Thursday, June 23, 2016

2016 Year in Review

With another year under my belt, its time to look back and think about what I've learned over the process.  (Here's my review from last year:   In many ways this year was easier than the first one. I had a much clearer idea about what to expect, what kinds of activities I would use and what  I wanted to work on specifically with my own practice.  But there were plenty of new things to discover. For one, the kids were mostly fourth graders rather than fifth so I knew from the start I needed to gear things a bit younger than last year.


This was my original plan for the year: which I stuck with. Overall, the problem of the week format was successful. On average I had about half the kids work on a problem every week and I never lacked kids to demonstrate solutions on the whiteboard. An unexpected side benefit was that parents often worked together with kids over the weekend on the problems.   That was great for two reasons. First I think its awesome to have families think about (low pressure) math together. Secondly, and unexpectedly parents often talked to me about how much they enjoyed the problems.  I think this format ended up giving them more insight into what their kids were doing and deepened their connection to the club. I'm wondering if next year I should expand the idea and create some kind of visually appealing chart/graph to motivate everyone. 


This year I only provided snacks occasionally for celebrations like Pi day or the last meeting of the year and as a reward for the group when collectively enough problems had been completed.  This  was a complete improvement over last year.  First this was much less work for me both in logistics and in managing messes and cleanup. There's also a well known psychological principle that comes into play: . Unexpected treats are special and get everyone excited.  Even the uncertainty of when the kids would cross another threshold for completed assignments led to the kids encouraging each other to work on the Problem of the Week. 


I experimented with short videos mostly from Numberphile during the second half of the year. As expected these were effective at captivating the kids. I don't want to overuse videos but I think I'll continue targeted shorts that are thematically interesting. They are particularly good at bridge/transitions.


I'm still not really satisfied with the format of off-site contests. They still are too speed focused and I worry about the effect on morale when kids don't win. Once again, although I'm unlikely to make a change I wonder if I should just focus on MOEMS, the Julia Robinson Festival, AMC and maybe purple comet.  I'm going to try to more mindfully give out practice sets to do at home for the few other contest if we go.


I've been doing a lot of sessions with the following format recently.  First a whiteboard session for the problem of the week, then a warm-up puzzle followed by a choice of 2 main activities. This worked well at the end of the year when I felt the kids needed more choices to stay engaged. But I still have an aspirational goal that more sessions look like the Julia Robinson Festival with the kids going deep on a problem set. This cuts against many of my curriculum choices at the end of the year where I looked for different activities to engage the kids where I felt I was losing them.  Part of why I picked the Ulam Spiral investigation was for its graphic qualities for instance. I'm going to spend the summer brainstorming for more new ideas for next year. This is especially important since I won't be able to repeat many of this years material if most of the fourth graders return.  

Small tweaks

Small changes can be effective. After one session, a parent who had been helping out remarked I think it would be useful to have the kids put their backpacks on the side of the room so they aren't distracted by them.   I hadn't though much about this, other than a few kids were fooling around with their packs at the beginning of the club. But I thought the idea was reasonable and tried it out for the rest of the quarter. Sure enough, it did help establish  a starting routine and made things a bit smoother.

Future Goals

I  have some concrete ideas:
  • Register for both the Elementary and Middle School divisions of MOEMS. Most of the fifth graders should be capable of the higher level.
  • Participate in AMC8. I had toyed with this idea last summer and then decided the fourth graders were not ready for it yet.

And then some bigger practices to  work on:
  • Better questions.  There's plenty of room still to improve on what I ask the kids as I go around the room or when they ask for help. 
  • Getting everyone to participate. I've been tracking who I call on but its still not perfect. Drafting quieter kids works to some extent for this piece. The main part of what I use the other parent volunteers for is to focus on kids who are stuck and unengaged and draw them back in. I'd say this year I achieved on average 75-80% engagement per week at any point in time on activities but that leaves 20% to aim for.
  • Maximizing open-ended activities.  I want to figure out a structure to let the kids report back what they discover and build on each other's observations. Right now when I'm having the kids investigate something like Ulam's spiral I don't  consistently build in a time for the group to discuss what they find and/or go back and look for deeper patterns.  It's tempting to let kids keep working when they are productive but the larger conversations are easy to short at the end.
  • Fighting answer seeking and modelling persistence. The first, I'm going to attack directly in our charter talks to start off. The second I still have no great answer for.

