Wednesday, November 25, 2015

Winding Down, Filling Away?

Next Quarter

This shouldn't surprise me anymore but despite it feeling like I just started the first quarter is almost done and I'm about to start registration for the winter. I've decided based on the number of other volunteers that I've found to raise the club size to 18 kids. So hopefully everyone who wants to participate has a chance. If the other dad who was my main assistant can commit for the next quarter I will go all the way to 20. I allow the existing students to preregister so they can continue if they'd like to all year long since Math Club really functions best as a year long activity. So far 11 out of 16 kids have definitely indicated they will keep going.  I have a few kids among those that have not contacted me yet that I really hope rejoin.  Its hard not to take some of these decisions personally when I know objectively many factors beyond me come into play including busy parents who will still register when it opens up.  On the bright side, I'm confident we'll easily fill up and hopefully based on the wait list from last session I'll have a bit better gender parity.  Just to prime the pump I sent the wait list folks a reminder mail which amusingly resulted in an instant email back from one parent asking if they could sign up right now.

Are they Retaining Knowledge?

Every once in a while something happens that makes me take pause and worry how much the kids are remembering. Last week it was the RATS = 4 * STAR problem. Some of the kids had seen the problem at least 2 times previously over a year and yet looked oblivious when I mentioned that fact.  Then last night working on a fun problem with my son I had another similar episode.  During the summer we had visited the Field Museum:  Besides the Voronoi diagrams there was also a mirror maze that had various problems in it posted on the walls. One of the problems was interesting enough that we talked about it for 10-15 minutes.

Given a 4x4 grid: how many squares can you find?  Can you generalize this result?

Sure enough this showed up in last night's work. I asked him does this look familiar: blank stares. I hinted: remember this summer in Chicago: more blank stares. Sigh, at least he reworked the problem quickly so I assume the learning has occurred subconsciously in this case.  This is why I've stopped worrying about repeating any material from last year. The kids really don't seem to retain particular problems the way I assume they would.

Tuesday, November 24, 2015

A review of dreambox

This week is Thanksgiving break and there are no Math Club meetings. Instead I thought I'd write down my thoughts about, an online math app. I've experimented a little bit with computer programs with my sons in the past with everything from Splash Math, to Khan Academy. My favorite one so far by far is which doesn't attempt to be comprehensive but actually manages to be both entertaining and instructive.  However, two weeks ago my younger son's school started a trial license of dreambox and I've been watching him interact with the it since then.


This program attempts to walk a student through a complete curriculum and covers the years K-8.   The student gets an avatar and explores among four different themes: pixies, pirates, animal friends and dinosaurs. Each themed area has a a series of quests where usually 6 different items need to be found. By practicing different types of problems the student finishes these stories gaining additional bonus points and character cards along the way. The actual problem sets (at least in second grade) seem fairly mainstream and concentrate on numeracy via number bonds, place values,  regrouping etc.  After each set is finished a more difficult one will eventually be offered until the curriculum goal is mastered. This typically take 4-5  sets of approximately 6 problems interspersed among various curriculum strands.

Overall this is the best curriculum replacement I've seen so far. Its compelling. My son voluntarily asks to play with it and it gets him to practice a fair amount very painlessly. The story framework that they scaffold the exercises with keeps him interested and is a lot better than some of the previous apps I've mentioned . There's also a fairly high quality set of different exercises to work on the various skills.  On the downside, while it usually has enough repetition for mastery, its not particularly adaptive. Once going you work through the sequence of problems regardless of whether you could handle them at a different pace or just skip the easy ones. (I haven't confirmed whether it repeats if too many problems are missed but I suspect that's the case.)  I also find the narrative structure to be a bit repetitive but that doesn't seem to be an issue for my son so take that one with a pinch of salt. It also doesn't require much supervision. For instance, when doing Khan Academy I heavily edited the flow and interjected instruction etc.  That isn't really necessary here although I still like to watch him work on the problems to get a sense of how he's thinking and whether there are any issues to address.  If he keeps up with maybe I'll update in a few months with how the game progresses and whether some of the other topic beyond addition/subtraction are also worthwhile.

Part 2:

Tuesday, November 17, 2015

11/17 First Olympiad

Today was a really fun day in the math club. We started with me handing out whoppers to everyone as they arrived since the kids had reached our next goal for completed homework problems. I continued with my idea from last week and picked one of the kids who had turned in his work and not demonstrated in front of the group yet to show the solution. Interestingly this week there were at least 4 different solutions running around in the room (which I told the kids was a record for us).
The problem was from AMC8 and I original saw it referenced on

Points ABCD are the midpoints of a square and form a smaller square. If the larger square has area 60, what does the smaller one have?
This one can be determined visually if you notice the four squares the larger one is divided into and the 8 triangle that these are further subdivided by.  That means half the triangles are inside the inner square and half are out so the area is 1/2 of the whole. A lot of kids found the solution in variants on this observation. A few computed the side length of the larger square and then the area of the triangles which is a bit more work.  Better yet when I asked what everyone had observed when trying this out with other quadrilaterals many of the kids had experimented and found the basics of Varignon's Theorem i.e. the inner shape is always a parallelogram and has half the area. I definitely want to do more investigations during further club meetings along these lines. has a good proof that could be developed in class.

