So, Josh is 7 and doing 2nd grade work. We're using Singapore Math, which I'm really happy with. Josh, it appears, has a natural talent for math (which he SO doesn't get from me LOL). It's the one subject he really looks forward to each day. He picks up the concepts really quickly and I have to be really careful not to blow through the units too quickly. My "problem" (if it even is a problem?) is in getting him to show his work. He can work out most any problem in his head, sometimes (ok, a lot of times ) more quickly than I can, even. He gets really frustrated, though, if I try and make him write out or explain to me how he got the answer. For example, right now we're working on 2- and 3-digit adding and subtracting without remainders. If I give him, say, 123 + 234 = ? he's comfortable with writing it a different way if asked, like 123 +234 ------ and figuring out the answer by adding the ones, then the tens and then the hundreds. If I don't ask for the question written differently, though, he has no problem working out the answer written like 123 + 234. Oh, and when he does work it out doing ones first, them tens, etc... I get the distinct impression that he's just humouring me because he's already figured out the answer in his head LOL Taking that same example, if I give it to him as a word problem "Anna has 123 marbles and she buys 234 more marbles. How many does she have altogether?" he'll read the question, take a second second, and just write the answer. Asking him to write out 123 +234 ------ 357 is practically impossible. I'm open to the idea that he's just too young to put the concepts in his head into words (or on paper), and that it'll come more naturally later. I guess I'm maybe concerned that what if the way he's working it out in his head isn't the "right" way, and it messes up working out more complicated equations later on? And, don't know if it makes any difference, but he apparantly today figured out remainders without ever having been taught them :? I had written out a bunch of questions for him to do on the white board and accidently included "You have 75 strawberries and 132 peaches, how many fruit do you have altogether?" It only took him about 10 seconds to work out the answer in his head, and I haven't even touched on the concept of remainders with him yet. So, thoughts? Should I be forcing him to show his work and ensuring that he's working stuff out the "right" way? Or am I just creating worry where there is none? I don't want to kill his love of math by squashing any natural ability under a pile of "it has to be done this way".
I wouldn't be too concerned at this age. He seems to have a God given ability to do math. Later on when there are multiple steps it may be helpful for him to write it down, so it he misses a problem, you can see where he went wrong.
I wouldn't push him to write out the steps. Some people just have a natural gift for Math & making him slow down to write out what he's doing in his head may cause him to not enjoy doing Math. I hated it in school when my teachers would get on me about writing it out. Especially in High School. Basically, their thought was that since they couldn't do Algebra or Geometry in their heads, then I must be cheating if I didn't show my work. As long as you know he's doing the work himself, in his head, I wouldn't push him to write it out. My kids are both great with Math. I have them do a few problems with it all written out, so I have proof on paper that they know what they're doing. As long as they can do a few that way, I'm fine with them not showing their work on the rest, as long as they get the right answers.
As someone who has always been very good at math, this whole topic infuriates me. I've heard from so many people whose love of math was trashed by teachers who gave low grades because they didn't show their work. Just as bad are teachers who mark a question wrong because a student used a different method to come up with the answer. I loved math so much that I would take a bus to the largest library in the region just so I could check out math books. I'd learn all kinds of shortcuts and "tricks of the trade" that served me very well. Quite often, I would not use the method explained by the teacher, and it would take an age to explain what I actually did. Just a simple case in point: It's a matter of simple algebra to prove the following: (10n+5) squared = n*(n+1)*100+25. For example, 25 squared is 625 (6 is 2x3). 35 squared is 1225 (12 is 3x4). 45 squared is 2025 (20=4x5). 105 squared is 11025 (110=10x11). 995 squared is 990025 (9900=99x100). And so on. Once you know this, you can square some large numbers in under a second. So, when calculating a difficult problem, this little trick may be useful for just one step in the calculation. Imagine the tedium that comes from trying to explain what I did in that one step. Imagine the frustration when receiving a zero for not using the humdrum, basic, time-consuming method used in an education factory (sorry, school). Personally, I don't care a hoot about workings, and I never asked my children for them. I would sometimes ask how they did it, just to be sure they knew what they were doing, but explaining an answer verbally takes a few seconds only and doesn't cause frustration.
