Planning a New Program

BEAM is receiving funding to develop a new program: a non-residential program in New York City that will reach students from a younger age, beginning the summer after 6th grade.  Students will learn mathematical reasoning, build basic mathematical skills, and become part of an intellectual community.

Now comes of the work of designing that program for a launch this summer.  I’m going to do that design here, on the blog, so that others can follow along with the process of creating a new program and see the ideas get developed and change over time.

To begin, I’ve created an outline of the major topics I plan to think through.  Each of these bullet points will become a link to a post.  Please note that both this post and all of the other posts in the series are likely to evolve over time.  They’re likely to get edited to reflect the final state of thinking as we move to launch.

Big Picture

Before the Summer

  • How is the program communicated to schools and students?
  • How are students selected for the program?
  • How will we hire staff?


Social Environment

  • How do we create a vibrant community?
  • What structures do we need to manage student behavior?

After the Summer

  • What, if any, additional support is provided to students?
  • How does this connect to the existing BEAM program?


The program will be known (for now) as BEAM 6, and all posts about it will be labeled as such.

How hard should it be to pass algebra?

New York State is grappling with the difficulty of the Common Core Algebra test.  The intent is to raise the passing score to require real mastery of the material, but realistically speaking, most students are not reaching mastery.  (In fact, even with the original, very low standards, some students had to take the exam many times to pass or might not pass at all.)

The core of this issue seems to be: what is the purpose of teaching algebra?  For example:

  • If the purpose is to preserve the opportunity for all students to enter science/engineering/math, then the standard should be high.  It does no good to a student to barely scrape through algebra if they want to be a scientist.
  • If the purpose is to give everyone exposure to a beautiful subject, then the standard should be kept relatively low: it is the exposure, not mastery, that is important.
  • If the purpose is to give people access to math they need for life, then algebra should be dropped or revamped.  Many people do not need algebra in life, and a high barrier to graduation does them no good.

Right now, the grade required to pass is being used as a proxy for this kind of battle.  Those whose focus is on high school graduation want the required grade to drop.  Those whose focus is on preparing students for STEM careers want it to go up.  Without resolving this difference of goals, everyone will just keep shouting at everyone else and we’ll end up with a muddled policy that drags students in multiple directions.

Alas, that is not so unusual.

NYC’s high school selection system might be the worst system for choosing a high school — except for all the others

The New York Post published an opinion piece about the byzantine high school selection system.  As someone who’s helped well over a hundred students through this system, it really is byzantine.  There are over four hundred non-charter public high schools in the city, plus dozens of charter high schools.  In eighth grade, students can complete a “Round 1” form where they rank the high schools they are most interested in.  Each of those schools have different criteria for judging admission: some are based on your grades, test scores, or attendance; others, on the level of interest you show by attending information sessions; others, on a special test you take for admission; others on a portfolio you submit; others on an interview you do; others on geography; and finally many combine several of the above.  Just as you rank the schools, the schools rank you.  Then an algorithm is done to match students with schools.  Just imagine having to research all those schools, understand each one’s admissions priorities, and then complete all those applications!

Sound complicated?  We’re not done yet.  In a separate process, there are the specialized high schools (think Stuyvesant, Bronx Science, etc.) which have their own test, the SHSAT, which you take for admission.  Admission is based solely on this one test, and many people study for years.  (A sad waste of talent.)

No, we’re still not done yet.  Based on the SHSAT and your Round 1 form, you get a school assignment in March, which could have 2 schools (if you got into one of each, you get to choose), 1 school (if you got into one “normal” school and one specialized school), or 0 schools (if you were not placed with any of your Round 1 schools nor did you score high enough on the SHSAT).  At this point, you can complete the Round 2 form, in which the same matching process is used as Round 1, but with those schools that are left over.  (Usually, not the good ones.)

If the Round 2 process doesn’t work out, you are automatically placed into a local school that has space.  But it’s still not over.  You can appeal your choice if something went wrong.  You can also apply separately to charter schools (yes, they’re in another totally separate system), which are decided by random lottery, but the more you apply to, the better your chances.

All of which is to say, the process is truly, incredibly complicated.  (I also simplified it, leaving out LaGuardia School of the Arts, private schools, and many intricacies of appealing your placement.)  As the Post‘s opinion piece justly points out, this process dramatically favors more affluent students who have much better coaching and access to information.  (Let alone that they often speak better English.)

