The child was stuck on the intuition that the argument must change in some way (i.e. that the function was linearly typed).

A function with a linear type uses it&#x27;s parameters exactly once.  The concept of &quot;linear type&quot; doesn&#x27;t relate to a the concept of &quot;linear function&quot; except in an abstract mathematical way.

I think the problem is that our intuition is more or less based on a mechanical world in which the concept of a side-effect free or ineffective input is foreign.

I&#x27;m curious what you mean.  Is the problem that it is a constant function, or that you are asking them to ignore parameters?  Or is that missing the point, still?

One of those functions is to consume nothing and returns their (the programmer&#x27;s) current age.This is - or at least might be - ambiguous. Current age as of implementation time or current age as of function invocation time? And while a result in whole years is a pretty natural and obvious choice for the age of a person in absence of an explicit requirement, it is certainly not the only sensible choice.

I have a series of programming assignments in my introductory course that has students write simple functions in Python. One of those functions is to consume nothing and returns their (the programmer&#x27;s) current age. So for example, you might write<pre><code> def get_my_age():
 return 21
</code></pre>
The students&#x27; reaction to this in office hours can be... interesting. I am curious how many of them actually follow the comment at the bottom of the question, and what they think:&gt; Reflect on the difference between creating a variable with a constant value and creating a function with no arguments and a constant return value.Up to you to decide if this has any pedagogical value. The real learning objective was to &quot;write a function that has no arguments but returns a constant value&quot;, so getting them to draw big conclusions about the nature of functions and values and so on is outside the scope of this little activity.

That section reminded me of how constants are implemented via functions in Perl e.g.<pre><code> use constant PI =&gt; 4 * atan2(1, 1);
</code></pre>
under the hood becomes<pre><code> sub PI () { 4 * atan2 1, 1 }
</code></pre>
<a href="http:&#x2F;&#x2F;perldoc.perl.org&#x2F;constant.html#TECHNICAL-NOTES" rel="nofollow">http:&#x2F;&#x2F;perldoc.perl.org&#x2F;constant.html#TECHNICAL-NOTES</a>

What I find surprising is how easy it is to understand functions as being made up of other functions (+), as opposed to functions that are not (constant functions).

&gt;\lambda x.\, 6 was also surprisingly difficult for him to guess (though he did get it right eventually). I think he was just stuck on the idea of the function doing something arithmetical to the input, and was having trouble coming up with some sort of arithmetic procedure which would result in 6 no matter what you put in! It simply hadn’t occurred to him that the machine might not care about the input. (Interestingly, many students in my functional programming class this semester were also confused by constant functions when we were learning about the lambda calculus; they really wanted to substitute the input somewhere and were upset&#x2F;confused by the fact that the bound variable did not occur in the body at all!)Just goes to show, our intuition works on linear types.

And it is endlessly entertaining to see just how quickly their minds can wrap around a subject you would think is too advanced for them. Its a matter of establishing a common language. Seeing that light go on in their eyes is unbelievably satisfying.

It&#x27;s fun to talk to kids! Many of them are way more curious than most people are.

I&#x27;d think it would also be a great opportunity to have access to a young, malleable mind that&#x27;s unencumbered by a significant body of past experience and preconceived notions. Every answer is like a pure function...it&#x27;s derived from first principles rather than coming an existing corpus of &#x27;state&#x27;. It seems like a tremendous opportunity to expand your own viewpoint with perspectives that you&#x27;d naturally ignore based on your own past experience.A good friend of mine was the grandson of a couple that ran a school for gifted children. When they retired, there was a video produced about their careers and, at one point, they were both asked what they liked best about their jobs. His grandfather answered, &quot;Getting to speak to the children every day.&quot; His grandmother answered, &quot;Getting to listen to the children every day. I always liked her answer better than his.

Out of curiosity, did you mean you have eight kids, or your kid is eight years old?Because if it&#x27;s the former, I&#x27;m impressed you have even one second to be on HN!

If you do have 8 kids, that&#x27;s amazing.  I have 5 myself and most of the time I feel more like a zookeeper&#x2F;riot policeman than a dad.

And, kids are fascinating conversationalists

Dad of 8 here. It makes life so sweet when we acknowledge the child&#x27;s capacities. Notice it didn&#x27;t take that much patience...it&#x27;s almost just a matter of creatively overcoming a language barrier.

When my daughter was 10 we were waiting in line to checkout at a Home Depot. She asked me what was algebra. I think she had heard the older kids mention it. I responded with a question. &quot;A plus B = 10, and A minus B = 1. What are the values of A and B?&quot; She puzzled it over while I check out. Then her face lit up like a whole realm of knowledge had just opened up to her, and she proudly told me the answer. It&#x27;s a special moment that we will both always remember. She told me that as a camp counselor that she has challenged younger kids who seemed bored with the same problem.Do challenge your kids intellectually beyond their years and you might be pleasantly surprised. My daughter heads to CERN in two weeks to study anti-matter, and I have no doubt that our brief intro to algebra at a Home Depot has a small role to play in that journey. Maybe Star Trek did too :)

As a parent, I agree. One should not shy away from talking about &quot;advanced&quot; topics. Kids are naturally curious and bright and I think we should encourage their desire to learn and understand complex things.I worry that if you try to dumb down things for kids, they might become interested in dumb things. :)Also, as the OP mentions, it can be a fun &quot;pedagogical challenge&quot; to try to explain free theorems or turing completeness or MySQL sharding to a young child. And you may find a clever way to describe it, that they can easily understand, which is satisfying for both of you.

<a href="https:&#x2F;&#x2F;www.amazon.com&#x2F;My-Little-Big-Math-Book&#x2F;dp&#x2F;9198282603" rel="nofollow">https:&#x2F;&#x2F;www.amazon.com&#x2F;My-Little-Big-Math-Book&#x2F;dp&#x2F;9198282603</a>I have not read this, but it&#x27;s from the same person who helped create &quot;anchor modeling&quot; and I find that quite interesting.

Related, &quot;Math from Three to Seven&quot; by Alexander Zvonkin[1] is an excellent book on discussing math with young kids.[1] <a href="https:&#x2F;&#x2F;bookstore.ams.org&#x2F;mcl-5&#x2F;" rel="nofollow">https:&#x2F;&#x2F;bookstore.ams.org&#x2F;mcl-5&#x2F;</a>

I often tell my kids &quot;I&#x27;ll explain that when you&#x27;re 16&quot;, or &quot;you&#x27;re too young for that, go out and play&quot;. This has proven to be very effective in motivating them to think for themselves, or ask someone more collaborative (s.a mom)

No, he asked &quot;What are you reading?&quot; and followed up with the question “What are free theorems?”

No. Children like to copy their parents, and want to learn about what they do. If a six year old wanted to play frisbee, he probably would have asked to play frisbee. At the very least he would&#x27;ve made his boredom clear pretty quickly. Yes, he wanted attention from his dad, but best of all was being able to learn about dad&#x27;s grown up stuff by doing something they both enjoyed.

Yes, those are equally suitable choices as you say. Why are you implying that stopping to play frisbee would have been the more &quot;equally suitable&quot; choice in this situation?

The kid was crying out for a math education!On a serious note, you may be right in that it&#x27;s a simple call for attention, but at least with my kids those are the opportunities for the best instruction, because the kid is usually bored, curious, and wants attention. If you give said attention, and answer questions in a way that they can think through and reason about, they will grok stuff you never expected.

