Table of Contents
Education 16: Practice Teaching
Materials from Radnor Middle School 
Page 
Introduction 

Why I taught secondary level mathematics 
i 
Acknowledgements 
v 
Observations 

Cooperating teacher's classes 
viii 
Other mathematics classes and other seventh grade classes 
x 
Classes that are neither seventh grade nor mathematics 
xii 
Seventh grade resource room class 
xv 
Lessons for School 

Lesson plans written during the summer and autumn 

1. "Room Arrangement" 
xvi 
2. "Flexagon Exploration" 
xix 
3. "Cipher Construction" 
xxiii 
4. "Homework Revision" 
xxvi 
Lesson plans implemented during the school year 

1. "Fraction Twister" 
xxvii 
2. "The Guesser" 
xxxi 
3. "Fraction Comparison" 
xxxiv 
IV. Work: Representative Student Samples 

V. Reports: Narrative Reports 

I. A. Why I Taught Secondary Level Mathematics
During my first tenure in middle school, I was struck by two realizations that would propel me into collegelevel mathematics. Ultimately, these realizations would return me to middle school for a second tenure teaching young mathematicians.
First, I was very confused by the pointslope formula in eighth grade algebra. Now, my father earned his Ph.D. in statistical mathematics shortly after I was born; mathematics surrounded and permeated me from an early age, and I merrily waltzed through its elementary school incarnation. But then the pointslope formula hit me. I did not understand how to determine values for slope m and yintercept b. Indeed, obtaining these values seemed a mysterious process; and, worse, the simple graphs they generated almost mocked me. Exasperated, I guessed answers to all the problems assigned me in that section of the book and subsequently received my worst ever homework scores in a subject I had always mastered (even without my father's assistance). By the time I resolved my confusions about the workings of the pointslope formula, the damage had been inflicted: my academic standing was vitiated, and my adolescent pride, wounded. Although it would be only the first of many other cognitive and affective struggles in mathematics, this incident perhaps constituted my first selfaware experience that mathematics and education, whether taken individually or together, were "not trivial."
The second thing that struck me at this age was the popular distribution of mathematics books. In comparing my father's library to those of others, I recognized in his shelves a typical diversity of books, with the disciplines of English literature, music, history, political science, psychology, even economics all represented. However, his readings also included significantly more natural sciences and, especially, mathematics literature of all nature, from reference to recreational. It seemed to me that, simply because he "read math," my father read more diversely than did others. Indeed, I soon concluded, extremely few people outside of mathematics seemed to even know its seminal works; yet people toiled through "the classics" of other disciplines with comparatively little hesitation. Wishing to remain as broadly literate and fluent as possible, I resolved to include mathematics in my future pursuits.
So it was that, around my middle school years, I suffered from my first real spell of frustration in mathematics; and resolved that these were good and worthwhile struggles. This resolve was one reason motivating my pursuit of a liberal arts education in mathematics at Swarthmore College. Others would surface, including the enrapturing conciseness, precision, universal applicability, and elegance of mathematics. In particular, one dimension of my Swarthmore College experiences proved formative to my teaching middle school mathematics.
As I continued in the mathematics major, I gradually assumed positions assisting others with their mathematics. As a grader, Writing Associate, and tutor even as a friend on the dormitory hall or in the house I did not just work, on my own, at mathematics; I also communicated, to a diversity of others, its many faces. And among those whom I assisted, I found a relatively poor interest in, and a disquieting lack of appreciation for, mathematics. The great majority of them did not even exhibit much articulateness in their mathematical pursuits; they mainly seemed intent on "getting" some "answer" that ever lay beyond them. Indeed, their struggles and fears bore an uncanny resemblance to my middle school showdown with the pointslope formula. Through this mathematics study and practice, I realized that, even if I not understand the pointslope formula the first time I encountered it in middle school or the homomorphism theorems the umpteenth time that I stared at them in College I could share some qualities and applications of mathematics, and support others' mathematics achievements, interests, and appreciation.
Working with other peoples' mathematics helped me approach my goal, set during my middle school years, of achieving a fluency that integrated the discipline into the whole of my life. A parent of one of my natural sciences friends observed the results of this practice when she remarked recently, "I'm struck by how often you two start casually discussing something, and then end up talking about the mathematics behind it."
Aiming to honor and work with the next generation of adolescent mathematicians' cognitive and affective struggles, I returned to middle school for a second tenure.
I. B. Acknowledgements
I thank all the professors of the Swarthmore College Program in Education: Eva Travers, Diane Anderson, Wes Shumar, Lisa Smulyan, and especially Ann Renninger. Through their remarkable sensitivities and insights, they have patiently continued to offer me the gift of genuine education opportunity for growing.
I thank Peter Corcoran, Chairman of Bates College's Department of Education, who first welcomed me to, and continually reinspired me on, the path of education. May our efforts in environmental education bloom!
I thank Margie Linn, my classroom observer, for diligently holding a mirror to me and then gently suggesting how to change what I saw.
I thank my cooperating teacher, Woody Arnold, as well as our fellow educators at Radnor Middle School, Mark Springer and Ed Silcox. They introduced me to, and helped remind me of, the rewards of the exhausting calling we call "teaching."
I am indebted to a veritable lineage of mathematics teachers, especially Messrs. Pace, Gonda, and Braden; and Professors Shimamoto, Klotz, Grinstead, and Hunter. With verve, clarity, and patience and despite my many errors and obtusenesses they shared with me an art and science.
I thank my father, who, in teaching me mathematics, both lit up and moistened my eyes alike. I still describe "chagrin" as the my father's reaction when I pointed out an error he made in reproducing on a triangular face of an icosadodecahedron I crafted out of posterboard a curious pattern of numbers. I might submit as ironic that a decade later, during my second tenure in middle school, I would share with my students this same mathematical curiosity Pascal's Triangle.
Finally, to my younger peers in mathematics, and to my cohorts in student teaching including Mike Dower, Watershed Program student teacher from New Hampshire I dedicate this portfolio. Jactate aleam!
II. Guide to Observations:
II. A. Observations of Cooperating Teacher's Classes
Grade: 7th Subject: mathematics Date: Thursday, 5 September
sixth period, group one:
desks grouped in fours, two facing two
[responses elicited by "what is a rectangle?"] "height is half the width"; "foursided shape"; "two longer sides and two shorter sides"
of five tables, only boys occupy two, and only girls occupy the other three
"Any football players? . . . Any football players?"