And last but not least I'm working on expanding to 2 groups one for the fourth graders and one for the fifth. I have permission to use another room and about 5 people have expressed interest in helping. This will be a huge step for me and hopefully if successful ensure the club will continue after I'm gone.

Topic Maps for this year:

Saturday, June 18, 2016

Two problems I'm looking at

I was reading a fun post over @ where Lisa is brainstorming problem sets. She's inspired me to collate the material we did this year.  But in the meantime here's a teaser:

I don't usually include algebra problems but I ran into this one last night and its too good not to keep for future use when I have a group of older kids.  At first this problem seems like its missing enough information.  For example, you can't directly solve for $$x_1 .. x_7$$ since there are only 3 equations. However, if you assume that some linear combination of the 3 equations will equal the  target i.e.

$$f(x_1..x_2) = x_1 + 4x_2 + 9x_3  + 16x_4 + 25x_5 + 36x_6 + 49x_7 \\
g(x_1..x_2) = 4x_1 + 9x_2 + 16x_3  + 25x_4 + 36x_5 + 49x_6 + 64x_7 \\
h(x_1..x_2) = 9x_1 + 16x_2 + 25x_3  + 36x_4 + 49x_5 + 64x_6 + 81x_7$$


$$a \cdot f()  + b \cdot g() + c \cdot h() = 16x_1 + 25x_2 + 36x_3  + 49x_4 + 64x_5 + 81x_6 + 100x_7$$

This reduces to 7 equations in 3 unknowns The first and easiest 3 of them are:

$$(a + 4b +  9c) x_1 = 16x_1$$ $$(4a + 9b + 16c) x_2 = 25x_2$$ $$(9a + 16b + 25c) x_3 = 36x_3$$

Then these can be solved directly and it only remains to check if the solution (1,-3,3) works for the other 4 terms.

Or can you generalize and confirm:  $$ x^2  -3 (x + 1)^2 + 3(x+2)^2 = (x+3)^2$$

What's interesting is to examine the general problem afterwards for instance if there were only 2 equations is there a linear combination that works (nope)? What about four (this has multiple solutions)? I also find a hint of the third row of Pascal's triangle in the solution but haven't looked into whether that's a coincidence.

Update: its easiest to solve the general equation above  $$ ax^2  + b(x + 1)^2 + c(x+2)^2 = (x+3)^2$$ This reduces to:

$$a+b+c = 1, b+2c = 3, b + 4c = 9$$
Interestingly in 4 cubic terms the solution  to $$ ax^3 + b (x + 1)^3+ c(x+2)^3 +d(x+3)^3 = (x+4)^3$$  is {3,4,-6,4}. Once again close to the 4th term in the triangle and in 5 quad terms the solution is (1,-5,10,5). The binomial theorem is all over this problem so its not completely surprising but I'll keep thinking about it a bit more.

Folding problems are always fun. This one stands out because it uses the notion of proportionality twice in both direction i..e if you draw a line from one vertex of a triangle to somewhere in its base the ratio of areas of the two triangles are the same as the ratio of the two bases (since they share the same height).

Wednesday, June 15, 2016

6/14 End of the Year

And so we reached the end of another year.  I wanted to celebrate a bit for the last day so I ordered a supermarket cake and we had a small party in the cafeteria prior to starting. I read bad math jokes from the book in the picture while the kids were eating that they mostly tolerated.

I started the club off with a survey around the room to see which activities the kids liked.  Although a lot of the things we did were mentioned no clear favorite emerged this time around which either means I picked a good medley of tasks or nothing was quite as awesome as the fold and cut  day from earlier this year.

We then talked about the problem of the week: "What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 5?"  This one was very approachable. When calling on the kids, I again framed everything in the context of "show us what strategy you used" I then asked some followup questions along the lines of if I changed the numbers how would you change your strategy. For example what if the remainder when divided by 4 was 3?  I haven't yet done a formal modular arithmetic themed day but I think based on this and a few other problems that it would work out fairly well.

Finally, after seeing another glowing review of the moview flatland I decided to end the year with a movie day rather than a game day.  Flatland works really well because its only about 35 minutes so we could watch the whole film and have time for a small discussion at the end about the fourth dimension.

As you can see from the first picture I received a few cards and gifts from the kids which was also very touching.  I'm fairly confident I'll have the opportunity to work with many of them again next year and it will be really interesting to watch their growth over another year.

