Wednesday, June 24, 2015

Random Geometry Recursion

You can recursively keep halving the length of the diagonals in this fashion. Creating a series of smaller 30-60-90 triangles in the process.  (I promise I will switch subjects soon.)

Friday, June 19, 2015

Geometric Translations

Recently inspired by some posts on problem solving over at I've been combing through the geometry problems at  So far the first 7 or so have been really interesting. I spent several days on the first one or two and went down several blind alleys. I've become used to relying on similar triangles and basic angle chasing for most geometry problems.  Most of these problems require triangle rotation/translation instead to make headway.


My final solution required three such rotation/translations. First ABD is translated inside BDC. Then the smaller new triangle is translated twice more and you find an unexpected equilateral triangle.

The complexity here also puts this beyond the kids in math club's skill set. However, I think there's the start of an interesting exercise for the day present in these type of problems.  Translation is not done much in school but is very simple to describe and then opens up lots of interesting areas. One of my favorite transforms is to take the two triangles created by the median of a triangle and then recombine them by sticking the divided edge together into a new but related triangle. These also can be thought of as tiling exercises. So I'm off to see if I can find a range of problems or activities that would be approachable.

Thursday, June 11, 2015

The Year In Review

Now that the last session of the math club is done for the year it seems appropriate to look back and reflect on my experiences. Going into the process, I thought that while I had plenty of math experience and had tutored my own children one on one that group situations would be a lot harder. I've read elsewhere that teachers take two years to become seasoned and figure out how to manage a room and that is while being in the classroom full time. I have to say, that this year has given me a small taste of the challenge and I'm a bit in awe of how full time teachers accomplish as much as they do. I only had a room of 15 kids once a week for an hour and that afforded me the luxury of spending a week planning what to do.

After my first session jitters I found the process to be most similar to running meetings at work as opposed to working with my sons on a math problem. And just like meetings the more preparation, the better off things go. With that said I hope to accelerate my own improvement by reflecting on each session as much as possible to wring out the maximum feedback. If I were really bold I would videotape myself but I fear that would be too much for me to watch.

So without further ado and reserving my right to completely change my mind after another year:

Subject Material

1. Be prepared for finishing a task early but also be confident enough to plunge right into the main task if it looks like it needs the full hour.

2. Geometry is awesome.  What's not to like about problems that almost have multiple solutions and require the smallest number of techniques to discover fairly profound phenomena.

3. If an activity comes in multiple levels always take advantage of that.

4. Always try to do the activity yourself beforehand and near enough to the session that you don't forget anything, This is really key especially if you don't think as well on your feet in front of a group. Give yourself permission to say I have to think about that if you get stuck. :)

5. Organize your materials and double check them. In front of a crowd is not the time to sort your papers.

6. Figure out your relative pace vs. the kids. I find that I should usually multiple however long it takes me by 2x or 3x.

7. For now I'm sticking with a whimsical/quirky approach to programming. I basically go with my gut and don't attempt to be systematic or sequential in anyway. I've had some conversations with others that indicates that sequencing is less effective anyway in a weekly extra curricular.

Classroom Management

1. Be explicit. This is a hard one for me but you need to have upfront talks about behavior expectations. I found it especially useful to have the kids voice their opinions on why they were in the room and how they wanted it to operate.

2. Maintain boundaries. Early on, I let some of the kids free range in the room and touch some of the classroom objects that they really shouldn't have without immediately correcting the problem. This has taken two months to rein back in again. Once a threshold is crossed it is so much harder to repair.  Likewise in the idea that we are club not a class  I let the kids be as noisy as they wanted. This established a bad norm. Next year I'm going to insist more strictly on quiet talk so others aren't disturbed who like calmer working environments.

3. Interact directly with kids who are going astray. Most kids will be course corrected by a minute or two of one on one time.