Who&#x27;s to say you can&#x27;t talk about functional programming while playing frisbee? Who&#x27;s to say that 9&#x2F;10 times the kid gets some rough housing when he asks?And much more likely: who&#x27;s to say that the overstimulated little monster that asked 15 questions in 5 minutes doesn&#x27;t need a little dose of adult life to encourage some self-play?

The kid got the attention and some more. How would frisbee be any better?

Note kid asked &quot;dad what are you doing&quot;, not &quot;what is functional programming&quot;. That&#x27;s a simple call for attention, and it was equally suitable to close the book and play frisbee with him.

Huh, that&#x27;s the first time I&#x27;ve heard of unschooling, it seems like a really cool concept. If you don&#x27;t mind me asking, where do you live and what are the laws like around unschooling there? How are people&#x27;s reactions?

My problem is unschooling is that just because you’re not interested in something (as a kid) doesn’t mean you shouldn’t learn it.It’d be nice if traditional school had a better balance where personal interests could be explored in a structured and formally accepted way.

I&#x27;m a father of two girls, ages four and nine.Explaining things to them and seeing the dawn of understanding in their eyes in easily my greatest joy in life. As a parent, I basically get to relive this xkcd[0], over and over, every day.We take it to an extreme - we unschool[1]. We do our best to treat our daughters as any other member of the family, expecting to act as adults to the extent that they&#x27;re able to do so. My nine-year-old spends much of her days right now playing Roblox and Star Stable, but even that is punctuated by her coming to us with random questions about things she&#x27;s interested in. If I&#x27;m not head-down on a project I take the time to explain as best I can. If I&#x27;m otherwise occupied I always at least take the time to say &quot;Go search for &lt;insert keywords&gt;&quot; and follow up with once I&#x27;m free.The results have, so far, been incredible. Our oldest was a late reader by the standards of the testing done in government schools, but once she ran into something she wanted to do that could only be accomplished by reading, she achieved fluency more quickly than I would have ever expected. For what it&#x27;s worth, that &quot;something&quot; was an online RPG where she had to do quests to progress. In order to do that, she had to be able to read the quest text.These days she&#x27;s engrossed in the &quot;Pegasus&quot; series by Kate O&#x27;Hearn, and devouring them at a rate of ~1k pages per week. She&#x27;ll have finished the series by the end of this week and I&#x27;m hoping she&#x27;ll pick up Asimov&#x27;s &quot;Norby&quot; series next. If not, she&#x27;ll find something else that interests her and continue reading well into the wee hours of the morning I&#x27;m sure.0: <a href="https:&#x2F;&#x2F;www.xkcd.com&#x2F;1053&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.xkcd.com&#x2F;1053&#x2F;</a>
1: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Unschooling" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Unschooling</a>

My teenage son is a musician and is taking AP Music Theory right now.I ask him to tell me about what he is learning, and he patiently tries to explain it to me, but I just nod along because I have no idea what he&#x27;s talking about.

Yes, it&#x27;s notable that Brent&#x27;s son first was interested enough to ask him and then stayed interested as he explained.You can&#x27;t start telling your child about maths if they&#x27;re busy doing something else. All you can do is wait for a moment when their curiosity spots something and try to feed that.Even when you catch their interest it&#x27;s easy to break it if you can&#x27;t explain it, so los of kudos to Brent for navigating it perfectly.

&gt;And it gets harder with each kid, at least for me.That is the most depressing thing I&#x27;ve read in a good while.

&gt; One of the things you figure out is that some kids aren&#x27;t interested. And if your kid isn&#x27;t interested in thinking about prime numbers, you&#x27;re going to be hard pressed to change that. Conversely they will have interests that you don&#x27;t share, and it&#x27;s going to be harder for you to participate meaningfully in that beyond being generally supportive. They are individuals that way.I grew up in a family with 7 kids and my youngest sibling is just 12 years old. I remember when my younger set of siblings were born, I was thinking &quot;I&#x27;m going to teach them EVERYTHING I&#x27;ve learned at a young age, and they will be way ahead of their peers.&quot; I failed to grasp just how much of their personality is ingrained in them from their DNA and that they might not have all the same interests as me. It was a great learning experience though -- when I do have kids, I&#x27;m not going to try and shove my interests down their throat. I&#x27;m going to pay attention to what they are drawn to and give them as many resources as I can so they can pursue that interest as much as they desire.

It&#x27;s an immense amount of effort to do it consistently. And it gets harder with each kid, at least for me.One of the things you figure out is that some kids aren&#x27;t interested. And if your kid isn&#x27;t interested in thinking about prime numbers, you&#x27;re going to be hard pressed to change that. Conversely they will have interests that you don&#x27;t share, and it&#x27;s going to be harder for you to participate meaningfully in that beyond being generally supportive. They are individuals that way.

Maybe it helps being rich, but just as a counter example, my grandfather was never rich. He was a clock-in&#x2F;clock-out guy and still managed to spend a lot of time with me while I was growing up. He had a lot more patience than most of the &quot;white collar&quot; adults in my life ever did, and taught me lots of useful skills like how to fix a lot of electrical or mechanical things.

It&#x27;s not &quot;being rich&quot; that helps, it&#x27;s having a high hourly rate of pay relative to expenses. Many rich people work 60+ hr weeks with travel, and don&#x27;t see their kids much.  Many non-rich people are stay-at-home parents or extended family.
Of course, they can hire nannies, governors, and tutors to fill in the gaps...

We actually made a decision to cut our income considerably in order to be as involved as we can in our kids upbringing - we&#x27;re basically almost always able to go to school concerts, we always eat together, we practically never aren&#x27;t with them at bedtime, etc..It&#x27;s great. But I think we missed the balance a little - there are so many life experiences that are out of reach that we parents experienced when we were kids. Yes being tight with our kids is awesome - but when they get old enough to spread their wings a little, and when they&#x27;re inquisitive about new things, then supporting that is exceedingly difficult when you&#x27;re of low means.School skiing trip? Not a chance, way out of our price bracket. Holiday abroad? Same, the interest is there, they&#x27;re keen to learn geography&#x2F;languages&#x2F;culture. Piano lessons? Same, we&#x27;ve got a keyboard, one child is really keen and show some ability but we don&#x27;t have means to support him in that and let him find fulfilment through that creativity.But I do get to talk Fibonacci series; build fires in the woods; teach them about how aliens with 0, 4, 16 fingers count; but buy an up to date globe, or take them for a train ride, or go on a boat, or have a pet, or visit a mountain, ...My problem is not time so much, nor fostering inquisition, but resources to develop the questioning in to solid foundations. Our kids are not the free-spirits of knowledge-hunger we anticipated because we instead have to follow economics.

One thing that makes this easier is being rich. What I mean by that is for example, when my kid is asking me questions or wants to &quot;do it myself!&quot;, the fact that I have a flexible schedule helps immensely, and that flexible schedule is possible because I can turn down work that doesn&#x27;t give me that flexibility.If I had a strict clock-in&#x2F;clock-out job, and only had limited time to run errands, I&#x27;d have a lot less patience for waiting for my kid to do stuff or answering her questions.I consider myself very lucky that I have that privilege and can pass that on to my kids.

