The Chinese New Year ended on Thursday with the Lantern Festival (元宵节). Compared to last year, my effort at this year's Chinese New Year celebration is somewhat lacklustre. However, I console myself that we have at least managed to focus on having Chinese dishes this past week.
The recipes have been adjusted for our dietary requirements:
We also pulled out our copy of The Warlord's Puzzle to have a play with tangrams, a 7-piece puzzle from ancient China that Tiger had a go with nearly five years ago.
The appeal of the tangrams is its deceptive simplicity. There are only three shapes and seven pieces. However, as we have found, to solve the puzzles are not an easy task. Tiger has been working through a tangrams puzzle book last week just for the fun of it. At first, he was quite frustrated with hiimself for not being able to solve the puzzles immediately but after a few rounds, he started to be able to visualise how the different pieces can be combined together in numerous ways. From then on, he was able to solve a few puzzles on his own.
I used the study guide that accompanies the book for activity ideas and the one that caught Tiger's interest most was that of writing a cinquain, which is a 5-line patterned poetry form that bears some resemblance to the nature poems written by ancient Chinese poets.
It appears that the patterned format of cinquains has a strong appeal to Tiger, so not only did he enthusiastically write three of them, he even went as far as editing and rewriting each one a few times before he was happy with the final result. In the process of editing his various drafts, Tiger happily highlighted to me that he was employing the write-in-the-margin lesson that he has learnt from Tolkien not so long ago.
We had time to have another poetry tea with a Chinese snack (sesame cookies) in the week, so I taught Tiger to recite another Tang poem about spring: "Early Spring" (早春) by Han Yu (韩俞)
This poem describes the beautiful scene of a street in Chang'an (长安), the capital city during the Tang dynasty, in the early spring drizzle. After having the meaning of the poem explained to him, Tiger was attracted to its sound and rhythm so he learnt the poem easily and quickily, and he even sang the poem alongside the clip above! Poetry, regardless of the language it is written in, portrays a sense of linguistic beauty and mastery to its reader/listener. I am very glad to see Tiger showing much joy and pride in reciting and translating classical Chinese poetry very well. After all, having an intimate relationship with classical Chinese poetry is part of identifying with the soul of the Chinese culture.
If I remember correctly, the first mystery series that Tiger read was The Boxcar Children, when he was about six years old. We started with the first book of the series, and Tiger enjoyed the story so much that we bought him the first set (books #1-4), followed by the second set (books #5-8). Over the next few years, he has read and reread the children's adventures many times over, and I often wondered how I could bring the story more to life. Imagine my excitement when, on one of our walks, we chanced upon a disused railway carriage, much like that found in the Boxcar Children!
We went close to the carriage, but didn't explore it because although it looked disused, it didn't look abandoned so we figured it might be in the process of being restored by train enthusiasts so we had better leave it alone. Nonetheless, it's not everyday that we come up close to a disused train so stumbling upon it was quite an adventure in itself.
With his strong interest in all things mystery-related, he took it upon himself to learn all about being a detective and how to solve mysteries...
while I busied myself searching through library catalogues for mystery stories. Luckily, it seems that everybody loves a good mystery, so I didn't have to look too hard to find suitable stories for Tiger to read.
As I started paying attention to mystery-themed learning opportunities, I found that they are in abundance! Almost anything can be turned into a mystery!
Take geography for example. Tiger has had no problem working through the Great Map Mysteries where map skills were learnt through solving mysteries:
Even music-making can take on a mystery theme, as we discovered at a 'musical mystery' workshop at Wigmore Hall, where the children were first introduced to the idea of musical motifs and combinations of notes before they had to compose their own motifs in their own groups and putting the various motifs together at the end of the day into a combined composition.
The workshop was led by a few professional musicians who were assigned to each group to guide the children in creating their musical themes, in part to ensure that the final product didn't sound too "unmusical".
As we explored more into the realms of mysteries, we found ourselves getting drawn into the darker world of crimes and murders...
