# A Mathematician's Lament: An essay on mathematics education

atyy
Do you teach algebra to 8 year-olds? If you do, then I want to learn your secrets.

This IS teaching them algebra. It just takes a little time to transition from concrete examples (blocks) to abstract reasoning (x). There are other ways of teaching algebraic concepts to young children, but they almost all use something concrete as a bridge to 'doing it right.'

We could ask this: Why don't we just teach primary school kids about fields and rings? Well, in fact we are - using ideas that they understand and building up to more abstraction as they have enough experience to make sense of it all.
I haven't taught algebra to 8 year old kids, so maybe if those methods work they're ok. I have to say I'm skeptical though, why not hold off until they can do with a method they will still use find useful as adults (or do many adults still use bars in everyday life?). There must be easier problems they can do before that that don't require bars. My own personal experience is that I found word problems really, really hard to do until my elementary school teacher taught me algebra.

I have to say I'm skeptical though...
You are thinking either bars or algebra. This IS algebra. You are just not seeing the connection for some reason.

mgb_phys
Homework Helper
I haven't taught algebra to 8 year old kids, so maybe if those methods work they're ok. I have to say I'm skeptical though, why not hold off until they can do with a method they will still use
You start with, what number do you have to add to 3 to get 5
Then, what number do you have to write in the empty box, ____ + 3 = 5 to make the sum correct
Then you ask, x + 3 = 5, solve for x

There is a percentage of the population that will stare at you blankly once you mix letters and numbers - they grow up to be managers

while yes, at my school/college (UK), there is alot of "it's given to you in the exam, don't bother to learn it" in regard to formulae etc i can't believe half of the stuff said here about the US system.

regularly we go through conceptual as well as numbered question, have numerous practicals/demos etc, and the work is rarely if ever boring. i must admit while i can't remember much of the GCSE maths (2 years ago) in terms of lesson time, it certainly put me in good stead to step up to a-level maths & further maths....

in reguard to the link posted earlier by mgb_phys, i have no idea what examination board that teacher was teaching on, but practically the whole of that article is baloney. (certainly in my experience)

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Moonbear
Staff Emeritus
Gold Member
You start with, what number do you have to add to 3 to get 5
Then, what number do you have to write in the empty box, ____ + 3 = 5 to make the sum correct
Then you ask, x + 3 = 5, solve for x

There is a percentage of the population that will stare at you blankly once you mix letters and numbers - they grow up to be managers
We pretty much did that since kindergarten. We learned our shapes at the same time, because instead of a blank, we got a triangle or circle to fill in. When I got to algebra and was told all we were doing was substituting a letter for the circle or triangle or blank, I was surprised at how easy it was, and at the very beginning of the course, even felt a bit insulted that we were wasting time doing kindergarten level math!

After reading the comments in this thread, I think a lot of people really lack understanding of childhood development. Children are taught a lot of things by rote and with very clearly defined rules because that is the stage of development they are in. As you mature, you begin to understand more of generalizable concepts and processes and problem solving, but if you just jumped in with that too early, it would lead to horrible failure. Schools that are large enough have an advantage in that they can split up students into different level groups for lessons. This means that those who are developing faster can get more advanced material before they get bored, and those who are slower can continue to be given material at a slower pace to avoid overwhelming them. When you have classrooms filled with students of different levels, it's hard to teach to all of them without either losing the top or bottom of the class to boredom or confusion, respectively.

atyy
You are thinking either bars or algebra. This IS algebra. You are just not seeing the connection for some reason.
Hmmm, perhaps I should listen to my mother then ... she used to teach from a syllabus similar to Singapore math, and I complained to her, and she said pretty much the same things you did! (She also told me I'd fail maths exams in Singapore!)

I think a lot of people really lack understanding of childhood development. Children are taught a lot of things by rote and with very clearly defined rules because that is the stage of development they are in. As you mature, you begin to understand more of generalizable concepts and processes and problem solving, but if you just jumped in with that too early, it would lead to horrible failure.
There are different theories and approaches to elementary education, and there is a lot that is far from settled.