One Last Smile:  a few kids asked me if their was a problem of the week even though it was our last session. Maybe I should have come up with a monster "Problem of the Summer".

Wednesday, June 8, 2016

6/7 Hinged Polygons

I was pleased that this week ran much more smoothly than last time.
  • First I didn't serve candy which probably helped. 
  • I decided to give a little talk to start off about working on listening to each other when working on the whiteboard. This was similar to the one I did at the start of the club.
  • I swapped the warmup activity in front of  the problem of the week.  The theory was this would let everyone transition back to math after the bell. I don't really want to do this permanently since kids work at different paces and that means I have to stop some in the middle after most of them are done. 
  • The problem of the week was less procedural and easier to explain in a shorter period of time.
  • I also changed my tack a bit and instead of asking for people to demonstrate solutions, I asked them to talk about what strategies they used without necessarily showing and writing all the details.  I think this is something I'm going to return to next year.
Some combination of all this made everyone's focus improve.  For the warm up I picked an worksheet: shape times shape worksheet that I've had my eye on. Many kids finished it a bit quicker than I expected (under 5min vs my guess of 10) . This was the first discussion where I tried out asking for strategies explicitly.  The general results were fairly positive. I had lots of hands up and each one went fairly quickly and I think although I'll have to watch more that it was easier to digest for the group.

We then talked over the 5 pirate puzzle from last week.   The kids almost immediately jumped to the recursive solution so I only had to pick different students to go over each subcase one by one as we built to 5. Along the way we hit a snag at 4 pirates where there were 2 different solutions offered. So I had each presenter each explain why there was correct and then had the group talk about which one was better.  This was also a productive conversation.

For the main activity I was fascinated by this particular hinged polygon transformation:

I found the following worksheet:  to have everyone try the triangle to square transform.  During the start I mentioned that this was a general property of polygons but I did not go any farther. It would be possible to build on this a bit more next year if I redo.

I also copied another non-hinged dissection square to hexagon for the kids to cutout and try if they finished the first:

Then to round things out I brought a huge box of crayons to use for another set of matchstick puzzles:

Both activities were well received but I'd like to buildup the polygon dissections a bit more so they are a full session's worth of material and they go a bit deeper. I plan to brainstorm more about this.


What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 5?

Friday, June 3, 2016

6/1 Art Math Intersection

I farmed out picking candy for math club to my wife this week and she was very generous in her purchases.  So I started the day handing out skittles and gummy bears to everyone since they had completed another 20  problem of the week assignments collectively. This was probably a bit too much sugar and I could definitely see the effects in the first 15 minutes of Math Club.  On the bright side after participation waned a bit last month, its picking up again.

We then started with the math counts problem of the week worksheet:  Unfortunately, I found it harder to keep everyone focused on the students demonstrating on the whiteboard. These problems seem to take quite a bit more time for the kids to write out and they usually want to just scribe without talking.  I'm thinking a bit if I want to keep using them and how to improve this process. This might just be end of the year fever or perhaps I need to have the kids doing something else while one writes up their work and then have everyone stop and listen to the explanation. Another idea would be to just have the kids talk about their strategy and no calculation allowed.

At any rate for next week I switched tacks and went with a classic riddle instead:   My plan for now is to start with warm up for maybe 10-15 minutes and then have everyone gather to talk at the whiteboard.

For the main  activities I brought rulers, crayons and straight edges.  I gave the kids a choice between investigating Ulam's Spiral or modular arithmetic circles.

The spiral is really cool.  I had the kids take graph paper and create a spiral and then pick a  pattern like multiples of 5 and color in the boxes that matched to look for patterns.  The only thing  I found myself doing further was asking kids to generate more cycles on the outside so they could see the patterns better.  I had a few kids decide to look into primes so I'm planning to bring a picture of what this looks like and maybe do a short discussion next week.  Two things to do if I repeat. Rather than just giving the kids graph paper, a pre-labelled sheet with the numbers already in it would speed things up. See:  Also its important to have everyone check in and share what they've found along the way.

The modular circles were inspired by: For these I precut out some circle templates, had the kids trace them and label digits on the outside and then start compiling data for various multiple tables.  Again. I thought this went pretty well. The only hitch was its better to encourage higher numbers on the outside. A circle with only 6 points doesn't generate quite as exciting results,