Exactly. My 4 year old sometimes surprises me like that. Last week she asked me how you can talk into a mobile phone. So I said, there&#x27;s a tiny transmitter in there, and it sends the voice over waves in the air. So she&#x27;s like, &quot;OK&quot; and I thought it went over her head.Then yesterday she asked: &quot;Why couldn&#x27;t you call gramps when we were on the plane?&quot; I told her that the mobile phone wasn&#x27;t close enough to the receivers on the ground.It amazed me that obviously some facts had been stewing for a week in that tiny head. And out comes another question.

I have three very inquisitive kids and I love trying to explain complex topics, like reinforcement learning, to them in a way that they can understand. The best thing I get out of it is clarity for my own understanding, or gaps&#x2F;lack of clarity, so it&#x27;s a valuable exercise.People don&#x27;t give children enough credit for what they can really understand.I remember distinctly teaching my oldest daughter how to do a basic cipher - like direct substitution and she got it immediately. I also taught my son how to do pin bumping and pin counting&#x2F;picking on locks with my lockpick set. I even bought him a transparent set of master locks to practice on and he would sit for an hour at the age of 4 picking those locks. The obvious downside now is that he knows how to get into everything!Very cool to watch what are basic principals being applied at the very basic level.

When my mother died, I was asked to think of one word to describe her. The best word that I could come up with was &quot;teacher.&quot; One quick example: when my own child was 2 or 3, we spent some time with Grandma at Yellowstone National Park. Grandma walked with her grandchild through the park and explained geisers, plate techtonics, and many other topics of geology. I rolled my eyes and thought, &quot;Really? He can barely speak and you are delivering a college lecture?&quot; But then I remembered all the college lectures of my own youth. It&#x27;s amazing how much even the youngest minds can absorb.One of my biggest failings as a parent has been to assume that my children will learn things somehow without me teaching it to them. If anybody learned 1&#x2F;3 of what my mother taught them in detail, they learned a lot.

What is truly astounding about this is the patience to even attempt to explain free theorems to a 6 year old. Most parents would likely answer &quot;math&quot; and that would be the end of it. I don&#x27;t (yet) have kids, but when I do I hope I have the wherewithal to recognize and take advantage of moments like these.

EXAMPLES is one of the primary categories of man pages. Your broader problem is that &quot;The full documentation for * is maintained as a Texinfo manual&quot;, and most people never learned how to access and navigate those.<a href="https:&#x2F;&#x2F;www.google.com&#x2F;search?q=%22The+full+documentation+for+*+is+maintained+as+a+Texinfo+manual%22" rel="nofollow">https:&#x2F;&#x2F;www.google.com&#x2F;search?q=%22The+full+documentation+fo...</a>

This is my biggest complaint with manpages, historically. Many of them don&#x27;t have examples and only describe the command switches in a list.

&gt; “A function is like a machine where you put something in one end and something comes out the other end. For example, maybe you put a number in, and the number that is one bigger comes out. So if you put in three, four comes out, or if you put in six, seven comes out.”Examples are such a useful tool to explain complex subjects and make them approachable.

&gt; Does anyone collect these educational inspirational stories?That might be really useful. Maybe one barrier to improved education, is people having difficulty visualizing it?Last summer, I tried to explore teaching atoms in early primary, with a focus on nucleons and nuclei.[1] One big surprise to me in discussions, was a pattern of non-linear response to my saying &quot;teach nucleons to kids&quot;.One physics postdoc, one with a professional focus on introductory physics education, replied basically &quot;Absurd! How could you teach muons to kids!&quot; It turned out that for them, &quot;teach nucleons and nuclei&quot; meant understanding muons. &quot;Teach ... to kids&quot;, wasn&#x27;t enough to change that image. So they tried to imagine teaching muons to kids, and exploded.Another physics postdoc tried to picture kids chalking quantum math on some graduate seminar blackboard. And was &quot;WTF?&quot; baffled.A primary-school teacher heard &quot;teach atoms&quot;, and tried to picture early-primary students memorizing and regurgitating 4th-grade paragraphs of definitions. Because for them, that&#x27;s what &quot;teach atoms&quot; meant.And more. These weren&#x27;t outliers.I ended up using an analogy. That it was like proposing to teach kindergarteners about traffic lights. And having highway engineers declaring &quot;Absurd! There&#x27;s no way kids could understand those standards documents! And they&#x27;re expensive! And the Chinese ones are in Chinese! You&#x27;re think you can teach technical Chinese transportation standards documents in kindergarten!?!&quot;&quot;Red means stop; green means go.&quot; &quot;Atoms are sticky little balls that jiggle.&quot;I&#x27;d expected that crafting accessible leverageable descriptions would be hard. And that how well one could achieve transferable understanding, at various ages, would be an open question.But I didn&#x27;t expect that people would initially be so dramatically unable to picture the possibility of teaching some topics.People were fine after a &quot;kindergarten traffic lights ... green means go... not network theory&quot; story to set context. But I wonder...Might there be a badly underappreciated need for stories like OP? To provide &quot;Oh wow... I wouldn&#x27;t have imagined one could teach that... Now I can picture it...&quot;That&#x27;s an idea for helping to disrupt education that I&#x27;ve not seen before.[1] a crufty slow-loading page, intended merely to set up in-person discussions, but I&#x27;ve nothing better: <a href="http:&#x2F;&#x2F;www.clarifyscience.info&#x2F;part&#x2F;Atoms" rel="nofollow">http:&#x2F;&#x2F;www.clarifyscience.info&#x2F;part&#x2F;Atoms</a>

beautiful read, thank you. Does anyone collect these educational inspirational stories?

This brings me back to ~20 years ago when my dad was explaining me Bresenham&#x27;s circle drawing algorithm. I was 8 at the time I think.

<a href="http:&#x2F;&#x2F;worrydream.com&#x2F;AlligatorEggs&#x2F;" rel="nofollow">http:&#x2F;&#x2F;worrydream.com&#x2F;AlligatorEggs&#x2F;</a>

Very true, and I did actually explain that to him.  In my rush to tell a good story I left out the pedantic part.

This is something I saw over and over again long before I had a serious interest in technology or science at all. It’s an attitude that’s rampant in liberal arts courses.There are “kiddie” versions of everything from History to to Music Theory to Grammar. It used to really frustrate me in undergrad how many things taught as almost moral absolutes were upended the following semester.I sort of get it that even in junior high and high school, the teachers don’t really know anything beyond the kiddie version, so that’s what gets taught.But at the college level, there’s no excuse for that. There’s a certain amount of science envy that some liberal arts professors have. They want to be able to say that there is one and only one way to understand a historical event or way to compose music.Milton Babbit was famous for this in the music composition world and argued that music needed to be “elevated to a higher form like that of math or physics” and that no one expects advanced math to be accessible to lay persons. Therefore music shouldn’t be comprehensible by anyone who doesn’t have a PhD in music theory or composition. This was actually a pretty popular attitude in the “classical” music world from the 1950s to the early 80s, and the music written during that time reflects it.Sorry to get so far off on a tangent, but while that specific fad has come and gone in music composition, there remains an inferiority complex among liberal arts academia that is extremely problematic and a general attitude among professors of all stripes that some people just can’t handle the truth: that outside of religion, there are no absolute truths.Getting back to the Hendrix quote, I absolutely agree. Teach children as much of the truth as you know. They can take it. In my own experience teaching violin, I’ve found that I get the best response when I treat the students the most like adults. About 20 years ago I started bringing in little vignettes about music theory and history with my youngest students (4-6 years old) as a break between lesson segments where we were focusing very intensely on specific violin techniques. When I realized that they enjoyed both the context switch and the color commentary about why exactly we do these kinds of things, I brought more and more of that to the table, and it’s been very successful.Obviously, this isn’t a robust study, it’s not meaningful in a larger scale, and it doesn’t prove anything at all.But if the theory is that young students can handle a lot of uncertainty and a lot of advanced material at a young age, then my experience is, at least, not evidence that it’s false.Deep down inside, I think that we need to be a lot less careful, protective, and hand-holdy about what we expose our children to. They can handle it.

