Opinions on foundational math materials to teach 8th grade, 9th grade kids at a Summer Camp

Question

I have been asked to teach mathematics/physics to a few 8th grade/9th grade kids for a summer camp. I have been thinking about it and I realized that I could go about it in two ways: One of the ways is to give them random recreational thought provoking puzzles to encourage problem solving nature (This is the way I started my journey into mathematics). The other way is to give them radical ideas of mathematics disguised as thought provoking puzzles, in order to give them a feel of what mathematical theories are like. The latter is an attempt to bridge the divide between theoretical mathematics and fun problem solving. I finally decided to follow an hybrid of the two and teach a mixed bag of olympiad-like problems, puzzles and theoretical material.

I have coached students for math olympiads before and I have a good source of olympiad problems from my books, internet and so on, so I am not looking for olympiad problems.

I have seen ideas here, but as I said I am looking for foundational ideas disguised as problems. Since there are a lot of experts who visit this site, I am requesting to know simple non-trivial problems that motivate theories and those problems should be explainable to 8th/9th grade kids. So, how can I introduce Calculus, Geometry, Combinatorics and Algebra with a fun problem? Even if there exists a theorem with an involved proof, suggesting a special case that is easy to illustrate will do.

Finally, I have not seen anybody attempt to encourage mathematics in my locality, and this is a new idea for me too. So if you think this is a bad idea ("8th/9th grade is too early for deep mathematics" e.t.c), I would love to hear your criticisms.

P.S: From the feedback that I have collected and the way mathematics is taught according to the curriculum, the talented ones believe math is boring but an important tool and others are mostly scared of mathematics. Nobody seems to be enjoying it as much and many of them have an uninformed view that graduate/undergraduate math is boring. I basically want them to see the fun side of interesting graduate mathematics and then make an educated choice for their future. Of course 8th/9th grade might be too early to expose kids to undergraduate mathematics. However, I want to do the best I can. So, please help me.

Thank you,

Isomorphism

Answer 1

The book Mathematical Circles contains some excellent material.Talking about being "involved", combinatorics is right up there for the purpose.Though I am not sure "Graduate Mathematics" will have a profound impact on them.In fact, undergraduate mathematics like group theory which is interesting enough is quite engaging and I see a lot of 8th and 9th graders reading Artin and actually proving things while I have just started with undergraduate stuff.Mind you, they are a highly motivated bunch coming from the Mathcounts-AIME-USA(J)MO setup.But what is common in them is that they gained some taste of higher math and the Olympiads which served as a catalyst to their interest in the subject.

However, I am not sure this is the age where the decide to be number theoretists or topologists or any such field of specialization. When you try to guide them,you may end up killing their interest in some areas of math instead .So, why not allow them to make the choice? :P

By the bye, I remember how as a ninth grader, I was fascinated by what modular arithmetic can do.Graphs can be easily visualized.More importantly, you are a better judge than me;I am just a boy in high school randomly dropping by.Hope that helps, though.User21820 has actually included some mathematics in his/her post.

Answer 2

Here are two possible activities that illustrate two fundamental mathematical concepts, bijection and induction. There are many other important concepts such as contradiction, bounding, canonical forms and extreme values, which are in my opinion more fundamental than the "basic facts and methods" in each field.

Get each group to form any binary tree, starting with one person being the root, and where each new person can only join by being held by someone with a free hand. Then tell them to count the total number of free hands. Is it always the same for any tree? One way is to count the total number of hands minus total number of "used" hands, and the total number of used hands is exactly the number of people except for the root. What if some people can only use one hand or none? Can they come up with a way to count the total number of free hands in terms of the size of the group and the restrictions on each person? You can later tell them all the different ways to view the problem. The above way treats each person and the hand that holds him/her as one unit. Another way is to treat each person with his/her (initially free) hands as one unit and consider what happens to the size of the tree and the number of free hands when he/she is added. This relies on induction so it is a good place to mention that too.

For the next activity, get them all to form one big tree such that everyone is as close to the root as possible. In other words, minimize the depth. Then you can explain that this is equivalent to maximizing the number of people that can be attached to the tree with a certain depth. Then you can explicitly show how a proof by induction of the maximum size works, as well as make it clear why you cannot "start from a tree of depth $d$ and build a tree of depth $d+1$...", which is the most common mistake of students and even teachers. Instead you have to start by asking them to form any random tree of depth $d$, and then show that you can remove the root, resulting in two trees which must by the induction hypothesis each have at most $2^d-1$ people, thus the original tree must have had at most $2(2^d-1)+1=2^{d+1}-1$ people, so by induction the formula is correct. If the students are fast at grasping this, you can show an alternative proof by removing all those with two free hands. Then the depth decreases by one and the resulting tree has at most $2^d-1$ people, and by the earlier game had at most $2^d$ free hands, so you must have removed at most $2^d$ people, hence the original tree had at most $(2^d-1)+2^d=2^{d+1}-1$ people.

I am sure that if students fully grasp the above two examples, they will have a very much better understanding of counting and induction than by all the school-syllabus examples. For an even more advanced example, you can let them compete to re-draw a planar graph with only straight non-intersecting edges. Is it always possible? Also ask them to figure out the maximum number of edges in a finite planar graph and attempt to prove it. The fastest solution I know is to first prove that you can "triangulate" any planar graph by adding edges. Finally you could also explore the graphs where every vertex has the same degree.