A small exhibition about crime fiction at the British Library
It was at The British Library that Tiger got a first real taste of hunting for clues (by following a trail that took us to various palces at the library) and using the information he collected to reduce who the real culprit was.
Encouraged by Tiger's crime-busting, mystery-solving enthusiasm, I started to look for more mystery-related materials for our normal lessons at home. In our homeschool, theme-based lessons often provide the necessary variation and "sugar coating" required to get some of the fundamentals done. Maths is one of them.
Tiger tried out the above data handling murder investigation with much keenness. When given a purpose (the "why") to solving a numeric problem, Tiger is often more motivated to learn the skills required (in this case, data analysis for Year 9) than if I were to ask him to learn a maths concept without him understanding how that concept has any real-world applicability.
While Tiger needed more help with the above, he is currently happily working on his own through a more manageable set of maths mysteries (see below).
I am aware that there are different schools of thought with regards to the necessity of themed studies. Some theorists love the idea of using themes to connect all the diverse and seemingly disjointed areas of learning, while others oppose the idea on the grounds that having the teacher organise all the learning opportunities into themes will rob children of the initiative to make the connections themselves.
While I don't go out of my way to organise themed studies for Tiger, I don't oppose to the use of themes either, especially when the learning opportunities happen quite naturally and with little effort on my part.
Can maths ever be a pleasurable and sociable pursuit for a normal young child? The answer appears to be a positive "yes".
To start with, maths is done slightly unconventionally in our house. Instead of following a specific curriculum, we tend to work on a certain topic that interest us at any given moment. As it turns out, we have been spending a fair bit of time on simple arithmetic in the past few months, as I said we would.
The independent aspect of maths learning here takes place when Tiger reads maths-related books on his own, such as the ones pictured below:
He enjoys reading them repeatedly but he doesn't want me to get involved with his reading. When I saw what can be done with these books, I offered to work through the problems with him but my offer was politely turned down. Somehow, despite (or maybe, because of) very little involvement on my part, Tiger has found his own pleasure in maths and an inner confidence to put the following notice up outside his room:
Note the happy face on his selfie.
Since he became the self-appointed 'consultant maths guy' in the house, Tiger has been pestering me for "worthy" maths problems. Sometimes I show him maths puzzles from this book, which he attempts to solve on his own before working collaboratively with me to find different solutions.
At other times, he sets himself the target of working through the problems/puzzles in the following book:
Once again, he doesn't want me to get involved in his work here. The reason he gives is that he can check the answers at the back of the book, and since the solutions are also provided, he can figure them out by himself. Fair enough. Nevertheless, Tiger is happy to share his approach: he raced through the "Easy" section within an hour and is a quarter of his way into the "Hard" section. There are some concepts in the "Hard" section that he hasn't come across yet, which he wants to figure them out by himself so he is working on one problem a week -- he reads the question on Monday, thinks it through for the rest of the week, then solves the question on Friday. When asked why he needs a whole week for one question, his reply is that he wants to "think through all possibilities and get it right" (his words).
Tiger will not have engaged himself in the above activities if he didn't derive some pleasure from solving mathematical problems, mostly on his own but sometimes collaboratively with me. Recently he has progressed to sharing his mathematical discoveries (mostly applications of numerical patterns) with Tortoise and I by giving us mini lectures in the evenings. Again, to give lectures as a way of showing what he has learnt is entirely his own idea; Tortoise and I are usually given a few hours' notice to attend a lecture in the evening. It suits us fine since we are keen to know whether and what Tiger is capable of learning on his own.
When
learning to learn is given a higher value than the learning of
information, then the educational system will have made a big step in
the direction of enabling children to be autonomous students (in the
general sense) for life. By encouraging the exploratory aspects of
learning, its excitement and inherent satisfaction can be generalized
into an approach to all life experiences; learning then is not
associated only with school and the classroom, but becomes a part of
living.
