Using a nonsense maths question

On Monday, the children and I had an excellent maths / critical thinking lesson, based around a couple of YouTube videos that had popped up on my YouTube homepage.  (You know how it is -- you watch a few videos on a topic, and then YouTube suggests more videos on the same theme.)

The first video showed up because I had been watching Sal of Khan Academy introduce Common Core (the US maths standards) in a series of videos.  YouTube then suggested lots more videos about Common Core, including this one.

In the 3-minute video, the authors introduce a maths question, which they claim is a US 4th Grade Common Core question, asking a few "random people on the street" to have a go at answering it.

Of course, it is all fun and funny, because the random people fail completely.  These stereotypically-standard middle-class USers flounder around with half-starts and guesses to a question supposedly for 9-year-olds.  Most finally give up and conclude that the question doesn't have enough information to be meaningful -- that it is a nonsense question.

While nothing is said directly by the video authors, I assume they are trying to imply that Common Core is Bad and Wrong, as it includes crazy nonsense questions like this one that no normal adult could answer.

I am not going to get into a discussion here about whether this question is a genuine one, or a genuine Common Core one, or whether Common Core is Bad and Wrong.

What I want to do here is show that the question asked in the video is actually an excellent question to ask children, and it can form the basis of an excellent lesson.  In my opinion, it is definitely not a worthless question.

If you haven't clicked on the YouTube link above, here is the question:

Juanita wants to give bags of stickers to her friends.  She wants to give the same number of stickers to each friend. She is not sure if she needs 4 bags or 6 bags of stickers.  How many stickers could she buy so there are no stickers left over?

For those of us who have recently been doing maths at around the 4th Grade level, the question seems structurally pretty familiar.  There are stickers and there are friends, and we want to share the stickers evenly between the friends and have no stickers left over.

A lot of 4th Grade maths work is about getting familiar with using the basic multiplication and division facts, and learning to divide by one-digit numbers with and without remainder.  Word problems verbally similar to this one are introduced to check that the students understand the meanings behind the equations (one of the big aims of Common Core is to ensure understanding, not just rote memorisation).

But there are a few weird things about this particular question:

1. We don't know how many friends she has.
2. We don't know the relationship of bags to stickers -- is Juanita buying stickers and then putting them into her own bags, or is she buying bags of stickers directly?
3. How does the 4 or 6 bags fit in?

It is obvious that if this was a standard 4th Grade maths question we would say, as the adults did in the video, that there is important information missing.

So, knowing that this was a weirdly-worded question, albeit structurally similar to questions that Mulan is familiar with, I introduced the question to her and Miya, then sat back and waited to see what they would do.

Just like the adults in the video, Mulan wanted to know how many friends Juanita has.  She puzzled for a while over the ambiguity of bags/stickers, as well as the 4 or 6 bag thing.

But then Mulan came up with her own solution, and one that I hadn't thought of.

Mulan said that Juanita could rip the stickers to divide them evenly among her friends.  Then they could draw in the other parts themselves.  So, it really didn't matter how many stickers she bought or how many friends she has.

This solution is typical Mulan, seeing sharing, compromises and communal DIY pen-and-paper activities as the way to go.  She would not see it as important to buy more of something to make it even between everyone, but just jointly use whatever they have got to keep things fair.

I immediately agreed.  Solution number 1.

But then I challenged Mulan further by adding a new requirement that they want to keep the stickers whole and so won't rip them.

At this point the three of us discussed it together for a few minutes.  I can't remember exactly what any of us said, but one thing I did want to emphasise to both Mulan and Miya was the importance of not being fooled by distracting information.

We observed together that the 4 or 6 bag point was not phrased as a definite requirement, but simply that Juanita was unsure of what to do.  I hoped the girls would see that people can easily get into the habit of scanning a maths question for numbers and then thinking that any number mentioned must be part of the calculations.  But it need not be; extra, unnecessary numbers may be sneakily put in to test our understanding of the question.  And we all agreed that the 4 or 6 bags thing was surely there as a distraction.

Mulan then returned to the problem of the missing information about how many friends there were.  She said that, since she couldn't rip the stickers, she would buy as many stickers as there were friends.

I then suggested that we could let x stand for the number of friends that Juanita has.  Mulan quickly caught on, and said that then she could buy multiples of x stickers.

Solution number 2.

I then said that there was one more possible solution that I could see.  When they stalled, the girls asked for a hint.  So, I directed them to the idea that there is one number of stickers in which there will always be no remainder, no matter how many friends there are.

Still no bites.  So I wondered out loud if it is always necessary to buy things.

At that, Mulan's grin grew wider with understanding, and she said that Juanita could buy 0 stickers.  0 stickers divided by any number of friends will always have 0 remainder.

Solution number 3.

Mulan liked the question so much that she wrote it out by hand to show the cousins.

