@Peter: Thanks! Yep, not every analogy can work for everyone (no more than a food that everyone loves), but it’s at least a starting point. There might be other variations that people like if the first version didn’t click.
hey brother will u please explane us about sequence and series, as welll as its relation with complex no., i mean both in real and complex part
Hey Khalid, I really like the way you teach and explain things. Can you please help me visualize dot product and cross product of vectors?
I teach statistics grad students to communicate and collaborate with non-statisticians. One module in my class is explaining statistics terms to non-statisticians. I make a big effort to distinguish an “explanation” from a “definition” and I give examples, but many of my students still “explain” terms via a definition. Ironically, I don’t think I explain “explanations” well enough, i.e., I can’t think of any analogies or diagrams to differentiate the two, just examples and technical differences. Do you have any tips for explaining the difference between an explanation and a definition?
I am so glad that I came across this site. I think this is the place that would be for my 4 year old. So glad that he and me will learn together mathematics 
Hello Kalid,
Keep up the good work.
@Eric: this Ted talk by Tyler DeWitt partially addresses your concern. http://www.ted.com/talks/tyler_dewitt_hey_science_teachers_make_it_fun/transcript?language=en
@Ehab: Ah, thanks for the feedback! I often write conversationally, which means a lot of slang gets in – I’ll try to keep that in mind :).
@Dan: Really appreciate the note, thank you. I like the idea of trying to find some (any) visualization for the ideas being presented. Not all can be visualized, but the exercise gets you thinking about how to put a concept into your own words. I love helping out teachers, so it feels great if this article sparked something useful!
Hi;
I like a lot of your articles, but i have a notice, i am Egyptian, and i think that your writing style is nearer to English speakers than others, please take this point into your consideration.
Thanks
Thanks - love your writing, and the way that you express math ideas. Thanks for sharing.
With this particular one I feel compelled to share a corollary. I try to have kids transform all problems into visual problems (from word problems to picture problems). I want them to visually see the problems.
We do this with the operations as well. From basic arithmetic upward I have them draw their math ideas out. It leads to some amazing discussions and helps me know what math concepts students need to be retaught.
Students are often shy at first about drawing, so I model for them what they say, and over time they become better at it. I teach HS math, and I find that they often want to revert to the symbols since that’s all that math has typically been reduced to for them.
Thanks again for sharing!
Kalid,
You are…AWESOME.
I loved these words of yours"So let’s be cartoonists, seeing an idea — really capturing it — without getting trapped in technical mimicry. Perfect reproductions come in after we’ve seen the essence." Your ‘idea of
mathematicians really being cartoonists’ is really great.
LOVE YOUR WEBSITE!!!
Thanks for the posts. Keep them coming. These are opening my eyes and I have been teaching Algebra for 6 years. Thank you.
Thanks Ryan, really glad to hear :).
Nice article as usual Kalid. Learning a lot everytime I get your newletters…thank you!
Hi Kalid,
Thanks for the article. Interesting and thought-provoking, as usual.
In the example of multiplication, wouldn’t it be more accurate so say “multiplication changes the size of things”? This would then apply better to negative numbers and numbers between 0 and 1. Less need for future “concept refactoring”
My point is that some caricatures are better that others…
@sb: Thank you!
@Massimo: Good feedback – I’d say multiplication changes the properties of a number. Normally we think numbers only have one property, their size, but they can have another (a direction: positive or negative), which can be changed by multiplying by -1. Later, we learn that numbers can have other directions (up/down, with imaginary numbers) and multiplication changes that too.
That said, totally agree that some caricatures are more expressive than others! (A stick figure caricature is too reductive to be useful.)
Good point.
I think we should never forget that we are human beings, and human beings are different from computers. Our brain can integrate things in a way we still cannot figure out till now. We should follow the way the brain works. Clearly, simplification and abstraction is one of the special properties of our brain. It will make us absorb things easily and efficiently.
Khalid,
Your mail has now replaced my early morning coffee , much more delicious !!
Love it and request to please keep it going.
Thanks!
Thanks Malini, glad you’re enjoying it!
I find ur teaching methodology quite fine and impressing by the way you argue out even in tough ideas you make them look simplified than ever. I do enjoy ur coaching. I would like to learn mre abt series an finding sum to infinity. Thnks.