Is math by hand better than math by keyboard?

Sorting books this afternoon, I came across Tahir Yaqoob's What Can I Do to Help My Child with Math When I Don't Know Any Myself? and found this passage:

The actual process of using your muscles to write something is a powerful long-term memory aid. The more that you write out things (and in different ways), the more your long-term memory will be etched out. It is not good enough simply to read and think (although this is important for reviewing large amounts of material shortly before taking an exam, but only if you have done the long-term ground work). Writing out full solutions to problems in math is especially important compared to other subjects, whether it is part of reviewing for exams or whether you are learning new material.

Writing things out can also help you to understand difficult problems. For example, if you see a fully worked solution to a problem in a textbook, but don't understand one or more of the steps, try simply writing out the solution yourself. You may be surprised that while you are doing that, you suddenly understand something that you didn't before. Sometimes the brain has a strange way of working. Despite its enormous capacity , the. brain can really benefit from an external "scratch pad." When you come across something that you don't understand, sometimes just writing out the steps in a brief form can make a great deal of difference.

What Can I Do to Help My Child with Math When I Don't Know Any Myself? Paperback – February 7, 2011 by Tahir Yaqoob - p133

I've always found this to be true, both for C. and for me. I don't know why. One of these days I'll get around to reading The Hand: How Its Use Shapes the Brain, Language, and Human Culture, which I hope will explain the phenomenon.

The OECD report on students and technology (Students, Computers and Learning: Making the Connection) found that using the computer for drill was associated with reduced achievement:

The decline in performance associated with greater frequency of certain activities, such as chatting on line at school and practicing and drilling, is particularly large (Figure 6.6). Students who frequently engage in these activities may be missing out on other more effective learning activities. Students who never or only very rarely engage in these activities have the highest performance.

Given my experience, the "other more effective learning activities" these students are missing may be drilling by hand.

The law of universal linearity

Interesting discussion

I was struck by this passage:

In terms of implementing this in practice, I think that college is way too late, and also quite difficult because college math (and STEM) courses tend to be mostly about transmitting massive amounts of boring technical content and technical skills, leaving little to no room for actual ideas or ways of thinking. Nevertheless, I do think it would be an interesting experiment to have students keep something akin to a "vocabulary notebook" where they record the meaning (as opposed to the formal definition) of the various kinds of expressions they run in to. For example, a fraction ab is supposed to mean "a number which when multiplied by b gives a"; it is short and illuminating work to figure out from this (using distributivity of multiplication over addition, which we definitely want numbers to satisfy) that ab+cd=ad+bcbd, that there is no number meant by a0, and that 00 can mean any number). This of course, presupposes that somebody takes the time and makes sure that the language in which these meanings are explained is coherent, so it would be a lot of work to design a course around this method.

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I did in fact once successfully disabuse a(n Honors Calculus) student of "the Law of Universal Linearity" using these ideas. The particular instance concerned manipulating the Fibonacci sequence, and the student had made the error of writing something like Fx+Fx=F2x. What I did is explain the stuff above and had the student apply them by analyzing the meaning of the various expressions he had written down was, and then ask whether that equality was justified based on what he knew the expressions meant. That seemed to make an impression on the student, but I personally believe it was an impression made ten years too late...

The UK regains some sense in math

I've heard it remarked that what happens in Europe is just a preview of what's coming to our shores here in the US.  If that's true, then this makes me smile:

Pupils aged 11 will be given extra marks for employing traditional methods of calculation in end-of-year Sats tests, it emerged.

Children who get the wrong answer but attempt sums using long and short multiplication or adding and subtracting in columns will be rewarded with additional points.

Ministers insisted the changes – being introduced from 2016 – were intended to stop pupils using “clumsy, confusing and time-consuming” methods of working out.

This includes so-called “chunking” and “gridding” where pupils are encouraged to break problems down into numerous component stages before an answer is reached.