Yesterday, with great delight, Mulan and Miya presented the question to two of their cousins (ages 11 and 9).  Without any adult initiation or involvement at all, the four of them discussed it together in a very systematic way.  I didn't catch all of the conversation, but I overheard Mulan clearly and accurately articulating the points that we had made the day before.

At the same time, I asked the question to Mama.  Mama immediately said that Juanita could buy all the stickers in the shop.  After all, there would then be no stickers left over in the shop.

Brilliant.  Solution number 4.

When our two discussion groups came back together, I pointed out Mama's new solution.  11-year-old cuzzie immediately said that she had said the same thing.

So, there you have it.  Excellent discussions and four possible solutions from a maths question that at first glance looked like silly nonsense.

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The second YouTube video we watched on Monday was this one from Derren Brown.

For those who don't know, Derren Brown is a UK TV personality who has been doing TV shows for several years centred around hypnotism, mind reading, etc, but from a psychological / scientific perspective.  After the impressive trickery, he points out some of the main psychological techniques.

This particular video that I showed to the girls showed up the unconscious aspects of advertising.  (I thought it fitted in with the theme of appearances and question misdirections.)  It is pretty impressive.  I highly recommend it.

After watching the video, Mulan made the connection between this and the political advertising that we are seeing around our home these days (local body elections).  This then turned into a discussion about how the politicians use advertising to try to influence us unconsciously to vote for them.

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UPDATE 29/10/2016: I see over at Math Mammoth they mention this question (Example 2 in the section titled Bad examples of "common core" or "new math").

They focus on the 4 or 6 bags of stickers sentence, linking to a conversation on The Math Forum and agreeing with Bart Goddard, who writes:

Presumably she's giving one bag to each friend (although this is a bit ambiguous, too) so she is expecting either 4 or 6 friends to show at the meeting. (I suppose that there's a set of twins who have a habit of not RSVP-ing, but crashing the party anyway.)

This means that they treat the question as a common multiple one.  Since common multiples of 4 and 6 are 12, 24, 36, ..., they think that any multiple of 12 is the correct answer.

While I can see that this is one possible interpretation of the situation -- that the reason she is not sure how many bags she needs is because she is friends with unreliable twins -- I think it is important to remember that this was not what was actually written there.

What they have done is give a possible interpretation, which includes additional made-up information that has consequently constrained their answer.  But this possible interpretation is not a necessary one (Juanita's uncertainty may have been for other reasons than what they have presumed), and so the constraints they put on the answer are also not necessary.

To put it another way, I think it is important to always read exactly what is written in a question, and not invent extra requirements through our own presumptions and interpretations.  The Math Mammoth answer is wrong because it adds requirements and constraints which were not part of the question, as written.

Maths update: Khan Academy

Every once in a while I like to write a post on what we are doing with our maths.  It is useful to keep a record of where we are at, at specific moments in time.  And who knows, it may be helpful for the one or two readers who happen upon this blog.

Both Mulan and Miya signed up at the online Khan Academy just over a year ago.  We played with it at the time, and for several months afterwards they did the odd day dotted here and there.  But just over three months ago, in mid-June, we started working on it more seriously, using it daily as our primary maths resource.

What follows is our experience of Khan Academy, used as a main maths resource for two primary-aged children.

For those who are unfamiliar with Khan Academy, it is a free online educational resource with several different subjects, but primarily focused on maths.

The maths at Khan can either be approached by subject (ie topic) or by US grade.  Each subject or grade level contains dozens of different skills (grouped together by type), with questions for each skill.  Skills are Practiced (by answering correctly a few questions specific to that skill) and then eventually Mastered (by answering correctly questions for that skill when presented together with questions from several other skills).  Progress is made by completing a combination of Practices and Masteries.  Once all skills for a particular subject or grade have been Mastered, then that subject or grade is completed.

Most skills are also linked to a short video lesson, of a few to several minutes, which teaches that skill.

Incentives are given in at least five different ways: (a) daily login acknowledgement and a count of the daily "streak", (b) points awarded for questions answered, (c) leveled mastery progress for each skill, (d) percentage progress for each section, and (e) "badges" awarded for achievements.

Mulan (9 1/2 years old):
When we first started on Khan, we headed for the subject section rather than the US grade section because, well, we are not in the US and have no special attachment to their schooling system.  The "Arithmetic" subject section looked interesting, so we tackled that first.

We got into a routine of logging in every day and working on it for maybe about 45 minutes or so.  Almost always, I sit beside Mulan as she works on the questions.  If she has any problems, I step through them with her, discussing and explaining as needed.

Occasionally (maybe once every week or two), we watch a video lesson.  Typically, we will watch a video if either (a) we are not completely sure of the question meaning or terminologies, or (b) we want a second opinion on how to solve a problem.  As we watch the video we typically stop and start as needed to discuss together what is being presented.

In our opinion, the videos are clear and teach the points well.  They are definitely a useful, though minor, supplement to our in-person teaching, and they give us confidence that we are on the right track with what we are doing.  Occasionally, however, we feel that the way they have taught things in certain videos is not the easiest way of solving some problem.  So, while we watch and understand their approach, we sometimes choose to solve problems differently.

