CAIS Math Day 2008

Minutes from Ernie Chen, Head-Royce (Henri’s talk):

CAIS Math Day 2008

KEYNOTE ADDRESS: Henri Picciotto (Friday 3/21/08 10:15AM- 11:05AM)

“Serving Our Strongest Students 6-12”

Henri started by saying that he has focused much of his energy on “ACCESS” for the weakest students. Why are the strongest students important? It is the strongest students who have political clout and if they aren’t being served, it is “unacceptable” from an administrators’ point of view. Philosophically, it is important to serve these strongest students because they are the future teachers. Pedagogically, they are important because they make your teaching better.

TRACKING From the teachers’ point of view, it’s GREAT. The issue then becomes you are taking the “engine” out of the other classes. Tracking “lock” students into a path because the courses don’t move in parallel but more often diverge. For staffing, the equity issue becomes who gets/has to teach the higher track classes.

ACCELERATION Value is given to doing math at a younger age. It’s an odd characteristic of Mathematics that taking a class earlier is a badge of honor.

THE POWER OF CULTURE Societal expectations Standard Curriculum School Tradition Parental Pressure Strong Students Themselves (push themselves)

KINDS OF STUDENTS “Racers” versus “Diggers” The former get excited by doing something first, before someone else. They actually don’t seem interested in the Math, but in the achievement. The latter group aren’t rushed, they are open to challenge and willing to delve into the ideas and connections. Access and Challenge can COEXIST in the same classroom. Some techniques for teaching heterogeneous classes: Alliance with the strongest students, make them your ally. They can’t resent being there. The strong students can propel the class forward. The tradeoff is to provide support for the weakest.

THE ELEVATOR STRATEGY (Goldilocks) You stop on all the floors, EVERY DAY. Need to present something too difficult, which will challenge your strongest students. Also present something too easy. Of course, provide something that is “just right”.

PACING You have to have “constant forward motion”, this helps keep alliance with strongest students. The weak students can’t hold the class hostage. Simultaneously, you have eternal review. The review helps the weakest students. Assign homework that isn’t from material that is learned that day. Stretch it out by assigning homework problems from material introduced the week before.

CURRICULUM Find activities that have “no threshold and no ceilings”. For example, you can order 6,9 or 20 McNuggets. What numbers can you NOT order? Then find an equation, which describes the largest number you can’t order. MORE of these types of problems at “http://www.picciotto.org/math- ed”

Try to get more depth. For example, have students come up with the theorems about quadrilaterals

Schools should provide more electives. The road should not always lead to Calculus.

EXAMPLE OF ELECTIVES AT THE URBAN SCHOOL SPACE •Transformational Geometry Matrices •Symmetry Abstract Algebra •Dimension 3D Polyhedra 4D Introduction INFINITY •Infinite Sets Cantor •Proof By contradicion By mathematical induction •Dynamical Systems Iteration and Chaos •Fractals Minutes from Shahana Sarkar and Kenny Ewbank, Head-Royce (Henri’s talk, Q and A, Upper School roundtable, Large Group Discussion):

March 21st, 2008 CAIS Math Day

Opening remarks: Chris Davies, HRS Math Dept Chair

Naoko Math trajectory,

Serving our strongest students Henri Picciotto, the Urban School

1. What about all the other students. a. Because aren’t the strongest students the easy bit? b. Issue of access – applies to all 2. Importance of strong students a. Politically – wield a lot of power thru their parents, as it is not acceptable to not serve them – parents who claim their kid is not being served will not be allowed b. Philosophically: we need math teachers/ science/ engineers in the next generation. Social responsibility to support and enrich those students, and challenge them. c. Pedagogically (see below on Pacing) 3. Tracking: one of the ways we have traditionally supported the strong students. Some schools have a few tracks, some have many tracks. a. From the teachers p.o.v. : this is great to teach the upper track. b. But if they aren’t in the other tracks…you are missing the engine that helps drive the class. c. Equity issues: not so easy for a student to switch to a higher track. i. And tracks are not parallel! They tend to diverge, ii. Equity among the math teachers – some get to teach the upper track. 4. Acceleration a. Value given to do doing the same math younger. Which is ODD. No issues of acceleration in English (Shakespeare in 10th grade)! But math acceleration “allows” him to take something else. But there are odd understandings, b. At Urban, they’ve discouraged acceleration, but every year, more parent/student pressure. 5. The Power of culture: we can’t fight all of these, a. Societal expectations b. Standard curriculum c. School tradition d. Parental pressure e. Strong students themselves! 6. Two types of strong students: Racers vs. Diggers a. Racers – exciting to do something fast, or before someone else – they are interested in something, but maybe not exactly math? They get a lot of credit for being fast. b. Digger (because they dig math!), and because they enjoy digging – they are open to challenges. c. The same challenge problem are thought of differently by these two types.