We then jumped into our first MOEMS Olympiad for the year. In my tryout beforehand I noticed a few things. First, by coincidence it has a simple decoding problem in the middle which meshed really well with the examples we had done that last few weeks. Secondly the final problem, arrange the 8 integers 1..8 around a grid (no center) with some specified sums for the rows and columns and then find the sum of the numbers in the four corners looked like it would take the most time.  It was solvable algebraically but as predicted everyone who was successful ended up using a guess and check strategy. Many more kids this year took the full ~30 minutes to do the problems than last year which is probably a function of age i.e. fourth graders vs. fifth graders. Incidentally, the cool part of having some returning students is the ability to see how much they've grown in a year via their performance on these contests.

Secondly on review I still have kids who given a simple sum will plow through it from left to right despite me trying to stress looking for shortcuts via rearranging the numbers. As an aside, with problems like this I'm taking more advantage of class surveys. So I'll start by just asking for a show of hands of kids who added the numbers in order and then ask for anyone who did something differently to bring out the alternative strategies a little more efficiently.  I should definitely stress this point more and do some examples of regrouping in club.

My favorite part of the whole experience by far was the wrap up when I had the kids show their solutions to each other. All our work on listening seems to be paying off. Despite there being 5 problems rather than our normal one, everyone was rapt during the demonstrations and I had lots of volunteers wanting to show how they had found the answers.

Also new this year, I planned for kids finishing early and brought another exercise from "This is not a Maths Book". This time we did the page on Pascal's Triangle which involved building your own triangle and colouring in evens and odds to search for patterns. This proved really fascinating for the 6 kids who were done early and kept them busy (and non-distracting) to the others who were still finishing.  Again this year, I plan to do a full unit around Pascal's Triangle some time in the winter. There's too much good stuff here to not touch again.

Finally, I'm trying a problem from a new source for the take home problem. I'm hoping again that at least some of the kids will notice that 1001 is a common factor in all numbers formed this way. We'll see what they come up with when everyone comes back from Thanksgiving break.

Friday, November 13, 2015

That article in Atlantic about explaining your work

The Atlantic recently published an interesting article about requiring students to explain their work

"If a student can consistently solve a variety of problems, that student likely has some level of mathematical understanding. Teachers can assess this more deeply by looking at the solutions and any work shown and asking some spontaneous follow-up questions tailored to the child’s verbal abilities. But it’s far from clear whether a general requirement to accompany all solutions with verbal explanations provides a more accurate measurement of mathematical understanding than the answers themselves and any work the student has produced along the way.  At best, verbal explanations beyond “showing the work” may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone."

This has produced quite a bit of chatter across the blogosphere. I initially had a strong instinctive reaction.  Personally, I find these written explanations to be tedious and not very useful so its not a technique I would  use with others either. However, I acknowledge that my own feelings are often in the minority and more interestingly I tend to be more progressive in practice when dealing with a room of the kids than I am in private when imagining these type of pedagogy questions.

Thinking about this some more, I think one can disregard the straw man position where these type questions are overused, or not handled effectively. One can find bad teaching any where that misapplies a good tool. So let's imagine, a teacher is only asking these questions occasionally to truly confirm understanding, is generous in reading the answers, provides appropriate feedback and adjusts based on the information discovered and doesn't mandate a particular format.  Reluctantly I think there is probably a place for such a technique. The key is to use it carefully. I also think that a lot of the same effects can often be had just through regular classroom interactions where good questions are asked back and forth and where you listen carefully to what the students are saying. Likewise, where ever possible I think understanding should be demonstrated through problems. Usually these discussion are framed around fairly simple problems that  are mostly straight procedure like:

"A coat has been reduced by 20 percent to sell for $160. What was the original price of the coat?”

I'd much rather see conceptual understanding of percentages pushed through a series of more difficult and interesting problems prior to requiring any written queries.  Likewise, any written explanations ought to be directed first towards proof building skills.  Its a lot more valuable to have students explain how they made a break through, than to parrot back a procedure they've already learned (and this is subject to blind memorization just as much as anything else).

Bringing this back to my experiences with the math club because my first priority is making a fun exploratory environment,  I will never require such explanations.  But I do find myself having many conversations about why a technique works. For example, our recent dive into why the divisibility for nines works which involved me asking a series of questions while working through the basic idea i.e. what does it mean to be a multiple, if you add a multiple of 9 to another multiple of 9 why is it also a multiple of 9 etc.?

I'm always looking at the kids work and questions for evidence of what areas they don't quite understand yet and thinking about how to circle back if necessary. Having said this, of course, I have the immense luxury of not needing to hit any curriculum and if I miss something I can rely on their regular teacher's to fill the gaps.