4. Ask for help. Additional parent volunteers are a really easy way to make things run more smoothly.

5. Think carefully about transitions (and minimize them).


1. Don't provide regular snacks. I thought this would build camaraderie but instead it was a bit of a logistical nightmare. If I delayed serving the snacks, the kids would focus on them to the detriment of the math. I always had to worry about messes. And somehow food brings out a weird competitive streak in a bunch of otherwise  normal children. As soon as you have portions, someone is going to worry about if they are all the same size. Bring two varieties of a cheese cracker and heaven help you if one box runs out too soon.  Occasional unexpected treats are the way to go.  They buy love but are not taken for granted and you can provide them on a convenient schedule.

2. Sequencing is not important. That's the great thing about being a club. Math class follows the curriculum. We just need to concentrate on joy.


1. Pay attention to everyone in the room. I'm not really methodical about this yet and its something to work on. Likewise as I posted elsewhere I'm musing about how much to hint and what kind of hints give.

2.  Find more games.

3. Find the right approach for take home activities. My current idea is to try again with a problem a week and track how many are done and give collective treats when we reach certain thresholds. I.e. after 15 completed problems from the groups bring in cookies.


Here's my topic map for the year showing what we covered:

Tuesday, June 9, 2015

6/9 End of the Year

Today was the last meeting of the math club for the school year. I feel a bit elated to have made it through an entire year and sad at the same time to see some of the kids move onto middle school. The card above from one of the girls was definitely a highlight of the day. If you like a teacher make sure to write or have your kids write a note at the end of the year.

For the last session I had the kids do a straight out algebra readiness assessment from Art of Problem Solving:  I did this mostly for all the parents who are deciding whether to have their children take sixth grade algebra or wondering if there are any holes to work on over the summer. The results were interesting to me. Generally basic skills were there although some of the trickier exponential and square root simplifications tripped up some of the club. I've never tried any kind of curriculum alignment but I  may want to insert some skill building sessions into warm up. For example have 3 or 4 single variable expressions to simplify. Another option might be to build up more complicated problem sessions around isolating variables and expression manipulation. This definitely deserves some more thought.

Secondly the problem solving section proved trickier. These again are mostly the type of problems I avoid since they closely mirror school topics. The MOEMS contests and practice probably get the closest. I'll have to consider whether to stress these more or just assume this reflects the need for more time/maturation for most 11 year-olds. As of right now my potential wait list for next years skews towards fourth graders anyway so I'm probably going to have to retool my problem levels and expectations.

One other small discovery. Moving the kids who had finished early to a mat on one end of the classroom where they ended up playing cards made it much easier for the rest of the test takers to focus.

Wednesday, June 3, 2015

On Hints

I've been reading the following thought provoking post by Michael Pershan @

and the one he references by Annie Fetter @

Both are concerned with whether teachers should provide any hints when students get stuck. If I could summarize the thinking it goes something like this. Any hint cuts off mathematical thinking that the students would otherwise have done. It also encourages bad problem solving habits basically undermining future thinking if nothing is obvious and even further can lead to shallower understanding along with an undue emphasis on just answer-seeking. 

I think these are really interesting perspectives and I've been musing how they relate to how I conduct the math club sessions. First, a full confession, I provide lots of hints and guidance during the hour. Among the things I do is walk and chat with the kids to see where they are in the problems. I then will often offer a suggestion: "Have you tried X?" Part of my motivation has been the worry that kids will just get stuck and make no progress when I want everyone to stay roughly on pace. There's also the exhaustion principle. If someone gets stuck for multiple minutes and doesn't see a way forward they are more likely to give up and start socializing or some other extraneous activity. On the other hand, if they can maintain flow, they will stay excited and focused on the math.  I've also found kids can be redirected back into the problems by one on one conversation which makes it useful for general classroom management. 

The other general motivation is that I don't have the experience yet to know how each problem will go. By selecting a set of problems that are at the edge of their abilities I feel like I need to channel the kids to some extent to achieve what I've selected or alter my tasks over time.  I usually pre-plan some scaffolding ideas if the kids don't perform as I expected. 

Finally  all the kids are not silently working away. They chat with their fellows and ask each for help and share strategies. So essentially if I'm not giving hints, they would propagate anyway through the room. Anything I say is generally going to be much less direct i.e. more of a hint/guidance than the entire process which is what they will usually tell each other.