When you have classrooms filled with students of different levels, it's hard to teach to all of them without either losing the top or bottom of the class to boredom or confusion, respectively.
Depends on the educational approach. If you follow the educational theories of Lev Vygotsky, you want classrooms with students of different levels and different abilities, because the students and interact and teach each other, and the students that are more advanced can help the students that are less advanced. One thing that I like about Vygotsky's theories is that it very closely approaches how I saw students learn physics at MIT, and the successful way that I've seen college students learn Algebra.

Also schools in Taiwan don't generally split students into different math groups.

jasonRF
Gold Member
I missed this post last month (november was a bad month at work!).

I have two daughters, 5 and 8. Lockharts essay is a charge to us parents! It is a wakeup call for me to do right by them by thinking up / finding such "problems" that should encourage their natural mathematical curiosity. We cannot expect the schools to do it all - we need to do our part.

Do any of you educators know of any resources to help some of us parents? I can probably do okay on my own, but resources would help!

My 8 year old loves number problems, for the same reason my wife likes crosswords - they are a challenge and fun to solve. She naturally thinks scientifically, trying to understand why things work the way they do. Recently she was telling me about "proper" and "improper" fractions, just like Lockhart mentioned. I had no idea what the difference is, even though I do technical work for a living (electrical engineer with a phd). Her teacher is forced to teach this stuff. My daughter actually hates her math homework - I have convinced her to rush through it, so she can do the things she does naturally. She is recently into sewing - the geometry of the simple doll-clothes she puts together is great! If she has cut out and sewn the pieces, and needs to make the pants-legs longer for the doll, should she let out the seem or take it in? Without thinking my first guess was wrong (although the right answer only works if the doll legs are skinny enough), but she figures it out, sometimes after mistakes, of course. She is great with numbers, thinks negative numbers are cool (think about borrowing from future allowance!) and sees patterns in numbers a lot, but filling out her math worksheets over and over is dreadful to her. Her current sewing kick is more mathematical than her math homework! I am worried that her education will stifle her amazing enthusiasm for learning and doing!

My 5 year-old is very artistic - I have a painting in my office she did when she was 4 that constantly amazes me. She is now starting to get interested in shapes, numbers, etc., and truly understands what addition and multiplication mean. It turns out arithmetic comes soooo easy to her. I fear for what 1st grade will do to her!

Yes, as their father I am biased!

One scary thing is that Lockhart's concern doesn't go high enough. In college I remember when I was short on time I could use my "math skills" to punch through problem sets without really learning the subject at hand! Intermediate micro-economics was this way for me - I could maximize a profit function subject to constraints (simple Lagrange multiplier stuff that I could do flawlessly every time, although I didn't actually understand why Lagrange multipliers work!) without really understanding the ecomonics! The econ department was bamboozled into thinking this turning-the-crank "math" was economics. I treated the engineering courses that I didn't like the same way - my "good math skills" allowed me to learn nothing in them when I chose. Of course, the classes I loved I really worked on to really understand, and played with ideas on my own. Electromagnetic theory really caught my imagination and being lucky enough to take four consecutive semesters was like a dream! And the semester after that I took three courses had serious electromagnetic theory content! I was depressed as a grad student TAing the only required electromagnetics course for majors. The professor I TAd for didn't ever ask the students to really think about or understand anything! He would do an example problem in lecture, force me to work an almost identical one in recitation section, put another one on the homework, and then ask it again in the exam. Dreadful.

ideasrule
Homework Helper
(relating to previous comments about abstract reasoning) I don't think children are inherently unable to use abstract reasoning. They can understand concepts like friendship, love, revenge, randomness, time, space, etc, none of which are physical entities that can be touched or seen, yet they can't grasp the concept that "x" stands for an unknown number? That's like saying they can't understand why "Alice" and "Bob" can stand for arbitrary people.

It's a lament on mathematics education in the North America. Schools in East Asia have *VERY* different mathematical education systems.

One thing that I find weird is that when it's been noted that students in the US generally do worse on math tests than students in East Asia, you'd think that the logical thing to do is to translate East Asian textbooks, find some East Asian math teachers to give talks, and basically change the system to work like the system in East Asia...... But no.....

What seems to end up happening is that people look at the low test scores of US math education, and conclude that the thing to do is to teach the system that doesn't work, even harder....
wholeheartedly agree with this.

mgb_phys
Homework Helper
wholeheartedly agree with this.
Remember schools are to education what lawyers are to justice.

Schools are there to look after the kids while parents go to work, to give local politicians something good and wholesome to support and to give teacher's unions a reason to exist.

