also a part of mathematics in which strand or areas intersect. While ~ have labeled this area geometric shapes, location, and spatial visualization, algebra becomes a too} for specii~ring both location and shape. Even the spatial visualization aspects of geometry are enhanced by experiences with transformational geom- etry and its related algebra. This is also an area with connections to measurement which means rational numbers are essential. We cannot talk about any aspect of measurement without being able to deal with a continuous number line where rational numbers and irrational numbers fib in the spaces between the whole numbers. So again we see connections among strands of mathematics. Another area that has much to offer to both the world of work and to higher education is chance. CHANCE Everyday humans are faced with reasoning under uncertainly. Many

adults make wrong decisions as a result of not understanding the difference between random variables, correlated variables, and variables that influence or cause certain behavior in another variable. Probability is an area of mathematics that is important across the board. Itisa very challenging subject and involves a practical every- day kind of mathematics and reasoning. Here, as in other strands we have discussed, the rational numbers play an important role. Probabilities are num- bers between O and ~ that express the degree of likelihood of an event happen- ing. In experimental situations, prob- abilities can be estimated as relative frequencies again a use of fractions and proportional reasoning. In other situations, probabilities can be found by analyzing a situation using an area of ~ square unit to represent everything that can happen. This makes a connection to geometry and measurement. So much of the information with which we are surroun(le(1 in our modern worI is of a statistical or probabilistic nature. Here the conception of mathematics as a discipline that comes as close to truth as we know it in the modern world and the idea that what mathematics predicts may not in fact happen in every instance clash. Students learn to deal with the notion of random variables anti the seeming contradiction that there can be a science of predicting what happens in the T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S long run when what happens in an instance is random and unpredictable. These are hard ideas that take a great deal of experiencing before they seem sensible and useful. ~ would argue that the study of chance or probability should be a part of the curriculum for all stu- dents in the middle grades. It is in these years that students have the interest and we can provide the time to experiment with random variables. Students need to collect large amounts of data both by hand and with technology simulations, to see that data can be organized to build models that help predict what happens in the long run, an(1 our own theoretical analysis of situations can help us build predictive probability mo(lels. Now an argument for one last area algebra. ALGEBRA AND ALGEBRAIC REASONING ~ think it is pretty clear that few workers meet a quadratic equation to solve on a (laity basis, or for that matter few ever meet higher (legree polynomial equations to solve. But they do deal with formulas on a regular basis that are quite complex. Figuring the exact amount of a chemical cocktail to inject into a cancer patient (leman(ls complex measures anti uses of formulas. The consequences of a mistake raise the stakes a great (leal. There are so many

examples of such in the world of work that a strong case can be made that all students do need algebra, both from a functions point of view and a structure or equation solving point of view. In fact algebra that integrates with other areas of mathematics and that uses applica- tions as a driving force can reach all students no matter what their career goals. For the future mathematician or scientist, algebra is fundamental. For the future office worker, algebraic thinking can mean the difference between remaining in a low-level job anti advancing to much higher-paying, more demanding jobs. ~ do have to express one caveat. If this is the dry memorized algebra of the past, then it is as likely to damage middle school students as help them. However, an algebra strand that at some grade levels has a real concentration on algebra and algebraic thinking can benefit all students. This means that the ideas wait develop over time. This means that applications wait drive the learning of algebra, not just be the problems that you try after you memo- rize the types of problems in the section of the book. This means that the funda- mental ideas of algebra wall be taught in such a way that students have a real chance of (leveloping (jeep un(lerstan(l- ing as they develop technical profi- ciency. It should also be an algebra that takes full advantage of calculators and MATH EMATI C S I N TH E Ml D DLE computers as tools to explore ideas and to carry out procedures. Rather than belabor my point, let me just make it outright. We need to develop a set of mathematical expecta- tions (luring the mi(l(lle years that are for all students. At some stage of high school, students can benefit from options of continuing in a statistics, discrete mathematics direction or toward calculus and more formal math emetics. These should be real choices. equally valued and equally valuable depending on the post-secondary directions a student aims to take. But in the middle years, we need to develop curricula that preserve student's options for those future choices. However, such a curricula alone will not make an excellent mathematics experience for all students. Teachers and teaching matter. What teachers do with the curricula, what the expectations are for students, how students are expected to work, what conversations are expected and how these are conducted matter. Teach ing in the middle grades requires knowledge of the subject matter and also of students at this age. TEACHING MATHEMATICS IN THE MIDDLE GRADES What are the challenges associated with teaching mathematics in the