 *snip*

Elizabeth Truss, the Education Minister, will outline the plans in a speech to the North of England Education Conference in Sheffield on Thursday.
Speaking before the address, she said: “Chunking and gridding are tortured techniques but they have become the norm in recent years. Children just end up repeatedly adding or subtracting numbers, and batches of numbers.
“They may give the right answer but they are not quick, efficient methods, nor are they methods children can build on, and apply to more complicated problems.

Everything old truly is new again.

Computerized teaching: the feedback gap

Yet another breathless account of the wonders of computerized learning appears in this weekend's New York Times Magazine in an article entitled "The Machines are Taking Over: advances in computerized tutoring are testing the faith that human contact makes for better learning."

The article opens with a scene of an actual human being tutoring a fellow species member. While her tutee works on a problem (calculating average driving speed), the tutor provides lots of interactive feedback. Neil Heffernan, the tutor's fiance, catalogued the various different types of feedback she gave under such categories as “remind the student of steps they have already completed,” “encourage the student to generalize,” “challenge a correct answer if the tutor suspects guessing”). According the the article, Heffernan then "incorporated many of these tactics into a computerized tutor," which he spent nearly two decades refining. Now called ASSISTments, it is used by by more than 100,000 students "in schools all over the country." The article describes the experience of one of these 100,000 students with the program's interactive feedback:

Tyler breezed through the first part of his homework, but 10 questions in he hit a rough patch. “Write the equation in function form: 3x-y=5,” read the problem on the screen. Tyler worked the problem out in pencil first and then typed “5-3x” into the box. The response was instantaneous: “Sorry, wrong answer.” Tyler’s shoulders slumped. He tried again, his pencil scratching the paper. Another answer — “5/3x” — yielded another error message, but a third try, with “3x-5,” worked better. “Correct!” the computer proclaimed.

In other words, it's the same old binary right-or-wrong feedback that nearly every educational software program has been using for decades. As the article notes:

In contrast to a human tutor, who has a nearly infinite number of potential responses to a student’s difficulties, the program is equipped with only a few. If a solution to a problem is typed incorrectly — say, with an extra space — the computer stubbornly returns the “Sorry, incorrect answer” message, though a human would recognize the answer as right.

True, the program is still a work in progress. But what's being refined, according to the article, isn't the feedback. Rather, it's the program's ability to detect when a student is getting bored, frustrated, or confused (via facial expression reading software, speed and accuracy of responses, and special chairs with posture sensors "to tell whether students are leaning forward with interest or lolling back in boredom."):

Once the student’s feelings are identified, the thinking goes, the computerized tutor could adjust accordingly — giving the bored student more challenging questions or reviewing fundamentals with the student who is confused.

Or "flashing messages of encouragement... or... calling up motivational videos recorded by the students’ teachers."

Also being refined is the "hint" feature, which users click on when stumped. Human beings (particularly teachers) track common wrong answers and have other human beings (particularly students) come up with helpful hints. These hints are then incorporated into the next generation of ASSISTments.

Cognitive Tutor, a more established software program that is "used by 600,000 students in 3,000 school districts around the country," also limits its feedback to hints and right-or-wrong responses.  And it, too, is being refined based on data from human users:

Every keystroke a student makes — every hesitation, every hint requested, every wrong answer — can be analyzed for clues to how the mind learns.

Ultimately, this data will be put to use not to refine feedback on particular student responses, but to help decide how to space out material and schedule periodic reviews.

But it's carefully tailored feedback on particular responses by particular students that makes human tutoring--the inspiration for all these programs--as powerful is it is.

In my earlier post on Cognitive Tutor, I wrote that programming sufficiently perspicuous feedback for mathematical problems "strikes me as even more prohibitive" than the feedback I labored for years to provide in my GrammarTrainer program. Last night I ran this impression past a mathematician friend of mine who cares a lot about effective math instruction. She emphatically concurs.

