Miya's maths: Khan Academy 7th Grade

13 months ago I wrote that Miya had finished Khan Academy 6th Grade maths.  Today she finished 7th Grade.

She'll start 8th Grade maths tomorrow.  (If she was in the US she'd be starting as a young 8th Grader in September.)

We're almost certain that Miya will finish homeschooling and start secondary school (Year 9) at the beginning of next year.  We're not expecting that she'll finish 8th Grade maths by that time, but our attitude is that the further she gets through it the easier it will be for her as she adapts to school life.

Mulan's NCEA results

The NCEA school exam results came through a few days ago.  We looked eagerly at Mulan's numbers.

And well done her!

As always, as with any on-the-day performance, be it athletics, music, dance or academics, there were some bits where Mulan performed smoothly, while other bits were a little under what we know she can sometimes do.  It's a snapshot in time, not an overall judgement.  Important, interesting and useful, but only in its context.

Having said that, overall it was a very satisfying result and something to be proud of.  I think it appropriately reflects what Mulan put into it -- her time, effort, seriousness, diligence, and, most importantly, investment-balance with the various other life areas in which Mulan is involved.  She put time and effort into her academic studies, but it didn't take over from also equally putting time and effort into her dance, music and athletics, to also achieve well in those.

So, Mulan passed NCEA 1 with excellence endorsement.

She got excellence endorsements in Maths, Science and Music, while she got merit endorsements in English, Accounting and PE.

She passed all 121 credits she sat, with 79 at excellence level and 42 at merit level.

Along with doing NCEA level 1, in her extension English class she sat a level 2 unit.  We were delighted that she passed this with merit, as in some of her earlier practices she hadn't done quite so well.

On the other hand, she didn't do quite so well in one of her Accounting units, getting merit rather than excellence.  Even though the rest of her Accounting was all excellence, with the bad luck of the points calculation system this dropped her down to merit endorsement overall in Accounting.  A little sad after she did so well in her school Accounting results, but that's the idiosyncratic nature of on-the-day performances.

Well done Mulan.  Super proud of her!

Miya's maths: Khan Academy 6th Grade

I last wrote about Miya's maths nine months ago.

Since then Miya has been steadily working on Khan Academy US 6th Grade maths.

Miya has now completed 6th Grade, and yesterday started on 7th Grade (if she was living in the US she'd be starting as a very young 7th Grade in half a year).

One of the sticking points for Miya is her accuracy.  Khan is a mastery-style course, and so it requires students to get everything correct before moving on.  That's great in many ways, as it means that students have a solid foundation before attempting the next level.  Mastery-style reasons that if a student is still not quite sure about one thing, then it is going to get even less clear if they move on before getting that sorted.

However, a downside of mastery-style is that some students may make occasional mistakes because they jump into it and try to do it speedily all at once and forget to carefully think through each step.  They mostly know it, and could probably get it all correct with the occasional suggestion.  But it may be hard for them to get 100% entirely independently, and so it will be hard for them to move on to the next level.

I think Miya has been a bit like this, and this is why she has taken longer to get through 6th Grade (she started 6th Grade one year and 10 months ago).  Miya also strongly prefers to do it entirely on her own, and doesn't like me to teach/correct her.  She doesn't even like me giving her hints/pointers to remind her how to start a problem.

We'll see how 7th Grade progresses, but if it is as slow as 6th Grade we may need to change the way we are doing things.

Our tentative thought for now is that Miya will start school (Westlake Girls) at the beginning of Year 9 in two years.  It would be great if Miya could complete (or near complete) US 8th Grade before she starts.

Maths update

I last wrote about the girls' maths work back in May of 2020.  It's time for another update.

Both girls continue to do a little maths work every day, year round.

Miya:

As of today, Miya has completed 63% of Khan Academy 6th Grade.

If she was in the US she'd be starting 6th Grade in September (at the very youngest end).

Mulan:

Mulan has completed Khan Academy 8th Grade and Algebra 1, and is 41% through Geometry.

I think I am right to say that Algebra 1 is taught in 9th Grade and Geometry is taught in 10th Grade.

If Mulan was in the US she'd be starting 9th Grade in September.

However, Mulan has now officially started school in New Zealand, entering Year 10 at Westlake Girls High School.  (I'll probably write more on this later.)  After a few days in a normal class, Mulan has observed that they are working on stuff that she covered a couple of years ago.  We're now talking with Westlake about options for keeping her challenged in maths.  It seems that either she'll transfer to an extension class or her current teacher can give her other work.

Census at school

Yesterday we completed the Census at School.

This is an online survey that all Year 3-13 school children may complete.

It's a non-profit educationally motivated project hosted by the Stats Department at Auckland Uni and with the help of the Ministry of Education and Stats NZ.