If the children learn anything it's pretty much a happy accident.

I have read Lockhart's essay twice as well as this thread in its entirety and I'm still do not know what a "good mathematics education" is. I do know that what I am now is what Lockhart calls a "trained monkey", that is, I can identify what I'm being asked to solve and find the right method(s) or formula(e) to solve the problem; basically, I can follow the steps. Sometimes I understand why the method works, other times I don't. These gaps in my understanding frustrate me because I've become very passionate about mathematics simply because I do not understand mathematics.

I think I know why 1/2 + 1/4 = 3/4, why x^2 + y^2 = r^2 is a circle, or why the graph of y = x is a line but after reading through all this I have more doubts than ever.

I read the essay, and although mathematics education has its flaws in the United States, the essay emphasizes education should be learning for learning sakes. Interesting it points out how many of the subjects taught in secondary education is not used by the majority of the public. It then concludes that mathematics is really a pointless subject except for a few people and therefore should be taught for enjoyment rather than practicality.
American public education is extraordinary expense, so I expect that the money that taxpayers pay is an investment. Such output from education like economic progress as a whole, lower crime rates, and a chance of economic mobility are all outcomes that are worth paying for. A chance for students to 'learn for enjoyment' is not worth paying for and it's just plain impractical.

But let’s say for one moment that education is ‘learning for enjoyment’. There are many joys and hobbies that people have. There are hundreds maybe even thousands of activities or hobbies that somebody might do. It can be something traditional taught in school like writing, or mathematics, not traditionally taught in schools like martial arts or learning to play the guitar, or something that might require any skills or learning like watching television. If learning is supposed to be a recreational activity how are we suppose to determine which subjects are to be taught? How is ‘mathematics’ any more important than any other hobby, or at least an ‘important hobby’? Perhaps those hobbies that a significant number enjoy can be free to the public. However, I do not seem mathematics being on the list that a significant number enjoy. Furthermore, if we adhere to ‘learning for learning sakes’ there is no way that grades should even be considered, because I cannot think of any recreational or even a competitive activity that involved grading.
To look at a true example of mathematics done for fun there exists academic clubs (math team, science Olympiad, fine arts group, etc.), and academic camps. However these groups do not adhere to the same formula as compulsory education. There are no grades but sometimes competitions, if any teaching it is usually done informal, social interactions are a more important factor, and there is often food and other activities done aside from the main pursuit. The guys model can be used for these types of clubs and camps, but not for school.

The fact is, that in the applied mathematics and applied sciences, sometimes it’s not necessarily important to understand ‘why’ or the ‘beauty’ behind why ‘area of triangle= .5*b*h’. It is more practical to know the formula. It becomes increasingly more complex to ‘show the beauty’ behind certain mathematics concepts.

We are just training others and should be training others. Few will ever enter pure mathematics or even into a job one truly loves. I don’t mean that everyone hates their job. I’m just stating that not everyone will love it. How much somebody likes their job is also factored into how much they make. Those with ‘fun’ jobs will have low salaries, because a high supply of people would want them. Likewise those with ‘boring’ jobs will have high salaries, because a low supply of people would want them. I would pay you to eat cake, but you would have to pay me a decent price to drink urine.

American public education is extraordinary expense, so I expect that the money that taxpayers pay is an investment. Such output from education like economic progress as a whole, lower crime rates, and a chance of economic mobility are all outcomes that are worth paying for. A chance for students to 'learn for enjoyment' is not worth paying for and it's just plain impractical.
You don't think people (of all backgrounds) learning to enjoy academic pursuits and being given the opportunity to pursue this is worthwhile? I would argue that this may very well help improve social mobility and eventually reduce crime rates.

I know that many social projects aim to reduce crime by giving children with a troubled background something they find worthwhile to do.