Alan arrives at solution of 1.27 to "98 goes into 125" by erasing, placing zeros, and otherwise using whatever hints teacher drops
sixth period, group two:
(second teacher, entering classroom): "Did you make the big announcement? . . . Who's playing pound football? . . . How much do you weigh?" [asking only the boys, and chiding those who are too heavy over 110 lbs.]
"If you need to have an excuse, be more creative than that" [citing field trips]
"What happened to that person in class [when he or she was absent]?"
"Would you write '13 out of 20'?"
seventh period, group three:
"Do you people realize? . . . Do you realize?" [referring to academic grooming inherent to "advanced math" placement]
Stephanie: "I have to be my mom wants me to be in Honors Algebra 1 so that I can be in Honors Algebra 2 next year . . ."
the two students who distributed textbooks kept the newest textbooks for themselves
"You are the lucky people. You get a textbook today."
working with Polly to calculate 98÷125: a multiplication problem preferable to a division problem? yes, I convince her; then she arbitrarily performs 98x125 . . . guess doesn't compare with answer! reconsider 125; can you make it "get to" 1000? . . . multiply 98 and 8 [then move a decimal over three places to the left]
working with Andres on another problem requiring long division and resulting in decimal:
"What would make this easier?"
Andres [after correctly setting up 99/105 as a long division problem]: "A calculator"; comfortable, but unwilling, to muck around in long division
Grade: 8th Subject: mathematics Date: Thursday, 5 September
third period:
children sit, separated by sex, into groups of threes and fours; last vacant seats always filled by girls
"make high school course selection"; "it's only fair to know" what we're preparing you for
"Keep a neat notebook. Tests will be cumulative."
"homework to be brought the day it's due."
lots of "we knew how"s to communicate to students that last year's good habits and expectations are also this year's
during silent inclass assignment at seats, Megan M.'s eyes meet David and Zack's wandering eyes; then whispers to them answer of ".784" [she had a calculator; David and Zack did not]
fifth period:
one foursome of girls, one of boys, two mixed groups of three
again, emphatic pronouncement of students as "bright, intelligent, hardworking"
"They [authors and publishers] intended it [the textbook] to have all the pages. You know what I am saying."
again, lots of fillintheblank monologue
Grade: 8th Subject: mathematics Date: Monday, 30 September
"Uh, ladies and gentlemen"
[nearly immediate silence]
"It worked! . . . I have some information for you." [and so class starts, five minutes behind schedule]
FA: 6, 9, 3, 6, 5, 6, 9, 6, 4, 8, 4, 7, 4, 8, 6, 4, 8, 4, 7, 9 [groupwork]
during groupwork, eyes and conversation roam everywhere
three students surround teacher
student: "I said 'I'm sorry' for disrupting Kathleen as she went back . . ."
[interrupting]: "Ladies and gentlemen, be mathematically constructive for the next 20 minutes . . ."
"Are we supposed to write it down?"
"I think that would be a good idea."
"Are you sure?"
[sarcastically]: "No, the test is on chapter nine [not one]."
Eleanor had hand raised for one minute, demonstrating little mind or interest in whether anyone noticed, and eliciting as much response
talk of cheesesteaks and farts persist
not wearing glasses, teacher sits at desk; receives Kathleen; then Eleanor and Caity wait for attention; Alex approaches, too
FA: 9 [groupwork], 9 (as teacher continues, while at desk, to prod Kathleen mathematically; Eleanor and Caity continue to wait for a total of two minutes), 9, 9, 9, 9, 9, 9, 9, 6, 7 (eliciting attention to the assignment on the overhead, which evokes "O compare! Do we have to hand this in?")
over ten minutes, receives four different students; Court and Meg wait three minutes to reach teacher
about eight (one foursome, two pairs) consistently ontask; ten minutes later, a second foursome gets down to work
"Why should I do it for you? I've already done my part."
Ling, completing in mock exasperation: " . . . and we've done absolutely nothing."
sarcastic? "O, my gosh" and "O, I see now"
FA: 5, 6, 8, 5, 7, 4, 8, 9, 5, 9, 3 ["O; okay, you're wrong."], 5, 4, 10, 7, 5, 5, 6 ["Think what you're writing about. What are you saying?"]
"Have I talked long enough to fill all your time?"
II. B. Observations of Other Mathematics Classes and Other Seventh Grade Classes
Grade: 6th Subject: mathematics Date: Tuesday, 10 September
desks grouped in fours, two facing two
(reminder): "Has anybody figured out the problem for yesterday? . . . Ashley, did you get it? Wanna come show us?"
looks to teacher for cue . . . "She's using an interesting [approach]. What would be the opposite of going outside the design?"
"I'm sure you . . . and I'm sure you have your homework out."
reviews, prompts mnemonic: "M.A.T.H."
"What do I expect for you to do with the homework?"
"The first problem has a new word: dimension."
"The only thing [troublesome about your iterative method] is, if we're [doing the 700th in the series], I don't want to figure out [the 699 previous cases] . . . What you're doing is fine, but . . ."
"Who could tell me the 50th [case] using [René's] method?"
Lindsay whispers "101" to boy who is calledupon and struggling to evaluate 2n+1 for n=50
Varaj and another boy offer diverging solutions; teacher points to the other, saying "Explain it [I don't understand it]"; finally, teacher writes it down and nods, smiles her approval
writes down student's formula that should be equivalent to (2n+1)n:
(n1)*2+3*n
assigns failure ("If you want to bring that in for math help during CRT . . ."); but simply missed order of operations that student likely wasn't even aware of, viz.,
[(n1)*2+3]n = (2n+1)n
". . . then you're solving an algebraic problem that some kids in high school . . ."
"When I was in high school [we never had any pictures]; we were taught to move things around [i.e., manipulate algebraically]."
(note: here, most of the bottom righthand third of my original page is missing I used it to write a note encouraging and assigning competence to the student who committed the order of operations oversight)
very little buzz from groupwork; 3 out of 21 are talking
perfect sexselected seating
Ashley gets another chance
René is halfway to overhead, then index fingercounts, realizes formula doesn't account for example, says "Never mind," finally stands up next to teacher and talks it out
"But I think we could change it a little bit," failing to hear boy's different formulae one for even numbers, another for odds
Grade: 8th Subject: mathematics Date: Monday, 9 September
desks grouped in fours, two facing two
homework was to cover textbooks
"Coming around is a seating map . . ."