middle grades? Not all students reach the same levels of cognitive growth at the same time. This means that the middle school mathematics classroom teacher has the challenge of creating an environment that supports students' mathematics growth at many different levels. Since middle school students are at many stages of cognitive, physical, and social development, the teacher needs to understand where students are in their mathematical growth. Typical paper and pencil computation driven tests give little specific understanding of student's thinking. Thus, an additional challenge for the teacher is creating and using many opportunities for assessing student understanding. Teachers are experimenting with many new forms of assessments varying from partner or group tests or performance tasks to projects or student portfolios. Middle grades students have many social, emotional, and physical chal- lenges with which to deal. In order to capture their attention and direct their growing cognitive powers on mathemat- ics, the mathematical tasks posed for students must be focused on things they find interesting and important. Things that we as adults think of interest to middle school students are not always on target. This argues for opportunities for choice within the mathematics classroom. Choice of project topics, choice of problems of the week to work on, and even dealing with problems that T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S allow different strategies for solution, are ways of allowing students to fee} the power of choice. In addition, the kinds of problem settings used for tasks need to take into account the interests of students at this level. In or(ler for students to make sense of concepts, ways of reasoning, productive procedures, and problem solving strate- gies in mathematics, and to develop skill with number and symbolic operations and procedures, they must be engaged in exploring, investigating, inventing, generalizing, abstracting, and construct- ing arguments to support their ideas and conjectures. New mathematics materials for the middle grades are problem oriented. Interesting contexts for investi- gation are posed. Within these contexts students bump into mathematics that they need but do not yet have in their tool kit. This need drives students to invent, to create strategies for solving their problems. Out of these student ideas come the conversations that, with the teacher's guidance, help make the underlying mathematical ideas, concepts, procedures, skills, and arguments more explicit for the students. As you think of the socialization of the middle school level, there is a natural match between the mathematical need to deeply under- stand ideas and the social need to inter act with ones peers around tasks that are of interest. We can use this energy and need to socialize to advantage in math- ematics by refocusing our instruction to

be more on mathematics as an experi- mental science where seeking to under- stand why is the major quest. Another too} that we have available to us in teaching mathematics is technology. Technology is engaging to students. It also allows access to mathematics that could not be explored in the past. It helps to create environments in which students can see change, can engage in a dynamic way with mathematical conjec- turing. It allows students to tackle real problems with messy data. It allows students control over different forms of representation of mathematical relation- ships. It allows students to invent new strategies for tackling problems; it allows students to reason through problems in ways that are different from the strate- gies we would use. Again this freedom to experiment and to create makes mathematics of greater interest to many more students. It connect mathematics to the real world and to the interests of these emerging adolescents. SUMMARY While mathematics in the middle years is cognitively demanding, it is suited to the developing capabilities of MATH EMATI C S I N TH E Ml D DLE middle school students. New forms of classroom interaction, more emphasis on experimentation, on larger, more challenging problems, on seeking to understand and make sense of math- ematics and mathematical situations, on using technology as a too} these are the ways in which new middle school curricula are seeking to engage these social, energetic, (leveloping a(loles- cents in learning for their own future. Mathematics can be a key to opening doors or to closing them for students. Our goals has to be to help preserve future options for our students until they reach a level of maturity to understand the consequences of their decisions. We cannot do this unless we can make mathematics interesting and relevant to our students. REFERENCES Gal, I. (1997~. Numeracy: Imperatives of a forgotten goal. In L.A Steen (Ed.), Hey numbers count: Quantitative literacyfor tomorrow's America (pp. 3~44). New York: College Entrance Examination Board. Steen, LA (1997). Preface: lithe new literacy. In LA Steen (Ed.), Hey numbers count: Quantitative literacyfor tomorrow's America (pp.xv-xxvii0. New York: College Entrance Examination Board.