As long as he has the "right" answer, he's doing it the "right" way. I wouldn't ask him to show his work unless he misses something. If he misses something, at that point, "showing his work" would be helpful in figuring out what he's doing wrong. But right now, he's doing it "right" because he's getting the right answer. Good for him!
I've thought a little more about this. If your son has a natural talent for math, introduce an element of competition. Not meaning to boast, but I later learned that I'd become something of a legend in my local school for some of the things I did in math. For example, the teacher would ask a trick question and ask everyone in the class (of about 50) for an answer. Not infrequently, the other 49 would select one answer and I'd give another. Maybe it's my personality, but I loved this and took great pride in getting things right and earning a reputation. The teacher didn't mean to put anyone down but, to his credit, he was giving me the chance to shine. To this day, I can still remember some of those questions. How can you spur your son in the same way? Can you give him problems when out in the car? "We're doing 55mph, and we leave the highway in 11 miles. How long's it going to take us?" "I have a bucket that holds 5 pints and another that holds 8 pints. Can you use them to give me 6 pints?" Let him earn a reputation among his siblings for being the resident math professor. If he's anything like I was, he'll rise to the occasion, and it will spur him to succeed more and more at math. Feed success.
I argee with you about the tips & tricks. My dh & I teach our kids tricks & shortcuts, and encourage them to use the ways they most comfortable with. I think it's great that you never made your kids show their work. The only reason I have dc show work on a few problems is so that I have proof, on paper, that they know what they're doing. The way the laws are set up in my state, full burden of proof falls on me. If someone were to report us, saying that we aren't doing school, I would have to prove that we are doing school & following the laws. So, I'm big on keeping proof. At this point, if I had to prove what we're doing, I could bury someone for months with all the work & records I've got. After looking over everything I've held onto and/or talking to my kids for 5 minutes, no one could doubt that my kids are getting a proper education.
Ok - ds is now in algebra and still can do some of the problems in his head once he "sees" the concept a couple of times. What I started doing with some of the problems when he was younger was he would have to redo any he got wrong on paper. Mistakes didn't start happening until stuff like 36x423 because of the way his mind was lining things up...now though he can do some of it in his head (not as quickly but he can) DH has this annoying talent as well I say annoying because from the teaching stand-point I have a hard time teaching him the REASON it was wrong because I CAN'T see what he did! But I wouldn't push it or else he may start not LIKING math later. If he starts making mistakes that's when you change the approach.
I think I'll refer this to my math teacher husband! I do make Faythe "show her work", because she's doing pre-alg. While she can do the easy problems in her head, I think it's important that she be able to write out each step for when the problems get harder. Phillip will do the adding in his head, so that he's not actually writing out the number he's carrying. But he'll "forget" about it half the time and get it wrong. One advantage of having them show their work is that YOU can see what they did wrong. It drives me crazy when they have a wrong answer, no work, and you ask, "How did you get that?" and they have NO idea!!!
I do understand the value of being able to retrace their steps. On the other hand, there's no way I want to cause frustration or slow them down. I guess it's a balancing act. In my case, I asked them from time to time how they got their answer rather than require them to document everything.
I make my dd do it only so that if it's work we use for sample week we can prove she's actually working at her assignments. She can write the answer on one page and then show me her work on another if she wants. Graph paper actually really helps keep her columns right for when she's working as well.
I agree at this age for these types of problems, it's not all that important for kids to show their work. However, I disagree that it doesn't matter; especially in higher maths. When students start math like Calculus, Trigonometry, and courses like 'Discrete Mathematics', it becomes very important to show work and the shortcuts used in simple mathematics often won't work or they actually become the hard way to do things. In other words, the standard and often long/hard way of doing things with simple math really about the principal than the result. In fact, most of math education is about learning the proper way to do things, because does anyone really care if you know that 3x + 5x = 8x? No one cares and it makes not difference to anyone in the world, but it's about knowing 'how' to do it. When you take higher mathematics and physics with very complicated problems it becomes critical to break things down step by step and you can't just do it in your head. I've had problems that often take 1/2 a page to 3 pages for a single problem and by showing work it becomes easy to go back and discover errors.