4 out of 20 are seated in mixedsex group
reviewed "Wizard's Towers" (Towers of Hanoi) problem; students suggest patterns of addition, of (2n+1), of doubling previous case
at board, student gives fifth in (2n1) blockbuilding pattern
(drawing ellipse): "Keep going as far as we want with that."
reigned in and clarified student pattern perception
"Is everyone familiar with that notation?"; pauses for aside: 2Ñ1 = 2x1 . . . diverges to explain why 2x1 will be confusing in algebra
[interrogatively]: "It doubles every time."
student, "self"correcting: "Adds two."
"How could I double this?"
another student: "Squaring."
"You said squaring before . . . something entirely different . . . (diverges to explain difference) . . . so don't confuse [squaring with doubling]."
only numbers, expressions, on board; English limited to two words "pattern" and "blocks"
"Any other [patterns]? [two second pause] What about . . ."
most eyes attentive, bodies still; remarkably wellbehaved
"do something like this" mess of numbering for problemsolving
"Variables are x." (writes "x = pattern #")
"Try the next three in this one" cues students to write, which elicits "Oo! We're going to write it down."
"You're a softball player; you can handle pressure."
Fibonacci numbers are "not a very recognizable pattern yet"
complete sentence formulation?
completing trivial pattern of "A, B, C, D, _, _, _" evokes only one halfhearted response
groupwork: "I don't hear a lot of discussing."
humor at "how dumb" is the pattern behind J, F, M, A, M, . . .
"Could I mark that [defensible, but unorthodox, interpretation] wrong on a quiz?"
"Always be confident in your answers."
girl who didn't care to go to board speaks to board, inaudible; "Can I just read you my answer?"
assignment always posted, thereby announcing its availability
beyond the two words "pattern" and "blocks" on otherwise number and expressionfilled 4'x12' board, by end of class there is still almost no English on board and certainly no complete sentences
Grade: 7th Subject: Watershed Date: Monday, 9 September
desks grouped in fours, two facing two
"Anytime there is a new seating arrangement, that means a new project."
[overlapping one another]: "What is a presentation?"
[repeating and reformulating a student's response]: "[A presentation is when you] take information and share it with people. That's a great way of putting it."
refer to own presentation on rocks
model humorous bad presentation
joke with rock tossing, eating
brainstorm qualities of good presentation; students develop evaluation criteria
"Any other? [two second pause] Well, . . . [you students will develop evaluation criteria]."
"So we've already solved Scott's problem [which he raised earlier]."
one teacher forcefully, emphatically presents while seated
". . . 'cause when you pave a big area like this [indicating bus loop beyond the classroom walls] . . ."
"Yeah. We got 'em [i.e., covered all housekeeping points]"; then waves hand, moves on to another subject
"You guys have to figure out for yourselves."
both continually refer to previous Watershed students' works
(as children begin discussing in groups): "You don't have to come up with an answer, but you have to begin the process."
children allowed to work comfortably; as if told "make yourselves at home"
one teacher interrupts to encourage more positive dynamic and to give some expectations to groupwork; "mention something positive about [what the previous speaker contributed]. Please be positive with each other. Be kind."
students doodle, rummage through soccer gear, pop spitballs at one another
Alex: "Guys, c'mon, let's make an idea. We have to get some good idea [sic]."
teachers are absorbed at photo montage; in 45 seconds, all students have joined them before the photo montage; two minutes later, group has dwindled to seven; one teacher invites them back to their seats
Grade: 7th Subject: Watershed Date: Wednesday, 11 September
as one teacher has class, the other is carrying on a conversation with someone else
"We need to amend that [not erase it]."
gives Steve the "gong"/buzz, but this does not return him to his seat
although many looked at map, more looked for Skunk Hollow as label, or measured distance relevant to field trip, although some researched elevation
offers better, synthetic statement of location
"For the benefit of Mr. Tchen," digress into a "math problem" what distance did we walk?
"Who can review?"; reworks every contribution
teacher leads student ["Did we do . . .? And we did . . ."] through ticking off of accomplishments; "Any other accomplishments?" [waits five seconds]
salvages Ashley's definition of "aquatic" by refocusing her contribution; backand forth is heavily guided by teacher, with lots of leading questions
Sammi: "That's a dictionary definition."
Steve [sarcastically]: "Really?"
"That plant can do something that none of you can do . . . make its own lunch"; then renders this imaginatively "[If you could photosynthesize,] you could walk out into the sun and be full."
students persist in photosynthesis challenge, suggesting all of the following as not relying on photosynthesis: Fruit RollUps, hamburgers, lobster, shrimp
"A tributary is like the opposite of effluent"; then listens to Flora's definition; returning to own definition, with little validation of Flora's, asks "What is [effluent]?"
"Are there any questions on vocabulary? We will not have vocabulary tests . . . "
Grade: 7th Subject: Spanish Date: Tuesday, 17 September
four rows of desks, all facing forward
coaxes recitation with hands
FA: 6, 8, 6, 8, 6, 8, 6, 2, 6, 4, 8, 2, 6
girls around the outside; only Rosa surrounded by boys
walks through days of week; then has students copy vocabulary down for game; directs students individually through textbook's organization
". . . the many vowels you know . . ."
paper ballthrowing and penswiping Pedro protests teacherdeemed inadequacy of Saranwrapped book by throwing another paper ball right behind the teacher
17 of 25 students are Watershedders: Ashley, Adam, Megan, Andres, Anthony, Polly, Rosa, David, Alan, Logan, Flora, Steve M., Stephanie, John, Marc, Claire, Stacy
"They'll come back to you . . . I'm sure their meanings will pop into your heads."
Steve excited to share delight in having played "Simón disse" last year
calls on different students to attempt pronunciation and meaning, but does not write on chalkboard these definitions
"Oh! we're supposed to write 'em down?"