Most of this talk applies to … DIGGERS

Other approaches. The Urban approach: not tracked within a level.

Access and challenge can coexist: it takes some doing – training in pedagogy, curriculum. Important to aim for that (after understanding that)

1. Pedagogy: Techniques for teaching heterogeneous classes – ALL classes (even tracked) are heterogeneous. a. Alliance with the strongest student: they can’t resent being there… b. Support for the weakest

 The elevator strategy: stop on all the floors. (the goldilocks strategy 1. Every day: a. something that is too difficult – don’t be afraid that will discourage the kids. b. something that is too easy - build up the participation, c. something that is just right. d. Navigate all the levels, rather than aim for the m idle 2. Pacing: COMPLEX… but aim for a. Constant forward motion b. Eternal review. Maybe when you assign problems from a chapter, skip a bunch so that oyu can use them for review later. i. Ex: If your homework is NOT on today’s stuff, but on old stuff, you are able to do interesting challenging stuff today in class, but you can extend a topic for 3-4 weeks, doesn’t hurt the stronger student, and they don’t mind because in weeks 2-4, weve brought in new material. See above: Strong students are also important C. Pedagogically.

2. Curriculum: a. No threshold, no ceiling activites in core classes. Ex: Macdonal’s – nuggets, how many can you order? More examples: www.picciotto.org/math-ed. ; Center for innovative teaching. b. Ex: geometry – Map of theorems about quadrilaterals.  What conditions make a quadrilater be a … kite, helps give an understanding of what proof is. Rather than present the students with a proof.  Pick a few things that are important, because due to time you can’t go in depth on everything. c. Breadth thru More electives: rather than all math classes being a freeway to calculus. Offer a wider range of electives. Urban model: Math 1, 2, 3… Left side of more accessible courses(MMaPS, programming), right side of more challenging (infininty, space)  Space: transformation geometry(matrices), symmetry (abstract alg), Dimension (3D , 4D intro)  Infinity: infinite sets (cantor), Proof (by contradiction, mathematical induction) Dynamical systems, Fractals

Q&A Value of projects, activities, manipulative with stronger based students? Manipulatives are great remap for strong students – allows them to understand things in more than one way. Allows them to translate between multiple understanding. Projects: great, time penalty,but I have obsereved if you give a wide open project, they tend to drop back 2 or 3 levels in math. They Do you believe in tracking?: Can’t really answer: I have helped in continuing not tracking at Urban. In many countries tracking doesn’t begin until high school. Here in the US, tracking in MS seems like a bad idea – you can’t really know. HP’s wisdom: Track later, and track less! Do all entering 9th graders take math 1? Perhaps a form of tracking, half take math 1, half take math 2. Difference is that everyone sees the same material, it is just earlier or later. Challenge probs: tarnished by inclination to cheat. Assignments: write down who helped you, and who you helped. Gives us some info. We do count it, but it is different from “can you do this on your own” which a test would tell you. It is part of the alliance with the strong students, because they are the ones who help the others. The explaining is as important (and it counts, towards the grades) as much! Give a quiz on the key ideas, then give the project because they still learn a lot. Grading challenges: relative value? Yes, give major points – because I (HP) value the explanation, they need to see it as important. We don’t do that many challenges, we still have tests and quizzes, and it is important to collaborate well. Resources: (no) Extrinsic vs. Intrinsic motivation At Urban, grades are secret. (unless you are heading for D or F), how am I doing: “you are ok at this, but struggle with that”. They are shifting the focus from grades, to where it matters. That helps the intrinsic motivation. But how do you get there? Finding stuff that’s interesting, challenging, Value of strong students modeling their understanding w/technology?