The original is correct, it says &quot;apples&quot; not &quot;apple(s)&quot;.

&gt; I told him that a number X is composite if you can fit X apples into buckets and end up with the same number of apples in each bucket.Wrong; X is composite if you can fit X apples into buckets and end up with the same number of apples in each bucket, with at least two apples in each bucket.&quot;You gotta tell the children the truth. They don&#x27;t need a whole lot of lies. Cause one of these days, baby, they&#x27;ll be running things.&quot; - Jimi Hendrix, &quot;Straight Ahead&quot;

&gt; Do you have a link to this paper on arXiv?This part was sarcasm.

&gt; Who knew?!?Your six year old came up with a new recursive prime generator over the course of a conversation introducing him to the concept of primes, and then submitted this as a paper to arXiv? Do you have a link to this paper on arXiv?

FWIW, &quot;rectangles&quot; is a visually&#x2F;geometrically natural way to express &quot;buckets of equal size&quot;.And &quot;1xN doesn&#x27;t count&quot; can be expressed nicely as &quot;it can&#x27;t just be a line&quot;, or &quot;the line must be folded up into a shorter rectangle [because more compact shapes are nicer]&quot;

When my son was around five or six, I did a similar exercise to explain the difference between prime and composite numbers.I told him that a number X is composite if you can fit X apples into buckets and end up with the same number of apples in each bucket.I drew 6 apples and asked whether they could be divided into buckets with an equal number of apples in each. It was immediately obvious that they could. My son circled one group of three apples, then another group of three apples.Then I drew 7 apples. He realized that you couldn&#x27;t divide 7 apples into baskets and have each basket contain the same number of apples.Voila! A prime number.Then we started talking about the frequency with which prime numbers occur on the number line, how important they are to fields such as cryptography, and the existence of dense Bragg peaks among primes in judiciously chosen intervals. By the end of the conversation, he had discovered a new recursive prime generator and submitted a paper on it to arXiv. Who knew?!?

I actually really like the look of that. One of my friends is learning to program (in Python) and has recently taken an interest in FP. Planning to show this to her.

One of the folks behind that is also involved in this:<a href="http:&#x2F;&#x2F;www.bootstrapworld.org" rel="nofollow">http:&#x2F;&#x2F;www.bootstrapworld.org</a>They&#x27;re claiming to get good results teaching young people a number of important topics, including coding. They&#x27;re also either using Pyret or planning to use it.

Maybe Pyret?<a href="https:&#x2F;&#x2F;www.pyret.org&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.pyret.org&#x2F;</a>

Almost anything in the End User Development research field.

I don&#x27;t think state is the problem. I think the visibility of state is a problem. In most programming, state is some nebulous thing that you really only get to see in a debugger. Otherwise, you&#x27;re left reasoning about what you think the states are and can be. How many bugs trace down to statements like, &quot;This data shouldn&#x27;t be here.&quot;That&#x27;s why Excel works for so many people -- the internal and intermediate states can be as visible as the user needs or desires them to be. And those users seem to have no problem chaining together multiple cells into what is essentially a very complex function.

Familiar? Seriously?!? That syntax is not familiar to non-programmers.&gt; Math should be taught in prefix notation if u ask me.Perhaps it should, but it&#x27;s not.Or was this whole post sarcasm, and it was done so well that I missed it?

Lisp is perfectly suited for this task due to the minimal and familiar syntax to non-programmers. Math should be taught in prefix notation if u ask me.

Can you please explain why Joy is simple to a non-programmer ? because all i see is verbose language, and i know some programming.

Yes: Joy.<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Joy_(programming_language)" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Joy_(programming_language)</a>(Shameless plug for my own implementation: <a href="http:&#x2F;&#x2F;joypy.osdn.io&#x2F;" rel="nofollow">http:&#x2F;&#x2F;joypy.osdn.io&#x2F;</a>)

Side question: maintenece of functional programs is easier, because you don&#x27;t need to think about state.Could it be used to create a functional programming language, easily understood by non-programmers or beginners , so they too could take a program and adapt it to their needs ? Does such language exist ?

This is the best thing I&#x27;ve read in really long time.

The link to the article is not working for me right now, so I cannot see how they define functional programming. In my years of programming I have seen the concept of functional programming defined in one of two ways: in context to call&#x2F;response or in context to containment&#x2F;scope.The call&#x2F;response perspective emphasizes that functions are instantiated against a reference that can be called and that such a call returns a value. The containment&#x2F;scope perspective emphasizes that functions are a bag of instructions containing a child scope and thus a lexical structure if nested. While both descriptions are valid they produce wildly different explanations and code examples that are often completely unrelated.

Yes. There is an old piece of sage advice: learning Lisp will make you a better programmer, whether you actually end up using it or not. Any paradigm you can think of is really just something you could do in Lisp...

I&#x27;ve grokked programming only after being exposed to Lisp. Before Lisp, learning how to programme seemed like a never ending exercise. After, I realised I already know everything I need to know - data in, data out…

The comment above is condescending and misleading.Being precocious doesn&#x27;t mean autism, and no professional SLP would &quot;freak out&quot; at an early reader.Being smart is an &quot;abnormality&quot;, and autism is not &quot;handicapped general intelligence&quot;.Warning signs for autism include obsession with letters and numbers without understanding meaning, not being able to reason about math and discuss mathematical topics.

I mean... honestly, understanding functions is not overly difficult. The only reason we don&#x27;t expect children to understand them is because we don&#x27;t teach it to them. On the other hand, we teach them bizarrely detailed and extensive algorithms for doing all sorts of computations on the rational numbers, and expect them to be able to do them perfectly.

Such precociousness is often[1] an indicator of abnormalities and handicapped general intelligence. For example, if our son independently began to read too early the initial reaction of my spouse, a Speech-Language Pathologist, would be to freak out rather than to be filled with pride.Perhaps you know this and used &quot;gifted&quot; in its more euphemistic sense. In any event, I mention this not to correct you but to address the parental anxiety engendered by examples such as you gave.[1] Depending on the activity and age of onset, they can be so strongly correlated that attentive educational professionals would recommend diagnostic tests to identify any condition (often some category of autism) and permit early intervention to minimize the negative developmental aspects.

I had a somewhat similar encounter with a 4 year old (!), who blew me away with his ability to solve arithmetic puzzles:<a href="http:&#x2F;&#x2F;heinrichhartmann.com&#x2F;blog&#x2F;2016&#x2F;06&#x2F;12&#x2F;Box-Counting-Arithmetic.html" rel="nofollow">http:&#x2F;&#x2F;heinrichhartmann.com&#x2F;blog&#x2F;2016&#x2F;06&#x2F;12&#x2F;Box-Counting-Ari...</a>It&#x27;s a great joy to work with such gifted young minds.

