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Output vs. exploration and discovery
My husband tends to show the math shortcuts: adding columns before my 8yo is really clear on place value. Times tables. Multiplying numerals with "0" in them.
I tend to place more value on exploration. Let her work through basic multiplication so she is understanding the "fourness" of 2x2, for example.
DD1 likes both approaches, but I hesitate when dh goes straight for the memorization and technique. She loves what he teaches her. I'd like her to take the path of deeper understanding.
I know the best thing is in the balance: at some point, those "shortcuts" can be mighty useful. Can they?
Thoughts?
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There is a place for using shortcuts  they provide a quick way to get a result when needed, and learning them can provide a sense of accomplishment. However, I generally prefer that shortcuts be taught after the underlying concepts are understood. I don't see the point of teaching an algorithm before a child is ready to understand the concepts at all, and teaching only the algorithm without understanding concepts could result in more problems later as the child develops her own rules in an attempt to make sense of things, and misapplies the poorly understood shortcut. That said, if a child is ready to learn the concept, working from the shortcut to understanding the concept can be a good way to learn new material  that quick sense of accomplishment, then the challenge of figuring out how it works.
I've taught secondary math courses, and have had students blindly apply algorithms they had learned in earlier years without understanding those algorithms or the problem that they were now trying to solve. I had to spend a lot of time with some students going back over what their shortcut really did and when it could properly be used, and some students really wanted to use shortcuts they "knew" even when those methods would never get the right results. (One recurring example I can remember had to do with isolating x in an equation involving two fractions  students would always want to crossmultiply, and didn't consider other terms in the equation or even understand that crossmultiplying is really just a shorthand for multiplying the whole equation by the denominators of the two fractions)
What I've done with shortcuts is either challenge people to figure out how they work, or work through the shortcut with them to show what is really happening. I think that looking critically at a shortcut is a very good way to learn some of the underlying concepts.
With my son and daughter, we mostly explore and play and I haven't done much with standard algorithms until recently, as ds has been encountering them in various places. With addition, my son has seen the adding of columns, but we've looked at other ways to write down the question so that it's clear what is being added. So with 183 + 471 in a column, beside it we'd write it in expanded form (100 + 80 + 3 with 400 + 70 + 1 below it). We'd add the hundreds, tens, and ones in the expanded form, and show how the overflow from the tens increased the number of hundreds. Then we'd go back to the original problem (and usually the example at the top of the page) and show that the left to right addition with carrying was just a shorthand for what we'd done. With multiplication, he's memorized a few facts from a table in a game he loves, but he can show me how multiplication works as area or as groups of objects. I don't worry that he's currently more interested in just getting the answer quickly so that he can get on with his board game  it's a shortcut he's using as a tool and not blocking his understanding.
Hi SweetSilver. I'm going to come at it from a slightly different viewpoint. I don't know if this is helpful so feel free to ignore if its not.
We're a very mathfocused family, just because my kids and partner and I all really appreciate maths. However, I think the work of the under 10 years in math is really to become enthused and intrigued by it, rather than necessarily complete a set curriculum.
What I'd say is that what you and your partner both seem to be doing is to communicate what you find fun and interesting about math. For you its the exploration. For your partner its the cool shortcuts.I think both are great. My kids think both are great. I think exploration, of course yes. But I also think that teaching shortcuts is cool too. No they might not remember them but they will probably remember that there is a shortcut to be found, which might well lead to them trying to figure it out themselves or asking you or whatever. He's teaching him to take control of math and make it work for them. That's pretty powerful IMO.
Seriously, you are both talking about math with them and what you appreciate/love/is useful about it. It can't go far wrong from there, IMO
Miranda
Mountain mama to two great kids and two great grownups
Thanks all, especially for the comments about enthusiasm. Yes, dh and I enjoy math, though probably not as much as the girls do. Of course, they have not had any school to make some parts seem like a drudgery.
I did have one thought after I posted, and that is that dd1 already has a clear concept on multiplication, something she has understood for years. She hasn't worked out every combination, but she understands what's happening perfectly. I shouldn't be so down on times tables at this point, then, and she could even explore multiplication further if I mention creating a table for herself. She already did something very mathematical with her rainbow Sharpies when she first brought them home she drew dresses in every color combination. First a black top and purple skirt, then every combination with black top. Then a purple top, and onward and etcetera. I'm sure she could come up with a times table, especially if it involves her beloved Sharpies (I wish I had never introduced thosethe fumes!)
I'm not sure she's 100% clear on place value yet. We do the occasional "100+80+1" as suggested. I see inklings of understanding when we talk about ityes, we really do talk about this stuff randomly! Crazy! This was never dinnertable conversation when I was growing up! Thankfully, dh has not tried teaching her "carrying" or whatever the proper mathematical name for it is. And I'm a bit nitpickity when one of them says "14+20 is 34 because 1+2 is 3 and 4+0 is 4." I feel compelled to say "you do understand that it's really because 10+20=30 and 4+0=4?" Perhaps it's time for an abacus around the house? They'd have fun with that!
The funny thing is, I wish I understood this Base 10 system we havereally grokked it (Hey! My Spellcheck recognized that word!). Why "10" is not represented as some other random symbol, more like Roman numerals. I'd really like to hear the history of the number system we use. That would help me clarify the brilliance of place value and the elegance of the system. I suppose even Roman numerals had special symbols for 5 and 10, 50, 100, etc. Interesting that humans seem to place particular importance on the number 10.
6.5yo dd2 tells me "But you have to understand me when I say 2+2=22." "I get it, Boo Bear. You mean that a 2 and 2 written together make the number 22." "Yes. People should understand that." "I do, Boo Bear. I do!"