Review of questionnaires:

PEDAGOGY: Small group (Upper Schools) Steve Gregg (HRS): HRS culture: our top math students are all racers. How can we teach them to be diggers, rather than racers? Henri P (Urban): teachers have a huge influence on a school culture. What are teachers rewarding? Research about kids who are praised for doing well in math, don’t try hard things because it is risky. Kids who are praised for taking up challenges…will try do stuff. Patricia (Sac C DS) NAIS: article by Dwek how we praise our students has a huge impact on … shared w/ kids and parents. Scott (Urban): The problems that you can finish quickly, vs. those that take a long. Real probs take a long time to solve. Give them problems to roll around in their head. Problems that aren’t so packaged. Shahana (HRS): how do you that? Scott: think ahead, what am I going to ask “the ultimate digger”, and I let him know that I am going to ask him. In day to day stuff: sponge problem, at the end of a worksheet, ready at the end of a worksheet connected to something. Identify which “everyone needs to know” vs. which is extra. OR this is connected to something you need in Math 3B, or later on down the road. Fern (HRS): helpful to know what’s coming next. Except I don’t know what’s coming next… Scott: teaching Math 1 and calculus to see the connection. Naoko: comes from teaching the whole range? Scott: what is math? Especially in the age of Calculators? What is the student supposed to learn when the calc does alg, better than the student (often) can. Ernie (HRS): computational part (taken out of their hands). Knowledge, and where does it fit into the big picture. How they handle new information and place in context and know where to use it. Chris D (HRS): Practical / practices: Checking HW: kids put up solutions, rather than boring the strong kids. Group work checking hw: strong kids are the engine. Naoko: Here are the highlights, or the nexst questions that arise from this. Henri: do a problem like 12, instead of doing the exact problem, do a problem similar to it. Patricia: give them an answer, make up a problem that makes it work. Or use English to explain it. Chris (Castilleja): find the teacher’s mistakes. Chris D: three wrongs and a right? Math embedded in games: Board racing, group racing, jeopardy, Grab the marker: pair by ability level, circle the answer, and grab the marker. Allows for all abilities to play. Patricia:send up groups of varying ability, and allow them to choose level of difficulty or random question. Shahana: the number line game : everyone gets a number (e, e-1, sqrt(3)/2, etc. ) taped to their back, put yourselves in order. Fern: Problem set, put an answer on the board, I’ll leave the room. Correct answers = +2, or -2 for wrong. Everyone is working. (Challenge problems must be included). Whole group gets the same points on or off. Henri: bonus problems, optional on the test, required on the test corrections. Scott: tests are never boring --- no one gets 100. Expectation isn’t that you get 90%. Lots of Bonuses. Scott: my strong students loved the hard problem sets, and it counted. It’s ok that there was collaboration, the other students still get something out of it. Henri: 85% A,B’s, 15% C’s. at Urban Alfie Kohn: Chris D: potential extra credit. A 15 pt problem set will be graded out of 10. : So everyone tries it all… Steve: equity issues on grading papers with lots of collaboration. All the solutions looks the same. Scott: give them a quiz on the day the paper is due, to see who really gets it. Naoko: Assign a 5 problem set, and next week, 2 (randomly) will be collected. Scott: from Henri’s book: little report on something. Limited to 1 page, and each paper is unique. You are writing the textbook. And their work reveals what are the salient points.