However, I do think there is a fundamental truth in Michael's ideas. Beyond solving individual problems, one goal for the club should be to develop problem solving and that includes how to handle getting stuck.  One easy change to make is to focus on asking questions when I'm checking on work. I think I do this anyway but I'm going to be more reflective about it.  I won't have a chance to apply this until next year. But I like the idea of making sure to always query what they're thinking first. I'm not sure I'm brave enough yet to try out offering no direct help at all.  We'll see next year.

I haven't done a lot of talks about work habits directly preferring to model problem solving behaviors. But this seems like something that should be made more explicit at the beginning sessions. I'm imagining a group discussion about "What to do when I'm stuck?"

Also I've been moving towards thinking more about progressive tasks which build on each other over the hour and leveling puzzles. I think this has been promising in letting the succession of math problems guide the kids towards deeper understanding. I'm going to continue with these moving forward. Its also worth realizing that my natural tendency is towards picking too much per session   (out of fear of running out of activities) and to tackle harder problems that I think will be fun. Balancing those instincts or at least being honest about when I'm over ambitious makes sense.

Finally, I  mostly dropped the idea  of giving weekly problems to be done at home after the first quarter. The kids mostly didn't take a look at them and they seemed to not be successful. On the other hand, they offer the best chance to work on something over  a longer period time without any supports. The question is can I structure things to make them more utilized?

[Addendum: See for my continued evolution on the use of homework and problem solving.]

Tuesday, June 2, 2015

6/2 Pentagrams and some inspirational math.

Just in the nick of time for the end of the year, I received the notice that Jo Boaler's Week of Inspirational Math.  was released. I took a glance through and thought the first day activity four 4's looked like a decent warm up.  The basic idea is you have four fours and you can combine them using basic operations (add,substract, multiple, divide) as well as factorials and square roots. You then try to find a combination for all of the first 20 whole numbers 1 - 20.  Example: 4 + 4  + 4/4  == 9. Over all this went okay but not as well as I hoped. I'm going to look at the material again and watch some of the videos. It didn't seem to capture the kids imagination as much as I wanted even compared to just using game of 24 cards which we've done in the past. I could certainly add a competitive element to this which might draw more kids in and  have teams compete against each other to find the most equations. This is very anti-Boaler but sometimes effective in practice.  In any event, I may retry this again and see if I can frame it differently/more effectively.

We then transitioned into our main activity. I went with my own passion and pursed the pentagram geometry problem. To start I used the whiteboard to discuss how many degrees are in polygons of increasing size. We approached the problem by breaking them up into triangles.  I had kids volunteer to draw dissections and asked along the way if anyone could find a pattern.  We made it up to a 7gon fairly quickly. Unfortunately no one came up with a general strategy and we had already covered pentagons so I  made  an executive decision to demonstrate the technique of drawing lines from one vertex to all the other ones.

This was in the interest of moving along the sequence I wanted to follow but this activity could be broken out more. I also skipped proving why a triangle has 180 degrees and left that as an assumption. Interestingly the general technique did produce a few aha moments for a few kids which was gratifying.

I then had them work through a packet that terminated at the pentagram problem from my prev. post.

To approach it first I had a few more rote worksheet on angle tracing that I took from: I had them the work on the interior, exterior and interior with algebraic variable sheets.  As I walked along the tables I asked each kid if they understood the techniques and if they said yes I told them to skip to the next page.

I then had a few more complicated angle tracing problems that required applying supplementary angles, triangle angle addition and occasionally some lines intersecting other parallel lines. These are essentially mechanical as well but require using several techniques at once. Most kids made it at least this far with a few needing some prompting on which angles to look at next.

Beyond that I included two more fun angle tracing/algebra problems from the five triangles site that are closely related:


These were a real hit with the kids who made it this far. Finally, I terminated with the full blown pentagram problem. However, most kids had run out of time just when we reached that point. So overall it was an effective sequence and I think it kept everyone's attention but next time I'm going to have more confidence and skip any warm ups and jump right in. Unfortunately, next week is our last session and I've already committed to doing an algebra readiness assessment or I would just pickup where we had finished.