In other words, it's a legit survey for kids to learn about stats, not a commercially-driven data-gathering exercise.

The questions were enjoyable and mostly easy for Mulan and Miya to answer.  The games section was a great idea for a bit of fun competitiveness.

A small challenge for us was that the question-designers mustn't have considered homeschoolers -- a few questions only made sense if the kids physically go to school and then have an "after school".  Those don't apply when schooling is a lifestyle, not an event.

After completing the questions, the girls spent some time on the website going through the questions from previous years and examining the data.  They've now got the Live Dashboard open to look at every once in a while.

Khan Academy: Miya 5th Grade

Almost a year ago I wrote that Miya had finished Khan Academy 4th Grade maths.

Several days ago Miya completed 5th Grade, and has now started 6th Grade (as of today 5% complete).

Over the past year Khan Academy added a lot more questions at each grade level, and so Miya also spent some time re-completing the earlier grades that had more questioned added.

If Miya was in the US she'd be at the very youngest end of 4th Grade, so things are continuing to go smoothly and steadily at a year or so above her age level.

Well done Miya!

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At the same time I wrote that Mulan was starting to work on US High School Mathematics I.

At that time Mulan had been zooming ahead fast with her Khan Academy progress, and things were starting to get a little stressful and not so fun for her.  So it was great that Khan Academy added lots more questions, and Mulan has spent the past year relaxedly re-completing the earlier grades.

As of today Mulan has 98% completed 8th Grade and 78% completed High School Mathematics I.

If Mulan was in the US she'd be in 7th Grade, so like Miya she's doing work a year or two ahead of her age.

Well done Mulan, too!

Book review: Humble Pi

For the past few years we've been watching Matt Parker on his YouTube channel StandupMaths.

Parker is an ex-high school maths teacher who now works as a maths communicator.  He's perfect for the job.  Parker has a unique way of presenting maths that is laugh out loud funny while at the same time being inspiring and educational.

Humble Pi: A Comedy of Maths Errors is Parker's second book, and was just published this year.  After a wait of several weeks I finally had my turn with a copy from our local library.  It was definitely worth the wait.  I highly recommend it.

In Humble Pi, Parker tells lots of real life stories about how maths errors have caused problems in the real world.  There are plenty of engineering problems of wobbly buildings and bridges, or crashing planes and spaceships.  There are wrong street signs and advertising posters.  There are computer programming mistakes and ancient Sumerian tablet mistakes.  There are mistakes that caused billions of dollars in losses.  And there are mistakes which are just really annoying.

Along with the fun storytelling, Parker does a brilliant job of educating us with plenty of thought-provoking maths.

But I think the most important thing that Parker does brilliantly is to emphasise that maths is about making mistakes and learning from those mistakes.  It is about putting the effort in.  To quote Parker:

Mathematicians aren't people who find maths easy; they're people who enjoy how hard it is.

Finally, I love how Parker numbered the pages in the book.  The first page is 314; the last page is 1.  Awesome!  It's with a maths book that for the first time I have to do less maths as I read.  With this book I didn't have to do subtraction in my head to work out how many pages I had left to read.  After all, that's the main purpose of page numbering, isn't it?

Khan Academy: Miya 4th Grade

Today, Miya finished Khan Academy 4th Grade maths.  Congratulations, Miya!

She also did a bit in 5th Grade, and will continue on with that each day (currently 15% complete).

(Mulan is now 77% through US High School Mathematics I.  We've decided to slow down a bit with introducing new skills, and are doing more review work.)

I last wrote about the girls' maths a couple of months ago.

Maths: Khan Academy high school

A few days ago I mentioned that Mulan was almost finished Khan Academy 8th Grade maths.

Well, she finished it today, and we had a look at what comes next.

It's all very easy.  Next up is High school Mathematics I.  Mulan is already 47% through it, as there is some overlap between grades.

So, life will just continue on as normal, doing a little every day.

Maths: Khan Academy

Usually I label these posts "Congratulations Miya" or "Congratulations Mulan", but they are happening too often these days.

Miya has just finished Khan Academy 3rd Grade maths.

A sticking point for her has been learning her times tables facts, but she has got them for the most part now, and has now been able to complete 3rd Grade.  It is still a work in progress to get them all fast and automatic, but the ones she doesn't know automatically she can work out in her head fairly quickly.

Miya is also 68% of her way through 4th Grade.  Our gentle, no-stress goal is to finish 4th Grade in the next few months.  If she was in the US she'd currently be in 3rd Grade, so our no-stress goal would mean that she'd be over one year ahead of her age.

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Meanwhile, Mulan is 98% of her way through Khan Academy 8th Grade maths.

When she finishes this in the next few days I'll need to figure out the system Khan Academy uses for its "High school" level maths.  After 8th Grade it uses a different system that doesn't label the grades year-by-year.

