*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.

## Logic

This one is my favorite! How do we help kids think, and do it in a really fun way that will translate to thinking about math?

Here’s my crazy plan. Logic courses will begin by looking at a particular type of game or puzzle. For example, they might look at Ken-Ken or Sudoku (or other Nikoli puzzles); they might look at liar/truthteller puzzles; or they might look at puzzles like Einstein’s riddle. These are different lenses for gaining the same skills: deductive reasoning; case analysis; careful organization; and, to some extent, proof by contradiction, because these puzzles all encourage trying one path and then, if that path fails, eliminating it and trying something else. All of the classes should also build careful reading skills. This is easy with liar/truthteller puzzles or the riddles. For the Nikoli-style puzzles, introducing a new puzzle type and having students read the rules will work well.

Once students have learned about deductive reasoning and case analysis, we transition to mathematical arguments that use those same skills. There are plenty of math problems where a case analysis can be extremely helpful. (One, which might be a tiny bit too sophisticated, is constructing all 3×3 magic squares.) By drawing explicit analogies, students can learn how to reason about mathematics. We can also help them write their own mathematical arguments, going step-by-step and saying which lines follow from which other lines (but without cumbersome rules of logic).

If time allows, I would like the logic classes to include looking at arguments (from other students, but also provided by the teacher) and deciding if they are correct, or if they are not correct, deciding where they go wrong. This reinforces how to think about deductive reasoning. One example of this is the Missing Dollar Riddle, although I will find more mathematical arguments as well.

All in all, I think this is a very solid ten hours (plus homework time).

## Math Foundations

The rest of these courses draw on more established areas of math teaching, so they don’t need quite as much detail to flesh out. For example, Math Foundations reviews mathematics from school but in ways that lead students to be more reflective on the mathematics. We’ve already developed modules designed for self-study that draw from problems in the Art of Problem Solving *Prealgebra* book as well as problems of our own design. Modules questions require thinking in new ways about math students know, which triggers self-reflection. In the spirit of Arnold Ross, I want students to “think deeply of simple things” — so simple that they’ve seen them in school! (It feels rather sacrilegious to take this quote from Ross and apply it to a program that is not the Ross Program, but perhaps this is the first step to getting there!)

Modules can form the basis of a solid course that reviews some aspect of school mathematics. Students would choose which course area they want to work on: fractions (probably the most important), uses of the distributive property (which could be pitched as mental math tricks), exponents, basic geometry, etc. Then the course would go through modules based on those ideas, led by the instructor.

However, there are some topics so essential and so frequently misunderstood that I believe *all* students should study them. These could include:

- The meaning of the = sign. We should also cover the different ways that = interacts with variables.
- Different representations of the same number. Fundamentally, 1/2 and 0.5 are actually the same mathematical object. Possibly also touching on infinite decimal expansions, depending on how feisty we feel.
- Division by 0, and
*why*it is not possible. Ideally, we would lead this from the standpoint of deciding as a class what the definition of division is, and then analyzing both 7/0 and 0/7 according to our definition to see which makes sense. As a bonus, we can divide 7/(1/2) and see what happens. - Is 0 even or odd? This can again be approached from the standpoint of definitions.

I don’t think we will have time for all of these. Covering the equals sign seems essential. The others are not critical from a content perspective, but seem very helpful from the perspective of understanding how mathematics is done and how to think about meaning in mathematics. Readers, if you have anything else that would go here, I would be very excited to hear it.

## Math Team Strategies

This course would prepare students for competitions like MATHCOUNTS. Most students have team meetings for years. We’d just be getting their feet wet, although it’s early enough in their mathematical careers that they could keep pursuing it.

I’d like each Math Team Strategies course to pick a mathematical focus area and to really develop students in that focus area. They might work on number theory, geometry, or combinatorics. As one example, a number theory course might develop prime factorizations in some detail and show students how to use prime factorizations to solve challenging problems. In this way, students see what it’s like to learn and use a mathematical subject in some depth, and they can feel a deeper learning of the material. Plus, this is pretty stuff!

Math Team Strategies might have some sort of built-in contest within the program. I would also like the course to introduce students to the Art of Problem Solving website by using Alcumus or even posting on a forum. In this way, students get started on a tool that can carry them forward mathematically. Of all the courses, this one is the easiest for students to stay engaged with after the summer, and I would like to encourage that as much as possible. The ease of staying engaged is my biggest reason for including the course in the program!

## Applied Math

This course track will require getting faculty who can really develop their own courses. The most popular is likely to be programming, and we will have to recruit heavily to find great programming faculty, perhaps finding people in industry who want to teach for an hour a day for a couple of weeks. Other applied math courses could include biology, astrophysics, engineering, etc.

These courses don’t have explicit mathematical goals. Instead, we want students to gain a stronger idea of what math can do, and to learn new ways of thinking about it by seeing genuine uses of math, rather than artificial short problems from school. The programming course should equip students to continue programming after the summer, because this can be a tremendous source of self-directed learning.

## Seminars

Finally, it wouldn’t be a math program (at least not one that I’m running) if there isn’t something to explicitly show students how big and beautiful and wonderful and fun math is. Math Explorations will be the final period of each day, a math circle where students explore something new each afternoon before going home. We can get a variety of presenters to give students many pictures of what math can be. It will also reinforce the thinking strategies they’ve been learning in the rest of the program.

Sometimes, we might replace the math circle session with guest speakers who use math in their careers. Those role models can be very powerful for the kids.

## Some Concluding Thoughts

There’s obviously a long way to go here in developing all of these materials. I’m excited to put together logic materials, and we already have the modules for Math Fundamentals. I also have some basic materials on Math Team Strategies. Applied Math will require people who can put together great courses. In any case, we’ll need teachers who can improvise; we’re not going to be handing anyone a fully-formed curriculum with handouts ready, at least not in our first year.

The other big challenge is time. Especially for programming, but really for all of these courses, there just aren’t that many classroom hours and there’s not a whole lot of problem set/homework time built in to the program. I’m still not sure if we want to assign work for students to do at home, since it’s summer vacation, after all! Can we actually get students far enough in programming that they can continue on their own with so few hours?

It’s a lot to think about and a big challenge for our staff. Fortunately, as a young program, we can iterate as we discover what works well. Sometimes it’s hard to remember that the first year is just the *beginning* of many journeys.

I think your applied mathematics piece could have some more specific foundational goals. For example, estimation, such as Fermi problems. I think applied mathematics is more foundational than many of us pure mathematicians give it credit for. Another idea, along the same lines, would be error analysis in some 6 grade appropriate way. Super exciting, and wonderful to see your thoughts written out.

Oooh… I really like the idea of estimation or error analysis. The challenge for me there is giving up programming as a course, because it would be so valuable, and we are terribly over-stressed for time.

Perhaps what we can do is work in both as options and let the students choose, then see what effects it has down the line for the students. (BEAM 7 is moving towards “every first week course explicitly teaches proof” because we’ve seen that difference depending on what course students take.)