It’s been some hours. Some hours since I last slept (almost 20, to be precise), and I still have stuff to do. Some hours since I’ve created this blog (maybe around 30) and I’m already writing a third post – and they say drugs are addictive. Some hours since math educator Bryan Penfound started following me on Twitter.

Can’t say it wasn’t a big moment for me. I’ve took a deep breath and done my best in not tweeting “@BRYAN’S FOLLOWING ME!!!1!!1 #EMOTIONS WEE HOO”. I thought, “wait, I hold a great respect for this man’s work and his posts have been very helpful in guiding my current research. What will he think of me if he knows I reacted to this like fandoms reacts to pop superstars? Is this how I would like to introduce myself to this group?”

So I’ve chosen you, Mr. Penfound, to be my first direct contact on this community of math blogs. As to many other blogs recommended to me, your’s is one I’ve been following silently for some time. So if you would give me a few minutes of your attention, I would like to share a personal story that I remembered today and I think it’s somehow related to your last post.

I was in Brazil’s fifth grade. It was 2007, I was eleven years old. We were currently learning about different number systems (binary and hexadecimal as alternatives to our decimal system). The ones in class who were using it for fun were making coded messages in the hexadecimal (something similar to ASCII code), and some of the others were just not caring much. I, though, was deeply fascinated by the binary system.

Maybe because it’s look on paper reminded me of *Matrix* and made me imagine myself like Neo, fighting the machines. Or maybe because it reminded me of computers and made me imagine myself like a hacker. But it had one thing that the hexadecimal had not: it could represent all the numbers with only two symbols. I’ve thought to myself, how can this be? It seems to be way simpler than the decimal system, so why don’t we use it?

My school had a very, well, “traditional” tradition. Even since young, we were prepared to pass on college admission exams, so our classes were lectures on the concepts and then lots and lots of practice exercises. For the younger students, this was less rigorous, but in high school was almost a dogma. But this doesn’t mean that I haven’t had (is this really correct? because it sounds very wrong) inspiring teachers.

One of them was called Mr. Daniel. Everybody loved Daniel. People could hate math, but they loved Daniel. He taught us concepts like number factorization, geometry, geometric constructions, and many others. He was also our friend and taught us how to be better people, or at least less little-devils. He wasn’t the one teaching number systems though, that one was Mr. Paulo, another very inspiring teacher (but because of other stories). But after class, I would always go to talk about math with Daniel.

After another class on the number systems, I went to Daniel to explain my thoughts on the binary system. I would say, “They seem to be so simple, and yet we have none of the arithmetic tools of the decimals available to us. Why don’t we use them for counting, and making operations?”

Daniel proposed me: “Well, we don’t we start developing those arithmetic tools right now?”. What followed next, was Daniel and me doing a very detailed study of what made the decimal much more popular than the binary. We did not discussed matters of history, why we have ten fingers on each hand, electric circuits and none of that. I was very naive and believed that people didn’t used binary simply because they haven’t thought much about it before. So we started looking for deficiencies of the binary that were healed in the decimal.

The first one was reading the number itself. We we’re so adapted to the Indo-Arabic digits that we couldn’t look at a binary number and relate to the quantity as quick as we can do with a decimal number. So we started thinking of how we could make the ‘reading’ process easier.

We thought that the problem with binary was that we counted it from right to left – first converting the digits into powers of two, and then adding them together. I thought of the following way to read them left to right: The first digit you read will be a 1 (let’s not think of the number 0). From then on, when you move a digit to the right, you double the number you’re thinking and add the value of the new digit. So reading 1011 would be:

- First digit is a 1.
- Next digit is a 0, so my value is 1×2 + 0 = 2.
- Next digit is a 1, so my value is 2×2 + 1 = 5.
- Next digit is a 1, so my value is 5×2 + 1 = 11. My number is 11.

This was simply a reversal of the first process. But dude, it felt **awesome**. I was probably that fastest binary reader in the whole school, though no one except me and Daniel knew it and we were probably the only ones that cared. I was like “**This** is what Paulo should be teaching us during class. Let’s make a revolution and drop the old system!”. This skill is something I never forgot and I have always used it since then to read binary numbers. Back in High School, I could read binary numbers of several and several digits fairly quickly – though I must say I’ve done some little tweaking to the method along this years.