You might think, at this point, that I agree with the author that we should abolish this system.  After all, it clearly favors affluent students and takes a lot of time for everyone.  But there is one key problem: nothing better has been proposed.  At least the current system gives students a chance to go to a better school.  At BEAM, we can coach students on how to get into a great school that will challenge them, and we have tremendous success.  Some families find their way on their own.  Compare to a system that just places students by geography, in a city that is highly segregated (by race and income) — what chance would low-income students have then for access to these schools?

This is a subtle issue.  Maybe a system based only on geography would help, because then there might be some mixed-income schools (although I find that unlikely).  Maybe the current system has another downside, “creaming” the best students from low-performing schools, leaving them worse off.  These are interesting questions, questions that deserve study.  But it’s not worth changing a system that offers some kids incredible opportunities unless you’ve done a very careful examination of the trade-offs, and we frankly have no idea!

So what should we do?  If more students had real advising on navigating the high school system, they could be vastly more successful at getting into great schools.  There are so many big mistakes that students make all the time: not filling in all 12 spots on the Round 1 form, for example, or incorrectly judging what school to apply to.  (We had a student who decided by looking at schools’ graduation rates, not realizing he’d ranked a dual-language school specifically for English Language Learners, a decision that he is now stuck with for 9th grade even though he does not speak fluent Spanish!)  Just as we need more guidance counselors for college applications, help with the high school application process might create a system that offers tremendous opportunities for all students, regardless of background.

Algebra for All in NYC

Mayor Bill de Blasio has announced a plan for every New York City middle school student to have access to algebra in 8th grade, and Chalkbeat is out with a great story discussing the challenges facing the initiative.

I work with many exceptional middle school students at BEAM, and for many, the lack of algebra in their middle schools is a tragedy.  No access to algebra closes doors; it’s as simple as that.  That’s because most people who pursue science degrees have taken calculus in high school, so you’ll be significantly behind if you don’t have that background.  But getting to calculus in high school requires taking algebra in 8th grade, unless you double up on math in some year!  There are ripple effects throughout high school and college; no 8th grade algebra makes it much harder to pursue a science major in college (which is why BEAM offers an online algebra course).

Yet simply providing some algebra classes opens a host of other issues.  First of all, algebra must be taught well, which will be hard for teachers who are not experienced teaching high school level math and who may not have the mathematical background to offer a high quality course.  Moreover, in many schools a teacher will have several classes of normal 8th grade math, and one algebra class—where would you put your prep time?  But the bigger danger is that kids who are not ready for algebra will be pushed into it.

The Chalkbeat article seems to treat this as a good thing, talking about “lower-performing students who could use the early exposure to a subject that trips up many students in high school.”  But if a student does not deeply understand earlier material, then accelerating is a mistake.  A study in California (also, paradoxically, in the same Chalkbeat article) points out that when California mandated algebra in 8th grade, those students got lower scores on 10th grade math two years later.  Really understanding algebra requires really understanding math.  Otherwise, you’re just memorizing formulas and not learning anything.

In fact, algebra is something of a debacle right now in New York.  On the old, easier, pre-Common Core Regents exam in New York City, roughly a third of students had to take the test multiple times to pass.  (And let me tell you, a pass does not demonstrate mastery.)  A quarter of students had to take it at least three times.  There were 66 students in New York City who took the test ten or more times, and half of them still didn’t pass!  Even for those students that did pass, did doing so on the tenth try really mean that they got something out of algebra class?

Fundamentally, there is a mismatch between what we are aiming for, namely that all students take algebra, and what we are achieving, which is that many students take the test repeatedly, focus on memorization, and don’t learn mathematics on a deep level.  Algebra should open doors.  There are no doors being opened by memorizing formulas, which (barring exceptional teachers) is all you can do when you don’t have the mathematical thought processes down going into the class.

It is good that algebra will be available to everyone.  It is a critical equity issue.  But for students to really take advantage of this opportunity, there is a lot more groundwork that must be put in place.

Should a test determine your high school?