Tiger's maths lectures so far have been very interactive and engaging. He usually starts off with an example of how he applies his method to a problem, then he explains how he arrives at his method, which is followed by a very lively Q&A session where he is often challenged to defend his new-found techniques. He is thrilled when his discoveries withstand the grueling challenges thrown at them from the floor, yet he is also able to find the grace and courtesy to accept that a few of his conclusions have been wrong due to careless calculation mistakes or a logical oversight.
Gifted children can jump to conclusions by a process of brilliant mental leaps, which are wrong because of lack of information. In a reassuring 'safe' exploratory classroom, mistakes are part of the process of learning; they are not 'failures'.
I find it interesting to observe that Tiger appears to be more receptive to learning maths in this way rather than in the traditional instructional way. To my mind, the traditional way is a very quick way to learn something -- someone tells you the 'right' way to do things and you just practise until you master it. However, that's not how Tiger likes to learn, at least where maths is concerned. His preference for learning through discovery takes a lot longer, and can sometimes lead him down the wrong path where he has to back track and relearn some of what he thought he knew, but he won't have it any other way. Luckily for him, being homeschooled gives him the time, space, and support to do just that. Can you imagine how his approach to learning would be interpreted in the mainstream schools? "Unteachable" is a word that comes to mind, but that applies only if the teacher has very fixed ideas about what teaching and learning look like. A child like Tiger may appear to resist formal instructions, but that doesn't mean he lacks the ambition, enthusiasm or capability to master his chosen subject matter. The challenge, then, is for the adult to get to know the child so well as to be able to provide the right kind of support at the appropriate level.
The teacher of the gifted child is in a particularly important position - not there to demonstrate her superior knowledge or to show what a good actress she is, but to enable children to grope and leap towards understanding. It is important that the bounds of that understanding are determined by the characteristics of the pupils, not of the teacher or the school.
Besides the in-house lectures, the more obvious social aspect of Tiger's mathematical pursuit comes in the form of maths circle.
We don't attend these very often due to their infrequency, but we enjoy them immensely every time we go. The maths circle informs us of a number of things:
that mathematicians work both alone and collaboratively with others to solve problems;
that real maths problems are multi-dimensional and require an ability to connect various topics that are currently taught in segregation;
that it's fine to spend a long time on a single problem;
that sometimes the solution doesn't come even after a long time of thinking.
At the latest maths circle, Tiger spent the entire time there trying to solve one maths problem, and that was with the help of two maths undergraduate students. I watched from a safe distance (far enough not to interfere) as the three of them racked their brains for nearly two hours, trying to grapple with that problem using various hypotheses, numerous discussions, and multiple experimentations. Although he did not solve the problem by the end of that session, Tiger came away feeling exhilarated and asked to attend more of such events.
Between this and a method that produces children who "can't wait to be done with boring maths", I'd be happier to take the former even though it looks nothing like the time-honoured (although not necessarily successful), conventional way of teaching and learning maths.
The following documentary addresses directly the American public, but the issues discussed are just as applicable to the UK and indeed, to anywhere in the world that subscribes to the "standards" of mass schooling:
Ball #3: Mathematics - Addition and Area
The way that we have been learning mathematics here seems to be increasingly topic- or themed-based rather than curriculum-based, which suits us fine. We started off following a maths curriculum, which worked very well, but was starting to get a little tedious for Tiger. Once we started experimenting with learning-what-we-want-to-learn-when-we-want-to in maths, the approach feels a lot more logical and natural to us and is what we do now.
This term we are going to focus on two topics:
addition
area
What About Gaps?
It seems that many homeschoolers are uncomfortable with the idea of not following a curriculum in maths. Usually their biggest concern is gaps in their children's maths skills or knowledge. Following an established curriculum certainly gives some people -- notably adult and children who are naturally predisposed to a systematic, highly organised way of learning -- a sense of security that they have "covered everything" that a child of a certain age/level needs to know.