Cool game; used to do something like that with kids, but more in context of sequence patterns. still, the game is limited to functions that are specified intensionally. Extensionally specified functions with no readily identifiable intension (like a function whose pairs are the result of a random pairing) are not really guessable. Oh well those are boring anyway...

You might enjoy <a href="https:&#x2F;&#x2F;graphicallinearalgebra.net&#x2F;" rel="nofollow">https:&#x2F;&#x2F;graphicallinearalgebra.net&#x2F;</a>. The author uses Legos to describe algebras and diagrams to describe symmetric monoidal categories.

I wish someone would explain all of Category Theory like this to me.

You did CS at Hendrix College? I did too (&#x27;11) and can&#x27;t help mentioning it even though I have little to say... because hey, small school, tiny department, it&#x27;s not often you run into another Hendrix CS alum, even virtually. And after years in &quot;industry&quot; I really appreciate the surprisingly good CS education there for a not well known schoolHow&#x27;s the dept doing these days? Is Ferrer still there?

This really makes me wish there was a way to watch his courses online! Anyone know of any available?

I&#x27;ll second this; he is a fantastic teacher. I had the incredibly fortunate opportunity of learning from Brent in my undergrad at Hendrix, where he currently teaches.There are other comments here that praise his patience with regard to teaching these topics to a child. I can say without a doubt he maintains the same patience with all of his students - not just his children. If you ever have the opportunity to learn from Brent (be from a course, blog, or talk), I would highly encourage it.I&#x27;m glad to see a post of his making its rounds on HN.

I&#x27;ll third this! Brent&#x27;s class on the art of recursion really taught me quite a lot about how to think about problems in a different way.<a href="http:&#x2F;&#x2F;www.cis.upenn.edu&#x2F;~cis39903" rel="nofollow">http:&#x2F;&#x2F;www.cis.upenn.edu&#x2F;~cis39903</a>

His code is also impeccable. Diagrams [0] is a pleasure to read.[0] <a href="https:&#x2F;&#x2F;hackage.haskell.org&#x2F;package&#x2F;diagrams" rel="nofollow">https:&#x2F;&#x2F;hackage.haskell.org&#x2F;package&#x2F;diagrams</a>

Agreed! I taught myself programming, and Haskell was my second language. I was at Penn so I emailed Brent out of the blue, and he responded and was so helpful with understanding helping me understand Monoids, not at all judgmental, and so different from all the educational experience I had up until then. And it was serious gateway drug to mathmatics. Funny thing is a few years later I TAed an advanced Haskell class with Stephanie Weirich (<a href="https:&#x2F;&#x2F;www.seas.upenn.edu&#x2F;~cis552&#x2F;current&#x2F;index.html" rel="nofollow">https:&#x2F;&#x2F;www.seas.upenn.edu&#x2F;~cis552&#x2F;current&#x2F;index.html</a>), who was Brent&#x27;s thesis advisor. All started with Brent!

Took a class with Brent at Williams College when he visited for a year. Brent is awesome!! :)

This is the text Christopher allen recommended to beginners as the best reference, until he decided to write his own book on Haskell.After completing that one, he recomended the NICTA github repo <a href="https:&#x2F;&#x2F;github.com&#x2F;data61&#x2F;fp-course&#x2F;tree&#x2F;master&#x2F;src&#x2F;Course" rel="nofollow">https:&#x2F;&#x2F;github.com&#x2F;data61&#x2F;fp-course&#x2F;tree&#x2F;master&#x2F;src&#x2F;Course</a>Disclaimer: All subject to IIRC

There is an updated set of lectures from fall 2016 [0]. Although I just noted that it&#x27;s taught by Joachim Breitner rather than Brent Yorgey.[0]: <a href="http:&#x2F;&#x2F;cis.upenn.edu&#x2F;~cis194&#x2F;fall16&#x2F;" rel="nofollow">http:&#x2F;&#x2F;cis.upenn.edu&#x2F;~cis194&#x2F;fall16&#x2F;</a>

Just read Week 1 of this course - very easy to follow.
Although with a different instructor, there are later versions - <a href="http:&#x2F;&#x2F;www.cis.upenn.edu&#x2F;~cis194&#x2F;fall16&#x2F;" rel="nofollow">http:&#x2F;&#x2F;www.cis.upenn.edu&#x2F;~cis194&#x2F;fall16&#x2F;</a> seems the most recent, with plenty of updates since the spring13 course. I&#x27;ll be reading more!

Brent is one of the greatest teachers I&#x27;ve ever experienced. I used to take classes from him (and TA some others he taught) and was consistently blown away by his enthusiasm and capacity for sharing concepts. He has a great essay on pedagogy as well (<a href="https:&#x2F;&#x2F;byorgey.wordpress.com&#x2F;2009&#x2F;01&#x2F;12&#x2F;abstraction-intuition-and-the-monad-tutorial-fallacy&#x2F;" rel="nofollow">https:&#x2F;&#x2F;byorgey.wordpress.com&#x2F;2009&#x2F;01&#x2F;12&#x2F;abstraction-intuiti...</a>)If you&#x27;re interested in learning some of the beautiful foundations of functional programming, I highly recommend checking out the lecture notes and assignments from his Penn CIS 194 class (<a href="http:&#x2F;&#x2F;www.cis.upenn.edu&#x2F;~cis194&#x2F;spring13&#x2F;lectures.html" rel="nofollow">http:&#x2F;&#x2F;www.cis.upenn.edu&#x2F;~cis194&#x2F;spring13&#x2F;lectures.html</a>)

I for one really like reading about Alice and Bob. Amongst other benefits, it helps me consider how, even if currently just a theoretical algorithm, that particular protocol could end up being used in real world communications.

(A)lice and (B)ob set up a channel. 
(E)ve is an evesdropper.When the only thing you change is to personify A, B, and E in your narrative description of a logical protocol, you are not making an analogy or an abstraction.Sequence diagrams, BAN logic, UML, pseudo code and other protocol notations are not teaching analogies, they are specifications. Some more useful than others.Teaching analogies and abstractions are more of a mix of use cases, literally analogous functional relationships, black boxing, edge cases and boundary conditions. The &quot;why&quot; behind the &quot;how.&quot;That Alice and Bob can be replaced in every situation with (A) and (B) means they are not abstractions or analogies, and are therefore patronizing cartoons that obfuscate useful information.

What about a definition of protocol that allows sending messages to yourself for recursion.

I disagree with your example being a bad and reason is right there in your comment: the word &quot;protocol&quot;. From wikipedia we find that a &quot;communication protocol is a system of rules that allow _two or more entities_ of a communications system to transmit information...&quot;, where I have underscored the essential phrase and cryptographic protocols are just a subset of communication protocols.The reason that Alice and Bob are introduced is because they are different entities and each has different powers, restrictions and goals. If you want to do a correctness or security analysis of a protocol, introducing and defining these parties is hugely important and useful. Otherwise, its very easy to make mistakes about where information is in the system and how it is being manipulated. More generally, there is a philosophical approach called operationalism [1] that demands that all science theories be couched in terms of protocols so we don&#x27;t make mistakes about what they say.[1] <a href="https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;operationalism&#x2F;" rel="nofollow">https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;operationalism&#x2F;</a>

This is fantastic. Reminds me of Hofstadter&#x27;s conversations between Achilles and a Tortoise.The trouble with most math I&#x27;ve encountered isn&#x27;t the concepts, it&#x27;s the quality of the abstractions we use to represent and interpret them.The best(worst) example I can think of is the Alice&#x2F;Bob description of cryptographic protocols, where somebody&#x27;s idea of teaching is, &quot;let&#x27;s take these symbolic representations and instead of just shouting them at you slowly, I will obfuscate them with a set of conceptually unrelated words and repeat the representation word for word in a more patronizing way.&quot;Great piece of writing.