"If I were you . . ."
models mouth shapes
pulls Pedro aside "for a few minutes," during which he misses review of 13 vocabulary words
FA: 2, 5, 4, 8, 2, 9, 2, 5, 5, 4, 8, 5, 6, 8, 4, 8, 4, 8, 5, 9, 5, 9, 5
Steve's synthetic, affirmationseeking statement "So there's two [informal] 'How are you?'s and one formal 'How are you'?" receives absentminded, vague "hmm"
FA: 5, 6, 6, 5, 5, 4, 8, 2, 3, 5, 6, 5, 6, 6, 5, 6, 4, [apology], 8, 2, 4, 8, 2, 4, 2, 5
on board, references glossaries only by their textbook page numbers "414/421," where 421 is circled
FA: 6, 6, 6, 4, 8, 6, 4, 8, 2, 6, 9, 4, 6
II. C. Observations of Classes That Are Neither Seventh Grade Nor Mathematics
Grade: 8th Subject: social studies Date: Monday, 9 September
four rows of desks, all facing forward
only girls in left three rows; mixed fourth row; only boys in right two rows (closest to the door, belying later arrival?)
had students silently copy assignment because "it's hard to write and think at the same time"
[on board] "Essay: Chesapeake 20 points
15 content/5 mechanics . . ."
left room twice as class went five minutes in silence, copying
announced that any time video watched, students should expect essay on content
girl hesitates [ultimately] six times to raise hand that might interrupt . . . finally, recognized upon pronouncement "Any questions?"
refers to Mr. Fisher's boatings and Sandy Hill field trip, occurrences familiar to students
watching TimeLife/The Nature Conservancy video on encounter between Native Americans and settling Europeans:
at least 18 out of 20 students do not record narrator's opening remarks
at least 12 out of 20 do not record fact statement: "It was the English and the French who first settled . . ."
at least 18 out of 20 do not take down less expository description of Sir Walter Raleigh's contributions
at least 8 out of 20 do not record "Native Americans' use of land . . . fire to thin"
student "o"s elicited by water snake; frog croaking [seen from behind] is "ugly"
"TERN: tern. TERN." not recorded by at least 5 out of 20
at least 18 out of 20 do not take down James River definition
"List 'eagle' under 'animals.'"
0 out of 20 take down Native Americans' view of eagle's role ("flies near 'breath giver'")
"crab" should be down; "Heron: HERON."
"Under 'plants,' 'cordgrass': CORD."
0 out of 20 record "food chain of the bay" and role of blue crab
spells out "sturgeon" over narrator's description of ecosystem energy exchanges
"How do you spell 'weir'?"
"WEIR."
"Under 'climate,' put 'hot and humid.'"
at least 15 out of 20 do not record malaria and typhoid . . . but categorize as "animals"?
how comprehensible is the narrated text "It pleaseth God to send us [enemies] who giveth victuals such as . . ."?
[whispered to a neighbor]: "What time is this class over?"
Grade: 6th Subject: chemistry Date: Wednesday, 11 September
three rows of desks, all facing forward
opens by correcting clocktime; "So what does that [i.e., the clock being ahead] mean on the other end [i.e., for when class ends]?"
"I think there's some Sesame Street character named Beeker."
"If my lab directions said I need 175mL, would I use this [150mL beaker]?"
out of the blue and in the middle of teacher's explanation of volumeappropriate use of beakers, one student asks question about previous day's mercurywater experiment in a bottle behind teacher
boys bunched in third row, the farthest from the teacher; only one girl as far back as the second [of three] rows
persists with referring to seating chart rather than asking students for their names
when turns on overhead to list lab equipment, students take cue: binders pop, papers shuffle
"That's pretty obvious. Hopefully, we'd all recognize that." [in response to student's observation that beaker has a spout]
says "another definition" of eyedropper is pipette, but writes them down, without explicitly comparing and contrasting, below graduated cylinder
student explanations of meniscus phenomenon: Ushaped bottom; optical illusion . . . finally, teacher dispels with water strider illustration of high surface tension
"Exactly, but I gave you two numbers [by which to quantitatively determine volume]," shelving Courtney's Archimedean explanation of determining volume "by how much the water goes up" [i.e., solid's displacement of surrounding liquid]
"Is this a reasonable number? You should know."
attempts density formula in last instants of class, as students are packing
II. D. Observation of Seventh Grade Resource Room Class
Grade: 7th Subject: math/resource room Date: Tuesday, 10 September
desks grouped in four, two facing two
Khara, in singsong voice: "Kevin walked in late."
ticket exchange poster: "25 = candy  pencils"
[asking a student other than the one who volunteered the response]: "How much is that?"
first student, continuing to add items: ". . . and another soda . . ."
"Khara, good morning, I'd like to hear from you."
[prompts]: "What if I only want to get soda?"
"If you want, please take out your homework."
Khara: "If we want?"
"If you would . . ."
because one student felt not so good about it, whole class goes through photocopied homework problems about large numbers
"Anyone disagree with her? [one second pause] She's absolutely right."
Frank, drawing a "lessthan" symbol in the air: "It goes this way."
"Absolutely right," writing down "<" symbol, with no reference or context in small nook of the board
Frank, in full explanation of the utility of rounding: "If you're shopping and you don't have enough money."
Michelle's "I know that if the ones place is five or more, I round up; if the number is five or less, I round down" gains teacher's approval, despite having included five in both sets
at least one student has trouble finding current problem on the board
"I think you're going to find this [worksheet on rounding] as an easy review."
"Is that in the tens?"
Michelle, correcting on cue: "O, the five [is in the tens place]."
no number line representation
sign up for extra help
(reviewing Frank's work): "You went to a smaller . . . This one's not smaller, this one's not smaller." [leading intonation] "Is this one smaller?"
Khara: "Do we get tickets for this?"
"Compare with your neighbors."
"Math" in upperleft hand corner heads day's activities
Chris, after two calls to do so: "Do we have to write it? Aw, man."
"One and then one. And then two."
Chris: "And I was thinking, if eight was there . . . nine? I don't know."
"Then up by . . .?"
Chris: "Ten."
"Why did this go up by one?"
[together]: "Twelve."
"Sometimes I have to put it [the number being added to] underneath to add it [sic] up."
"What's the biggest number [of the Fibonacci series] going to be?"
"I'm not sure that that's right."
circles room with tickets
[to one table of boys] ". . . and you guys are like me just workertypes."
President's game: you gain by outracing the person next to you in mental arithmetic
III. A. Lesson Plans Written During the Summer and Autumn
"Room Arrangement"
Learning Objectives:
To feel welcomed in the physical space of the classroom.
To work with area and distance.
To generate a room arrangement map and implement it.
Material(s) Required: chairs, desks, and other classroom furnishings; measuring tape, paper, and pencils.
Time Required: one or two class periods.
Lesson Plan and Extensions:
Welcome the students to their new classroom, and announce to them that they will be in charge of planning and then arranging the furnishings in it. Begin by refreshing students' memories about their previous year's math classrooms. How were the room and its furnishings laid out? What did the students like and not like about the ways that chairs and desks were organized? Did they feel as if the style of instruction worked well in that space? If they do not offer quantitative measures of their former classrooms, invite the students to do so, whether they are estimating or just guessing. If their classmates this year are different from the ones they had math class with last year, this quantitative comparison may be particularly helpful for communicating the floorplans. Remember to keep the focus on spatial layout (e.g., where chairs were), as opposed to more fixed parameters (e.g., chair height).