Thomas Dickinson Professor of Curriculum and Instruction, Indiana State University SMALL STORIES' Successful middle schools have small stories "because the school has time for small incidents before they become large" (Lipsitz, 19841. ~ have one such story to share. It was one of those early spring days that promised clear skies and warm weather for weeks to come. School had just let out, and ~ was sitting at my (leek, gra(ling papers. Wrappe(1 up in the process of marking and recording, ~ didn't immediately see the small figure at my door. "Mr. Dickinson, are you busy?" the voice asked and my grading reverie was broken. It was Amy Bounds, a seventh grader in my homeroom and a student in one of my social studies classes. Amy was one of those students that make up the bulk of our classes, best described by phrases such as "good student" or "neat kill," one of those in(livi(luals that we honor with the label of "nice to have in class." The question repeate(1 itself, "Mr. Dickinson, are you busy?" ~ shook my hea(1 anti wave(1 her on into the room where she took the seat next to my desk. 'what are you doing here Amy? You shoul(1 be outside on a (lay like this." ~ tol(1 her. Her reply abruptly change(1 the (letache(1 moo(1 ~ was in. "Mr. Dickinson, I've got a problem." ~ attempted to fight off my immediate a(lult questions what kin(1 of problem coul(1 Amy have? An(1 as my min(1 race with possibilities, it (li(ln't slow (town with her next comment. 'Nou're a man, Mr. Dickinson, so you'll understand my problem." Gathering my strength, ~ i"Small Stories" was originally published, in an expanded version, in the November 1988 issue of Middle Schoolloavrnal. Used with permission.

asked Amy, 'VVhat's the problem? How can ~ help?" 'VVell," she sai(l, "where (lo you get your ties?" "My ties?" ~ asked, trying to put two an two together, unsuccessfully. "My ties?" 'yeah, your ties. My father's birthday is tomorrow and I'm not going to let my mother buy my present for him. ~ want to buy him a tie like the ones you wear. And I've been saving my lunch money. Do you think ~ have enough?" And with that Amy extracted from her backpack a plastic bag stuffed with change anti ploppe(1 it on my (leek. Amy Boun(ls was growing up. She was, to use her phrase, "no longer a baby." And this translated most immedi- ately into a problem with her father's birthday. She wanted to buy her father a present, an appropriate present, anti she wanted to do it herself. But who could she talk to, who could she ask? Not her father. It was his birthday, and she wanted very much to surprise and impress trim. Not her mother. She might not understand, or she might try to interfere. So ~ was chosen, partially because ~ was her homeroom teacher, partially because ~ was a man, and mostly because she liked the ties ~ wore. Before we did anything we sorted the money and talked about her father. ~ asked about his taste in clothes, colors, and fabrics. Amy could answer all my questions because she'd THE MIDDLE SCHOOL LEARNER checked his closet, several times. During our talk Amy related that she didn't have a regular allowance, that she had been saving "extra" from her lunch money for the last three months. She'd also been lugging it around in her backpack since she didn't want to risk it being discovered at home. She also knew better than to trust her hall locker. And once a day she'd been han(ling it over to her best frien(1 for safekeeping while she took gym. Most interesting to me was Amy's revelation that she'd been counting it every night, hoping that she'd have enough before her father's birthday. We talked ties for quite a while rep stripes, paisleys, solids, Italian silks, and knits. She ha(1 a list of the ones ~ wore that she like(l, anti we went through her list and discussed all the possibilities. Finally Amy decided on something silk, something blue or blue/gray, anti something with a small conservative pattern. And then she asked her origi- nal question again, ' Where (lo you get your ties?" ~ ha(1 been buying clothes from one men's store ever since I'd had my own money. ~ drew a map for Amy of where the store was located. Would she be able to fin(1 it? She no(l(le(1 with a quiet "yes." Did she want the present gift wrapped? They would (lo it for free, but she ha(1 to ask. Did she have a car already? No, well, there was a card shop two doors down.

