BEAM 6: Designing a Curriculum

This is part of a sequence of posts developing a new project for Bridge to Enter Advanced Mathematics called BEAM 6.  BEAM 6 will be a non-residential, four-week summer program for underserved 6th grade students in New York City.  You can find the other posts about its design here.


At BEAM 7 (a program many BEAM 6 students will attend the summer after 7th grade), we tend to throw students off the deep end when it comes to doing math.  I mean it: they come in and we teach them about proofs, and we have them solving MATHCOUNTS problems, and we have them learning number theory and combinatorics and even group theory… and a lot of them are kind-of still weak on fractions, y’know?

It varies by school, of course.  Not surprisingly, some schools tend to give us more prepared students, while others don’t.  BEAM 7 has had seventh graders (top of their class at school!) who were not comfortable multiplying negative numbers.

I’ve been asking myself what I wish our BEAM 7 students knew.  They’re held back constantly by foundational math knowledge.  They also need to learn how to look at a problem and focus on what it’s asking, rather than guessing at a solution mechanism.  Finally, I want them to have more skills in deductive reasoning and case analysis.  It’s a little bit crazy to be thrown into a proofs class without that!

In the end, there are five course tracks that I really want to work into the program.

  • Logic
  • Math Foundations
  • Math Team Training
  • Applied Math
  • Seminars

I want students to have choice, so each of these topic areas will have different courses within it.  During their summer, each student will take on course from each track.  The exception will be Seminars, where each will be independent (see below).

It’s a minor nightmare to fit all of these classes into a four-week program.  Right now, the best I can do is 10 hours of class time each, plus some homework time depending on the course.  So they have to be compact and get to the punch quite quickly.  Scheduling will be covered in detail in a future post, but lack of time is a huge concern.

Before I get to describing the course tracks, readers who know about BEAM 7’s courses will see right away that this is super different.  BEAM 7 is basically a playground for our faculty to develop all kinds of interesting math courses and then teach them.  These courses are much more structured and targeted.  Why?

The primary reason is simple: BEAM 6 has a very different goal.  BEAM 6’s main goal is to remedy specific gaps that students need to succeed both in BEAM 7 and in their future mathematical studies.  In contract, BEAM 7’s goal is to transition students to other programs for advanced study where they will have to do more abstract thinking.  Hence, BEAM 7 invites faculty to rock out in courses similar to what students will do at future programs.  In contrast, BEAM 6 is a laser aimed at skills and knowledge that students need.  BEAM 6 courses will be lots of fun, but they’ll also have much more concrete goals.

There are advantages and disadvantages to both approaches.  One big advantage of BEAM 6 is that I can develop a strong curriculum for students.  A second advantage is that it opens our program up to more potential instructors, because they do not need the same experience designing enrichment classes.  However, BEAM 6 is still open to those who want to create their own crazy classes through both the Seminars and Applied Math topic areas.

Great!  Let’s figure out what’s actually in the courses.

Continue reading “BEAM 6: Designing a Curriculum”

BEAM 6: Goals

This is part of a sequence of posts developing a new project for Bridge to Enter Advanced Mathematics called BEAM 6.  BEAM 6 will be a non-residential, four-week summer program for underserved 6th grade students in New York City.  You can find the other posts about its design here.

In about seven months, there will be 100 sixth-grade students all ready to learn math.  Almost exclusively, their mathematics educations so far will be designed around memorizing procedures and passing tests.  We have four weeks to change their lives.  What should we do?

No pressure or anything.

It’s rare in education to get an opportunity to work with motivated, talented students with no outside requirements.  We can design the program that is best for them.  This is the first post developing BEAM 6, and so we will set down the program goals.

Goal: Teach Them to Think Deeply

If students leave the program and they have not learned about logical reasoning, I will feel exceptionally disappointed.  I want students to grasp ideas of deductive reasoning.  This might be my single biggest goal.

I also want to change the way they think about mathematics.  For many students, math problems are defined by the solution method.  “Oh, this is an addition problem.”  “Oh, this is a related rates problem.”  “Oh, this is a Pythagorean theorem problem.”  This thinking leads to oversimplification and memorizing procedures.  It makes it more difficult to solve multi-step problems.  Students should engage with the question, understand the problem independent of its solution, and accept or reject solution paths because they do or don’t solve the problem.

This leads to the broader question of mathematical communication.  For example, the equals sign.  Students often interpret the equals sign as asking a question.  In elementary school, it is always used as “2 + 5 = ?”.  By algebra, the question changes — “2x – 3 = 15” means “solve for x” — but the equals sign is still primarily used to express a question.  Students don’t realize that “25 + 7 = 32” is a statement that can be true or false; that the purpose of = is not to ask a question but rather to give a statement.  The result is a failure of both communication and conceptualization.

These goals are less mathematically sophisticated than BEAM 7’s goals.  This is in part because the students are younger.  It’s also to build synergy with BEAM 7.  Students often come out of BEAM 7 with a strong grounding in abstract mathematics but still well behind peers in school-based math.  For example, students often do well taking a number theory course at CTY or going to a program like MathPath, but do relatively poorly in a contest like MATHCOUNTS.  BEAM 6 can close that gap and set students on a path to deepening their facility with school-based math.

Goal: Help Them Love Math

People love math because it is beautiful; because it is thrilling to challenge yourself with a hard problem that you finally solve; and because it is interesting to see how it applies to the real world.  We must show students what math really is.  That it is not about memorization or following procedures.  That it is beautiful and creative and exciting.  A love of math will carry you far, and we should develop it in the students.

Goal: Develop Their Self-Identities

In my experience, self-identity drives a lot about a person.  More than just thinking something is “cool,” self-identity can push someone to pursue an interest; it can create resilience to failure; it can drive life decisions.  If we can develop self-identities in our students as scholars, and furthermore as scientists and mathematicians, they are much more likely to succeed on that path.

What contributes to developing self-identity?  Here are some thoughts:

  • Interest/passion for a topic.
  • A feeling of self-efficacy; confidence in your abilities.
  • Membership in a distinctive community.
  • Role models.
  • A sense of future (where will it take you?).

We should harness all of these within the program.  We have special expertise in creating a mathematical community.  To drive students’ further engagement, creating a very strong community will be essential.

Goal: Develop Independent Learners

A summer program cannot alone cover the mathematical education of all these students.  If they will be successful, they must continue to pursue learning after the summer is done.

Students should be connected with resources for further study, such as Art of Problem Solving.  They should get used to these tools during the summer and be encouraged to continue using them when they’re done so that they continue to get better.

Concluding Thoughts

These goals feel right.  They cover what I feel is very important to develop in young mathematicians.  However, they are not complete.  While program elements will be tied into these goals, as the program development continues we will also find new goals that we want to achieve.  These will be included below as updates to this post.

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.