Month: December 2015

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?

Curriculum

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?

Logistics

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.