NOTE - This post may include some mathematically technical material. I will attempt to give warning by bolding parts that need some extra background.
(continued from Part 3A)
Facebook (like other social networking sites) is a great technology. It enhances personal relationships, allowing you to do so much more than you could do otherwise. The ability to share photos, write on people's wall, send people messages, comment on stuff, etc. allows you to do a lot of interactions with people who are physically distant - things you could never do without the technology of the internet.
But imagine the following scenario. Two friends, Alice and Bob, are sitting in a coffee shop, having a conversation. At one point, the conversation turns to Bob's recent vacation to Bali. Alice is curious about what it's like. Bob says, "hey, I've got some pictures on my iphone, do you want to see them?" When Alice says yes, Bob says "hang on, I'll upload them into a private album on Facebook only visible to you and me. Once they're finished uploading, you'll be able to see them with your iphone and tell me what you think!" And the conversation goes silent as he sets to work selecting the photos to upload. A couple minutes later, he finishes, but the photos won't upload for another 15 minutes, so he puts the phone down, and they keep talking. When they finally finish, the conversation is far from Bali, and so there is a jerk in the flow of dialogue as Bob informs Alice that she can see the photos now.
The jerk in the flow of dialogue happens to be named Bob.
Why do I tell this story? Because I believe it might inform our thoughts about the way math is learned - in particular, how to deal with the technology of counting.
As mentioned in Part A, once you have counting down, you can learn just about everything else in mathematics from there. It truly is a powerful technology. But what I'm wondering is... is it the best way? Is it the most efficient way?
Take division - which might just be the root of all the conceptual problems people face in elementary school math (problems with memorization and following rules are a different story). How do we get to division? well, we start with ratios, install counting so we know about whole numbers, learn that adding is extended counting, learn that multiplying is repeated adding, learn that division is the opposite of multiplying, and that from division, we can get ratios.
What the function.
I don't know about you, but that seems a lot like Bob relying on Facebook to show Alice photos when he could have just held up his phone to show her.
My big question is this - are we relying too much on the technology of counting? Are we trying so hard to base everything on the discrete, when the continuous is just as, if not more, natural to us? Just because a technology works doesn't make it the most efficient way to get to every point. Maybe we should bring the ratio into school much earlier - not as a consequence of division (which is a consequence of multiplication (which is a consequence of addition (which is a consequence of counting))), but as something much more fundamental.
I remember reading a blog post once about multiplication, and how it's always taught as "repeated addition," when it's really something more basic than that. In fact, if ratios are the basis of our mathematical reasoning, then multiplication is actually more fundamental than addition is! It's actually an amazing truth that multiplication can be achieved through repeated addition - but this isn't a definition. Defining multiplication as repeated addition is like defining trees as "a source of building material" or defining typewriter as "the longest word you can type using the top row of keys on a typewriter" - both of these are in fact true, but neither count as definitions. Just look at it geometrically - adding is "sliding" the number line, while multiplication is "stretching" the number line. Is stretching equal to repeated sliding? No! In fact, looked at it this way, it's pretty incredible that repeated adding would give you multiplication.
And if multiplication could be made more basic, so could division - as it should, since division is where we get ratios, the most basic mathematical facts we know. Perhaps we should do it the other way around - from ratios, introduce multiplication and division? And introduce the "repeated addition" approach only after the concepts are solid and it's time to learn how to actually compute things? No idea what this would look like in the classroom - but I'm sure some basic ratio games and activities could be introduced very early on, not just to prepare them for "when they'll learn multiplication in second grade," but to actually teach basic multiplication and division before addition has even settled in. Teach it from the geometrical side, stretching, shrinking, perhaps even give them basic number sense activities like the one I showed in Part 2A - show the connection to discrete "five groups of three" questions later on. This will appeal to students' basic number sense and develop that, giving them a better intuition about ratio, and hence about multiplication and division, which depend on ratio; then, once it's time to actually calculate things, like adding two fractions, that intuition about how it should work will be there much more strongly.