Now that the students are thinking critically and quantitatively about room arrangement, ask them to each sketch quickly and in silence their previous year's math room layouts. Encourage the students to dwell not on exact details of individual chair and desk shapes; rather, emphasize the area that different furnishings occupied. To this end, squares with circles on one corner suffice to convey chairs with their attached desks. After five minutes, or whenever the students are ready, ask a few of them from different previous math classrooms and with differing sketches to present and describe their work briefly before the rest of the class. Encourage each presenter to focus on where things were in the classroom, then say how they felt about this layout. Follow each presenter by facilitating discussion: do others familiar with the same classroom have different sketches? Do the former classmates understand the presenter's map? Do they think it fairly represents their old classroom? If it does not emerge from the conversation, point out that the edge of standard 8.5"x11" paper may not correctly represent the edge of the room. Finally, discuss with students any perspectives they used other than the overhead or "bird's eye" view, and their merits and shortcomings, in terms of describing area.
Now, divide the student population in half: a chairsanddesks division ("CD"), and an aislesandfloorspace ("AF") division. Break down these two large groups into heterogeneous groups of four students each, drawing on any observations of individual student competencies provided by the previous exercises.
For both divisions, each group must arrive at one proposal, by consensus. Also alert the groups that they should generate a map like the ones everyone just did the ones representing their math classrooms from last year; but that this time it should be complete with relevant measurements. Remind them that these maps should include only those objects that are part of their division plan. Finally, each student should be prepared to explain their group's proposal. Distribute tape measures and other measuring apparata, to use only on their division's objects chairs and desks for the CD groups, floorspace for the AF groups.
Announce to the CD groups that their role is to figure out the best way to arrange chairs and desks so that students can, with equal facility, group them into fours (for groupwork) and break them out into rows (for viewing overheads and the chalkboard, assessing students individually, etc.). One constraint they might have to meet is the distance separating students in their respective chairs and desks: for groupwork, no one should be farther than the shortest student arm's length; for individual work, no one should be closer than one and a half arm's lengths. Another constraint to work within might be the time and effort it takes to move chairs and desks from group to row formation and back again. Encourage students to think about staggering chairs and desks within the rows for individual work, which would minimize the ability to see others' work; or about what might be too close for comfort or manipulating paper and so forth in groupwork.
Announce to the AF groups that their role is to figure out the best way to use the floorspace so that it allows students to enter and exit the classroom. One constraint they might have to meet is the minimum aisle width that allows for safe passage in the event of a fire drill. How little space can safely be preserved in the area of the door? around the windows? How much area could backpacks consume? How far can students be from the chalkboard or overhead projector to see them well? Are other classroom resources, such as books and supplies, within easy reach?
Reconvene the students once each group has a map and each member is prepared to present their group proposal. What did groups within the same division do differently? Do some CD proposals work better with certain AF proposals? Facilitate consensus on one room arrangement, and then, if time allows, implement and test it out.
As an extension into following class periods, the [three] classes of students could compare one another's floorplans and, if appropriate (e.g., if the classes are the same size in number), agree on one for the remainder of the school year. This comparison could occur concretely, with students rearranging the furniture according to others' floorplans; or more abstractly, with the students working entirely from the floorplans to determine what is different about the classroom setups.
"Flexagon Exploration"
Learning Objectives:
To exercise mathematical reasoning and writing in groups.
To explore flexagons (hexaflexagons suffice) and then demonstrate their geometrical and topological properties.
To carefully define the geometry term "face" (or "side").
Material(s) Required: one strip of paper and transparent tape; at least one flexagon, as well as six differentcolored markers for every four students; rolls of thin, wide brown packaging tape, the kind that requires moistening to activate the glue (optional).
Time Required: one class period.
Lesson Plan and Extensions:
The object of this activity is to describe a flexagon (cf. Countryman, 67). Due to the topological complexity of this object, begin with the Möbius strip. While the Möbius strip is another topologically curious object, it offers a much more graspable introduction to two and threedimensional geometry than does the flexagon.
Announce to the students that in this activity, they will think pretty hard about faces of physical objects. Begin with a simple strip of paper. How many faces (or sides) does it have? Mark each of the two faces in a different color. Now hold together the two ends of the strip, without connecting them, so that the paper forms a hoop and the colors remain continuous one on the "inside" of the hoop, another on the "outside." How many faces does this have? Finally, rotate one strip end with a halftwist and fasten it with transparent tape; this should yield a Möbius strip with the colors no longer "matching up," and offering no obvious "inside" or "outside." How many faces does this have? Discuss any conflicting views of what constitutes a face, distinguishing between two (e.g., the strip of paper) and threespace (e.g., the hoop and the Möbius strip) objects.
Show the students a flexagon, preferably a hexaflexagon. Warn students that this object may look geometrically innocent, but they'll spend the remainder of the class time in groups discovering how it isn't. Display the flexagon flat and, without naming or flexing it, ask students to offer their immediate geometric observations about it. The critical question to discuss, of course, is "How many faces does it have?"
Once students are satisfied with an answer, group them (heterogeneously, if possible) into fours, and then designate a different role to each individual in the group: flexer, recorder, reporter, or facilitator. Both these steps may be accomplished by simply having the students count off by fours and making all the "ones" flexers, the "twos" recorders, and so forth. Write on the chalkboard the functions of each role, as well as an example action; for instance:
Flexer: Slowly and gently flex the object. Balance your own explorations with suggestions from the recorder and reporter, as directed by the facilitator.
>For example, you might execute a series of flexes on your own, memorize the procedure with the recorder, then "replay" them for the reporter to practice.
Recorder: Systematically record the flexer's moves, and color faces as needed.
>For example, you might watch a series of flexes, ask the flexer to repeat them, and then plan out and use a coloring system so that you know exactly what the flexer is doing.
Reporter: Test and retest flex procedures so that you can lead the flexer through a demonstration of one sequence of ten flexes for the rest of your classmates. Answer the question: "How many faces does this object have?"
>For example, you might work with the recorder's coloring scheme and notes to then test running a bluegreenbluered flexing sequence with the flexer.
Facilitator: Keep your team members focused on their respective roles, and on the larger group goal of the tenflex sequence. Also, pace how your team members spend their time.
>For example, you might see that the flexer is growing frustrated in attempting to execute a difficult sequence given by the reporter. Work from the recorder's notes to pinpoint and then resolve the exact steps that are confusing the flexer.