In New York City, there are several highly selective public high schools called Specialized High Schools.  You’ve probably heard of some of them, like Stuyvesant, Bronx Science, and Brooklyn Tech.  They’re excellent schools.  The way you get in is by taking the test that might have the most frightening acronym I’ve ever seen: the Specialized High School Admissions Test, or SHSAT.

New York has been embroiled in a conflict over this very policy.  The SHSAT was created to provide an unbiased measure to try to combat racial discrimination in admission and to create greater diversity in the schools.  It hasn’t worked.  Out of 952 students admitted to Stuyvesant last year, only 7 were Black.  Seven.  In a city where 25.5% of the population is Black, less than 0.7% of Stuyvesant’s admitted students were Black.  (Stuyvesant had the highest cutoff scores; the results for other schools were significantly better but nowhere approaching representative.  For example, 6.9% of admitted students to Brooklyn Tech were Black.)  The situation is so bad that the NAACP has filed a lawsuit over specialized high school admissions alleging that the SHSAT is racially discriminatory.

Now, I run a program for underserved New York City middle school students with talent in math.  We work to create a realistic pathway for our students to enter into careers in science, mathematics, engineering, programming, and more.  As you can imagine, admission to a good high school is critical to their success.  It matters a lot if you have the opportunity to study calculus, to be on a math team or a robotics team, or to do independent research.  Our students are 88% Black or Latino.  Over 25% of them get into specialized high schools, and another 25% get into other highly selective schools in the city.  Getting our kids into good high schools is of primary importance to me.

I wish that changing the SHSAT could possibly address the problems faced by our students, and students like them across all of New York.  It won’t, and we shouldn’t change the system without a viable alternative.

Why?  Because the SHSAT is just reflecting a stark reality: minority students in New York City are less prepared, and they are not given a chance to excel.  If we drop the SHSAT, we might see an increase in diversity.  But it will stratify the schools, creating two tiers of students, those with adequate preparation (who will be taking advanced classes) and those without it (who will be taking more basic classes).  Can you imagine a school with such obvious racial segregation?  This would send the wrong message to the students, the teachers, and the outside world, and it would paper over the truth, which is that these incredibly brilliant, incredibly promising kids are not getting the education they deserve at a much younger age than high school.

Let’s see if we can understand just how stark this difference is.  There is a test called the National Assessment of Educational Progress (NAEP).  It is a low-stakes test given every few years to a nationally representative sample of students, with sub-samples in major cities.  Because it is low stakes (nothing depends on the outcome), no one studies for it.  It’s just a measure of what they’ve learned.  While there might be biases—some curricula might be more closely aligned with it than others, for example—within one city it’s a good measure.

NAEP has a wonderful online tool where you can run queries on its data, so I did.  So let’s see what percentage of students score at the Advanced level on this low-stakes test in New York City in 8th grade.  Asian students?  26.20% score at Advanced.  White students?  17.72%.  Black students?  1.40%.  Hispanic students?  0.93%.  That’s not a typo.

No one is studying for this test.  There are no cram schools for it, no tutors.  Much as I dislike tests, as far as measures of what the students have actually learned, this is pretty good.  And what we find is stats which, when scaled for city population, match the overall specialized high school admissions pretty well. (*)

Now imagine that you artificially admit more underrepresented minority students.  What happens?  They are not as well prepared.  Obvious differences appear in the student body.  And it doesn’t fix the real problem, the one that no one seems to want to address, which is that the students are not adequately prepared when they are younger.

I’ve worked with a lot of our alumni who go on to specialized schools.  It’s a tremendous opportunity for them.  It’s also very, very difficult adjusting to a much more rigorous academic experience than what they had in middle school.  We provide a lot of support and tutoring, like what more affluent students would be able to get.  Broadening admissions simply does not address the real issues.  Moreover, there are several great schools in the city (such as Bard High School Early College, or The Beacon School) which are not based only on tests, but consider grades, interviews, portfolios, and more.  We recommend each student to the school that is the best fit for them both for admission and for attendance.  There are other options.

There are serious, legitimate questions to be asked about the SHSAT.  The test should be properly validated.  It should be examined for racial bias.  We should consider other admissions mechanisms that might be more fair—but not jump to them willy-nilly!  Right now, there is no serious alternate proposal that looks likely to accomplish the goals of the specialized schools, and there is no evidence that the SHSAT is discriminatory given the prior academic achievement of the students who take it.