I also have no illusions that I've mastered the technique of lecturing yet. I tend to minimize the amount of work I do on the whiteboard in front of the kids as a result. Every time I engage in a longer talk I think about it afterwards and worry about whether the kids followed along with me. Asking questions during a talk helps and having kids demo parts of the problem on the board also helps but neither is perfect. In sum, I don't have the perfect answer yet here and its something I will worry about and continue to think about how to improve.

Tuesday, November 10, 2015

11/10 Relay Madness!

After my bad activity complexity estimation last week it was a relief to properly size the planned exercises for math club this time around.  We started by going over the problem of the week:

The number 1,525,354,555,657,585,950 is, as it happens, evenly divisible by 99. Fix all the 5s digits where they are in the number, and rearrange the other digits randomly. What is the chance that a rearrangement of this form is still evenly divisible by 99?

As I posted earlier, I sent home links to divisibility rules to help everyone out. In practice, most of the kids found the solution based on this help even without covering it in club. This time while checking kids in I tried something new.  I scanned everyone's sheets and encouraged a boy who hadn't done so before to do the demo on the board.  The rules themselves seem to be fairly easy to master (compared to applying the distributive law for instance). I even sneaked in a quick demo of why the divisibility rule for 9's worked using the standard regrouping technique because the student proof went so quickly. I'm continuing with my theme of emphasizing the distributive law during such examples.  AoPS standard proof

So for the main task we practiced doing math relays using some old ones from the Knights of Pi competition: old contests   The way these works is you group the kids into teams of 4 and then they work on the four part problems one kid at time. After the first kid finishes, their result plugs into become part of the second problem and so on. This means an early error will invalidate the entire team's work. My main goal was to practice the format for those that are going to do the contest later and hopefully have it be a different and fun way to try out problems for everyone else. I originally had planned to stop after each round and correct the problems as a group but I quickly changed my mind.

1. Relays leave 3 kids without problems to work on. So I quickly changed tacks and had multiple rounds going simultaneously. That meant less down time for everyone.

2.  Secondly, I and my assistant ended up individually helping on problems rather than doing the group corrections since everyone was going at a different pace.

Overall watching the kids work I found a few areas to focus on at a future time.

1. There was a smattering of kids who were unable to easily do operation inverses .i.e if the average is 1.5 and there are 6 items, what is the total? 

2.  One of the problem sets I chose happened to use basic statistics which I don't think anyone has covered much.  In this case, this is exactly what might happen at the contest so I wanted the kids to have that experience. But having just gone over the stats chapter in AoPS pre-algebra I could also definitely plan a themed day around it.

3. Another similar triangles problem came up. I really want to do a day doing an inquiy based investigation of how this works i.e drawing triangle and measuring to watch dilation / expansion in action.

Finally based on some twitter traffic I picked  a problem based on Varignon's theorem for the problem of the week:
I've added on some open ended experiments for the kids after they finish the concrete example.

Thursday, November 5, 2015

11/3 Oops I did it again

This week I did a great job planning to ensure continuity, rolled with how things played out in the room and didn't finish nearly as much as I wanted to. *sigh*  My originally thinking was last weeks practice Olympiad ended with the STAR * 4 = RATS decoding problem which was the most difficult part for most of the kids. So I thought we'd warm up with another example and keep working on the strategies to use. So I found


I gave this a test before I started and thought it was a bit difficult but I could scaffold the room through the problem and still move onto a divisibility talk with some problems. In reality almost the whole math club finished the problem but it took almost the entire session and I'll have to save the divisibility exercises for another week.   The stickler seemed to be reasoning about where you need to carry and then testing the possible number that could fit. I think taking time to do a harder problem was probably worth it and none of the kids wanted to quit and move onto the second half early so I think were still having fun. We'll see if I bring one these back later this year if a 3rd sample will go quicker.

What was unfortunate about my planning was that I picked a problem of the week that was meant to pair with the second half:

Problem of the Week
Via Dan Finkel:
The number 1,525,354,555,657,585,950 is, as it happens, evenly divisible by 99. Fix all the 5s digits where they are in the number, and rearrange the other digits randomly. What is the chance that a rearrangement of this form is still evenly divisible by 99?

We didn't go over testing for divisibility by 11 and  I'm unsure  if the kids all know it or not. I had preprinted the problem so I couldn't change anything on the fly. So as a result I plan to send out a mail with some hint links to: and

and hopefully this will be enough for some of the kids to work through this problem which looks much more complicated than it actually really is.

I was pleasantly surprised that 3 or 4 kids solved last weeks problems. When going over it in the room rather than deriving an algebraic equation based on similar triangle and the area formula I opted for a visual proof where you divide the triangle into 16 congruent sub-triangles.  My hope was that this would be more accessible. If I repeat it again I think I'll take more time and pre-print the triangle so the congruence is very obvious and perhaps even have the kids measure the triangles with a ruler to confirm they all are the same.