Big Group Discussion

-Low threshold hi ceiling problems: -trisecting a segment (geometry) -Locker room problem (MS Math) -Draw a shape on graph paper, find the largest area possible for this shape/smallest possible are, generalize to a formula-Henri -Geoboard problems? Henri uses those in geometry and moves from the physical to pencil and paper -Problem of the weeks in MS create strong students, leave open to different levels of instruction -Ask students to search for old challenging math problems (have kids present on different Fridays) Students share how they might have shown the problem differently -Hands of the clock problem (what are the angles created between two hands?) move slowly at first, and then ramp up difficulty (1:00 o clock to 12:07) -Number of heads, hands, and feet in a group of animals problem -Cheese problem, cut it into cubes and smaller and smaller pieces then find the relationships -Color mixing problem: assigning values to primarys, then creating numbers for each possible derivation of color after that -Shahana asks; what are areas that we can frontload in lower levels to preview material for later on. -Algebraic modeling can happen in 6th grade, start with guess and check and then move on to variable expressions -Henri mentions that colleagues are the best source of this type of information -Fern makes BC calc students teach difficult concepts in 8th grade -CD: To what extent do you have strong students teach other students? -7th graders teach to 5th graders at prospect sierra in small groups -Classroom partnerships (chosen by teachers) -Formal peer tutoring program –teachers identify those in need and tutors are chosen and trained, starts to become ingrained in school culture. (11th and 12th grade tutors mostly) -In Middle schools, its hard to find maturity in 8th graders to remember to show up, and the timing of the schedule doesn’t permit much -An SF school is having its 8th graders put on a math day for the K-5 students -Scott has his students present information (they know ahead of time) and present their solution to the class -Have students pick out one topic from the year that was salient to them and teach it -Use the white board to have 5 students present multiple solutions, takes the pressure off just one kid going up there (use lifelines in the crowd) -MS teachers are finding that adding “explain” components really serve all students well -Round robin activity, create different stations using technology, lab gear, pencil and paper activities, real world problem solving, and move them quickly Games -Mixing students during game play so that they are not attached to any one team (safety) -Strong students can help during gameplay -Review questions that are given to groups of 2-3 where all students have to be able to explain to get the points (strong students share knowledge with weaker ones) -Board game from RAFT that involves cubes that represent different variables, evaluate an expression based on these and you move that many spaces (self regulation of answers) -The four fours problem, create any value from 1 to a 100. Accesible to all students but can ratchet up the level of difficulty Math Clubs and Teams -MCDS has a thriving math program, open to everyone, then we have tryouts to see who can represent the team in mathletes program. Sign ups occur during activity periods. -Steve asks, does it have to do with the personality of the teacher who runs it? -Yes in some ways, and the culture is in place at the school so parents and older sibs encourage. -Racers are very excited by math club activities -Snacks certainly help for the middle schools (of course) -Marketing the math team program can be essential, sell to parents and others

Minutes from Scott Clark, Head-Royce (Henri’s Talk, Middle School Roundtable, Large Group Discussion):

Serving Our Strongest Students 6th-12th CAIS Math Day, March 21, 2008

Henri Picciotto (Urban School)

Political, philosophical & pedagogical justifications for making sure all levels are served.

Possible Responses: 1) Tracking a. Great for top students. b. “Engines” that could drive other classes not present. c. Becomes increasingly difficult to switch tracks. 2) Acceleration a. Value assigned to doing the same math at younger ages. Seems to be a math-related phenomena (as opposed to English, etc..) 3) Cultural Power Drives Demand for These a. Societal expectations b. Standard curriculum c. School tradition d. Parental pressure e. Strong students themselves

Strong Student Profiles/Tendencies 1) Racers – excited to do things fast, ahead of others (not necessarily interested in the subject so much as the race) 2) Diggers – they “dig” math, enjoy going in-depth with ideas, challenges, etc.

Alternative Approaches 1) Assumption – access and challenge can coexist; even tracked classes are heterogeneous 2) Alliance with the strongest students & Support for the weakest 3) Elevator/Goldilocks Strategy – every day, stop on all the floors! a. Something too difficult b. Something too easy c. Something “just right” 4) Pacing (helps to keep all levels engaged) a. Constant forward motion (weak students cannot control this) b. Eternal review 5) Curriculum – a. Search for questions that have no threshold, no ceiling. Easy access, but rich in possibilities. (ex. McDonald’s 3, 6, 9 McNuggets – what quantities can we not order? Can you express this algebraically? Beyond a certain number, any amount can be ordered. What is that number?) see www.picciotto.org/math-ed for examples. b. More electives – is math “the freeway to calculus?” Consider wider range of choices (see Urban curriculum map) and levels of challenges

Hints/Ideas o Try giving quizzes on big problems, rather than grading the problem itself (allows for collaboration in problem solving process). o Try shifting conversation away from grades (ex. keep grades secret until the end of the term); response to “How am I doing?” is “You need to work on persisting through non-routine problems, etc..”