The way that I handle the question of gaps is very simple: I make a mental note of it when I notice it, then I find an opportune time to introduce it. The next question then is, what is an opportune time? Well, my definition of an opportune time is:
when I have gathered the necessary teaching materials/resources; or
when the topic ties in with another area that we are studying, e.g. there are many cross-curricula opportunities to be created around the topic of geometry, symmetry, tessallations, Islamic art, etc; or
immediately - we are prepared to drop everything else if it is a matter of urgency to learn a certain topic. So far, we haven't come across any situation that requires us to adress our mathematical deficiencies immediately.
My confidence in our approach comes from the realisation that:
Time is on our side. Tiger is only nine years old so he still has quite a few years to go before he has to worry about standardised tests or entrance exams. I might start panicking if we were still on the topic of addition when he is 19 years old.
Tiger is capable of buckling down to do the work when it becomes necessary.
Once I got over my initial fear of gaps (yes, I experienced much doubts and fears in the early days as well), I have come to view the learning of mathematics to be more than the acquisition of multiple sets of facts, short-cuts, and formulae. It is a language of its own. As soon as I realised that mathematics is a language, I decided that, as with any language-learning, establishing a strong foundation is very important. Just as a strong language user needs to have a contextual base to understanding and mastering the subtlties of any language, a strong maths user needs to have be able to apply mathematical skills and knowledge for them to become meaningful. Merely memorising mathematical formulae without the experience of applying them to solve real problems (as opposed to clinically designed textbook problems) is akin to memorising English idioms without knowing how to use them in the right context.
What About Practice and Mastery?
Having explained (I hope) above why I'm ok with Tiger's maths progress not following exactly in the same order or at the same pace as determined by any curriculum, you can see why I can tolerate us dwelling on any topic -- addition comes to mind immediately -- forever. However, the time that we spend on any one topic is not used on repetitive drills or learning the same set of skills at the same level over and over. An example is the latest addition game that we played. Different maths skills were used at the same time to solve that set of problems -- addition, long multiplication, counting, number sense. When we play maths games or riddles like the example that I have just referred to, both Tiger and I can easily and clearly determine whether he has any gaps in his knowledge (in which case we will work at closing those) or where his gaps are (so that we can direct our effort more effectively)
A tool that I find to be effective for practice is the Khan Academy's maths section. I've mentioned before our sporadic use of the website whenever we feel Tiger needs to have more focused practice on any topic. Most recently, we discovered a new feature on the website called Mastery Challenge, which is very similar to a series of short composite tests on various maths topics. Tiger doesn't mind working on a few rounds of these each day.
To me, it works in the same way as what we have been doing at home to identify gaps and areas of improvement. For example, one of the recent challenges shows that Tiger needs more work on the long division (which we had touched upon briefly in November but not formally worked on yet). With this information, Tiger watched the two-digit division video:
and worked through the example alongside the video, before attempting several rounds of two-digit divisions on the site. Working on these exercises enables Tiger to solve a recent maths challenge, but it is not the way in which I would like him to learn long division (even though he certainly can do them mechanically now), so we will spend some time later in the year to learn division in a more visual, hands-on way.
I don't claim that our method of learning maths is superior to any other method, but it is one that suits Tiger's need for variety, applicability, and minimal repetition.
Now that we are very familiar with straightforward addition, let's try our hand at finding the sum of all the numbers in a range, say from 26 to 765, without using a spreadsheet or calculator?
In order to learn the technique to work out sums of the above nature, let's start with the basics. Q1: Find the sum of all whole numbers from 1 to 10.
The answer (55) can be easily worked out mentally or by writing the sums out on a piece of paper, but we are trying to learn a logic that will enable us to work out sums of any range.
To do this, I used a piece of cotton string and attached each individual number card 1 to 10 with paper clips.
I then asked Tiger what he would do, besides using mental calculation, to find the sum of all the numbers from 1 to 10. He removed all the numbers from the number line, so that all ten numbers were accounted for, and paired them up into equations with the same sum total of 11 each: 1 + 10 = 11 2 + 9 = 11 3 + 8 = 11 4 + 7 = 11 5 + 6 = 11
Once Tiger set the five equations out on the floor, it was obvious that the sum total would be 5 x 11 = 55, which tallied with his mental calculation.