Jesus that shows you how uneven of a baggage children have when they start going to class.. this little dude has a computer science prof as his father, &amp; is doing abstract logic problems at 6 yo, for fun no less!!my cabinet maker father can&#x27;t spell a sentence without making four different mistakes in his native language, and my social worker mother literally grimaces with pain anytime something resembling math is distantly mentioned

Actually, for a rather unconventional understanding of the term &quot;machne learning&quot;, which is basically anything that makes living and working with machines easier, you don&#x27;t necessarily need to make the machine&#x27;s functionality more versatile. It might be enough to boost its user&#x27;s understanding. With more effect, too, because an informed user by far beats any RNN-based AI.What I&#x27;m actually trying to say is that brains are simply better at learning to grok machines than machines are at figuring out how to brain.

Functional programming? What he&#x27;s describing is machine learning! :)

And that is how a programmer is born.

I love to hear about conversations with people who aren&#x27;t exposed to these kinds of concepts. I think it&#x27;s interesting to hear a more simple opinion. Sometimes I talk to my wife about ideas I&#x27;ve been thinking about and she can give some helpful feedback despite not having experience with any of the underling concepts.

Mines 1 1&#x2F;2...Better have &quot;The Talk about FP&quot; soon.

I&#x27;ve failed as a parent. My oldest is 9 already, and we still haven&#x27;t discussed functional programming.Well, maybe the 3 year old still has a chance.

This is so awesome. I&#x27;m consistently amazed at how much kids can understand intuitively, even when they can&#x27;t express their understanding in precise language. For example, my partner teaches lower elementary at a Montessori school. Not everyone understands it, but she&#x27;s managed to teach some basic algebra, including to several of her first-graders.

It&#x27;s not, but do you really think that&#x27;s going to stop true FP believers from upvoting this article?

I was programming functions in my brother&#x27;s calculator when I was 8..

&gt; ...who told you that lie? An educational psychologist?<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Piaget%27s_theory_of_cognitive_development#Formal_operational_stage" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Piaget%27s_theory_of_cognitive...</a>Fine, I was off by one year.

...who told you that lie? An educational psychologist?More like &quot;at age 12 even the most incompetent teacher can teach things involving abstract&#x2F;counterfactual thinking to even the least intelligent child&quot;... the bad ol&#x27; &quot;lowest common denominator thinking&quot; and the other horrors of our standardized educational systems :| Unfortunately it&#x27;s not economically feasible to lecture to children in a classroom unless you calibrate on the lowest common denominator.Also, it&#x27;s the huuuge difference between &quot;understanding&quot; and &quot;being able to communicate efficiently in words&quot;. People can &quot;grok&quot; very advanced stuff even if they don&#x27;t fully command the language for expressing it. Unfortunately most incompetent teachers are totally incapable of &quot;trying to explain something beyond the level of the language&quot; but the human brain&#x27;s pattern matching engines are not always tied to language...

Counterfactual reasoning is a subset of abstract reasoning which is a subset of abstract thinking more generally. Children are capable of abstract thinking (e.g. pretend play) and even abstract reasoning (e.g. basic conditional reasoning) very early on. I haven&#x27;t studied this field (though my spouse has), but a cursory reading of this paper shed light on the distinctions: <a href="https:&#x2F;&#x2F;www.ncbi.nlm.nih.gov&#x2F;pmc&#x2F;articles&#x2F;PMC3582172&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.ncbi.nlm.nih.gov&#x2F;pmc&#x2F;articles&#x2F;PMC3582172&#x2F;</a>

Don&#x27;t underestimate how big a difference is made by actually being interested, not worrying about how you&#x27;ll look to the other kids in class, etc.

Uh... what? Counterfactual thinking develops well before the age of twelve. Kids that I know are asking &#x27;what if&#x27; questions in pre-K and kindergarten. It&#x27;s a natural thinking mechanism that children engage in all the time.

Functions? Algebra? How is comprehending any of that, possible without the sort of abstract&#x2F;counterfactual thinking that develops around age 12?

I&#x27;ve actually never gotten to play real Zendo, but you can make up your own class of koans, like Lisp programs or regexes. Maybe the real-worldness of arrangements of pyramids does give them more of a Zen feel?

Come to think of it, I find Zendo much harder than the game described in the article. I think it’s because the inputs (arrangements of shapes) are more complex than numbers, and the outputs (Booleans) give you less information than numbers.

If anyone wants to try, I wrote a small game like this. Input is 3 (ordered) digits, output is a Boolean.
<a href="http:&#x2F;&#x2F;laurent.le-brun.eu&#x2F;games&#x2F;rules&#x2F;" rel="nofollow">http:&#x2F;&#x2F;laurent.le-brun.eu&#x2F;games&#x2F;rules&#x2F;</a>

This game can be played in a group, too: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Zendo_(game)" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Zendo_(game)</a>&gt; once the guesser thinks they know what the function does, the players switch roles and the person who came up with function specifies some inputs in order to test whether the guesser is able to produce the correct outputs.In Zendo when the guesser offers the wrong rule, the &#x27;master&#x27; must tell them an (input,output) pair where the guess disagrees with the rule the master had in mind. Then play continues.

What a great and heartwarming story! It really puts a great example to the old rule of thumb that &#x27;you can be sure you understand the concept once you successfully explain it to a five-year old&#x27;.

I always liked to imagine a scenario where my daughter would invite me to her class to talk about what I did, and I would try to explain the concept of programming to young children. I would pretend to be a robot, and they would have to command me to do things, but I would only do them exactly as they described them.I had a similar experience when I was in school and it really stuck with me. Kids love this shit, they just need it in a way that lights up their minds a bit.

Obligatory &quot;Mindstorms&quot; recommendation: <a href="https:&#x2F;&#x2F;www.amazon.com&#x2F;Mindstorms-Children-Computers-Powerful-Ideas&#x2F;dp&#x2F;0465046746&#x2F;ref=sr_1_4?ie=UTF8&amp;qid=1525736070&amp;sr=8-4&amp;keywords=mindstorms" rel="nofollow">https:&#x2F;&#x2F;www.amazon.com&#x2F;Mindstorms-Children-Computers-Powerfu...</a>

The parent is also superlatively gifted.<a href="http:&#x2F;&#x2F;ozark.hendrix.edu&#x2F;~yorgey&#x2F;" rel="nofollow">http:&#x2F;&#x2F;ozark.hendrix.edu&#x2F;~yorgey&#x2F;</a>

If this anecdote is credible (and I don&#x27;t see any reason it wouldn&#x27;t be) then in four years this kid will ask, &quot;What are you reading?&quot; and then the father will say what he&#x27;s reading - and then sheepishly as for help from the ten year-old on the part he&#x27;s stuck on.I mean this kid is clearly superlatively gifted.

