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Why Memorize Math Facts?
From AutoSkill, a provider of software (not an endorsement -- haven't yet looked into the products)
I think there's a basic problem here that we as the parents of gifted children must come to terms with. Not all useful learning is intrinsically interesting. Our kids have a right not to be bored in that they should not be held down, but they do not have a right not to be bored such that they have a right to skip anything that isn't fun to learn. Math facts are boring. Absolutely. But that doesn't mean that our precious children who don't tolerate boredom well shouldn't have to learn them. We have to teach our kids the difference between being bored because you are being taught something you have already mastered and being bored because the work is intrinsically boring but still important. We can turn our fertile brains towards making the practice fun and interesting, if we don't tolerate boredom well, but we don't get to just declare ourselves to be so brilliant that no one should ever make us do anything we don't feel like doing.
Why Automaticity in Math Facts?
The notion is that the mental effort involved in figuring our facts tends to disrupt thinking about the problems in which the facts are being used. Some of the argument of this information-processing dilemma was developed by analogy to reading, where difficulty with the process of simply decoding the words has the effect of disrupting comprehension of the message. Gersten and Chard illuminated the analogy between reading and math rather explicitly:
"Researchers explored the devastating effects of the lack of automaticity in several ways. Essentially they argued that the human mind has a limited capacity to process information, and if too much energy goes into figuring out what 9 plus 8 equals, little is left over to understand the concepts underlying multi-digit subtraction, long division, or complex multiplication (1999, p.21)."...Practice is required to develop automaticity with math facts.
"The importance of drill on components [such as math facts] is that the drilled material may become sufficiently over-learned to free up cognitive resources and attention. These cognitive resources may then be allocated to other aspects of performance, such as more complex operations like carrying and borrowing, and to self-monitoring and control (Goldman & Pellegrino, 1986. 134)."
I have a new academic therapy client, a delightful 4th grade boy on the spectrum. For various reasons, he missed a lot of 1st and 2nd grade. Academically, arithmetic is his weakest link. As nearly as I can determine, the only math facts he has to automaticity are adding and subtracting 0, 1, and 2 for even numbers, and multiplying by 1 and 2 for even numbers. Everything else requires him to actually do the calculation in his head or worse yet, his fingers. Calculation--> frustration--> anxiety--> decreased cognitive ability--> frustration--> anxiety--> decreased cognitive ability ...
I have three challenges: convincing him that putting the work in to getting to automaticity is going to be worth it, finding the methods that work most efficiently for him (flash cards aren't it), and keeping his anxiety low enough that he can learn.
Any and all suggests for the three challenges gratefully accepted.
10 comments:
Some have commented that worksheets (rather than flash cards) work well. I used to leave worksheets on a table and my son would come by and work on them. That didn't last too long, however. But don't assume that kids hate learning. They hate failing.
If you start with really simple problems that you know he can do, and then work your way up, that might do the trick. If there is no time limit (at first), then his brain won't stress out as with the flash of a card and the hard to hide impatience of the teacher. (I know this from personal experience.)
You will still probably have to force the issue, but some sort of reward system could be useful. I remember when I was growing up that our public library had a summer reading "contest". They put a big map of the world on the wall and each child got a picture of a suitcase. Every time we read a book and wrote a book report, we would go to a new country and get a sticker to put on our suitcase. Another year (in the sixties!) we got space ships and traveled through the solar system. We also got a little extra knowledge out of the map and solar system.
I think kids really like to know stuff, even if it is rote learning. My son picked up a list of presidents a month ago and started to memorize them. He was bound and determined to go to school and recite them to his teacher in spite of my warnings. (I haven't got any notes home yet about "mere facts".) I can tell that they are not "mere" because he now knows that when he reads something about Teddy Roosevelt, they are talking about the end of the 1800's and early 1900's and he will be able to match it up with other "mere" facts. Lot of "mere" adds up to real knowledge.