After that, came the problem of writing binary numbers. That algorithm was good for reading, but terrible for writing: multiplying by two and adding one in each step? You could get out of control and miss your number going that way. So our solution was extending the division algorithm to give us the number. If I wanted to write 11 (it’s in decimal, OK? very poor choice of number) as a binary number, I would do something like this:

The number in binary would be the remainders written backwards! But this method, sadly, took up a lot of paper space. We wanted a more compact one. So we thought one that required knowledge of powers of two, but after that it was easier and took up less space:

On the top left corner we write our number. On the middle column, we write powers of two less or equal to our number in decreasing order. For each row, if the power of two was smaller or equal to the value written on the first column of the row above, we would subtract the power from the value and write the difference on the first column of the row, and mark 1 on the third column as well. If the power was greater than the value, we would just repeat the value and mark a zero. We stop when we mark a zero on the first column, and then zero out all remaining spaces on the third column. The number in binary would be written on the third column!

You see, this was also simply a reversion of the way we were taught to read binary, by converting to powers of two and than adding the results. Yet it felt like this was going to be adopted in days as the next official system by the S.I.

I bought into the pretext that I was discovering something new. That I was working on something that was groundbreaking and no man has ever devised before. I bought into a story were I was the professional. I was tasked with improving the world around me and my tools to understand it (by exchanging the decimals for the binary, can you imagine?), and I knew Daniel was expecting me to succeed. Not to fail so he could show me why is it bad or impossible or worse than the actual standards. He believed that I could reach whatever answer I wanted, and gave me the desire for answers. Forget Wiles and his Fermat proof, I was doing **real** work back then.

After that we moved on to algorithms for the basic operations, and when we reached algorithms for factoring I’ve already understood that underlying all my methods I was using the decimal system, in my head. My calculations were written in binary, just as much as my answers. But once we left the mechanical world of the basic operations, all the metal effort was being done in decimals. We were trained since young for using them, and today I would say, personally, that they yield a good ratio of representation of quantity and paper length.

I still have a draft from the conclusions we wrote that day. The final paper got lost somewhere in the void. But I still keep the reading technique sharp. And my memory of this experience alive, just as many other experiences where I was motivated by my teachers to pushing my knowledge and understanding to a critical, limit point, where repeating what I’ve seen wasn’t efficient anymore and I would need to reinvent myself. Experiences were I was instructed to try something different than drill. It was from that moment on that I was always on the look for an alternative answer (“This could be could be converted to cubic meters, but the calculations are easier in gallons of oil!” – another story that gave me vivid memory and a 0 on a physics test).

As a student, when I felt like I could take on a more advanced task, I would crave for it. Then, I would love to be given a situation where I would need to think as a professional. I would also lose interest in the drill exercises. Getting my Middle School me to do repetitive work was hard, and at High School, impossible. But asking me to try and discredit the decimals? Right away. Find a divisibility digital criteria for arbitrary numbers? Why not, let’s try. Organize a school riot in response to a S.I. debate on a physics test? Probably the best thing I’ve ever tried and failed.

Providing motivation, opportunity and a damn good feeling of reward. I’d say these are things I’m learning from the math education blogosphere/blogoball/blogowhat? and that are indispensable principles for every teacher.

I always find it amazing that people can connect so easily across continents by simply writing down thoughts and ideas. Thank you for sharing your thoughts on teaching and learning. I am honored that you feel so strongly about my writing/thinking. If you ever want to discuss mathematics, or teaching, feel free to message me on Twitter and I will give you my email.

When it comes to your algorithm for converting a decimal number to a binary number, I had a moment this morning when I learned something. I often teach base-5 to my future teachers, so I am aware of the division procedure to convert a decimal number to base-5 (divide by decreasing powers of 5).

For example, if we want to convert 176 to base-5, we first divide by 125, then by 25, then by 5 – keeping track of the remainders to get (1201)_5. Notice that your algorithm won’t work (easily) for base-5 (can you see why we would use division instead of subtraction in base-5?). Your subtraction procedure intrigued me, and I wondered why you didn’t use division – then I realized that in binary you either divide by the power of 2, or you don’t! How beautiful! Division is not needed because either we get digit 1 or we get digit 0 – there are no other digits like we have in base-5.

I also find it interesting that you would do the mental arithmetic in base-10 – this is often how I teach my teachers as well! It is interesting that base-10 is so ingrained in our body that it is difficult to truly part with when learning another base system. I wonder if anyone has raised children in binary! Now THAT would be interesting to hear about 😀

As you teach more you will find that the “damn good feeling of reward” is pretty vital to most – this, I think, makes us remember certain teachers. We can recall when we were given opportunity, and when we were rewarded for it. Of course, it is well-known that this “feeling of reward” leads to more motivation! Ideally, it is a beautiful circle of learning.

Looking forward to reading more!

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