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The whole point of a place value system (any of the basewhatever systems, as opposed to say, Roman numerals) is that the same symbol can have different values depending on its place. That's why we don't have a separate symbol for 10. That's the point at which we've said "okay, no new symbol here: we're going to just push this numeral 1 into a new position, and it will have a new value." The reason we do it this way is (a) it allows us to write huge numbers like 6.2 x 10^23 without having to invent a huge variety of symbols and (b) it allows us to compute with large numbers using exactly the same basic arithmetic facts (the "4 x 3 = 12" fact is useful even when it's four million times three tenthousandths).
To my mind it's a little like writing music with those oval blobs on a staff: if the blob is at the bottom of the staff it stands for a low note, and if it's at the top of the staff it stands for a high note. Same blob, different meaning, depending on its position. Imagine if text worked the same way! I can imagine a cool phonetic system whereby short vowels would be superscript and long vowels would sit ^{o}n th^{e} lin ^{a}s usu^{a}l.
You may understand all this perfectly well. One might also ask "why isn't there a symbol we can use for ten which we can then agree can be *re*written by pushing a 1 over into the nextleftcolumn. I think that's a good question, and at times it might be worth inventing one to make sense of a particular algorithm. "Okay in this column we added 6 plus 9 and got fifteen, so lets write Flower for the ten, and 5. Now the next step is to change the Flower into a 1 in the tens column  because Flower is our symbol for ten."
Your family might enjoy "The Story of 1," a documentary about the the history of human numerical systems. It's presented by Terry Gillian of Monty Python fame and is really well done.
Miranda
Mountain mama to two great kids and two great grownups
Free documentary! This is awesome. I was going to check the library, but having it right here is grand. I want to watch it right now, but I've wasted enough of the morning already.
ETA: Copyright infringement. Oh, well.
Off to the library!
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Interlibrary loan. Try it out. I used to feel guilty for using this service, especially since it can cost the library money, but sometimes it is worth it.
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"Let me see you stripped down to the bone. Let me hear you speaking just for me."
As for related topics raised upthread, here's some possibly interesting info.
Roman numerals are somewhat positional, as IV is four and VI is six.
Base ten means we have ten digits, the numbers 0 through 9, with ten represented as 10. For hexadecimal, base sixteen, sixteen is represented by 10. The number ten is represented by A, eleven by B, and so on, to fifteen represented by F.
If you take that to base eight, then 10 represents eight, the digits 8 and 9 are not used. Binary, base two, means 10 represents two, the only digits used are 0 and 1.
All of this can be confusing, or interesting, depending on the individual.
Also for number history and younger kids I cannot recomend this set of videos highly enough, they are very funny http://www.bbc.co.uk/bitesize/higher/maths/revision_videos/ . I'm hoping you can get themits a bbc site but not afaik a UKonly one. Seriously my kids (around same ages as yours SS ) love this.
Title is "The Approximate History of Mathematics"
Drat, sorry. We rented it a few years ago on the Canadian equivalent of Netflix, back when they mailed out DVDs. Sadly Fillyjonk's Approximate History videos are "Not available in your area" for me.
Miranda
Mountain mama to two great kids and two great grownups
Our library has the "History of 1" and I have it on hold. No luck with the other title.
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Awwww sorry about that. Its really a great set of videos.
We are exactly here with my son who is 8.5 years old. He can read very well but not so much on math. We realized with the Montessori math manipulables that he can easy you know, add 3+4 beads= 7. He can count just fine and because he reads well he can also read the numbers just fine. Writing is another story :) .
Recently someone in our family (name's are omitted) decided to just teach him addition tables. I was a bit miffed. Where is the independence in that? The free thinking? The DIY approach we UnS for?? But then I was feeling like yeah, at 8.5 he should be able to answer what's 2+1... I was confused, feeling a little embarrassed even, unsure about our unschooling in math. So I thought and thought and mostly ignored it a few days or maybe weeks. Then we sat down again. I told him let's play a new game. We sat down. I wrote down a number and adked him if he could get me the right number of beads. So for say 256 he got 2 hundred squares, 5 bars of 10 beads and 6 unit beads. Then I asked him to add 5 unites or add 3 ten bars (no carrying) and he was able to do it which showed me a) comprehension b) willingness. Then I ignored the whole math thing some time longer and thought about how I do math. do I always creatively think each time I add or do I sometimes just use my memory of the addition table (i.e. 5+3=8, bang, not too much thinking about real numbers, integers, the essence of numbers, do numbers really exist, etc). I realized that to tackle "higher" math, he would need to really know these addition tables in depth. He would need to know the table of 1 (1+1, 1+2, 1+3,1+4) and the tables of 2 (2+1), 2+1, 2+2), etc etc before we could count by 2's, 3's or start multiplication. So now we play games with adding and repeat a lot. I ask him in the car about the ones he knows (and he then feels pretty good about it). Every few days (or when I think about it) we add a few additions. I put them on a paper to keep in his pocket and let him know I think it would be a good idea for him to memorize them (millions of others ways this could be asked / done, I chose the easiest for me gah). He wants to be smarter than me so he does read the paper then smartly asks me as I'm doing dishes, mom what's 7+4? I know without even looking at my paper! I act like I'm thinking really hard. 10 I tentatively ask? Hah nope it's 11 mom. Hey mom would you like a paper you can keep in your pocket too?
Khanacademy.org also has some math resources and now they have 3rd grade+ "curriculum" which could be helpful for any worried Unschooling people. I mostly try to ignore it but sometimes I do really wonder so what are they learning in school :)
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