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.
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.