But let’s say for one moment that education is ‘learning for enjoyment’. There are many joys and hobbies that people have. There are hundreds maybe even thousands of activities or hobbies that somebody might do. It can be something traditional taught in school like writing, or mathematics, not traditionally taught in schools like martial arts or learning to play the guitar, or something that might require any skills or learning like watching television. If learning is supposed to be a recreational activity how are we suppose to determine which subjects are to be taught? How is ‘mathematics’ any more important than any other hobby, or at least an ‘important hobby’? Perhaps those hobbies that a significant number enjoy can be free to the public. However, I do not seem mathematics being on the list that a significant number enjoy.
People do not enjoy mathematics due to the way it's taught (i.e. only something resembling math is taught). Math should take precedence over playing the guitar because the skills learned are universally applicable. Learning how to spot patterns, think critically, simplify, generalize, form abstractions, think deeply about a concept, etc. are all skills that should be taught in math and which are widely applicable to other fields. However where I went to school it was possible to take classes on playing the guitar (well music with a focus on an instrument), and physical education (i.e. playing various sports) was mandatory. People should be encouraged to pursue their hobbies.

Furthermore, if we adhere to ‘learning for learning sakes’ there is no way that grades should even be considered, because I cannot think of any recreational or even a competitive activity that involved grading.
You cannot think of any sport whatsoever where people are ranked? Or perhaps graded on a scale 0-10? If you watch for instance the olympics quite a lot of the disciplines are graded. Also school is in itself a competitive activity (who has the highest grade? Who got into the best school?) Also the science olympiads which you mentioned yourself are very competitive and graded. I really see no reason why you can't grade someone who is learning for learning's sake. The important thing is not to let the test taking and grading scheme decide how the subject is taught (for instance teaching how to write grammatically correct English is good, but teaching how to receive a high score on the SAT is stupid).

To look at a true example of mathematics done for fun there exists academic clubs (math team, science Olympiad, fine arts group, etc.), and academic camps. However these groups do not adhere to the same formula as compulsory education. There are no grades but sometimes competitions, if any teaching it is usually done informal, social interactions are a more important factor, and there is often food and other activities done aside from the main pursuit. The guys model can be used for these types of clubs and camps, but not for school.
Why can't this approach be used? Only do infrequent evaluations (these are also used at academic clubs aiming to compete because the best to represent the school have to be chosen).

Were I live it used to be policy not to grade students until after 8 years of study, and in my opinion this worked well (to clarify: people were still given feedback by teachers, but the teachers were forbidden to use a standarized scale and the results were not used for anything except to tell the student how good he was doing). Recently these policies have been changed due to political pressure and everyone pretty much agrees that the change is for the worse since people are to some extent taught how to take tests, not how to understand the subjectmatter.

...I cannot think of any recreational or even a competitive activity that involved grading.
You mentioned martial arts yourself. However, HOW to grade things is a big issue of its own...

People do not enjoy mathematics due to the way it's taught
You are delusional if you really believe in this. It might apply to a small minority but in my experience most actually likes the pseudo maths taught early better than the real maths taught later, people in general do focus on the real world and the further away something is from it the less people likes it.

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I disagree with some of the things said about math education in Asia. I won't quote exactly which, but I'll just dispel some myths, or clarify some details. I speak from an educational background in Singapore, and having competed in some mathematical competitions, done my SATs etc.

If you take a look at the math notes for high schools in Singapore, you'll find them heavily condensed, to the point that they are essentially formula lists that could just as well be the appendix of a physics textbook. These notes cover much depth and breadth, but don't do anything to cultivate one's interest in mathematics per se. Asians do very well on aptitude tests not because of a better education system, but simply because much more time is spent on rote memory and computational exercises (or 'mechanizing' the problem solving process in math).

We have something called "http://en.wikipedia.org/wiki/Ten_year_series" [Broken]" down here. Most high school students complete all of these papers within 1-2 months before the final examinations. On top of that, each high school will come up with a 'preliminary' examination paper with predicted questions based on the trends from previous years. The high schools then share their preliminary papers among each other: and it's the student culture here to finish all of these papers. So you have high school students doing at least 15+ sample examinations before the actual thing. That's our weekly homework: each examination is set for a 6 hour sitting. Besides these, many of my schoolmates went for biweekly lessons at tuition agencies to supplement their classes; these are very similar to the "cram schools" in Japan. "Spend at least 30 to 40 hours outside of school studying." That's the advice given by a teacher to my friend's class in another school. At least. And this really happens. If you ever have a chance to land in Singapore past 12 midnight local time, take a look at the cafés at the airport. Many of the seats will be occupied by high school-ers in their study groups

China, India and the rest of SE Asia send many of their top scholars to Singapore too, and I can tell you from being in the same classes as these people for more than 4 years that they have the same formulaic way of doing well at math.