Once every group has some flexing sequence wellrehearsed, reconvene all the groups to witness reporters direct their flexers. After a few presentations, invite any observations as to the rules that govern flexes. Is every conceivable sequence possible? Does each possible face have only one arrangement or do some of the triangles in a face rotate with a flex? Return the students to their groups and have them, still working within their roles, write group definitions of a flexagon "face," as well as any rules they understand as governing flexing sequences. Encourage students to develop symbolic notation and to invent terminology ("fold," for instance, is the more common verb for "flex"), so long as they first carefully and uniquely define these.
Finally, wrap up the groupwork with a brainstorming session that would give an appropriate name to this object. To conclude, tell the students that a 1960s Princeton University graduate student in physics developed the flexagon as a toy. If they are interested in reading more about the goof genius who created it, Richard Feynman, refer them to Surely You're Joking, Mr. Feynman!; The Beat of a Different Drum: the Life and Science of Richard Feynman; What Do YOU Care What Other People Think?: Further Adventures of a Curious Character; and No Ordinary Genius: the Illustrated Richard Feynman.
As an extension, take the instructions for building the hexaflexagon and extend them to those necessary for building the hexahexaflexagon. What other flexagons could be constructed? What rules do they observe? How do these compare with those of the hexaflexagon?
Alternatively, return to the Möbius strip. Encourage students to conduct experiments on what occurs to a strip when it is repeatedly cut lengthwise. While flexagons have yet to find much practical application, inform students that Möbius strips find frequent use in belt drives and some conveyor belts, as the halftwist extends the life of the belt.
"Cipher Construction"
Learning Objectives:
To construct and use a cipher.
To learn student names.
Material(s) Required: papers, pencils, markers, and displays of several elementary cipher systems, such as the Cæsar shift.
Time Required: one class period.
Lesson Plan and Extensions:
Teachers (and sometimes students) frequently struggle to remember students' names during their first few school encounters. In lieu of yet another "name game," this activity uses elementary set theory and algebra to reveal how readily applicable mathematics can be.
Inform the students that, instead of using English to communicate their names to every other English speaker, they will spend the class period using mathematics to try to restrict communicating their names to select individuals.
Group students into fours, and announce that each group will become a secret agency. By creating and using their own ciphers, each secret agency will be able to communicate with others in their agency; but those in other agencies will not.
To begin with, the secrets that agencies will share are group members' names. Demonstrate a simple alphanumeric cipher system by writing out the string of twodigit numbers corresponding to the ordinal alphabetic position of the letters in your name. For instance, RICHARD would be enciphered in the digits 18090308011804; as a string, careful to preserve the zeros, this could be read without the dashes: 18090308011804.
Ask students to encipher, without the help of other secret agents, their first names in this straightforward fashion, and then to write these big and in marker on separate pieces of paper. Now have secret agents shuffle their papers within their agency, so that each agent does not necessarily have her own name enciphered before her. Bring two whole secret agencies forward, with each agent holding an enciphered and shuffled name, and challenge them to match agents with their true enciphered names.
Facilitate discussion between the students after this deciphering round. Was it easy to figure out which enciphered name belonged to whom? One observation might be that relatively uncommon letters, such as "z," or very common combinations of letters, such as "th," are easy to pick out because of their scarcity or frequency. After a little practice, it is easy to recognize the numbers "26" and "2008," and so infer the complete names they represent. Note that these numbers are also unambiguous even when buried in long strings; they cannot occur "coincidentally," since there is no letter corresponding to 60something, on the one hand, or to 00, on the other.
A second, much more immediate observation might be that one could count pairs of numbers in every string, which must have an even number of digits, and so gain a good idea of how long the corresponding enciphered name must be. That is, in this cipher system, short names remain short, and long names, long. Facilitate this realization with questions that compare encipherings of short and longnamed students. How can we get around this and keep identities secret?
These two observations of symbol frequency and "word length" are central to cryptology: the science of cipher systems. Even in the simple demonstration of this classroom exercise with student names, they may prove to be two critical keys to deciphering.
Now, challenge the secret agents to create unique cipher systems that will better protect their identities. Working in pairs within their agencies, have agents practice making better ciphers. Since symbol frequency is not so helpful without frequency statistic tables or for the sometimes odd orthographies of given names, the main challenge should be to minimize the clue of word length. A very easy way to work around this is to pick a standard word length that is no longer than any single agents' first and last names (ten should do); encipher all of the first name; then fill the remaining spaces by enciphering the letters of the last name. For instance, say Christopher Smith and Joy Johanssen are collaborating secret agents. They choose eleven as their "word length": so Christopher's name is enciphered 0308180919201516080518; and Joy's as JOYJOHANSSE is enciphered 1015251015080114191905.
The remainder of the secret agent challenge should be devoted to insuring that enciphering and deciphering are unique. That is, every number string has one and only proper deciphering. Secret agent pairs should test one another's codes for this uniqueness property. Here, introduce any other available simple cipher systems; encourage students to mix letters and the numbers 0126 to form their cipher systems. Once secret agent pairs have secure cipher systems, try them against the other secret agent pair in their agency. Students should discuss how they assigned numbers to letters, and how they could better encipher names that do not lend themselves to secure enciphering with this simple system, such as "Lilly" and "James James." Finally, each secret agency should decide on and carefully write up instructions for using one cipher system probably settling on one "word length," too from which each secret agent makes up his new "enciphered name" page.
Reconvene the secret agencies and engage them in deciphering wars. Which cipher system is hardest to crack? Other questions to ask of the whole class include: What are the steps or shortcuts to deciphering? What is necessary for a cipher system? Which enciphering problems, such as symbol frequency, persist? How could these be overcome without sacrificing the necessary uniqueness?
As an extension, introduce students to keywords and modifications of the Vigenère cipher system. Additionally, cryptology is an excellent avenue for practicing modulo addition.
"Homework Revision" (ten minute lesson plan)
Learning Objectives:
To work in pairs to solve, in three minutes, one homework problem that, individually, students struggled with.
In pairs, to present at the chalkboard a solution to this homework problem.
Other Objectives:
To value the importance of explicitly stating solution methods.
To appreciate the utility of homework one's own and others.
Material(s) Required: one complete set of the class' homework, photocopied so that names are omitted; pencils, chalk, and chalkboard.
Lesson Plan Outline:
1. Praise homework efforts
(Note: given more time, praise specifically by, for example, sharing problemsolving techniques that some students used and that peers might also find helpful.)
2. Cite specific homework shortcomings.
highlight discrepancies between the statement of the homework assignment and what was submitted
introduce the proposal of allowing one student's homework, selected at random on the day of a quiz, as the only notes allowed for that quiz
highlight one homework problem with which all students struggled
3. Assign classwork: pairs of students solve the difficult homework problem and prepare it for presentation in front of the class.