I put it at <a href="https:&#x2F;&#x2F;drive.google.com&#x2F;open?id=1hfzm3Rvm4xpwp_xHUKBHxuIEgfecZkiI" rel="nofollow">https:&#x2F;&#x2F;drive.google.com&#x2F;open?id=1hfzm3Rvm4xpwp_xHUKBHxuIEgf...</a>. Please let me know if that fails, and I’ll try to fix it.

Scribd is the worst thing on the internet. Please stop using it.

Is there a non-scribd (Google Drive) way to download the PDF?

Thanks! I think my 6yo would like doing these (though he doesn&#x27;t have a strong grasp of multiplication&#x2F;division yet).

This is fantastic, good job!I&#x27;m currently teaching an introductory programming course using Python (for people who have never programmed), and the level of abstraction you are using in your worksheets inspired me to actually update the exercises on my worksheets I&#x27;m using for the course (e.g. break down concepts even further and use more repetitive tasks to actually help students get the hang of these concepts).Thanks a lot for sharing your ideas!

This is super fun and such good timing! Thanks for sharing this.

Fun(ctional)!When my daughter was in 4th grade, I volunteer-taught a group of pull-out 4th–6th graders, so that she could have some peers to do math with.You might enjoy this set of math worksheets I created for them. <a href="https:&#x2F;&#x2F;www.scribd.com&#x2F;document&#x2F;15720543&#x2F;Squarrows" rel="nofollow">https:&#x2F;&#x2F;www.scribd.com&#x2F;document&#x2F;15720543&#x2F;Squarrows</a> [EDIT: now also at <a href="https:&#x2F;&#x2F;drive.google.com&#x2F;open?id=1hfzm3Rvm4xpwp_xHUKBHxuIEgfecZkiI" rel="nofollow">https:&#x2F;&#x2F;drive.google.com&#x2F;open?id=1hfzm3Rvm4xpwp_xHUKBHxuIEgf...</a>]I was learning category theory at the time. I gave them some element-chasing problems so they could work on the same areas I was studying, although at a different level of abstraction.We also did some stuff on permutahedrons, and relating single-character-replacement paths (I think Nabokov calls these “rooks path”, when each step is a word) from AAA → BBB, to geometric cubes, and 2³, etc., but I didn&#x27;t collect the worksheets for those into the same document.

I enjoyed it.I spend a lot of time listening to people ramble in meetings or troubleshooting, where I wish they understood the subject matter enough to explain it as if talking to a six year old.

And it works with procedural programming also! I&#x27;ve been teaching programming in schools since I was in school myself - teaching the kids who know more than their teachers, as well as the kids that teachers don&#x27;t think are ready to code.Procedural thinking is something that anybody who knows socks come before shoes can do. Conditionals can be understood by anybody who wears different clothes on a weekend than on a schoolday. Loops can be understood by anybody who can eat a bowl of soup by drinking a single spoonful and then doing that again and again until the soup is gone.Kids can definitely engage in computational thinking from a young age. We just need to offer them the opportunities to do so.

I enjoyed reading this story :) Nice buildup to the gate of complex problems.I remember that at elementary school we turned on a bunch of C16-s at the same time and we tested random number generators and some showed the same value, cool demo of determinisminstead of xλ6 I would have chosen xλ3.14159265359 
it might lead to different questions, but who knows, just deeper rabbit holes :)

&gt;&gt; if 4 is prime then all numbers are prime&gt;Is a true theorem (and things along these lines)Saying &quot;4 is prime&quot; &quot;primes are non-composite, ...&quot; reduces to error; at least one of these premises is false.?

The difficulty in conceiving of a constant functions seems similar to the difficulty in recognizing that&gt; if 4 is prime then all numbers are primeIs a true theorem (and things along these lines)

my favorite snack is &quot;&quot; ! (pronounced &quot; &quot;)

&gt; It’s called the function machine game. I will think of a function machine. You tell me things to put into the function machine, and I will tell you what comes out. Then you have to guess what the function machine does.Cool, this sounds like a general form of the picnic game[0].An input to the &quot;picnic&quot; function is a food (or other object), and the output is whether or not it can be brought to the picnic. The picnic game&#x27;s hidden function that needs to be uncovered is a predicate function. E.g. if the hidden function is &quot;Things that start with the letter A&quot;, then &quot;apples&quot;, &quot;anchovies&quot;, and &quot;albatrosses&quot; could be brought to the picnic, but not &quot;bananas&quot;, &quot;cameras&quot;, nor &quot;dessert&quot;. In that example, the function would look like:<pre><code> lambda x: x.firstLetter() == &#x27;a&#x27;
</code></pre>
[0] <a href="https:&#x2F;&#x2F;www.thegamegal.com&#x2F;2011&#x2F;06&#x2F;11&#x2F;going-on-a-picnic&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.thegamegal.com&#x2F;2011&#x2F;06&#x2F;11&#x2F;going-on-a-picnic&#x2F;</a>

&gt; (and to be fair, free theorems are only interesting when polymorphism is involved which I definitely didn’t want to go into).&quot;there&#x27;s a thing called polymorphism, and it means &#x27;of many forms&#x27;, what it means is that if you put the thing in, you&#x27;re guaranteed to get a specific type of thing out, but it might be different depending on what you put in. So you put words into the machine, and letters come out, but if you put lists of numbers in, numbers come out.&quot;

It is a difficult course! We are doing graph theory right now.

The &#x27;\lambda x&#x27; is just another way of saying &quot;f(x)&quot;So &#x27;\lambda x., x-3&#x27; --&gt; &#x27;f(x): x-3&#x27;Also, I really enjoyed my discrete math courses! The proofs were.. tedious.. but the amount and breadth of &#x27;stuff&#x27; covered was excellent.

It&#x27;s just jargon.  I really don&#x27;t know why the title isn&#x27;t just &quot;Conversations with a six-year-old on functions&quot; and &quot;f(x) = x - 3.&quot;  You lose nothing, increase your audience 100fold, and it&#x27;s more intellectually honest.  It is a less hackernews clickbaity title, though.

&gt; &quot;How is this read: \lambda x.\, x - 3&quot;I&#x27;d read it as &quot;Lambda x maps to x - 3.&quot;In Python, you can rewrite lambdas as regular functions if you want.<pre><code> lambda x : x - 3
</code></pre>
is the same as<pre><code> def f(x):
 return x - 3
</code></pre>
This runs in the IDLE as such:<pre><code> &gt;&gt;&gt; f = lambda x : x - 3
 &gt;&gt;&gt; def g(x):
 ... return x - 3
 ... 
 &gt;&gt;&gt; f(0)
 -3
 &gt;&gt;&gt; g(0)
 -3
 &gt;&gt;&gt; (lambda x : x - 3)(0)
 -3</code></pre>

I&#x27;m taking a discrete math class right now.How is this read: \lambda x.\, x - 3 and what is a simple example?In a discrete math perhaps it is the same as &quot;All x, x is an integer such that....&quot;?

Good find! Looks like he said something similar, but never those exact words.

Einstein never said that, though: <a href="https:&#x2F;&#x2F;skeptics.stackexchange.com&#x2F;questions&#x2F;8742&#x2F;did-einstein-say-if-you-cant-explain-it-simply-you-dont-understand-it-well-en" rel="nofollow">https:&#x2F;&#x2F;skeptics.stackexchange.com&#x2F;questions&#x2F;8742&#x2F;did-einste...</a>.