What do I think about this education setting? It's definitely great for basic applied math skills: that's why our physics and chemistry papers can afford to be heavily quantitative; and give any average student here something like the SAT II and he/she will slaughter it. 11% of us have a perfect score on the SAT II Math paper.

But if you give the average student something like the AMC papers, or any Olympiad paper, you'll find that he/she will underperform, because the questions on these don't have formulaic, or formulaic sequences of, solutions. As someone has pointed out, that's why many Asian countries' Olympiad teams have poorer medal tallies than USA's consistently. (Note: China's performance on the IMO, on the other hand, should be attributed to its huge population rather than the strength of its education. They have the resource pool to pick the few dozen people who can be bothered to go home, on top of their heavy workload from school, spend hours cracking problems on http://www.artofproblemsolving.com/" [Broken].)

Now, what's the point I'm driving at? I think that no one should be deceived that implementing an 'Asian' education in US/Europe will solve the problems laid out by Lockhart: these problems are just as pertinent over here.

It's true for the high schools in the States that a more rigorous/intensive curriculum that requires more time devoted to homework/rote memory will indeed grease the cogwheels of the economy and industry to meet growing competition from China. But this doesn't address the problems of declining appreciation for math as an art, language, field, or community that has a rich history. Even at the tip of USA's Olympiad program, many high school students are merely taking part to earn them a place in the most prestigious colleges - this for example, can't be solved by implementing a more challenging syllabus. This growing challenge besets all of today's educators in math, not just in the USA.

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There's a lot of research on what makes a hobby/job fun - see the positive psychology literature. That 'fun jobs' will not be well paid is just plain wrong. Surgery has been found to one of the most fun jobs, it has a nice balance of challenge and novelty that neatly matches improving skills, given some talent. Surgery, performed by a skilled practitioner, is perfect for achieving the state of flow that is essential to fun in a job or hobby. But don't worry, physicists, you don't need to become doctors. Solving mathematical problems can provide flow, as can designing and setting up experiments. But, then again, so can almost anything, as long as it is suffciently challenging, but not too challenging - from yacht building to reading a novel. So which to choose? (Nice problem!)

You are delusional if you really believe in this. It might apply to a small minority but in my experience most actually likes the pseudo maths taught early better than the real maths taught later, people in general do focus on the real world and the further away something is from it the less people likes it.
Maybe I'm delusional, but until something different is attempted we really can't determine whether that is the case.

When people prefer "pseduo math" to "real math" I feel that this too is because they have never been taught how to understand math. They still see math as a mechanical activity, except now it's a mechanical activity together with a bunch of "tricks". People may memorize entire proofs word by word, or remember very specific tricks. A good student will when he encounters something he couldn't think of himself stop and ask himself what kind of motivation went into this and what the general thought behind it is, and then when later asked to reconstruct a proof he can reconstruct it on the fly because he understands all the techniques needed.

Take something such as integration by parts. People learn mnemonics for it, write it on cheat sheets, etc. and in my experience it's one of those formulas that people in calculus 1 are most scared of. However people should realize that it's really just the product rule of differentiation being integrated, and few people seem to have trouble with the product rule. If people just understood this they wouldn't need to remember a proof since it would be obvious.

Similarly in linear algebra people set great store by finding a formula for a linear transformation given the corresponding matrix, and conversely finding a matrix given a linear transformation. The important part is not the procedure, but that there is an isomorphism between the space of linear transformations $F^n \to F^m$ and the m x n matrices with entries in F. Once you know this the procedure should be pretty obvious (for instance just evaluate at the n basis elements F to get the columns of the matrix). People are presented with this viewpoint, but due to exercises and exams testing them on the procedure not the theory behind it, often people only remember the procedure and not why it works or what's so important about it.

