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 4: The Shape of Numbers 1: Addition is commutative

An important idea in math is commutativity. Addition of numbers is commutative. In the simplest of terms, a + b = b + a. And though this may sound obvious, and a child may "know" it, it may not be so obvious as to why it is true. So, my son and I explored this idea a little. 

You pick a number A and you add it to number B. You get a number C. 

Now you do the reverse. 

You take the number B, and you add it to number A. You again get the number C. Why?

There could be several ways to understand it. (Also prove it formally). But intuitively, here is some of the stuff we did, as "geometric" proofs. 

First, we look at the "shape". Here is 4+3 on the first two rows, and 3+4 on the next 2 rows on an abacus. 

And we see that the "shape" of 4+3 is the same as the "shape" of 3+4, except that one is the flipped or mirrored or reflected version of the other. So, we see that the "shape" has not changed, it is the same shape, we have only somehow moved it, and therefore, 3 + 4 = 4 + 3.

And what if now we increase the number of numbers being added? Here is 1+2+3 and 3+2+1. Again, the "shape" is flipped, but since it is the same shape, we can say 1+2+3 = 3+2+1. 

Of course, this does not immediately show that 2+1+3 will also be the same "shape", so we need to explore it in another way too. 

What if we look at another shape way? Here are 7 rectangles forming a large rectangle. 

And now, you can cut up 3 rectangles leaving 4, or cut up 4 rectangles leaving 3, like this. 

And now it becomes clear: because to make the larger rectangle, you have to put together smaller numbers of rectangles, but the order of how you put them together does not matter, you can either put 3 first and then 4, or 4 first and then 3, both make the same "shaped" larger rectangle of 7 smaller ones. And this idea scales now to more numbers than two: using this idea of cutting up a large rectangle into several pieces, you can show that 1+3+3 = 3+3+1 = 3+1+3. And so on. 

Another similar "proof" is that you have a bucket that needs to be filled up by say a certain number of balls. Say the bucket can contain 10 balls altogether. Then, it does not matter whether you throw in 4 balls followed by 6 balls, or throw in 6 balls followed by 4 balls. The end result is that the bucket gets filled up. 

This property says that addition of numbers is commutative. 

And before we as adults start saying, "this is obvious", we should think up some counter-examples, where changing the order of operations does not lead to the same final result, and we find that most of our everyday processes are actually non-commutative.  

Eating your food and then digesting it is not the same as digesting the food and then eating it. In fact, it is not possible. In the same way, putting a ball in front of you and then kicking it, is not the same as kicking a ball and then putting it in front of you. However, when you are in Manhattan, going down 1 block east followed by 1 block north gets you to the same point as going 1 block north followed by 1 block east. This is commutative.

I asked my son to think up of one commutative example and one non-commutative example. He said, putting the engine on and then flying a plane is not the same as flying the plane and then putting the engine on, because you cannot fly a plane without first putting the engine on. Whereas, if you have a plane and a helicopter, putting on the engine of the plane followed by putting on the engine of the helicopter, is the same as putting on the engine of the helicopter and then the engine of the plane, because now you have both engines on.  

So, the important idea to grasp is that when things commute, the order does not matter. In whichever order you choose to do things, you reach the same end result. Whereas, in things that do not commute, the order matters. If you change the order of doing things, you get different end results. 

We had good fun discussing examples and counter examples of commutative behaviour: including this one that made us both laugh. He said, you know, at school yesterday, the computer had my name as Last name, First name, while I know my name is First name, Last name. And I asked him, so does your name commute? He said yes, because it is the same name! :) 

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.