Now can we please, please, please support the very promising students in elementary and middle school who are ready for more math in their lives?


(*) It does seem like a higher than expected percentage of admitted students are Asian, and lower than expected percentage are White, but this could be due to any number of factors: extra studying on the part of Asian students, data based on city-wide demographics instead of based only on school-age children, NAEP’s “Advanced” ranking is in the wrong place on the bell curve, or because a larger fraction of White students go to private schools.

A Reminder That Personal Relationships Matter

I wanted to briefly point you all to an article by Atul Gawande in a recent New Yorker.  Gawande is writing about the adoption of medical advances, but his remarks about teaching rural nurses in less developed nations are as relevant as anything to education.

Gawande asks: what drives a nurse to internalize that hand washing or warming the baby are important for safe childbirth?  He has a fascinating description about how one nurse was able to persuade another nurse to change her practices by becoming her friend.  Not because the mentor nurse’s training was impressive, nor because she had the force of law behind her.  Because the two of them sat down to tea.

Gawande says that success at getting nurses to adopt improved methods—especially those (like hand washing or warming the baby) whose effects are only visible after the child has left the hospital—come from personal connections formed by mentor nurses.  Otherwise, no matter the law, no matter what classes or informational videos or glossy handouts you offer, change comes slowly if at all.  To me, this sounds a lot like convincing kids to learn mathematics or to want to go to college.

Indeed, I think that these observations, hardly a surprise to anyone who’s seen the success of individual tutoring and mentoring, have implications across education.  I suspect that a difference between successful charter schools and unsuccessful ones is while both shout “college! college! college!” from the rooftops, only successful schools forge persuasive personal relationships.  While MOOCs make great resources available, they still have to persuade people to invest time in their classes.  How much did you learn from your best teachers because you felt like they knew you personally, or because you admired them and wanted to be like them?

Anyway, it’s a great article.  Read it while thinking about teachers—especially the difference between great teachers and merely good teachers—and it will give you provocative new thoughts about education.

The Latest Updates from SPMPS

I periodically send update e-mails to our supporters.  Here’s the latest:

Dear friends of SPMPS,

What happens when almost 40 middle school students from low-income backgrounds get to learn mathematics topics like Logic, Number Theory, and Combinatorics?  Amazing, amazing things.

In Group Theory, instructor Ben-Blum Smith invented a dance.  As Ben played on his guitar, the students responded to calls like “East-West Switch”, “LAX to JFK”, and “Jam in Place” to learn about symmetry groups.  In Proofs, Shelley taught her students about the pigeon hole principle and how to phrase a formal mathematical proof.  In Numbers, Sets, and Mappings, Marcus helped students prove that the quantity of natural numbers is the same as the quantity of even natural numbers, but that the real numbers are a higher order of infinity.  In Circuit Design, Sage helped her students construct circuits to add two numbers in binary.  In Digital Communications, Taylor’s students designed their own methods of sending images across a room using nothing but sound.

Thirty-nine students came to the program and got their first entry point to advanced mathematics.  Of course, the summer is just the launching-off point: over the coming year we will be connecting students with selective high schools and summer programs, the New York Math Circle, math contests of all kinds, and other opportunities that will open a new world up to them.

We’re still crunching the data, but here are a few outcomes that we’re already seeing:

  • Students took the AMC-8, a contest given to advanced students nationwide.  By the end of the program, the average of our students’ nationwide rankings grew by 21 percentile points!  Six of our students, exposed to serious mathematical study for the first time, saw their scores rise by 40 percentile points or more.
  • Students reported huge changes in their dispositions towards mathematics and challenges.  97% agreed that the program “showed me that I can learn more than I thought I could”, and 82% strongly agreed.  Students also learned the importance of hard work (not just natural brilliance) and of frustration and challenges: every student agreed that “Working can improve one’s ability in mathematics”, and 95% agreed that “time used to investigate why a solution to a math problem works is time well spent”.
  • Students raved about the program in their summer evaluations.  “A life-changing thing,” wrote Edson.  Math is “the best subject in the world” wrote Faith, who also said that she learned to “stop being shy”.  “I find math as a more common language than the language I speak”, said Seth.  Tiffany summed up something that many students said: “I want from liking math to loving it.”