I then reset the number line with numbers 1 to 10 again, and asked Tiger whether he could solve the problem in a slightly different way, i.e. using a total that is not 11. When Tiger couldn't think of any other way after a while, I suggested changing the total to 10. This way, the logic behind the solution is the same as before, except that there are a few extra steps:
When we set the total to be ten, the number 10 is removed first from the
number line. That is followed by the pairing up of numbers that make
up ten. When the pairing is completed, you will find that the number 5 is left unpaired.
The pattern will look like this: 10 1 + 9 = 10 2 + 8 = 10 3 + 7 = 10 4 + 6 = 10 5
Therefore, the final summation to find the answer is: (5 x 10) + 5 = 55
Between the two methods (sum to 11 versus sum to 10), I personally lean towards the one with sum to 10, but Tiger prefers what he considers to be the more straightforward method of summation to 11. As you've seen from the above, both are equally valid so the choice depends on what makes the most sense to the child.
Q2: Find the sum of all whole numbers from 12 to 30.
We tried another exercise to check whether Tiger has understood the logic/technique to apply it to another question.
Tiger had a choice of using whatever method that worked best for him. He chose to do the first method, i.e. the one where the first number (12) was added to the last number (30), and so on.
By doing so, he found that he had the 'middle number' (21) left unpaired, so he put it at the top of his layout so to ensure that he did not forget to add it to the final answer later on.
Therefore, the final summation to find the answer is: (9 x 42) + 21 = 399
At this point, I introduced another concept to Tiger: how to find the
number of items (in this case, the number of pairing) from one number
to another.
By counting or just looking quickly at the layout above, we determined
that there were nine sets of pairings that summed up to 42. Counting
is
fine until the range gets too large. Therefore, I asked Tiger how he
would determine the number of pairings without counting. When he started to experience some difficulty, I asked him to
think about how he would determine the number of pages he has read in a
book, say from page 1 to 10. After first he said the asnwer would be
nine pages since 10 - 1 = 9, but when he quickly realised that the correct number was ten. That was a bit puzzling. I then asked him how many pages he would have read had he read from pages 2 to 11. Again, answer is ten, but if he had applied the simple subtraction method of 11 - 2, he would get only nine.
By now, Tiger saw the logic behind this: that the number of items in a range is the simple subtraction of the last and first numbers followed by adding one more to it. I helped him with the next step by putting his explanation into a simple formula: the number of items in a range = last number - first number + 1
We then verified our formula against the two examples above by first applying the numbers to the formula followed by verifying our answers with visual counting.
Q3: Find the sum of all whole numbers from 27 to 68.
Now it's time to see whether Tiger really knows what to do, so I set him the following exercise and left him to solve it himself:
Tiger proceeded with it methodically:
First, he laid out all the numbers in pairs so that every number on the number line was accounted for. The sum of each equation was 95. The pattern that he made was: 27 + 68 = 95 28 + 67 = 95 29 + 66 = 95 30 + 65 = 95 31 + 64 = 95 ...... 45 + 50 = 95 46 + 49 = 95 47 + 48 = 95
Using the formula he learned above, Tiger found out that there were 21 sets (48 - 68 + 1 = 21) of equations that each added to the sum of 95. Therefore, the final answer was 21 x 95 = 1995.
Once Tiger has got the hang of it, he is now able to find the sum of all whole numbers from different ranges. I gave him a few extra questions to work on:
Q4: Find the sum of all whole numbers from 5 to 100.
Q5: Find the sum of all whole numbers from 36 to 205.
Q6: Find the sum of all whole numbers from 12 to 156.
Q7: Find the sum of all whole numbers from 4 to 237.
Having said all of the above, it makes sense to acknowledge that we are dealing with elementary maths, which is essentially concrete, so perhaps not...