If Mulan was in the US she'd be in 6th Grade, so she is over two years ahead of her age.

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As I've commented before, in US/China Mulan is three school years older than Miya.  However, in New Zealand Mulan is four school years older than Miya.  The school year system is fairly arbitrary, so how they compare with their school year is at best merely a useful rough guide to check that we are doing the right amount of work here at home.

Practically, what is more important is that the children are learning to their own individual potential.  These achievements are a result of spending anywhere between 3 and 30 minutes a day, every day, on maths problems that are typically challenging but do-able for them.

(I last commented on the girls' maths in January.)

What happens when maths goes wrong?

A few nights ago I was relaxing in front of the screen, watching Matt Parker give a talk at the Royal Institution.

You know how it is.  It's about 10 o'clock at night.  The 8-year-old is in bed asleep.  The 11-year-old is almost in bed, but dragging things out quietly so you don't notice.  You just want to relax with a bit of maths.

Mulan quietly sat down next to me and started watching alongside me.  We ended up watching about half of the hour-long talk together.

But eventually it got too late and we had to stop it, unfinished.  Mulan informed me that I wasn't allowed to finish watching it alone, and I had to wait until the next day to watch the remainder with her.

What could I say?  Deprive her of learning some maths?

So, the next day we watched the rest of it together.  Miya joined us too.

If you want an awesome piece of maths advice, skip to 3 min 50 sec in the above linked video.

Matt Parker is always really inspiring with his maths videos.  His standup maths YouTube channel is excellent.  I've just requested from the library his two books, Humble Pi and Things to make and do in the fourth dimension.

Congratulations Mulan (again)

Yesterday, Mulan finished her Khan Academy 7th Grade maths.  Again.

She completed it last April, but then Khan Academy added lots of new stuff in many of the grades.  So, Mulan went back and for the past few months she has been doing the earlier grades.

She will start back again today with 8th Grade, which she is currently 49% of her way through.  Our aim is for her to work through this steadily, but do more problems to help with speeding up her problem-solving time.

A very relaxed and flexible goal is for her to finish 8th Grade by the middle of the year.  In the imaginary world in which she lives and schools in the US, she would then be two years ahead of her age.

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Meanwhile, Miya is continuing to work through both 3rd Grade (88% complete) and 4th Grade (67% complete).

Her main goal at the moment is to finish learning her times tables.  She now knows most of them, but occasionally she needs to calculate them in her head.  So, it is about speeding this up and making it automatic.

I think this would enable Miya to speed though the remainder of both 3rd and 4th Grade.  She might then start 5th Grade before mid year, which, in that imaginary world, would make her between one and two years ahead of her age.

(One of the strange things about the schooling system(s) is that here in New Zealand there are four school years separating Mulan and Miya, while in China (and I think the US) there are only three school years separating them.)

Term 4

We are now into the first week back in Term 4.

(Yup, even though we homeschool, we still have to follow the school term system.)

For most of the two-week holiday break we were over in Sydney, Australia.  It was my first time there in 25 years, while Mama and the girls had never been there.  I wrote a travel diary while we were away.  I'll probably post it here in the near future.

This term, our big focus (once again) is ballet.  The girls' ballet school will be doing a performance (Alice in Wonderland) in December, and they are learning dances for that.  Mulan is having 8 1/4 hours of lessons each week (four days a week), while Miya is having 4 1/2 hours of lessons (three days a week).  They often do ballet at home, too, including teaching the cousins.

As part of getting the girls more independent, last term we were often catching the bus to ballet rather than taking the car.  At first I went with them in the bus, but they are starting to go on their own now, too.  We bought a basic mobile phone for them to take while out and about.

The girls are still doing their music, though lessons don't start until next week.  Mulan is still learning flute, cello, recorder and ukulele.  Miya is still learning violin, recorder and ukulele.  They practice daily, alternating instruments.

Both Mulan and Miya still love the water, and are continuing with their swimming lessons.  In addition to lessons, we go to the public pools once a week for a fun/training session.

The girls are both still doing gymnastics once a week.

And finally, we have (sort-of) started athletics.  I say sort-of, because it clashes with ballet.  Both girls have chosen to prioritise their ballet, but they squeeze in a bit of athletics when they can.  Miya can attend the last half of Wednesday clubnight as well as some of the Monday coaching session.  Mulan may attend the occasional athletics evening when ballet is not on.

Oh, and our "schooling" stuff?  Yeah, that gets done quickly and easily, squeezed into our day:

For maths, both girls are still doing Khan Academy daily, and both are still about two years ahead of their ages.

Both girls are reading heaps -- both our own books and borrowed books from the library.

I continue to read aloud daily to the girls.  At the moment we are (unsurprisingly) reading Alice in Wonderland.

The girls still practice their typing daily.  Mulan gets up to 80 words per minute while Miya gets up to 60 words per minute.

We watch a YouTube video each day while eating lunch.  This varies depending on what jumps out at us.  Sometimes it is from Crash Course.  Sometimes it is ballet.

Both girls are still having Chinese lessons daily with Mama.  They have now joined Mama's Chinese teaching business as paid assistant teachers.  They work with the preschool students, doing Chinese language games and activities.

Our other subjects are typically done more "unschooling" style.

Congratulations Mulan

Today, Mulan finished her US 7th Grade maths at Khan Academy.

She is now starting 8th Grade.

(Mulan finished 6th Grade 10 months ago.)

Congratulations Mulan

Yesterday, Mulan finished her US 6th Grade maths at Khan Academy.

She is now working on 7th Grade, and as of today has finished 24%.  (There is some overlap between grades, so some skills mastered in 6th Grade are counted as mastered in 7th Grade.)

Miya is also doing very well, and has completed 70% of 3rd Grade.

I last wrote about the girls' maths work in January.  Since writing that, we have slowed down a little with regards to introducing new skills.  The girls are still doing some Khan Academy every day, but on many days they are consolidating skills by repeating already "mastered" "practices", rather than continuing on with introducing new skills.

My general rule of thumb with how much new stuff to introduce to them is that it is okay if they can't do the work completely independently, but of the bits of the problem that I talk with them about, they need to be able to understand my explanations and do the steps of calculations themselves.

Maths update

Here's another update to what I wrote four months ago on Mulan and Miya's maths learning.

Both Mulan and Miya are still using Khan Academy, logging in and doing a few problems every day, seven days a week.  Some days this might just be a few minutes each, while other days it may be 30 minutes or so.  But the important thing is that they do something every day.

I sit alongside the girls 90% of the time while they are doing their work.  Even though Khan Academy is theoretically a self-teaching resource, with video lessons, etc, I am still there alongside as the primary teacher.

Both girls have now got to their right level with their learning, both completing all the way from the beginning at US Grade K.  What they are each doing is challenging for them, but still doable.  As of today, Mulan is 92% through US 6th Grade, while Miya is 52% through US 3rd Grade (making them both, as far as I can see, about two years or so above their age level).

Now that the girls are at their right level, we have slowed down with introducing and mastering new skills.  What we now do for both girls is this:

1. We look through the skills at their grade level that they have not yet started, and choose one that looks good.
2. On one day, they complete the "Practice" problems for that skill.  Sometimes they may need me to explain things a bit, possibly working through some problems together before they can do it for themselves.  Occasionally we may watch the video lesson together, to give another perspective on how to do it.
3. For the next two or three days, they then complete "Mastery" problems, one per day.  They level up that skill until it is mastered, while at the same time review other skills that have previously been mastered.
4. When that skill has been mastered, we return to step 1, choosing another skill to work on.

At the rate we are now going, my rough guess is that the girls might complete a grade every eight to ten months.

Countdown

Yesterday evening, during my relaxing time, I watched an episode of 8 out of 10 cats does countdown.

If you are not familiar with the series, it is a bunch of comedians (sometimes) doing smart things with numbers and words.  Funny and intellectually stimulating -- the perfect match.  (Occasionally there are some more mature jokes, so I wouldn't show it to the younger children without previewing it first.)

One of the regular number games they do is that they are given several random numbers and, using basic mathematical operations, have to make another given random number.  Just to make it even tougher, they only have 30 seconds to do it in.

I try to play along at home, but mostly I never get it.  With some of the panelists, their numeracy skills are both impressive and inspiring.

But yesterday, I actually got one (!!!), and happily Miya happened to be sitting with me at the time and I was able to (hopefully) impress her.

These are the numbers they (we) were given (at 26:19 into the video):

100, 8, 6, 6, 4, 2

And they had to make 162.

Can you do it?  There are several possible answers.

Within the 30 seconds, I got:

8(6+6-4)-2+100=162

I paused the video, and stepped Miya through it, getting her to do all the addition and subtraction.

Miya then wrote down the numbers and took them to Mama and Mulan to do.

Mulan sat for a few minutes and got:

2*100-4*8-6=162

Within a couple of minutes Mama got:

100*2-6*8+6+4=162

and then:

8*6+4*2+6+100=162

UPDATE 18/11/2016: A few times now, Mulan has asked to do more of these; she says they are fun.

Well done Miya

Today, Miya completed the Khan Academy K-2nd Grade maths course.

This is the result of just doing a little bit every day -- no more than maybe half an hour or so, and often less -- seven days a week.

Well done to her.

She is now starting 3rd Grade.

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.