I'm sorry, but while I agree that not all learning is fun, for me, math facts ARE fun. Always were. This "math is not fun" overgeneralization is bogus. For SOME people, yes, it may be. And for many of those folks, it may be because it was badly taught. So instead of categorizing learning in terms of fun or not in the first place, or assigning such a value judgment, how about we just work on teaching kids as well as possible, so they aren't bored or anxious or upset? Because then learning isn't painful.
Liz, I certainly don't know enough about your new client to know what methods would work for him, but speaking as someone with a lot of anxiety issues and knowing how that interfered with thinking, I'd actually try the "fun" route. Play math games. Try puzzles. Play card games like Uno and War and change the rules slightly. So, in a variation on War, instead of having the higher or lower card, you need a card that "when you add 4" to it, is bigger than X... or "is a multiple of 3". Or play a form of war where you each put down a card and whoever says a true math sentence that uses both cards first wins. (you a 4, him an 9: 4 + 9 is 13, he says.
Teach him to play 21 (a simplified form of blackjack where you stand on 16 and hit on 17 always, and no splitting/doubling down/insurance.) Play for pennies! kids love betting. Perhaps visualizing the numbers and the count of items on the cards will help him to gain automaticity too.
For me, things which make me feel like the teacher could actually lose the game lower anxiety, because it's not such a performance thing where I'm feeling like I'm on trial. so games really help that.
re: convincing him that putting in the work is worthwhile: all I can say is that you have to make it be baby steps so he can see the success each and every hour you're there. that's REALLY REALLY difficult, I know, but it's really the only way that works, because he's got a success every day.
My cousin didn't learn her math facts until high school and her tutor's technique was to toss a ball which she would then catch with her left & then right hand, alternating with each math fact. I was told she is a kinesthetic learner. I have no personal training in this though so would love to know if this a valid/valuable method for some learners. Anecdotally, I can vouch for it but statistically, I have no clue. Sounds fun though...
I have been involved this year with a pilot program for a web based subscription 'skills trainer'. Its called ALEKS and you can learn about it at aleks.com.
In a nutshell it is DI on steroids. Kids are assessed then are fed only those tasks that are in the zone (ZPD). They manage their own progress by constantly being brought back to a pie chart of their progress. They are only allowed to work on things in their own ZPD so they might have a handful (4-8) tasks available to them at any given time.
Progress is made through mastery, no exceptions. If they get stuck on a skill there is an 'explain' button which brings them to a training session (usaully one screen) before going on to more practice.
Personally, I love it. The kids love it too. I have to rip them out of the lab when a session is out of time. They get mad at me when time is up. Don't ever tell me kids don't want to learn this stuff.
I think the key to its success is due to several things. First, it provides instant feedback and positive reinforcement. Second, by staying in the ZPD for each student it eliminates the grinding frustration one gets in a traditional setting if you don't get it. And lastly, it is a lot like the media experience that kids get today outside the classroom, fast paced, self driven (to a point), and fun.
It even has built in games that are triggered at various points as mind candy when you've worked hard. The best part (for me) is that the games actually step up the pace of the math making them work harder and faster but they can't wait to get a game to pop up.
I've brought in 'digital addding machines' that dumped the fingers after just a few hours on this system.
"No one ever told them they had to memorize this stuff before." That's an honest to God quote!
The best part for me is that it produces a student by student assessment aligned with my state standards to a grain size that is typically a fifth of each standard. And also for the really nifty part, you get a report that tells you, by substrand groupings, what kids are ready for next. Kind of a ZPD aggregator so you could concieve some really fabulous connections to your instruction.
I get giddy thinking about how I could use this for data driven instruction but sadly my district will not be using this anytime soon. Once the pilot is over it's toast. Gotta get the bulletin boards squared away first.
I know the math book costs are higher than this thing (suscriptions can be had for around $30 per kid per year). Maybe the math books can be used as seat height adjusters if I could get parents to cough up the 30 bucks.
I forgot to mention...
the ALEKS pilot was offered to us as 100 FREE seats per school in our district (Jan-Jun). It's totally web based so there is no installation mess and kids can use it at home as well as school.