(Note: in the actual classroom with the actual students, the teacher might rather invite the students to choose, within their groups, one homework problem they could not solve or about which they have further questions.)
4. Invite one pair to present their initial difficulties with the problem, followed by their eventual team solution and method.
5. Conclude by sharing some of the "other objectives."
III. B. Lesson Plans Implemented During the School Year
"Fraction Twister" (10 October, 1996)
Learning Objectives:
To determine, in less than thirty seconds, a fraction from a given combination of hands, feet, and Twister mat circles.
To represent, in the same thirty seconds, a given fraction value as 1) a fraction, 2) a division problem, or 3) a pie section or "fraction bar" drawing.
To distinguish from one another similarsounding, but mathematically distinct, fraction "statements."
Other Objectives:
To practice and value working in teams.
Material(s) Required: Twister game set, index cards, poker chips and container, and chalkboard and chalk.
Time Required: one class period.
Advance Preparation: Before class begins, write fraction "statements" on index cards; for suggestions, consult the sample index cards on the following page. The play of the game will insure that no single question will ever have the same answer. Thus, syntactically diverse statements (e.g., What fraction of ". . . all feet cover green circles?" and ". . . all circles are green and covered by feet?") will be more instructive than will syntactically similar statements (e.g., ". . . the green circles are covered?" and ". . . the yellow circles are covered?").
Also, post these instructions:
Cards: Pick a card. Beginning with the words "What fraction of . . .," read the card.
Chips: Pick a chip. If it is white, write your fraction as a fraction: numerator/denominator. If it is red, write your fraction as a division problem: numerator÷denominator or denominator )numerator). If it is blue, draw your fraction as a pie section or fraction bar.
Sample index cards for Fraction Twister lesson plan
"What fraction of . . ."
Lesson Plan Outline:
Select teams of at least three members each.
Announce goals.
for those on the mat, to outmaneuver one another without falling outright, or allowing knees or other body parts to rest on the mat
for those not on the mat, to help your teammate by correctly representing, according to the posted instructions, fractions statements
Outline procedure of play.
each team nominates a member (the "twister") to begin on the mat. Twisters position themselves according to Twister game instructions for beginning play.
beginning with one team (the "drawing team"),
one drawer chooses an index card from the deck, begins with "What fraction of . . .," and continues by reading the fraction statement written on the card; meanwhile . . .
. . . another drawer removes one poker chip from the container and then announces its color
all teams have 30 seconds to write their answers on the chalkboard, then cover them by standing in front of them
model steps a., b., and c.
"twisters" maneuver according to whether or not their teammates answered correctly (see 4.)
if any twisters rest on the mat or fall outright, then they leave the mat and join the rest of their team
rotate to another drawing team
h. repeat steps b. through g. until only one twister is left
i. a new game begins with new twisters from each team, so that everyone gets a turn as a twister.
Explain moving and "torturing."
· if all the teams answer correctly, all the twisters move according to the outcome of spinning the spinner
· if any team answers incorrectly, but at least one other team answers correctly, then the correctlyanswering teams' twisters move according to the spinner; and these correctlyanswering teams instruct twisters of incorrectlyanswering teams where to place a foot or hand, thereby disadvantaging, or "torturing," these twisters
· if all teams answer incorrectly, the teacher "tortures" all the twisters
Announce safe play: when "torturing" twisters of incorrectlyanswering teams, "torturers" may only put twisters into positions that they can comfortably reach; twisters should never be "tortured" to stretch beyond their own limits.
6. Begin play, and continue until every person has been a twister.
7. Return to objectives.
a. learning objectives
i. determining fractions: Ask students to recall how they determined the fractions they heard. Did they count? use some information, such as the number of all Twister mat circles, again and again? What ways did they find more and less useful for determining fractions?
ii. representing fractions: To show a fraction, which did students prefer: writing it as a fraction, writing it as a division problem, or drawing it as a fraction bar or pie section?
iii. distinguishing fractions: Consider the two questions "What fraction of the green circles are covered?" and "What fraction of all circles are green and covered?" Do these ask the same question? What other questions sounded alike, but asked for different information?
b. other objectives: Recall students' teamwork, and praise any methods that seemed very effective. How did teams succeed or struggle? Brainstorm how students might determine fractions more quickly. Did any teams assign one person to determine the numerator, and another, the denominator?
"The Guesser" (17 October, 1996)
Learning Objectives:
Given a fraction, to use a computer to display it as a pie section by first entering a denominator, then a numerator.
To determine a fraction with value between two unequal fractions, given their respective pie section representations.
To list at least two ways to evaluate fractions.
To determine the strategy to playing "The Guesser" that, on average, most radidly guesses the randomlychosen fraction.
Material(s) Required: MECC's "Fraction Concepts" software and supporting computers (e.g., Apple IIe and IIGS), paper, and writing instruments.
Time Required: one class period.
Lesson Plan Outline:
Motivate lesson by reminding class of its earlier conclusion that it is difficult to draw, freehanded, pie sections.
Describe "The Guesser": a computer game that randomly chooses a fraction. The objective is to determine this fraction. For each guess, "The Guesser" draws a corresponding pie section and then states whether the guess was "too big," "too small" or exactly right. It keeps displaying only the best previous "too big" and "too small" guesses and their pie sections.
Outline procedure for playing "The Guesser," emphasizing points c., d., and e.
enter first name and then last name
b. choose play with denominators between one and eight (as opposed to those between one and four)
c. skip the directions for playing and the rules for scoring; evaluation will not be based on score, but rather on what is learned and shared about 1) ways to evaluate fractions and 2) strategies to play the game
d. guess the fraction; here, invite students to consider, rename, and modify these four common ways to evaluate fractions, and to brainstorm others:
· visually, i.e., viewing pie sections
· arithmetically, i.e., calculating decimal equivalents
· using prior knowledge, i.e., recognizing common fraction values (such as 1/2 = 0.5)
· outright blind guessing
e. state the way used to evaluate the fraction.
4. Construct charts to record ways of evaluating fractions used during game.
a. make a fourcolumn chart to record these strategies. What are the fewest number of folds needed to divide a piece of paper into fourths?
b. treating two middle columns as one, write "guess" in the leftmost, "way evaluated" in the middle, and "response" in the rightmost columns
c. model charting a response, using the above four (or more) ways of evaluating fractions
5. Pair students, beginning with one playing "The Guesser" while the other records the player's guesses and ways of evaluating fractions, as well as the computer's response.