Next year, SPMPS is looking to grow to a second campus.  We’ll be able to serve twice as many NYC-area students.  This expansion is possible thanks to a three-year grant from the Jack Kent Cooke Foundation which expires after next summer.  We’re fundraising now to raise the difference to make the second site possible next year, and also to help us transition smoothly as this grant concludes.  If you are able to contribute or you can make an introduction to someone who might be able to, please do let me know!  We are working to raise an additional $150,000 over the next twelve months to help support summer 2014 and summer 2015.

There are two other ways you can help us out:

  • We’re seeking meeting space in NYC on evenings and weekends.  This year, we will be greatly increasing our year-round programming to help students enter future programs for advanced study, and space costs may become an issue.  If you have access to free or low-cost space, please get in touch.
  • We are also looking for someone to help redesign our website to match our growing profile.  Please drop me a line with any connections!

Finally, you might enjoy this Bard news article about this summer:

Thank you so much for all your help and support with the program!  Until next time,


There are lots of exciting things in the coming year with much stronger year-round programming and connections to other mathematical offerings.  (What this post doesn’t tell you, for example, is that we’ve seen an explosion in SPMPS alumni registering for the NY Math Circle.)  This is a very exciting time for SPMPS!

The Many Uses of =

The equals sign is a sophisticated, subtle tool.  You may not think of it as such, but it is deep.  It is hard to learn and use and understand.

I first understood the challenge while teaching a calculus class.  I had two expressions, something like (x – 2)(x + 3)(x – 7) and (x – 2)^3 – x + 50, and our goal was to show that they were equal.  I took one, and I simplified it to x^3 – 6x^2 – 13x + 42, a chain of equals signs spread across the blackboard.  I took the other, and simplified it to the same thing.  So I proudly declared that they were equal, and thanks to the magic of my boardwork, the two identical expressions had ended up next to each other.   I drew an equals sign between them, creating a giant chain of equals signs connecting the two original expressions.  I thought this was a very clever presentation.

They didn’t get it.

I’d heard before that this might be a challenge.  They’d all been drilled in elementary school on problems like 5 + 2 = ?.  They thought of equals as meaning the outcome of an operation, which is why they felt comfortable using it sloppily.  (You know what this looks like: asked to compute “five squared plus two”, they write 5^2 = 25 + 2 = 27.)

I thought, then, that it was time to correct this misunderstanding.  So I speechified at length about how they had been tragically mis-taught about the equals sign.  It actually means that two things are the same thing!  I pointed to (x – 2)(x + 3)(x – 7) = (x – 2)(x^2 – 4x – 21) and said “these two things are the same thing!”  Then I pointed to (x – 2)(x^2 – 4x – 21) = x(x^2 – 4x – 21) – 2(x^2 – 4x – 21) and proclaimed, “these two things are the same thing too!”  Since the first expression was the same as the second, and the second was the same as the third, the first and third must be the same, and a pointed and said it.  I went on, through all the equals signs, until I had discovered that the two things on opposite sides of equals signs were the same thing.  I discussed why I didn’t like the way they did a problem like “five squares plus two”.  I was proud of my improvisation, but somehow, it didn’t seem like the students had the great moment of realization (and repentance for their mathematical sins!) that I wanted.

I say it again: the equals sign is a sophisticated, subtle tool.  Despite my oratorical skill, I had failed to bring together a deeper understanding of all the possible uses of equals.  There are at least four:

  • As the outcome of an operation (used in school, but mathematically the wrong way to look at it).
  • To declare that two things are equal all the time, as in 4(x + 2) = 4x + 8.  This is an equvalence.  There is a hidden universal quantifier.
  • To declare that two things will be equal for the right value of x, as in 4x + 2 = 3x + 9.  (This is disambiguated from the previous case by explicitly saying, “Find x so that…”  To a mathematically sophisticated reader, it is clear that this is the same = sign, just with an existential quantifier first.  This is not clear to someone just learning mathematics.)
  • As a definition, such as f(x) = 3x^2 – 4x.

Without any explicit guidance, we expect students to recognize these different situations.  To recognize quantifiers and definitions without any discussion of what “equals” really means.  All this while learning how to “solve” these problems.  Also without discussions of the other subtleties of equals: symmetry, transitivity, reflexivity.