Here is a contact person to get it going (Northeast locations)
Elizabeth Barnes [ebarnes.iak@comcast.net]
My daughter has ASD and flash cards didn't work for her either. She even got sent to the principal's office in 3rd grade for using a calculator under her desk during a timed fact test. In our district, once a child gets through all the multiplication facts time tests in the required time, they can get an ice cream treat from the cafeteria. Megan really wanted that treat. What finally ended up working was her case manager put up a list of all the facts on a bulletin board next to her desk. Lots of ASD kids are so visual and Megan usually had all the classmates' names and birthdays memorized pretty early in the year as they were always posted on a bulletin board. So we used the facts on a bulletin board along with her list-making obsession. She kept repetitively writing lists of the facts and learned them pretty quickly. Another thing that she likes (and that we are using with our now 2nd grader) along with the traditional timed worksheets, are the Nintendo DS "Brain Age" games. They have timed math facts tests and other games requiring quick calculation. My girls love these and they are really trying to beat me as I am the Brain Age pro in our house.
I have some suggestions, but no proof that they are effective. I have tried them with a student I tutored, but it's hard to know how well they worked. I only worked with her for a few months, and she was also being taught in school, of course.
Anyway, one skill that I think helps kids build facility with numbers is finding number pairs that add to 10. So, you could strip all the face cards from a deck, then turn up one card at a time. For every card, the student has to say out loud what number to add to get ten. For example, if you turn up a three, the kid has to say "seven."
You can do the same thing with a die.
A little more interesting activity with addition is to add bowling scores. Of course, you can do this at a bowling alley. It's a lot more convenient, though, to use a pair of dice to "bowl." The first ball of a frame is the sum of two dice. A 10, 11, or 12 counts as a strike. For cases where you did not roll a strike, the second ball of the frame is only one die. Again, any extra beyond what is needed for the spare still leaves you with just the spare. The activity involves adding the dice, but also the interesting kind of addition involved in the scoring of bowling. Who wins the bowling game itself is of course a game of pure luck, so a teacher or parent would be a fair match for the student.
I also think that backgammon is an excellent game for teaching simple sums and differences. It, of course, involves more strategy to actually win, so you might have to watch out for the frustration of losing.
Just my two cents' worth,
Dan K.
Try reducing the levels of abstraction.
I have kids visualize digits as dots in a 2x5 array. Then an addition problem is simply a process of rearranging the dots in two arrays(mentally, not a paper process). A full row is 5. Two full rows is 10. Two full rows plus leftovers (in the second array) is easy, just 10 plus a digit which is just appending the leftovers to a 1.
This works really well on the tougher, high digit, combinations that result in a 'carry'
I faced a similar problem. My son is gifted and did not take to memorizing his math facts-too boring from his perspective. I, however, do research on math cognition and knew that memorizing the facts freed up cognitive resources for higher-level understanding.
My solution was to take out the table of addition facts that he had to learn, which resulted in the usual groan. But then I explained commutativity, and told him that, because he knew this one property, he only had to learn half the facts. To underline this, I ripped the paper in half. That got his attention!
It seemed much more manageable to only learn half the facts, and he felt like he was getting away with something, so his motivation was higher than before. We did the same thing for multiplication. He learned all his basic facts to automaticity. I have shared this tip with other parents struggling to get their gifted children to memorize and they too saw good results.
http://www.redshift.com/~bonajo/mmathmenu.htm has one possible technique that may work, depending on what level he is at with his addition understanding -- my small experience with friends is that it works well and quickly with children who understand number bonds. (note: I do disagree w/Michele on the upper levels of multiplication, as I prefer the Singapore style of using the distributive law).
If the student doesn't understand how to add two numbers by counting up from the higher, he has a lot of making up to do and manipulatives will be beneficial until he's ready to go to symbols that he can manipulate in his head and on paper.
The card game War works well once he has his number bonds down. Denise at let's play math summarizes the variations here: http://letsplaymath.wordpress.com/2006/12/29/the-game-that-is-worth-1000-worksheets/
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