6. After an appropriate amount of time, switch players and recorders.
7. As a group, discuss the third and fourth learning objectives. Questions to stimulate discussion include:
· What was the most helpful fraction to guess first? Why? What fraction was most helpful to guess second? Did the computer's response to the first guess influence choice of this second guess?
· Consulting the charts, what ways to evaluate fractions were more successful?
· Was there a strategy or several for playing the game? What were these strategies?
· What is the largest number of guesses these strategies required?
· (challenging) Is the most successful strategy for "The Guesser" similar to the fastest way to fold paper in powers of two?
8. In individual notebooks, summarize the lesson.
9. Assign homework: draw a number line from zero to one, and show on it all the fractions with denominators and numerators each between one and eight.
"Fraction Comparison" (21 October, 1996)
Learning Objectives:
Given calculators, to solve, during the class period, 16 problems of comparing fractions by 1) converting to decimals, 2) drawing pie sections, and 3) drawing fraction bars.
To define "equivalent fractions," and give an example of two.
Material(s) Required: previous homework assignments, calculators, and Mathematics Structure and Methods Course I textbooks (Houghton Mifflin, 1988).
Time Required: one class period.
Lesson Plan Outline:
Check in homework.
Ask a student to read aloud her or his summary of the previous lesson; praise strong points of the summary and reinforce others as necessary.
Facilitate brief discussion about number lines: what did students find in arranging the fractions?
Define equivalent fractions (cf. pp.1845) and otherwise support students as necessary.
Assign classwork: drawing on homework and recent classwork, solve fraction comparison "Class Exercises" 14 and "Written Exercises" 112 on p.192. For each such problem of comparing fractions, show work by 1) converting to decimals, 2) drawing pie sections, or 3) drawing fraction bars. For every problem, use one of these three ways to evaluate fractions. In the end, exercise any one of the ways on at least three different problems.
Narrative Report of "M"
Recently, as a peer was writing his solution to a homework problem, M announced that he had solved the same problem differently. After a brief exchange, we concluded that M's method was not so different from his peer's. He continued:
M: "Can't I write my method on the board?"
Richard: "Why isn't your peer doing it for you?"
M: "But . . . I'm me."
This demonstrates M's persistence what his mother had earlier deemed, in a written reply to his weekly report, as M's "sticktoitiveness." It also reveals M's desire to express himself, which is a prize in our discussiondriven mathematics. For instance, M more often than his peers do seeks my approval and immediate feedback when I am making my "homework rounds" at the beginning of classes. He is also quick to volunteer for the dryerase board a careful solution to a homework problem. M has also benefitted classmates by his articulate and thorough peerreview feedback. All of this supports what I feel is M's investment to quality written products both his own as well as others'.
I've noticed that M tends to volunteer much more regularly when we are discussing assigned homework problems than when we are posing new questions that arise from class explorations. Similarly, he appears to be at his least confident when completing tests and quizzes when M must negotiate his own way, in a fixed amount time, through old material and new applications of it. Indeed, M's regular calls for my attention, especially during tests, have not escaped peers' notice: they have reported that M capitalizes much of my time during such assessments.
Although M's persistence and desire to share his work have, on occasion, frustrated some of his classmates, M's desire to "get it right" benefits the class on many more instances. He is quick to verify and clarify our homework assignments, for example. I look forward to helping M to become more attuned to his classmates and to learn to stand more independently on his own fine work.
Narrative Report of "S"
S frequently takes charge in our classroom. Whether presenting her homework problem solutions at the dryerase board; investigating a question at the overhead projector; or working on a smallgroup project; S almost always carries herself with a certainty and poise that command the situation.
She is wellversed and welladjusted for this leadership. S has an articulate voice and a good ear for discussion, so she knows the material she's talking about and hears what others' are saying, and balances the two. I am particularly impressed with S's ability to bridge our learning and our reallife experiences. She appears comfortable in both applying abstract knowledge to the Watershed Program's physical stream testing, as well as bringing practical reasoning into homework assignments and classroom assessments.
Similarly, S integrates mathematics and English well. On written tests, she has applied, in concise language, her good logical and reasoning skills. And as a classmate and I independently saw, S's "hard problem solution recipe" was very clear and easy to use, right from the very first draft. Still, she also incorporated our feedback intelligently to exhibit, in the final draft, even more of her strong understanding. So S is a selfconfident, articulate, and responsive student.
With S's other writing, however, I have had a little more difficulty. In particular, S seems to squeeze into her homework problems a lot of numbers and other information. Her solutions are, much more often than not, correct; but I tend to get lost or confused by them because they read more like notes rather than responses. I could read her work more easily if S were even just a little more generous in spacing and laying out these daily homework assignments. Reproducing, onto her homework, the textbook's questions might help remind her to flesh out and organize her responses better. I believe this will only help S later review from her own writing; and further enhance S's excellent ability and high penchant to communicate her good mathematics.
Narrative Report of "L"
Whether the assessment is done in class or taken home, and whether the mathematics is number theory or fractions and their applications, L has performed well. I am particularly impressed with L's descriptions of procedures: he writes directly to the reader, giving friendly, yet concise and practical, explanations and instructions. It is one skill to communicate such mathematics orally (and sometimes a challenge to do so even without time constraints!); but L shows that he can express, in writing, his mathematics competence even under pressure. So L's strong mathematics writing on these assessments typically earn him good scores scores that have improved over the course of the semester.
L's demonstrated skill on written evaluations does not, however, seem to transfer into the more social dimensions of school life. When we are together oneonone, for instance, he grumbles that he's "no good at math." Such low academic selfesteem flies in the face of his good written performances and, of course, belittles his scholastic potential. Around his peers, whether during class discussion or projectbased groupwork, L is not only frequently distracted, himself; but I have too often seen him distracting his peers, sometimes with spitballs. Even more disturbing, he seems to have made a habit of taunting two of his peers in particular each of whom, on separate occasions, he has reduced to tears.
Beyond disrupting the intellectual growth and emotional wellbeing of his peers, L's classroom demeanor has other implications: he shortchanges the value of oral presentation before a group, especially a group of his peers. Once, for instance, he faltered when performing simple arithmetic at the dryerase board, choosing instead to get carried away on a peer's flippant comment of "Go Logan."
The disparity between his individual written work and his work in social settings leads me to ask that L respect his peers and refocus on his presence and purpose in school. Journalling, for instance, may help L become more selfaware, and so improve his work with peers and raise his selfesteem.