Embedded here is also the understanding that two things can look completely different and can be the same thing, can be equal.  30/3 and 10 are different representations of the same number.  7 + 3 and 5 + 5 are all different representations of that same number.  All of these things are equal!  They are also equal to 2x when x = 5.  The mathematical symbols alone are not enough; the words nearby must be used to interpret them.

Worse, uses of the equals sign get conflated in different settings.  For example, suppose that you are trying to find an x so that 4x + 2 = (x + 2)^2.  You might write that (x + 2)^2 = x^2 + 4x + 4.  One of your equals signs is only true for a few values of x; the other equals sign is true for all values of x, all in the same math problem.  Imagine how confusing it would be if you wrote 4x + 2 = (x + 2)^2 = x^2 + 4x + 4!

The same thing can happen when you are defining a function.  You might say “Let f(x) = (x + 2)^2 = x^2 + 4x + 4”.  In one line, two different uses of the equals sign.

To me personally, these uses of = are all the same thing, stemming from the same definition.  To a student just learning algebra, however, there are layers upon layers of subtlety, and they must be addressed explicitly or students will not truly understand the mathematics they are doing.

Teaching Log: 0.9999… = 1

It worries me that as I move into leadership roles, I get fewer and fewer opportunities to teach.  Hence, I was excited to give a talk at the Sonya Kovalevsky Day at Barnard College.  The entire 9th grade of the Urban Assembly Institute of Math and Science for Young Women was at the event, and about a quarter were in my class.  It was a challenging talk for a number of reasons.  First, the students had not chosen to be there, and had no particular interest in math.  Moreover, I felt that it was important to do more than give a fun math talk.  I wanted the students to really learn something that would help them in their mathematics in school.

The topic I chose was “Does 0.9999… = 1?”  I did this for a few reasons:

  • I could pose an interesting question at the beginning and let the students think about it and come up with ideas.
  • I could vary the level of the discussion based on student background.  At one end I could always retreat to 1/3 = 0.3333…, multiply both sides by 3, and get 1 = 0.9999… (ignoring issues of how you multiply an infinite decimal); at the other end I could talk about infinite geometric series and convergence (although that was very unlikely).  In the middle, I could talk about how you prove things in general, give the overall idea of a series, and talk about what convergence means.
  • I would be able to emphasize ideas of “math makes sense”—that different pieces of math fit together and work in the same way—and I would also be able to reinforce basic mathematics skills.  This is as opposed to a topic like, say, combinatorial game theory, where I could talk about logical reasoning but wouldn’t be able to tie it into what they’ve already seen or build their ability to understand numbers.

I began by introducing myself and having all of the girls tell me their name.  I asked about their subway ride to Barnard and generally tried to bond a bit and be friendly.  Then I put up the question “Does 0.9999… = 1?”  I took some questions (“what does the bar over the 9 mean?”) and took a poll.  A bunch thought no, a bunch thought yes, several said they were unsure.  Great!

I next asked students to explain why they were or were not equal.  I got a few ideas but nothing very deep.  I knew that underneath, students were struggling with what this question even means, so I asked them straight up, what does this question mean?  I got some not terribly enlightening answers.

Now it was time to actually help them understand the question.  “What does = mean?” I asked (and wrote on the board).  We discussed the notion of equality for a while, and gave examples of things that were equal (1 = 1, 1/4 = 0.25, 2 + 3 = 1 + 4); part of my goal was to emphasize that = means “are the same as”, not “is the outcome of an operation”.  I asked them if x + 3 = 2x, and we had some debate before deciding that it is only true if x = 3 (I’m still not sure if they got this); then we had some more debate about if x + x = 2x before deciding yes.  Although I had hoped to doubly address the meaning of = and the hidden quantifiers in most algebra problems, I realize in retrospect that this probably muddled the picture more than it helped.

Once I felt like they understood =, I asked what 1 means (briefly, although I wish I’d had time to talk about it more) and then I asked what 0.9999… means.  I ended up getting drawn off-topic in the discussion, which is OK, but I wish I’d gotten more of a bead on “it’s 9/10 + 9/100 + 9/1000 + …”.