&quot;If you can&#x27;t explain it simply, you don&#x27;t understand it well enough.&quot;
- Albert EinsteinBrent definitely understands it well enough.

This reminds me of when I explained inverse functions to my kids. I was about to teach them to my grade 10 students and I wanted to use the Dr. Seuss book, The Sneetches to give an example. I asked my kids if I could borrow their copy. When they asked me what for, I told them and they ended up asking what inverse functions were. I similarly played a game where I told them what a certain function machine did and they had to come up with a machine that undid what my machine did.

Poor kid...I imagine how his classmates would react if he offers them to play machine function game

I was just being facetious. It was too easy to identify that  numbers are being doubled, so I&#x27;d contrive something else instead, that happens to be equivalent for the few examples given.

What&#x27;s to hate? You solved the problems correctly and creatively. Any given reasonable limits on complexity of solutions (simpler -&gt; better, Occam&#x27;s razor), there are only a few solutions to any given problem.

I remember exercise sheets at school (in the UK) like this - I think called &#x27;number machines&#x27;.I _hated_ it; felt patronised by the illustrations of these machines, so I rebelled by trying to come up with a different correct solution - aware that there were infinitely many answers given just a few example domain&#x2F;range pairs.But if anyone&#x27;s interested in them for a car journey or some time you can&#x27;t play this game aloud, there ought to be examples available online aimed at teachers.

This is really just a conversation about functions...

great stuff. what is consciously learnt and discovered by one generation can be intuitive to the next, precisely because of teaching.also, i love the idea of experimenting on your own children and then writing blog posts about it :)

For beginner who wanna learn Haskell and Purescript, I recommend those resources:<a href="https:&#x2F;&#x2F;github.com&#x2F;allenleein&#x2F;brains&#x2F;projects&#x2F;9" rel="nofollow">https:&#x2F;&#x2F;github.com&#x2F;allenleein&#x2F;brains&#x2F;projects&#x2F;9</a>

This sounds a lot like the card game &quot;New Eleusis&quot;: 
<a href="http:&#x2F;&#x2F;www.logicmazes.com&#x2F;games&#x2F;eleusis&#x2F;index.html" rel="nofollow">http:&#x2F;&#x2F;www.logicmazes.com&#x2F;games&#x2F;eleusis&#x2F;index.html</a>
<a href="http:&#x2F;&#x2F;matuszek.org&#x2F;eleusis2.html" rel="nofollow">http:&#x2F;&#x2F;matuszek.org&#x2F;eleusis2.html</a>
<a href="http:&#x2F;&#x2F;matuszek.org&#x2F;eleusis0.html" rel="nofollow">http:&#x2F;&#x2F;matuszek.org&#x2F;eleusis0.html</a>It&#x27;s meant to &quot;simulate scientific research&quot;, sorta: one player (&quot;God&quot; or &quot;Nature&quot;) comes up with a secret rule for what cards are legal to play in a given situation, and then the other players take turns playing cards to figure it out. It&#x27;s a lot harder than you think, even with quite simple rules. (The scoring system is meant to reward a &quot;God&quot; player for coming up with a rule that&#x27;s not obvious but still guessable.)

this is super expensive. How much do you need to earn to send your kids to schools like this.

I sent mine to a Steiner &#x2F; Waldorf school on the hope that this is more of a social &#x2F; confidence building environment. Then I surrepticiously feed her inner geek at every opportunity I can find. My personal current favourite is Wild Kratts [0] which teaches a amazing number of things.Also we send my daughter to a violin school where they teach the suzuki method which has a fairly nerdy type of group of parents that go there.So it means she&#x27;s in a school that accepts most types of people whilst exposing her to nerdy things outside of school.Curiously I found that a lot of the parents who send their kids to a Steiner &#x2F; Waldorf school tend to be nerdier. The best example was a couple where the mother was had a Philosophy degree with a Physics PhD and the father travelled around the world as a street puppeteer.[0]: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Wild_Kratts" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Wild_Kratts</a>

My informed but non-professional understanding is that: the good news is if your kids are biologically your own and you are naturally curious, they are predisposed to be as well. The bad news is that studies show that if you want to steer them towards being more or less that way through parenting style it is not going to have any long term effect. [0] I came to this understanding (which, I&#x27;m aware, is pretty different from the public&#x27;s understanding of child-rearing) from reading, among other things, The Nurture Assumption and the chapter on children in The Blank Slate.[0] That&#x27;s not to say that there&#x27;s no correlation in the real world between parents trying to steer their kid to be more curious and the kid growing up to be more curious, just that there&#x27;s no causation because both effects are caused by their shared genes.

I hope my kids want to learn and are curious about how things work like the OP&#x27;s kid!I dont want to be one of those parents that pushes their kid to the point that they dont enjoy learning or suffer socially.Asking for a future me: how do you give your kid the best chance &#x2F; environment to develop socially, but still enjoy learning &amp; embracing the nerdier things?

&gt; Or maybe there is a function machine where you put other function machines in and get function machines out!Made me think of higher order functions as self-replicating machines [0].[0]: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Self-replicating_machine" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Self-replicating_machine</a>

Ad redirect on mobile, beware subway commuters

Teaching CS by simple analogs is the most intuitive way for people to grasp it. I wrote lectures for a 8 week 3hrs a week CS50 level course and completely nontechnical people loved every analog. Example: Front end vs backend vs middleware development is like a Macys window display. The figures and placement are Html, colors are CSS, backend database is the store warehouse and the employees are RoR, PHP, Go or another middleware that connects the pieces. Just like any warehouse - you want most frequently accessed items in the front - so you need to design your database for quick and efficient access to the information you will need to show users most frequently.Since then I’ve been thinking of a wikihow- or w3schools- like site that explains a CS50 level class online and can expand to reach Python, or Go or another language in depth. A second parameter to teaching is the pace - the speed of lecture and the steepness of the learning curve. Maybe we could structure a slow or fast paced versions of each module and the viewer can pick or customize a track for themselves. Is anyone interested in joining or contributing to such an effort?

Mines 1 1&#x2F;2...Better have &quot;The Talk about FP&quot; soon. ;)

I loved reading this. When I was that age, I used to ask so many questions to my father and I was lucky that he was always happy to not just answer, but explain everything. If he didn&#x27;t do that, I may not have followed the path to where I am today.

I am surprised. Growing up in the 80s in France, this was a classic exercise for 7&#x2F;8 year olds. I remember doing it a million times in school. They presented the problem as 2 columns with numbers on the left and on the right, and a box at the top, which we also called the &quot;machine&quot;. It had a little arrow as well to show which way the numbers would go through the machine. The exercise was either &quot;find what the machine does&quot; or &quot;guess the input&quot; or &quot;guess the output&quot;, sometimes combined together.

When I was working as a math tutor I would sometimes have to explain composition to a student for an introductory algebra class. When it finally clicked to them that you could pass functions into other functions many other concepts seemed to come to them more easily. Those were some of my favorite moments at that job.

If you like this article I would definitely recommend this one too: <a href="https:&#x2F;&#x2F;hackernoon.com&#x2F;how-my-10-year-old-learned-javascript" rel="nofollow">https:&#x2F;&#x2F;hackernoon.com&#x2F;how-my-10-year-old-learned-javascript</a>

Conversations with a six-year-old on functional programming