When it comes to educational software developers--as opposed to educational software users--there is some somewhat perspicuous feedback on whether their answers (answers to students' educational needs) are on track. As I write earlier, that feedback isn't particularly encouraging.

(Cross-posted at Out In Left Field).

Being Tough on Tough

Though I haven't read it, I honestly don't get what the big deal is about Paul Tough's new book, How Children Succeed. From what I gather from various reviews and interviews, Tough's Big Idea is that persistence and curiosity matter more than IQ does for success. But was there ever a time or place when this statement wasn't obvious? Of course IQ means little if you don't apply yourself; of course intelligence leads nowhere interesting if you lack curiosity. Does anyone--especially in this Emotional Intelligence-obsessed world of ours--really think that the successful people out there--even the genuises--achieved what they did primarily because of their IQ scores? Didn't Malcolm Gladwell already write a book back in 2008 on the findings that what makes an expert is 10,0000 hours of practice? What is it about Tough's book that's garnering so much attention?

A slightly different take on Tough's Big Idea is voiced by Joe Nocera in today's New York Times:

Tough argues that simply teaching math and reading--the so-called cognitive skills--isn't nearly enough, especially for children who have grown up enduring the stresses of poverty. In fact, it might not even be the most important thing.

Notice how quickly Nocera slips from the obvious--that teaching teach math and reading isn't nearly enough--to the ridiculous. To say that learning to read and do math might not be the most important elements of success is like saying that adequate food and shelter might not be the most important elements of staying alive (after all one must also breathe oxygen). When it come to essential elements, it's pointless to quibble over what's most important.

In interviews Tough is careful to admit that, while schools need to do more to encourage persistence and curiosity, there are no clear studies on how to do this. Refreshing though this caveat is, it, too, raises the question of what this book has to offer that's new and plausible, or at least useful.

There is one disturbing answer to that last question. To the careless reader who approaches the book from the perspective of the dominant educational paradigm, it offers yet another reason to water down academics in favor of "the whole child." The connections between grit and academic rigor, and between curiosity and well-taught academic subjects, should be as obvious as the inherent importance of grit is. Indeed, I'm guessing these connections are obvious to most people. But they clearly aren't obvious to many of those wielding the greatest power over whether or not our children succeed.

Help Desk: algebra remediation

A friend of mine is a special ed teacher at a highly selective science-oriented magnet school. She has observed a number of incoming freshman who are unable to handle beginning (9th grade) algebra. I'm guessing that some (most? all?) of them have no inherent math disability, but have merely been poorly instructed (most come from elementary schools that use Everyday Math/Investigations and from middle schools that use Connected Math).

Anyway, when she asked me what I knew about math remediation programs that might help prepare these kids for algebra, I realized I had absolutely no ideas and should turn to ktm for help. Any suggestions?

Investigations Math in action: crashing and burning with large numbers

A third grade girl attempts, unsuccessfully, to add several large numbers using an Investigations Math strategy. She then adds them successfully using traditional "stacking" (disallowed at school) in a fraction of the time the Investigations method took her:




Filmed and edited by a fellow concerned parent who is a specialist in math remediation.

Math in Ukraine - some impressions from my trip

I spent July in Ukraine, visiting my family and friends. My son (now almost 9) finally realized that mom and dad are not the only relatives he has. Anyway, from our observations of life of kids (school age) and interactions with people, a few things really stand out. And math is one of them.

First, for my son, the trip was educational in many respects. Learning to multiply two- and three-digit numbers in his head seemed easy for him. My husband's grandmother, 73 years old, a teacher for 50 (!) years and still going - taught my son the method of mental multiplication in less than an hour. And mind you, she teaches high school language and literature (both Russian and Ukrainian). My son loved it!

Second, one really needs to calculate fast there...(I forgot that in all these years in the states). Food is bought primarily at the farmer's markets. And any "sales person" (most often, a farmer's wife or a kid), will calculate the price for "2kg and 400g of tomatoes" in their heads. And you'd better be as fast as they are, or you may be cheated. I felt pretty safe sending my son to do shopping - at least I was sure he can calculate the right change.