Instead, we ended up talking about what 0.3333… is.  None of them recognized it as 1/3, which took me rather by surprise.  OK, time to talk about 0.3333…

We spent some time discussing how to turn decimals into fractions in general, but quickly concluded that you couldn’t straight-up turn 0.3333… into a fraction; 333333…/100000… makes no sense.  So instead I suggested looking at 0.3, 0.33, 0.333, 0.3333, and so forth.  I was again a bit surprised: they seemed to have no intuition that these numbers are actually very close together.  So it was time to take out a number line and draw them.

Where, I asked, is 0.3 on the number line?  Only one girl knew how to put it up; she knew that 0.1, 0.2, …, 0.9 were equally spaced and so she found 0.3.  But when asked about 0.33, she didn’t know.  So we spent some time talking about how 0.3 is 3/10, and how to find 3/10 (divide the number line into 10ths; there’s 1/10, now we want three of them, so we go over here).  Then we decided that 0.33 is 33/100, and we talked about dividing the number line into 100 pieces.  Then we saw that 0.333 is 333/1000 and we talked about dividing the number line into 1000 pieces.  (I actually liked this part a lot, because it required a certain amount of abstraction to visualize dividing the number line into so many pieces!)

Then I asked, “OK, where is 0.3333333333 on the number line?”  The students really had no idea that it would be right by the other numbers.  Some thought it would be very far out indeed!  Others thought it would be close to 0; an original hypothesis had been that as you add 3s, the number gets closer and closer to 0.  Eventually I got them to write it out as a fraction: 3,333,333,333/10,000,000,000.  I asked them where this fraction was on the number line, and again, no idea.  (Although they were amused, perhaps even impressed, at my ability to rattle off “three billion, three hundred and thirty-three million, three hundred and thirty-three thousand, three hundred and thirty-three.”)  Eventually we realized that this fraction was close to the others, although I’m still not sure all of them were convinced.  I finally explained that if I offered to give you that many dollars, I’d give you 34 cents and then you’d owe me some money!  This seemed to help put it in perspective for them, and we briefly discussed how you might scam someone with this “trick”.  Note to my future self: give the example of 5,000,000,000/10,000,000,000; it should be much easier to see that it is just 1/2 despite the large numbers.

At some point we noted that these were getting close to 1/3, although many of them didn’t have an intuition for where 1/3 is on the number line.  We talked then about how to convert 1/3 into a decimal and I divided 3 into 1, at which point some of them realized that they had seen this before.  That said, I realize in retrospect that I didn’t emphasize that fractions are division and so something of a learning opportunity was lost.

We’d also gotten that 1/9  = 0.1111… in this discussion, but still no insight on 0.9999….  I had them discover the relationship between 1/9 = 0.1111… and 1/3 = 0.3333…, that in both cases you can multiply by 3 to go from one to the other.  Then we saw that 1/3 times 3 is 1, and that 0.3333… times 3 is 0.9999….  At this point I tried not to hint further, and despite having everything right there, the class went off on a wild tangent, thinking they could “add” something to 0.9999… to get 1.  (A perfectly good theory, if they were different, and I wish I’d emphasized that more.  We spent a while talking about what happens if you add 1, if you add 1/2, if you add 0.1, and 0.01, and 0.001, and so forth; I wish I’d had more time to allow them to do this themselves.)  Somewhat flummoxed, I did a poll.  Nearly everyone was now convinced that 0.9999… is not equal to 1!

Eventually I brought us back on track, and someone made the magical connection, and a student realized that since 1/3 times 3 is both 1 and 0.9999…, they must be equal.  This part could have used more discussion, relating it back to equality, but we were really quite short on time.  I ended the class by hinting at a graphical argument, and tantalizing them with the sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + …, and then we were done.

In the end, the class was engaged and had very high energy, but had such a deficit of basic facts that they really struggled.  I don’t think I managed to bring everyone with me as we went forward, nor was the class nearly as student-led as I’d hoped because it was hard to get them moving in this context.  I’m actually happy with the presentation itself, and I hope to continue to refine it and find ways to make it accessible to everyone.

I am reminded just how much students in schools lack, however.  The insistence on moving students forward to algebra and beyond when they don’t fully have the number sense to understand how things go on the number line seems to be doing them a great disservice, and I wish we could find a way to enable teachers to address these basic challenges with them.  Otherwise, they’ll be continually forced to memorize procedures and they’ll never understand why they do what they do.