And then everything was settled and in place. She left with a grin and a quick thank you. After she was gone ~ went to the office to make a telephone call. The next day ~ had a note on my desk at the start of homeroom "Everything is set. Thanks." That evening ~ received a telephone call from Amy's mother. Years later ~ only remember her 'Thank you." What ~ do remember as if it were yesterday was the grin on Amy's face as she came into homeroom the next day and her comment, "He wore it this morning, and it looked terrific!" DEVELOPMENT IN CONTEXT: THE CHILD, THE SCHOOL, AND SOCIETY You teach Amy. Or you teach teach- ers about Amy. Or Amy may sit across from you at your dinner table. Amy is an individual. She is a human being. She is a person. She is not a list of developmental characteristics. She is not a "typical" young adolescent (there aren't any). She is instead, Amy. And that is enough, if we would but see and acknowledge it. ~ began with a story about a young a(lolescent not in a mathematics context on purpose. That purpose was so that all of us would see development for what it is: a natural anti normal process, boun(le(1 by general psycho T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S logical guidelines, but embodied in individuals. And this embodiment is bounded, shaped, and influenced by a range of contexts the individual context of self, that mixture of heredity and randomness; the context of family, street, neighborhood; the larger social context, which today includes an over- whelming and disturbing mass media context that impacts young adolescents at their every turn. (Half of all young adolescents spend three hours or more each day watching television.) The average young a(lolescent is expose(1 to the mass media a total of about four hours per (lay. Only sleeping anti attending school occupies more time. ~ won't even go into the content if you'd like, watch MTV for an hour, or flip through the latest issue of any popular magazine. To try to un(lerstan(1 (1evelopment for young adolescents we have to frame these contexts with their in(livi(luality. To put it bluntly, we have to understand the in(livi(lual to un(lerstan(1 their (levelopment. Goo(1 mi(l(lle schools try to (lo just this. They try to un(lerstan(1 anti teach in(livi(lual children, not types or categories of children, but in(livi(lual chil(lren themselves. This is what This We Believe: Developmentally Responsive Middle sieve! Schools (National Mi(l(lle School Associations, 1995) talks about. It's what Turning Points: Preparing American Youth for the 21st Century

(Carnegie Council on Adolecscent Development, 1989) recommends. And it's what I've been seeing in a small school in San Antonio, Texas. DEVELOPMENT AND THE TEACHING CONNECTION Donna Owen looks out over the overhead projector at a sea of khaki and white and asks "What's my rule?" and points to the bright yellow poster on the board. Two columns of numbers lie waiting for her young charges: IN | OUT | 3 11 5 15 o 5 10 25 The students are already at work on this daily ritual. These are 6th graders, in uniforms (that's the khaki and white) at the American Heritage School, an Outward Bound Expeditionary Learning (ELOB) school-w~thin-a-schoo} that is part of Edgar Allen Poe Middle School in San Antonio, Texas. The vast majority of students are Hispanic. Their school, a new building, stands in stark contrast to its neighborhood surroundings. Kids, seated in pairs, are tackling the problem some with charts of their THE MIDDLE SCHOOL LEARNER own, others with series of problems trying to arrive at a formula that works for all of the four pairs, one student using a number line. Around the room there is a rich riot of color number lines, bought and teacher-made posters with vocabulary lists anti mathematics operations, and two computers (still not hooked up to the internet it's Novem- ber when this is being written) in the back of the room by the teacher's desk. The kids continue to work focused, on task, quiet. What first drew me to this teacher is outside. On one side of the hall is a large gallery (think about that wor(1 as oppose(1 to a "bulletin boar(l"), maybe 15 feet long. On it are student charts and graphs. But not just charts and graphs, but charts anti graphs about the students. In preparation for the team's first "expedition" where the topic was the Americas, stu(lie(1 through the lens of the focus question "How is the good life in the Americas nurture(1 anti challenge(l?" Donna had her sixth graders work on charts and graphs by collecting data on them- selves. Working in pairs the students queried their teammates, recorded the data and graphed or charted it, and then wrote narrative explanations of their findings. There were bar graphs, line graphs, pie charts. And the topics of the charts and graphs were wide ranging:

· the males and females in the class · the numbers of letters in students' names · the days of the week students were born · the right- and left-handedness of students · the number of lima beans that each student coul(1 pick up with one han(1 (!) · the height of individuals in the class, in centimeters · the circumference of individuals' heads, also in centimeters And all this work was displayed (and gra(le(1 too). The accompanying narra tives demonstrated: · a sense of audience (especially in their detail and completeness) · a concern for language, structure, and correctness · an appreciation of self and others (peers) as a source of information that could be used in mathematical understanding Donna Owen was teaching individu als. She was also teaching mathematics. And she was also helping students understand themselves. She was placing (levelopment within a context of in(livi(luals anti those in(livi(luals within a rich and involving context of learning, in this case mathematical un(lerstan(l ing. And if you're wondering about this marvelously insightful teacher, here's the bio: she is late to teaching; at 29 she T E A C H ~ N G A N D ~ E A R N ~ N G MAT H E MAT ~ C S went off to get a degree after serving as a teacher's aide in a classroom where she was getting more and more respon- sibility because of her talent, insight, and drive. Encouraged by her teacher anti principal to pursue a (1egree anti teaching license, she (li(1 just that anti became the first of seven children (she's the ol(lest) to go to college. An(1 she (li her degree in three years, cum laude, taking 18-21 hours of course work at a time, arranged on three days a week so that she could continue to do her aide work. By the way,Donna has taught middle school English and journalism for the past ten years. She thought she needed another challenge so she got licensed in mathematics and is in her secon(1 year of teaching that subject. Last year, her first teaching mathemat- ics, 91% of her students passed the Texas Assessment of Academic Skills. I'm back in the classroom again. The secon(1 expedition is un(ler way. Stu- (lents are working on problems involving space travel. Along one entire was, probably 40 feet, is a swath of purple paper, door to ceiling, with the sun at one en(1 anti off on the other si(le of the room (actually on the back of the door), Pluto. This is the entire solar system, to the scale of 1" = 10,000,000 miles. An(1 the sizes of the planets are to scale as well. The students are working on estimated travel time in hours to the planets. It's another example of 'VVhat's my rule"

where He kids are denv~ng the rule Tom data. ~ watch, amazed at Be degree of engagement. At Me end of We class my notes summarize the lesson: · focus on reading directions · finding necessary information · using available data · drawing conclusions Tom data working in cooperative groups recording information accurately · doing basic mathematical operations . working alone · making choices to investigate individually ~ could tell you all the developmental psychology connections here, but ~ hope you see them. ~ know my readers, THE MIDDLE SCHOOL LEARNER being mathematically skilled, can see what's happening here mathematically. This is another "small story." And like most small stories in good middle schools it goes relatively unnoticed, except by the teacher and students who are growing and changing together, as individuals. REFERENCES Carnegie Council on Adolescent Development. (1989~. Tavrning Points: PreparingAmerican Youth for the 21st Century. New York: Carnegie Corporation. Lipsitz, J.S. (1984~. Successful schools foryoa`ng adolescents. New Brunswick, New Jersey: Transaction Books. National Middle School Association. (1995~. This we believe: Developmentally responsive middle level schools. Columbus, OH: Author.