At about two thirds of the way through the "Arithmetic" section, Mulan got to the point where the going was getting too tough.  It had reached her limit.  So, since we were still enjoying Khan, we looked around the website and decided to give the US grade sections a go.

We figured at the time that if Mulan was in the US, she would probably be coming to the end of 3rd Grade, so we turned to that section.  After having been struggling with the Arithmetic section, Mulan found the 3rd Grade extremely easy, and completed it all within a couple of weeks.

Next, we moved on to 4th Grade, and Mulan finished all of that within another couple of weeks.  With both the 3rd and 4th Grades, Mulan almost never needed my (or the videos') help on anything.  Nonetheless, I have no doubt that she was learning a lot, and it gave her a good solid skills base as well as a lot of confidence.

At the beginning of August, Mulan started Khan's 5th Grade.  Getting into this section, we felt that this was more properly her level.  It was challenging, but often doable by herself.  Anything that I explained to her was almost always immediately understood and internalised.

Mulan finished 5th Grade within about five weeks, and a couple of weeks ago she returned to the Arithmetic section.  With a few more months of maths tuition under her belt, Arithmetic was no longer beyond her abilities.  And today, Mulan completed the Arithmetic section and started on the 6th Grade section.

We are not sure how much longer Mulan will be able to continue with 6th Grade before she reaches her limit, but we will keep going with it every day and just see what happens.

Miya (6 years old):
There is not nearly as much work available on Khan for Miya's level as there is available for Mulan's level, and Miya has very quickly reached her limit.

In the US grade section, there is one section called K-2nd.  This is exactly equivalent to the "Early Math" subject section.  Miya completed 80% of this, before reaching her limit.

At the moment, and since Miya still likes to do a bit every day to keep her daily streak going, Miya logs in and re-Practices, for several minutes, a few skills that she has already Mastered.  As I see it, it is all good, as it keeps her numeracy skills up.  But it would be much better if Khan Academy extended and developed the questions for younger learners.

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Some final thoughts:

1. Since Khan is an online teaching resource, the questions asked have got to have clearly answerable questions that can be marked unambiguously by a computer.  And it is pretty hard to mark open-ended discussions.  This means that I sometimes feel that, for a maths resource, too many of the questions are equation-style and too few are problem-solving style.  This is not only the case from a balanced teaching perspective, but it is also the case from a student-interest perspective -- Mulan is becoming a little bored with too many straight equations, and is not so keen to be bothered answering them.
2. Khan Academy is clearly from the US.  It uses US spelling, US school grade levels, US non-metric measurements, US maths terminologies (such as reversing trapezoid and trapezium), US money, US education standards (Common Core), and US cultural references in questions.  While it is understandable who their market is, it is still disappointing that there is little attempt to provide a more international feel.
3. Given that Mulan completed entire grades at her age level in a couple of weeks, and both girls are reaching their limit within a few months of daily use, it is clearly the case that Khan Academy has too few questions to last as a daily long-term maths resource.

Each of these reasons in themselves would be enough to want to use other maths resources alongside Khan Academy.  All together, it makes it pretty much inevitable.  Nonetheless, Khan is an excellent partial maths resource that has been very useful to solidify important maths skills in both children, and in a way that has most definitely kept the interest alive for both of them.  A big thanks to Sal and team for creating such an awesome educational website.

Solo dad

I've been trialling being a solo parent (again) these last few days.

Don't fear, Mama and I haven't split.  But Mama is on the opposite side of the world at the moment.

Laolao's (Mama's mama) cancer has returned, so it is back to the hospital for another operation.  On Tuesday, Mama went up to Beijing to be with Laolao, while the children and I stayed back home.  Mama will probably be there for a couple of weeks until Laolao has recovered from the surgery.

So far, the children and I are surviving okay without Mama.  The main daily-life things I have noticed are:

1. Mama usually organises dinner, and it is at that time of the day that I am most tired and like to rest on my own.  Without Mama, I have to keep going and force myself through my tiredness to do dinners.  Consequently dinners are fairly quick, easy and unimaginative -- steamed greens, boiled potatoes/kumara/carrot/pumpkin, eggs/canned fish, canned soup, etc.
2. Miya still likes to come into the big bed during the night, sleeping with Mama.  Without Mama in the bed, Miya is with me, and she does like to wriggle around.  Too often I get kicked awake, and these days I am waking in the morning not nearly as refreshed as I'd like to be.
3. Mama teaches most days here at home, so without students coming and going things are a lot quieter and more relaxed.  But on the downside, we are also getting less money coming in.

And I shouldn't need to add that we are all missing Mama simply being around.

Other than that, our life is continuing much as before, with lessons, outings to classes, meals and play.

More importantly, I hear that Laolao's operation went smoothly, and she is recovering well.