A math freshman at my college mentioned to me how he was confused on an algebra exam because question 1 said "Give an argument showing that ..." and question 2 said "Give a proof for the fact ...", but he was unsure exactly what to do. Basically he wondered how rigorous an argument has to be compared to a proof. The answer is of course that in math an argument and a proof is exactly the same thing. There is no such thing as a somewhat rigorous argument. There is a correct argument (aka a proof), incomplete arguments and motivation. I feel many people have this notion that a proof is something very mathematical and when they are asked to give one they start thinking what kind of proofs there are (contradiction, contraposition, direct, by cases, etc.) But ask the same person why tic-tac-toe can't be won against a good opponent, and he would just start experimenting and probably start with something like "well if I place my first piece in a corner, then, ..., if I place it in the center then, ..." this is actually a proof by cases, but it comes intuitively to them, and often a few arguments by contradiction are also used. People need to see math in the same intuitive manner, and not think of it as symbol manipulation even if at a technical level that's a valid viewpoint.

Actually I feel that in early grades analysis of various games is an excellent way to both excite young students and teach them what math is about, not what technical machinery has been introduced through the ages. Even people not interested in math seem to enjoy learning about a good game. For instance in the tic-tac-toe example there is a great deal of symmetry which will teach students how to exploit this kind of structure in an argument. It's also easy to miss explaining how you deal with a particular case and this kind of error will teach the students to think through carefully whether their proof really works.

Maybe your experiences differ, but in my opinion people enjoy a good thought-experiment as long as you remember not to call it a thought-experiment or math. They have learned that they do not enjoy math, but they can often enjoy a discussion which is essentially of a mathematical nature, but with the machinery removed.

Ask a senior in high school how to add two positive integers of 2-5 digits and he can probably do it and has probably done it hundreds of times, but ask him to explain why to procedure he uses is correct and he will probably not be able to give you a satisfactory answer. He will probably not even try, but what should be the case is that he should immediately start experimenting and thinking. Maybe start by thinking of the case where there are no carries, or maybe just consider 2-digit numbers. We need to add some creativity into math, because until college (and to some extent in many intro-college courses) it is severely lacking.

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As a mathematics teacher, this essay help me at this moment. I’m currently developing a set of intervention and extension exercises for grade 6 students. And now I feel so unclean, knowing that I’m not teaching maths at all, in fact I’m burying the maths under a method and calling this mathematics.

Remember schools are to education what lawyers are to justice.

Schools are there to look after the kids while parents go to work, to give local politicians something good and wholesome to support and to give teacher's unions a reason to exist.

If the children learn anything it's pretty much a happy accident.
How would you suggest students learn mathematics?

I'm not disagreeing with you, but it seems that if schools are not (or are unwilling to) educate students, that job would fall to the parents, who may not have the requisite knowledge to teach math.

Are you suggesting a less-rigorous, more informal curriculum would work (i.e. learn to solve problems as they come up, but allow students more freedom to learn what they want)? If so, I lean towards agreeing with this, although it would cause problems for teacher's unions, universities, working families...

It's an old complaint, in 1900 British scientists were complaining that British education in science,particularly chemistry and engineering was so far behind Germany's that British industry would never be able to compete.

A century later it's clear that British chemical and car industries have nothing to fear from Germany's. America is in the same boat - it was able to invent the atomic bomb and go to the moon without relying on foreign education.
I wonder how many of the people who made those things possible were "pure" Americans?

I think the fundamental problem is not the students (on some level, it is), but it is the teacher's fault and the media.

Media says doing Math is for geeks, delinquents thinks it is true, and they hate math.

Incompetent Math teacher discouraging a talented Math student, Math student hates teacher, Math student hates Math.

thrill3rnit3
Gold Member

Take a look. I think his method is pretty interesting.

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eof
I remember having read Lockhart's essay a while back. I think my biggest issue with math is that we do not tell kids why the boring stuff we are learning should actually be appreciated. With this I mean Alfred Whitehead's famous quote:

Civilization advances by extending the number of important operations which we can perform without thinking about them.

A case in point would be addition. It would have been great if in later classes someone would have e.g. told me how this boring procedure that was done over and over might not have always been so trivial. With this I mean the following observations:

1. The reason why addition is simple is because of the base-10 system in which we write numbers (any other base would do too, so the key idea was the invention of based numbers).

2. Try adding by using e.g. Roman numerals.

3. When doing (2) you will actually cheat, because in your head you add in base-10.

4. The name of numbers in our language are in base-10 which makes them easy to add. Some ancient languages had random names for different numbers and could only count easily to some limit (i.e. up to where they had words for these numbers). We can easily add e.g. 2+3 but if the numbers where e.g. 27+35 and all our numbers had distinct names without a pattern say up to 100, could you do it?