I have one kiddo that doesn't like to show work, so we have a deal. No showing and tedious writing it out on the ones you get right...but on the ones you get wrong, we go through together and he has to show me what he is doing. This makes sense and seems "fair" to him, so I don't get much argument... I have even encouraged some in the head stuff...we just did what is half of 488...and discussed with my younger son, how with easy numbers like that you just do half of 4, half of 8 etc. with it written out, he could see WHY that works. (he isn't quite as intuitive as the elder one.) we use the graph paper too.
I would encoruage it because as they get more difficult the mistakes do come and they can't tell you where the mistake is. So say every so many problems he has to show his work, thats how I do it. or did, we SOS math this year. with this we use a scratch paper that he uses and I use to explain it to him on paper. So if he doesnt do it maybe if you do it to explain how to he will see the usefulness in it?
I had a teacher once tell me to have them turn their lined paper sideways to keep the columns straight! My middle one used graph paper for a while, but not any more.
I think I might ask the child to "humor me" and just show the work on, say, every fifth problem, just so I can file it for the portfolio.... Incidentally, there are no "remainders" in addition and subtraction, although there is "regrouping", "renaming", or plain old "carrying and borrowing". Remainders are for division, when it won't go evenly.
I'm normally a stickler for it myself. BUT....if he's got a knack for math, I wouldn't hassle him on it unless he starts getting a lot wrong b/c he's figuring wrong. I used to have the same fight with my oldest (actually has aptitude in Math, but doesn't like it - she's very smart and picks stuff up right away....just hates doing it). I finally got to the point if a problem was wrong, she'd have to rewrite on paper and show me all her work. But only if she got it wrong. Now, we're using Teaching Texbooks, so I do barely anything on her math at all (and I love that).
Could somebody please explain to me what Discrete Math is, anyway??? All we had in high school was Algebra I and II and Geometry.... and all I took in college was college algebra and trigonometry. I never heard of Discrete Mathematics, until about maybe ten or fifteen years ago.
I also hated teachers in school who forced everyone to do exactly the same steps on every assignment. Some needed it. Fine. Some didn't. That's the great thing about HS. If he's getting all the right answers, have him show the work on 2-3 and then skip the rest. You know he's understanding it. If he's missing every 2nd-3rd problem, he definitely needs to show his work so you can back track. If someone shows up at my door asking for proof of learning, they're welcome to ask my child a question.
I have a lot of thoughts on this question jumbling around in my head and no clear-cut answer for you. One way I got around this was to require my son, who was also very good at math, to show the steps for every two or three problems. My daughter, on the other hand, NEEDED the steps. She had to see the steps to understand the concepts. I think it may come down to learning styles? An abstract learner may be more able to "see" the answers in their head, while a tactile or visual learner may need to see them written down? Just thinking out loud. I had a very, very good friend in high school - I dated him through my Senior year and Freshman year in college. He was a math genius. He was a lot like you, Steve. Whenever we were given math problems, he'd find a different way to come up with the answers, and they were right! In all honesty, I think he worked above the teacher's head. He went on to school for accounting, but couldn't do it. WHY? Because he hadn't learned the correct steps to follow in the math. And it was way too hard to go back and relearn it all. In accounting, you had to follow exact rules and exact formulas and couldn't go your own road. I think things would probably be somewhat similar in engineering. Architecture, maybe not so much so. (look at all the buildings we have that architects said would never stand!). So, what I am getting at is that there may be a place for learning the steps along the way. And even though you learn the shortcuts and the tricks of the trade, you would need to UNDERSTAND HOW those shortcuts work and why in order to thoroughly understand the subject. At young ages, I would think that would be of paramount importance. So, for me, anyway, and I certainly think everyone should teach as they feel led - but for me, I think I would, when each concept is presented, make them learn the steps in the concept - the "correct, book way" of doing it, and then allow them to discover the short-cuts and tricks to follow. It would be tedious to do this, because you would have to pick apart your text and be sure to be there with each new concept taught. Once they have mastered the concept - ANYTHING GOES! And I would also review frequently, or maybe include a few test or quiz questions that had to have the steps written out. Why would I do this? Because everything in God's universe has order. In order to thoroughly understand any concept we must attempt to understand that order as much as possible. All things decently and in order. Precept must be upon precept. As long as they understand WHAT they are doing (in the shortcuts) and WHY they are doing it, and not just that it works - then GO for it!