Counting is powerful - but are we pushing it too far? Perhaps it's time to bring the ratio, the continuous, back into early early education. Don't just teach students the names of geometrical objects - teach them how to interact with them. How they grow, how they fit inside each other. How every polygon can be split into triangles. Imagine the wonder on a first grader's face when they realize that incredible fact - a wonder totally lost on middle schoolers who have learned long ago that math is all about calculating and rule-following and are taught that same fact just as something else to be memorized. And this is much more possible now, with the ipad or other touch-screen technologies, than it would have been just five years ago. The discrete is incredibly important - but we can't let it be superior to the continuous.
(It's actually an interesting irony that our K~12 education system works almost in reverse of the way our minds do. Right away, from the outset, get students to be thinking about the discrete, and as they go through education, they'll start putting the continuous back in, little by little, but always basing it on the discrete - through decimals and fractions, through the real numbers, through geometry, through functions, until the climax: calculus, the study of continuity, of infinitesimal ratio)
On a very similar note, I've heard some people wonder if we should somehow introduce the concept of complex numbers from an early age, since they are in some way "more natural" than the real numbers, and the only reason people have trouble with complex numbers is being over-familiarized with the reals. I can't agree with that as easily, since complex numbers, despite their mathematical consistency, don't appear to have the same strong connection with our natural number sense as ratios do. But I'd love to hear evidence that such a natural connection to complex numbers does actually exist.
So those are some of my thoughts on the monopoly of counting. What do you think? To those who have been thinking a bit about the subject - Maybe I'm going too far? Maybe not far enough? Maybe my theory is good, but completely impractical? Maybe there just isn't any other way to develop advanced concepts without the firm, clear properties of whole numbers? Maybe my ideas would have been good a couple decades ago, but now that we're into the digital age, we have to focus completely on the discrete?
To those who don't think about learning math all that much, and somehow just survived through these posts - what do you remember about your early learning experiences? What concepts did you struggle with the most? How did you manage to work through some of your difficulties (if you ever did)? Do you think there's some other topic that's a bigger problem?
To those who study number sense in depth - have I got any of my research wrong? Am I missing any important results? Have I made some bad conclusions? Are there other people who have written about the same topic, confirming, denying, or just discussing the ideas?
Thanks for reading. Wow, this is long. O.o
Disclaimer - throughout this series of posts I've used certain terms in ways that aren't mathematically accurate. The main one being continuous, which I roughly use to mean "measurable; not countable," in contrast to discrete. For example, I state "volume is continuous" as an obvious fact, though this would actually be quite a controversial claim using the definition of continuous as being "infinitely subdivisible." I'm sure there are other words used like this as well.
Heeheehee… this blog post was excellent and very entertaining to read. Thanks for that. Word of caution: I don’t think I actually made any points in what I’m about to say. Some thoughts are incomplete, and there are probably tons of typos.
ReplyDeleteAs someone who doesn’t think about learning math that much, I found your argument eye-opening, not completely weird, but conveniently difficult to refute because one’s own deep-rooted experiences with learning elementary mathematics (especially if successful) gets in the way. I’m not nearly as eloquent, but here are all my thoughts in response.
Like you’ve acknowledged, your theory is acceptable but may not be very practical. Or at least, it’ll be some time before its practicality can be realistically tested. It is totally mind boggling to try and picture how it would look in the classroom – after all, I (and assumingly most people I know) learned addition, subtraction, multiplication, division, in that order. Your big question is whether we rely too much on the technology of counting, and whether we should modify this technology to make learning elementary school math more efficient. The process you outline about how we get to division from ratios sure seems to suggest that we might as well go straight to division from ratios (which theoretically should be tangible for young children because it’s related to how they can tell that something is “larger”, “bigger”, or “more” than something else); hopefully then we can harness their natural number sense and make it smoother for them to learn the other operations. But that implies completely getting rid of *counting* as a preliminary step to learning mathematics. Counting could be describing a quantity of finite objects using a ratio, but it’s also about matching the amount to a natural number. Like you said, once counting has been learned, it is a powerful technology that is used to learn more advanced stuff. As much as we want to avoid too much memorization in math, how else are you going to have a consistent system for a child to communicate amounts? Unless, you want to start from fundamentals not at all about absolute amount but relative amount. Oh man, it’s so hard to imagine how a child’s natural sense of ratios can be fine-tuned without using “addition by 1”, as counting does. But I can accept that we don’t have to begin a child’s mathematical education by labelling ONE finger (or pencil, or apple, or orange…) as “one”, or “一”, or a picture of “1”, then TWO fingers as “two”, “二”, etc. It just that learning the counting technology is universally pretty straightforward, even though its beginnings seem contrived if you describe it a certain way. Still, most people get past that and don’t feel like it’s unnatural at all. The conceptual problems students have in elementary school math – especially with division – may not be caused because the continuous is ignored and then brought in bit by bit. Learning division might just be a stumbling block, like any other stumbling block. Oh wait, then again, it’s sort of like when children hit grade 6 RCM on piano, and get so frustrated with their progress that they give piano up completely. Usually it’s because of poor foundations in technique and discipline. But then again, maybe going farther in piano just isn’t for everybody… Okay fine, it’s different with math. Math is so important to society that we really should find ways of improving our education systems to make it as enjoyable and accessible as possible.
[This is when I was randomly speculating to myself about how to teach children about quantities while avoiding “counting”] Number sense is essentially the sense of “more-ness” and “less-ness” that we’d like to refine so we can do stuff like addition and multiplication, which require a fine-tuning of how we naturally think. When we’re really young, we can tell a difference between two quantities so long as one is sufficiently larger or smaller than the other. The study you reference provides evidence that as we grow as children, we become more sensitive, and two quantities can be closer and closer in size to each other and still be distinguishable to us (the ratio of amount A to amount B can be closer to 1). So say a child can tell that amount A is larger than amount B and that amount A is larger than amount C. If he can tell a difference between amounts B and C, he can eventually order A, B, and C in ascending progression. If we have amounts A, B, C, etc. which, for any pair, he can distinguish between which is “more” and which is “less, he can then be taught to order them all in ascending progression, without any introduction to “counting”. Now, we try to maximize the number of distinct amounts, at least, until the child is bored and needs a nap. In the end what we’d have is a linear ordering of amounts, ascending say, from left to right. We’d like to think that if the child can distinguish between amounts that have ratio X (X>1), he can also distinguish between amounts of *at least ratio X*. Can we work with something here? What if that ratio X was 2 and the first thing the child learns with this vague, peculiar method is that if we split a quantity D in half, then the separate parts will each be indistinguishable from the quantity immediately to the left of D, which might happen to be C. Somehow we can get the child to understand that “D divided in half makes C”. Does this seem awfully contrived? I mean, counting is “adding by one each time” to a child [actually, according to my little brother], and a child who is taught it for the first time is just nodding his head trying to follow along... In this really odd and probably flawed example, the child is assumed to not know how to count (at least not with natural numbers yet) and so can possibly be taught simple division and stuff directly… Of course, I haven’t thought about it enough to see how all this would reconcile with math beyond basic operations or affect one’s learning about negative integers and stuff. But it’d be amazingly convenient if along the way we could avoid as much confusion as possible with rational/irrational numbers… Oh well, I’m done here. Hopefully this inspires better ideas from you.
ReplyDeleteI was just thinking… okay, skip forward in a hypothetical situation where a student knows division and multiplication concepts really well, and is proceeding to round out his understanding with addition and subtraction. When students learn multiplication in the modern system, it is often described as repeated addition, because teachers want to build on the existing understanding, which is that whole numbers added to other whole numbers make bigger whole numbers. Multiplication at this point rarely, if ever, means that an original number gets smaller – since the students are most familiar with whole numbers, multiplication always implies “big… BIGGER…” So now take a student who only knows division and multiplication and is being taught new operations called addition and subtraction using only his existing understanding. Suddenly, addition is so weird because it like involves multiplying some ratio (er, number) by another ratio such that the resultant ratio is larger than the first. And uh, if the result is smaller than the first, then subtraction has been performed. Would this process of learning be easier than learning addition, subtraction, multiplication, division in that order?
No matter how I ponder the possibilities, I still feel that starting with the natural numbers is most natural (hehe) when children begin their mathematics education. Essentially, I don’t think our counting technology is terribly inefficient, at least for the way our society is now and for how most people use math in their lives. The lame pun on “natural” made me think of the Fibonacci numbers. I think it’s reasonable to say that to find Fibonacci nature, addition has to come before multiplication (therefore, it’s simpler and more natural – whee!). Nature never reaches the “end” of infinity. Trees develop and go through stages in which the numbers of branches at certain heights correspond to Fibonacci numbers. We’ll never see physical evidence of the Golden Ratio on a tree because it’s a limit. No matter how many branches the tree has at a certain stage, that number divided by the number at the previous stage will always miss the golden ratio. We might think inherently in ratios and be able to discern between relative sizes of quantities, but adding is often a quicker way to
ReplyDeleteDivision in elementary school is when you start with a whole thing, and the number of separated pieces, and the sizes of individual pieces compared to the whole, before establishing something that relates all three. It’s confusing because it involves dividing up a group of things into smaller groups of things and being able to tell exactly how many groups of things there are and how many things are in each group. Adding seems... to take fewer steps. I don’t know how to express this, but when I read what you said about sliding a long a number line versus stretching it, I thought of how I prefer to repeatedly add amounts of yellow paint to a given amount of red to get orange, rather than start with calculated amounts of yellow and red to get to a desired orange. But maybe the analogy fails. Anyways, with addition, it just feels like less work while division has more steps. What’s wrong with learning the difficult thing after the easy thing? Okay fine, maybe I say this because I’m used to counting.
In the experimental world, most quantities must be measured on a continuous scale, like frequency, volume, length… Scientists use uncertainties to get closer to the truth. When we’re in 3th grade “math centres” and exploring measuring with a ruler, it doesn’t occur to us until later that 8 cm is never exactly 8 cm but is 8.00+/-0.01cm or something. As we grow, we notice more details not only because we are encouraged to in our education. We are more aware of finer differences in spaces and between things. If you look at it this way, it may make more sense that our education begins from the discrete and introduces more and more of the continuous, at least in loose senses of the words. Discretely, the jump from 0 to 1 is very small. Continuously, you can’t count the jumps from 0 to 1. For the people whose interest in math was salvaged in later education, this development in understanding isn’t something to particularly complain about, is it?
ReplyDeleteBut yeah, the discrete should *not* be made superior to the continuous, because after all, the discrete is fundamentally limited and the continuous sort of fills up all the spaces that the markers of the discrete miss. In conclusion, I think you are right in hinting that 21st century touch-screen technology can help immensely in perhaps bringing the ratio into mathematical education earlier. I mean, animations (strings of slides played before our eyes to make a more or less continuous idea instead of many discrete ideas) hold a child’s attention better than separate slides. Maybe the fascination with “oooohh, pretty moving pictures” comes both from our natural interest in “novelties” as well as affinity for the continuous. Even looking back just as far as our parents’ generation, education was largely done through dialogue, handwritten calculation, hand-drawn diagrams, etc. Think of the possibilities with touch-screen technology. Suddenly what you said about having a child “interact with geometry” and not memorize it has a hopeful ring. And I think there are many applications, games, and books out there that border on what you’re hinting at, and only the future will show how they might come together to streamline elementary school education and subsequently later education .
I am fascinated by the results of the studies on infants and their ability to distinguish between quantities of certain ratios. In my own counting habits, I tend to recognize one, two, three, and four objects without having to sum up portions of the group, but when there are five objects, I almost always group “two, one, and two”. If the five things are in a line, the middle one seems to stick out, and if they’re in a more random arrangement, there’s still one that seems to stIck out. For six, it’s “three twos”, although since I was first taught to slide on the number line, it’s more like “two and two and two”. Seven is almost always “four and three” – note: not “three and four”. Eight is usually “five and three”. Nine is “five and four”. Ten is “five and five”. I can’t really tell how or if I subdivide the steps and don’t know if this all means anything important.
I tried the approximate number sense game. Hehe, it was at first harder for the blue balls to convince me that there are more of them. The first round caught me off guard because when the balls disappeared, the brightness of the yellow balls was imprinted in my vision. Therefore, I picked yellow more often than blue for the next few rounds, even though I actually expected to be wrong. (The yellow balls leave a “bigger” impression on me! The number of them must be “bigger!”) It bugged me at first that the balls were always on a grey background, and that blue is definitely more similar to grey than yellow is. What if it was a white background? Then the blue would “stick out” more. Oh well; after a while, I got used to the game and had a near-100% score. Argh, human learning and depending on expectations.
ReplyDeleteAnyways! I think this can lend more strength to the idea that our natural inclination is to think continuously. A child (okay, maybe just me, because I haven’t checked with anyone else) might play the number sense game more as if it was asking “what colour of balls is there more of?” – I have awesome grammar, I know – even as they understand the simple challenge “are there more blue balls or more yellow balls?” For example, fourteen small blue balls can take up less area than eleven medium-to-large yellow balls, but there it seems size brings in an optical-illusion-ish component in addition to number sense. But the balls can’t all be the same size or else… yeah, we’d be like “uh, there are more yellow balls and there is definitely just MORE yellow.” How will your game “randomise” (no such actual thing) the size of balls and their arrangement? I haven’t thought through the more subtle ways this kind of game could be biased. I’m sure you have. I just enjoy how this particular test of number sense ties together one’s immediate visual and the sort-of analytical abilities.
It’d be interesting if balls of one colour are always bigger than balls of another colour. Would some people be more sensitive to the “continuous” part of it (the area taken by the amount of that colour) or the “discrete” (the number of spots of that colour, regardless of size)? Guhh. I’m going to stop now. Anyways, I’m eager to try out your version of the number sense test! Even more eager to see how you’ll make it more sensitive based on other (hopefully more researched and well thought out) feedback you get.
Great thoughts - I'm tempted to respond to all of them, but for the sake of keeping the discussion open, I'll just address one for now.
ReplyDeleteI probably should have made this more clear in the main writing, but I'm not by any means suggesting that we learn multiplication and division first, and then, with those under our belt, get to addition and subtraction. You're absolutely right, there are many things that addition works far better for than multiplication, and it's a lot easier to reach addition through the route of counting than through the route of multiplication. In fact, I highly doubt it's even possible to teach how to compute multiplication and division problems before knowing addition and subtraction - because computation is fundamentally discrete, based on counting.
The bigger thing I'm attacking is the concept that education must be linear. Why must we have a single route through all school math topics? Can't we start from our number sense and head out in two different directions - counting and ratio - and join back together later on? Teaching every topic just as a consequence of the previous thing seems, to me, like it just emphasizes the mechanical, rule-based, step-by-step side of math, not the lifeful, exciting, creative side.
Yeah, we shouldn't have to - and it's impossible to - settle on a best way of teaching. Ideally education should be designed to cater to each individual's needs and learning style. Sequential teaching according to a curriculum has some advantages, but when the direction and pace is strictly set for everyone without the opportunity to try different paths, many students give up on the main path altogether when they may have thrived otherwise.
ReplyDelete[Note: this is before reading Meng's extensive comments; they're almost as long as the blog posts! :D Yay!]
ReplyDeleteThe primary thought I had while reading, especially this last section, was "Why not try this?" Even outside of standard education, we can try constructing and introducing activities to our own future young children (assuming some of us eventually do). Also, I'm sure that there are daycares/preschools/kindergartens around that might be glad to try out some of these ideas; it doesn't seem too fantastical to design a study that could try to investigate this idea. It sounds reasonable to me! And fun! For some reason. :)