Middle School Session Tutoring/Enrichment Time 1) Upper school students paired with middle school students (for community service credit or compensation). 2) Math labs – once every other week 3) Unofficial buddy system has sometimes worked, but finding time in the day is difficult. 4) Options/Activities period – a. provides for a math study hall b. math enrichment group (elective) c. math team time – once a week for 40 minutes – work on NCTM monthly calendar problems

How do we prepare students to enter into an integrated high school math program? 1) Some schools are using spiraling curriculum in middle school. 2) Homework the day of a test – “Party Shuffle” – review of content from other units. 3) Warm-up problems on a variety of topics 4) Provide two grades/use rubric grading – a. Accuracy grade b. Score for showing strategy

Skipping routine work 1) Students can pick the problems they want to do (ex. Dolciani C-level problems) – different types of problems available, but students must do one of the choices. Sometimes requires telling stronger students that they need to do the enrichment problems.

Manipulatives 1) Many use them to introduce concepts. 2) Sometimes difficult for strong students who are more paper-and-pencil oriented. 3) Can be used to introduce/explain proofs. 4) Can be used on “mandatory manipulatives” day, and then optionally thereafter. Allowed for justifying solutions. 5) Good tool for advanced students to explain to peers.

Smart Boards 1) Outline of notes and printouts available. 2) Can be used to move back to different parts of discussion 3) Allows for shrinking/manipulating student work

Other Technology 1) Calculator Based Rangers – (CBR’s) 2) Graphing Calculators (see “Make These Designs” on Henri Picciotto’s website) 3) Website Content 4) Geometer’s Sketchpad 5) Fathom

Sources for Enrichment Problems 1) Books of problems (ex. Dale Seymour, Groundworks, Pearson, Carol Greenes) 2) Mathcounts 3) Math Olympiad 4) NCTM Calendar problems 5) Drexel website (problem of the week – costly!) 6) CMC publications 7) Math Boxes 8) Henri Picciotto’s website 9) Singaporemath.com (K-6th grade)

Tracking – How do you handle self-esteem problems with non-honors tracks (students label selves members of the “dumber group”). 1) Have frank class conversations about it (ex. – even regular Algebra classes are still ahead from traditional curriculum).

LARGE GROUP DISCUSSION:

Rich Problems - examples 1) Hands of the Clock - What are the angles at various times? How often do the hands coincide? How often are they perpendicular? 2) Ducks and Horses – I count three heads and ten feet? How many of each kind? Moving from guess-test to algebraic strategies. 3) Block of Cheese – dip in chocolate and cut into cubes. How many will be covered on one side? Two sides? Etc.. Try with rectangular prism. 4) Take four fours to make 99 (etc..). Take the year and find a way to make numbers one to one-hundred. Students As Teachers 1) High Schoolers guest teach classes in middle school. 2) Student presenter time. 3) Students are combined in partners/cooperative groups for regular dialogue. 4) Formalized peer tutoring program (sometimes in coordination with learning specialist). 5) Math Day – pairing up grade levels to do activities (ex. 8th graders lead activities for 7th graders). 6) Rotating the job of presenting select problems from the homework 7) At the end of the year, students present content from earlier in the year. 8) Multiple students put responses on board simultaneously. Include a “lifeline” student for each presenter who can help if needed. 9) Small groups rotate through stations, and help each other.

Games Alphas – stay with their group Betas – move over one group each problem Gammas – move over two groups each problem Work on problems together, and scores are collected for team that has rotating membership. Require 60 sec. of silence. Each group member must show their own work.

Review game Groups of two or three (mixed ability level) – Ten questions on little slips of paper Can’t start on next question until every member of the group can explain solution.

Raft – (Resource Area for Teachers) - http://www.raft.net/index.php? pg=idearesults&dtl=Math