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:

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.

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.

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.

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:

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:

This post is linked up to:

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.

-- Joan Freeman,

*Gifted Children*, page 272Gifted 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'.

-- Joan Freeman,

*Gifted Children*, page 273I 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.

-- Joan Freeman,

*Gifted Children*, page 273Besides 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:

This post is linked up to:

- Hip Homeschool Hop - 4/1/14
- Entertaining and Educational - West Africa Study
- Collage Friday: Como Se Dice? Dad's in Honduras
- Weekly Wrap-Up: The One In Which Spring Sprang... or Is It Sprung?
- The Homeschool Mother's Journal (4/5/14)

Your post is another reminder to be patient with my son. He sounds very similar to Tiger and I often struggle finding ways to encourage his discovery approach to math. We recently purchased the Murderous Math books, but haven't started using them much yet. Number Devil was another one I was looking into. Do you like it?

ReplyDeleteI agree in the fact that I think my son would struggle greatly with the mainstream approach to math and education in general. He needs to understand the application behind the math. Without application it's just a set of procedures that are difficult for him to remember.

Although my daughter learns well with a more traditional approach. I make sure she is learning math application as well. Once she completes a section of traditional math, we switch gears into more literature based math so she gains understanding of the procedures she has learned.

Thank you for this post.

You're welcome, Julie. I glanced through "The Number Devil" and it looks like a rather good story-based maths book. Tiger certainly enjoys it enough to have read it several times.

DeleteChildren learn in many different ways. It seems to me that the best thing to do is to find out what works for each child and support them individually. That's something that no school can do, given the sheer number of students in each class. :-)

Hwee,

ReplyDeleteI really enjoyed reading about Tiger's approach to maths. I am very impressed by him wanting to direct his own maths learning, and how he takes the time to think about things. He certainly knows how he wants to learn. It sounds like he is learning maths for its own sake rather than because it is expected.

Gemma-Rose (10) is very resistant to any structured maths learning. I must admit at times I just want her to sit down and learn all the concepts the usual way. I think it would be so quick and much easier if she just applied herself and memorised all the work. But she digs her heels in and won't do this.

Knowing the child and his/ her needs...yes! I am using a bits and pieces approach, finding interesting ways of presenting maths to the girls. We had a great week looking at Pi and ended up making some Pi artwork using both Windows and Gimp. Another week we explored Fibonacci numbers and music. Gemma-Rose might still not have her times tables memorised perfectly but that doesn't mean we can't enjoy 'the big picture' sometimes. With the basics, I will just have to be patient!

It's much easier (on the parent) to have a compliant child, isn't it? :-) Like you, I find myself sometimes wishing that Tiger would just get through the texts and workbooks "normally" so that we can move on to the next level. I find myself constantly conflicted between providing an interesting avenue for Tiger to learn and expecting him to know certain basic concepts already. It's a difficult balance to maintain, and one that forces me to be much more creative than I ever thought possible. :-)

DeleteThank you for sharing your experience, Sue. It is both comforting and reassuring to know that I am not the only homeschooling parent who has to find creative ways to guide her child. :-)

This is my favourite EVER post of yours, especially because you are pictured in it!! I love seeing how well this is working in your lives. We are still failing miserably - or at least that seems to be how it feels to me. I wish we could do a similar thing to you, but with five children all wanting to learn differently the reality of the girls learning this way is that it feels they are falling more and more behind. To be honest, if I could afford it I would bring in a maths tutor because I know I am not doing a good enough job with their maths education and worse still I don't know what to do about it.

ReplyDeleteThank you for such an encouraging post.

Thank you, Claire. I'm not particularly photogenic so this is probably the one and only photo of me to be up here. The reason it even gets to be put up here is because it's the only photo we have of the maths lecture. :-)

DeleteI wrote this post to encourage myself, really, and to put things in perspective, because Tiger's maths learning is becoming increasingly "creative" and I'm beginning to be slightly troubled by how differently it looks from what most people are doing. I'm glad to know that the post resonates with quite a few others as well. It's good to know that we're not alone in our struggles.

As for hiring a maths tutor, I honestly don't think it's necessary. You're more than capable in teaching the girls what they need to learn. I don't think hiring the tutor will give you that much more in terms of expertise. What the tutor can do is to give undivided, one-to-one attention for two hours straight, only because he's paid to do it and therefore during that time he doesn't have any other commitment. What you lack is a long stretch of undisrupted time for one-to-ones with the girls, not the expertise to teach them maths. If I were in your situation, I will try to carve out a 30-minute block every day to spend with the girls for focused maths instructions. You'll be amazed at how much ground you can cover with a consistent, undisturbed time of 30 minutes every day.

I agree with everyone - wonderful post. (And I would have got here to comment sooner but it's J's birthday and this is my first chance to sit down since I read it, what with all the preps and celebrations.) I too was excited to see you in the photo :-)

ReplyDeleteYou do such a great service to the home ed community by sharing what works for Tiger. Every child is different I have so much respect for you honouring his way of working. (And "working" definitely is the operative word, on every sense, from what I can see.) So many people get caught up in how learning maths (for example) "should" look, but the truth is there are as many ways it "should" look as there are unique children in this world.

My J(9!) is also extraordinarily independent, but he hasn't yet found Tiger's diligence. I live in hope! Meanwhile I am very appreciative that C(10) still likes to work alongside me - I would miss my maths if they were both as independent as J. You are lucky you get the benefit of Tiger's lectures :-) (btw, what a great life skill)

I'm very behind with my blogging but I'm hoping to do a piece about how maths has been working for us recently, too.

Happy belated birthday to J! Thank you for your very kind words, Lucinda. :-)

DeleteMaths seems to present the trickiest issue in education these days, home ed or otherwise. There are so many different opinions about how to do it *right* so as to ensure that our children are perceived to be 'competitive' in the league tables or in the job market. I am not totally convinced about any of the proposed solutions that are being discussed in various places (online and otherwise), so this post is written also to help me sort out my own thoughts about how maths learning has been happening in our home recently. The process may evolve in the future but I find the current process to be quite fascinating, from an observer's point of view. :-)

I look forward to reading about what's been happening in your homeschool, especially maths.

I really enjoyed sitting down and reading this this morning. I find that my own Firecracker has difficulty working with standard math, and I really needed some "math encouragement." Now to go look for a few books to strew onto the bookshelf for him :-)

ReplyDeleteGood luck with it! :-) Sometimes the child may return to standard maths after a short break, but even if he doesn't, we can still teach them maths as long as we are flexible about the format that the learning takes.

Delete