Watching the clip above led us to ponder more with about the following:
If one gets into higher, more abstract maths (as the following videos show), then the concrete number sense that we have just spent so much time learning are nullified, similar to how conditions of quantum physics defy many common laws of physics.
But I'm not worried about that just yet. I'm happy that Tiger is enjoying the concrete aspects of maths at the moment.
Algebra. Sounds like a really sophisticated maths idea, doesn't it? The truth is, most children already intuitively know the basics of algebra without calling it that. The concept of single-variable algebra can be very simply introduced using the following tools:
a balancing scale
several one-pence coins (or the equivalent of the smallest monetary denomination in any currency)
modeling clay of different colours
To prepare the materias, I first made a clay ball with the equivalent weight of a 1p coin.
I often like to get Tiger involved in making his own learning tools, so I got him to help me make the equivalent-weight clay balls.
That is, until it occured to me that I had to make the clay balls myself to make the game work, so I diverted his attention to a few slices of cake while I carried on with making the following clay balls:
orange clay balls ("O") - each is equivalent to the weight of one 1p coin
purple clay balls ("PB") - each is equivalent to the weight of two 1p coins
green clay balls ("GB") - each is equivalent to the weight of three 1p coins
Now we're ready to play.
The first step is to give the child an idea of balance, i.e. the one side should be the same as the other. Without Tiger seeing, I placed five 1p coins in the right-side tray and covered it with a tissue paper. Then I asked him to find the equivalent number of coins to balance the scale. When he found the scale to balance with five 1p coins, I lifted the tissue paper to reveal the answer.
Next, I repeated the step above, but this time I used two orange clay balls and four 1p coins on the covered, right-side tray. I then asked Tiger to find the number of 1p coins it would take to balance ths scale. He found that it took seven 1p coins to balance the scale.
Tiger immediately knew then that each orange clay ball represents one 1p coin. I asked him how he would represent the information on the scale as an equation. This is his equation:
7 "1p" = 5 "1p" + 2 "O" ==> 1 "O" = 1p
I did another exercise with him to make sure that he understood the concept. This time I used three orange clay balls and three 1p coins, asking him to find the equivalent number of 1p coins to balance the scale.
After he has found it to be six 1p coins, I asked him to write the equation down again:
6 "1p" = 3 "1p" + 3 "O" ==> 1 "O" = 1p
Just as Tiger was starting to think the game was too easy, I changed it slightly. For the next go I used one purple clay ball and five 1p coins (covered up using tissue paper) in one tray, and asked Tiger to find the number of 1p coins to balance the scale. He found it took seven 1p coins.
When I removed the tissue paper, Tiger was briefly stumped by the number of items on the right-side tray. He quickly figured out that the purple ball must weight more than the orange balls that were used earlier. He then went to to figure out that the purple ball must be equivalent to two 1p coins. When asked for the equation for this, he said:
7p = 5p + 1 "PB" ==> 1 "PB" = 2p
More variations on the same theme followed. Next up, I used two green clay balls and five 1p coins. It took eleven 1p coins to balance that. Using the same reasoning as the example above, Tiger worked out that each green ball is equivalent to 3p. Obviously there have been some mental calculations that involved addition, subtraction, and division to arrive at the answer but Tiger doesn't like to write out all the intermediate steps. Therefore the equation for this is shown as:
11p = 5p + 2 "GB" ==> 1 "GB" = 3p
For the last exercise, I used two purple clay balls and three 1p coins, which Tiger found to need seven 1p coins to balance.
Once again, he worked out that each purple ball is equivalent to two 1p coins, equation as below:
7p = 3p + 2 "PB" ==> 1 "PB" = 2p
This time, Tiger wanted to show his 'proof' of how he arrived at his answer, so he demonstrated by removing three 1p coins from both trays, leaving four 1p coins in the left-side tray and the two purple balls on the right-side tray. He did not write out the equation for this, so every step was done visually and orally. For those who would like to see his demonstration in mathematical form, this is how it would look:
