3 Cognitive Observations on Math Learning

I realize that this blog has been idle for a few months now. While there is always tons to write, work, home, and life take over, and there is no time left to write. Which is both good and bad. Good, because it helps you to see what you are prioritizing. Bad, because you have started this project, and should take responsibility for its continuity because you continue to feel passionate about the topic of learning, especially math learning for small children. 

Lately, I have been reflecting on the processes used in learning and how a child engages with their learning. These observations have helped me to reflect on some of my (university) teaching I do in design, especially human centered ways of thinking and learning, and translations of this understanding to the design of interfaces. Here is the critical question: while the online world is now flooded with that next app, that next digital platform, that next interface to make learning easy and engaging, and your children can pick and choose digital forms of learning that you never dreamt of, is there an evaluation framework? How do we evaluate the claim that learning, especially math, through a digital environment actually leads to better learning outcomes and a more engaging way of learning (as claims go)?

An observation: It is certainly having a big effect in practical terms. The dissemination of weekly public school homework (at least where we live in Sydney) is online, on a "social network" where kids, parents and teachers can interact. All the homework activities themselves are online, in the form of multiple choice questions and reading text passages sourced from various learning environments. When we look at something that is present, we must also comment on what is absent. Writing (with a pencil or a pen) and the "doing" of math (where you read a problem, break it up into parts, work out the meaning of the problem through drawing, text, or symbols, figure out the steps to solve it, and verify your solutions). This was the more traditional way but is completely missing in the homework pattern of today. 

I am being careful to not make value-based judgements here, but just noting the characteristics and attributes present or absent in these two different forms of learning. For example, one of the things my 8-year-old son is extremely fond of is making paper planes. He often comes up with intricate designs, which I know nothing about, and certainly won't be able to reproduce by observation unless he spends some time tutoring me. When he does, it is great fun because he becomes my "teacher". He claims that he has learnt it from a few You-tube videos on origami. There was a point in time when he stopped watching the videos and started to experiment himself with different designs, noting that if "I put a paper pin and increase weight on the nose, the airplane behaves differently" or "this plane is a model only and while it looks good it cannot fly very well". I think this has been one example, where people sharing their hobbies digitally on Youtube, has led to an extremely engaging form of learning for a child, which may or may not have been the original purpose of producing the video. Thus, I am not saying that digital learning is all bad, and the non-digital ways all good - just that the medium of interaction has a role to play in what and how we learn, and critical aspects of the non-digital forms of learning cannot be replaced by the digital environments of multiple choice questions that are currently all the rage in examinations, in worksheets, in homework. 

More importantly, the question is, if more and more learning is shifting onto digital platforms, what lessons in interaction can we learn from observing how a child learns? This is a design question. So, here are three observations from the recent work my son, some of his friends, and I have engaged in:

• As they grow, children need to discover that most problems in math cannot be solved "in the head" and that the journey or process of solving a problem is more important than getting the right answer. 

Since the screen presents them with 4 choices, and the questions are mostly straight off one-step questions, children tend to think this is what "normal" math is like. And if there is anything more challenging, that is just too hard for them. Taking them to a mental place where they will stop trying or give up soon if the problem you present them with requires them to think in multiple steps. More importantly, they don't come to appreciate that the same problem can be solved in different ways to get to the same answer, and the different ways in which "pleasure" or "reward" is realized in our brains (for example, solving a problem in three different ways and realizing that you have discovered a much faster or more elegant way the third time as compared to your first solution). They get pleasure from the clap sounds or "you're clever" statements on getting a "correct" answer, so the reward system in digital platforms rewards a correct answer, not the method that was used to get to the answer. 

• As they grow, children need to develop mastery in moving between concepts and their symbolic representations in order to be faster, more proficient and creative in their problem solving.

By this is meant, knowing that there are multiple ways to represent the same concept or moving between different concepts via a representation. For example, fractions and division, the relationships between symbols, and discovering parallel relationships between proper and improper fractions, mixed fractions, converting between forms, leading to understanding ratios and then proportions, and then percentages, and then...it could go on. Working with them on pen and paper, one realization was that if they develop the flexibility of moving between representations (and in parallel concepts), they tend to master higher and higher topics with amazing speed and clarity. On the other hand, the digital environment does exactly the opposite. By having sections on "fractions" where kids only see 10 or 20 similar questions on pizza slices, and then a section on "division", where kids see 10 or 20 similar questions on dividing cakes or lollies equally between a certain number of children, makes them concretize exactly that: that "fractions" mean this one thing, to be visualized as pizza slices, and "division" means this other thing, to be visualized as dividing lollies. Pizza slices are a great introduction to fractions, but if it stops there, we do have a problem. 

• Learning should not always be made "easy" for the child. 

The joy of working on a difficult problem, failing a few times, and then finally solving a problem, is not just a math thing, it's a life thing. Trying to engage children in math by making it easy for them is a bad idea. It teaches them to take pleasure from the final reward (usually in the form of "points" they earn, that they can "spend" to play games within this learning environment). It teaches them that the reward for doing math is not just that - the doing of math and the creative joy you get from it, but an external reward that then gets you other things (e.g. video game time). And yet, almost all the digital environments do exactly that. But there are simpler ways to reward the correct form of learning and making kids internalize the joy and fun of solving something, i.e., doing the thing for its own sake is the reward. When I work with the 8-year-olds on a traditional white board with colored markers, I introduce a concept, then give simple problems to solve, usually, I don't "tell them how" but see if they can "discover" it, with gentle prodding in the right direction needed sometimes. I ask them to write out or draw each thought carefully. Even when they have drawn correctly, I ask them to then write the same using symbols, encouraging multiple representations for the same concept. Finally, I ask them to "make up problems" after they have worked with a concept for some time, and to give these problems to each other to solve. They do not find this borning or hard, they enjoy it, and one little girl made my day by asking her mom when they would be having the "math playdate" again. 

To a person who does a lot of math, these observations would feel like "duh". However, I wrote these down because these are exactly the attributes I find missing in the current digital learning environments, that are often the primary mode of introduction to mathematics in primary school for both kids and parents (since there are no books anymore, and no "math notebook" in which to solve problems from a book). This implies that thousands of children (and parents helping with homework) are experiencing these environments as their only lasting introduction to mathematics, and therefore, they increasingly form the following impressions in their minds (opposites of the three observations above, and real statements made by real 8-year-olds when I began working with them, stemming from real confusions about concepts and their symbolic representation): 

1. "I should be able to solve this one in my head. If I am not able to, I am not really good at math."
2. "I cannot understand what these symbols mean, if we are doing fractions, why can't you show it to me as a pizza slice?..."Why do I need to learn why fractions and division can mean the same thing, fractions is about pizza slices, while division is about dividing lollies between friends, they are not the same". 
3. "This problem is way too long and hard, I can't do it, I am not really any good with math". 

So, while the multiple-choice model may be a great one for older kids (see the excellent Alcumus as example), when children have learnt the basic important lessons and know how to pull out pen and paper and work out problems, as an already internalized and natural way of working. But, a really bad idea for little ones who are, perhaps for their life, forming an impression of what math is at this age. I wouldn't mind digital tools being designed that respect the complex and deep mind-conceptualizing-hand-writing-visualizing-writing-solving connections so essential to math (it will be a super-hard design task though). But more than that, I think a lesson in design is, if a simpler tool (books, notebooks, pens, paper) gets good results, perhaps we should not be quick in discarding them by putting them into the category of "rote ways of learning" and replacing them with much more primitive digital tools that supposedly make learning more engaging but actually are introducing alarmingly simplistic and fragmented notions and learning in children. By labeling all traditional learning as rote, we routinely throw the baby out with the bathwater. 

Post 3: 2 ones are 2, 2 twos are 4, ...

I grew up in India. We had in school what were called "mental math" tests. A teacher would rapidfire at us 50 questions in quick succession, and we, ready with our pencils and paper, would have to quickly jot down the answer, and whoever did the most correct answers got the highest marks. The questions went like this:

Teacher, reading out: 

19 4s are ... (2-3 seconds pause)

1000 - 564 (2-3 seconds pause)

Does november have 30 or 31 days, yes or no (2-3 seconds pause) ...

... and so on. 

So, obviously, the only way you could do well in these tests was practice, practice, practice. And my grandfather (blinded by cataract, he would hold his transistor radio, and walk up and down listening to the news all day) took responsibility. Every morning before school was what we called "Tables time": Times Table Time!

I would sit and sing away the tables, from 2 to 20, and then there would be the harder practice: how you could use the tables and place value to quickly figure out higher numbers multiplying each other without resort to pen and paper. 

I started school at 3 years of age (in those days there were no rules on being too young or old, there was a test, if you passed, they let you sit in class :)). And this game continued in simple form till about the time I was in Year 2, and then became a lot harder and more pleasurable, because the nature of my grandfather's questions changed to: "Is this number X prime, or not".

So, of course, armed with this experience, as a parent I introduced the times tables to my son when he was 3. He loves his (ever-growing) set of hot wheels cars, and I thought this is perfect. So, the 2s, 5s, and 10s happened quickly (little cars arranged in rows of 2s, then skip count verbally, then the same with 5s, and then 10s, maybe a post on this later with pictures). Worked like a charm. Before he went to pre-school, he was chiming away at questions like, if there are 3 wires and 2 birds on each wire, how many birds altogether? 

By the time he was starting formal school, kindergarten at 5 years of age, (the age for starting school in Australia), he knew his tables from 2 to 10, maybe not the 8 or 9, but if he did not "remember" he knew how to work it out, and he knew the importance of "remembering" the answer, when say we are walking on the road, mum asks a question, and I have to quickly work out the answer.

And later, he did not even need to memorize or learn by externally provided parental support, because he began to figure out the number patterns. For example, here are multiples of 11: I did not make him learn it, he was not counting by groups of 11, he simply figured out the pattern of progression, especially after 110, all by himself. He knows he is counting by groups of 11, but that is not the "algorithm" used, the algorithm emerged from number play and pattern recognition. And to be honest, I realize, even today, the delight of this. To date, I discover a new sequence that will make me go, WOW! (sequences and series post later). And my hope is that it all starts here, in elementary school math.

So, to me, maths is as much about discovery and play and conceptual understanding, as it is about practice and making memory stronger. Only when you dribble a ball for hours, you master a game. Only when you run everyday or go to gym everyday, do you master fitness. Only when you run your fingers again and again on the piano, do you master playing it. Only when you sing repetitive patterns of notes again and again (the term in Indian classical music is called "alankars"), do you master melody. Only when you put in practice, do you master anything. 

By the way, I completely agree, that every child will have their own pace, and practice without conceptual understanding is not a good idea at all. But, in working with my child, I found that the reverse is also true: conceptual understanding without practice will never take root and flower in the brain. And sometimes, like my post yesterday, pattern based understanding comes not only from learning the formal rule or concept or schema or prototype and its application (deduction), it comes from doing some pattern over and over again, and then discovering the general rule or schema from the examples (induction). In other words, deductive learning is one way, inductive the other, (there is a third, but maybe later on that), and mastering something comes from both the deductive AND the inductive ways. 

Finally, I found a couple of days ago, that this seems to be the case not only in math, but also in language (see specially his answer to the last question by the interviewer): Steven Pinker's Sense of Style. 

Post 2: The importance of being visual and verbal

When my son was quite young, about 3-4 years old, there was a lot of scribbling and drawing. There still is. The main story in this post is that: we never know when and how they are doing and learning very important stuff in all that scribbling. It may not be all scribbles to them even if that is what it appears as to us. 

So, here is the story. He had learnt to count up to 10 and 20 quite quite quite early. And by simply talking about things, I think that he somehow latently and implicitly knew two very important things in learning to count things: 

1. Numbers and their symbols are not the same thing: The symbol is representational. Meaning every language writes the symbol differently, but the particular number in question, the concept of 1 thing, 2 things, etc. persists. (So, for example, we have three languages in the house, but another post on that). 
2. Numbers are used to count things: so, the larger the number of things, somehow the bigger or more complex the representation of that number, and smaller the thing, simpler the representation. 

So, imagine my surprise, when at 3 or 4 years of age, I found him "drawing" the following (this picture is a much-later version, and hence much clearer and the handwriting much nicer, the original versions were on a little blackboard and chalk, when he pretended he was "teaching" mum): 

The thing is, he is 6 now, and he can work easily with tens, hundreds, and thousands. He does not yet formally know place value beyond that (in the sense that I have not yet worked with him beyond that or introduced anything formally, but you never know how they surprise you!), but in this drawing play, I learnt something very important: almost always, pattern based, partial understanding precedes formal and deeper understanding, for children as well as adults. And we should encourage this play. 

I also think he had captured the idea that numbers can grow as big as you want them to, and so his question on the previous post, came after a lot of this kind of play, both on paper and verbally. For example, he would say random and wrong numbers, as follows: 

Me: How many rotis is mumma going to make today? (Rotis are Indian bread that we make fresh everyday, so dough making time is the Indian version of kids doing play-doh in the kitchen). 

He: 5 hundreds and 4 thousands, and 95 hundred. (followed by giggles). 

Me: Ha, funny funny, but seriously, how many should mumma make? How many do you eat? 

He: 1. 

Me: How many does mumma eat? 

He: 2. 

Me: How many for papa? 

He: 4. 

Me: How many should I make then? 

He: hmm, 7. 

So, when this kind of conversation happened, I never said, you are wrong, that is not a right number.  I never think, it is too early either. I let him babble away, and to my surprise, several days later, he would start saying the correct stuff. Statistical pattern recognition, statistical human learning, pattern based learning. Something to totally encourage. 

Post 1: What is the biggest number?

Children ask the most difficult questions. So, it is important that they don't get lost. I have this belief: If you can break down a difficult concept into stuff that the child can relate to, then many of the concepts that are acknowledged as too difficult for a young child ("it is too early for him to know"), can be made not only understandable but enjoyable. And the child learns it in a way that they will never forget. 

When my child was in Kindergarten, he asked one day:

Mum, What is the biggest number?

Me: What is the biggest number you can think of?

He: A million?

Me: And what happens when you have a million lollies? J

He: Wow, can I? J 

Me: Ok, so, imagine, just like Charlie and Mr. Wonka, that you have a million lollies.

He: Well, that would take YEARS to eat J

Me: Ok, what happens when I give you just 1 more lolly after the million? How many do you have?

He: I have a million and one lollies J

Me: So, you said 1 million is the biggest, but now you have 1 million and 1, which is bigger!

He: Oooooh, yes…so if you give me 1 more lolly I will have a million and 2, which is even bigger?

Me: Yes, you are getting it!!

He: So, if I have 2 million lollies, and you give me 1 more, that is bigger than 2 million?

Me: Yes! So, no matter how many lollies I give you, you can always add 1 more lolly to make a bigger number of lollies J

He: SO, there is no biggest number, because there is always 1 more than the biggest?

Me: Yes, and that is why the word we use to describe this is called “Infinite”. We say there are infinite numbers, because they go on and on and on and on…no matter where you stop, you take one more and one more from there, and so on.

He: So, the biggest number is called infinite?

Me: no, there is no number with the name “Infinite”. Infinite means this idea that the numbers go on forever. And the way we write it in math is we take the number 8 and ask it to lie down, like this (at this point, I write an 8, followed by a lying down 8, the sign for infinity).

He: Can I try it?

Me: Yes, sure (and he draws for a while).  And by the way, going from lollies to stars, how many stars do you think there are in the universe?  

He: Ooh, got it J So, that is why my book said the universe and stars and galaxies and all that stuff is infinite, because it too goes on forever?

Me: well yes, maybe yes J

He: Ooh this is mind boggling (he has learnt this new word, so he uses it liberally).

And yes, this was not a one-off conversation. We continue to explore other such pictures of infinity in the universe. He is in Year 1 now, and he has realized that infinity can go the other way too, meaning you can make a number as small as possible, by going the other way (negative numbers)!