Third, my son played with some random kids - in the streets, on the beach, -some were younger, some were older. But even the younger kids act and look more mature then my son. All of them could do multiplication, division, addition, subtraction fast and efficient in their heads. And most of the kids are really physically fit. They run, climb, walk. My son could not compete with them (and comparing to his classmates here, he is pretty skinny and well trained!)

Fourth, Ukraine changed the years of schooling to 11 (they tried 12 for at least 5 years); kids still can officially finish their schooling after 9th grade. Those who do not want to continue being in school, can go to vocational/technical schools or start working as they are.

Fifth, the official workload of a teacher (full-time position) counting the time in-from of the students is... 18 hours a week! The rest of the time is for planning, meetings, collaboration. The workday ends "as soon as everything is done". Kids are in school usually until 1 or 2 pm. No more than 7 periods a day (and that's more than I had when I was in school - never more than 6!) .

Schedule is different for every day of the week; the courses are still taught in vertical strands - physics begins in grade 6 and continues until grade 11, twice a week etc. Algebra begins in grade 6, along with geometry ( but taught as separate subjects) . I think they added more of calculus to 11th grade, but kept the earlier grades with the same sequence/pace as I had.

By any means, I was glad to learn that elementary/middle grades education is still solid in Ukraine. Because later -well, it too late!

(By the way, Ukraine does standardized testing - in 11th grade, math and language/literature. The results are submitted to colleges/universities. The results count for kids, but not for teachers. As my husband's grandmother put it, in the last grades, it is too late to teach things that were supposed to be learned earlier. But for the promising kids, the kids who showed that they want to continue their educations - the teachers ensure that the test are passed well (if you know what I mean).

As we returned to the US, unfortunately (or fortunately), I keep seeing and comparing. My son wanted to take karate classes. But I do not see the instruction/learning. I see "Mommy's treasure" - Good job! Did you have fun, my dear? And I look at that "dear" and at my son and I see them doing a lousy kiba-dachi and fooling around. And I really wish that instructor would yell or better hit my son (well, it's karate, you do things for a reason!) so he learns fast and effective - humility, obedience, and the right position. Well, we'll get home and my son will owe me 40 push-ups.

children's savings accounts & math scores

more from the National Affairs blog:

Math Achievement and Children's Savings: Implications for Child Development Accounts

William Elliott, Hyunzee Jung & Terri Friedline
Journal of Family and Economic Issues, June 2010, Pages 171-184

Abstract:
In this study, we propose that children who have a savings account may be more likely to have higher math scores than children without a savings account. We find that children's savings accounts are positively associated with math scores. Children with savings accounts on average score almost nine percent higher in math than children without a savings account. Further, results suggest that children's savings accounts fully mediate the relationship between household wealth and children's math scores. However, household wealth moderates the mediating relationship. We find math scores of low-wealth children increase by 2.13, middle-wealth children's increase by 4.36, while high-wealth children's increase by 6.59 points. Policy implications are discussed.

Forbes on the Uselessness of Math

Forbes has an interesting guest editorial on its online page. Mr. Joseph Tartakovsky has decided to write an op-ed on the uselessness of studying math. I suppose that Forbes decided to run this in the interest of equal time, given that they've linked to an essay by Diane Ravitch. Tartakovsky is a law student who has no use whatsoever for math, stating "Why teach math in the age of the calculator? The device is available everywhere, from cellphones to fashionable watches."

Like any good lawyer, he backs up his thesis with facts:

"Once a visitor to the Indian prodigy Srinivasa Ramanujan (1887-1920) noted that his cab number, 1729, seemed "rather a dull one." "No," replied Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways." He did that in his head. So what? Give me two minutes and my calculator watch, and I'll do the same without exerting any little gray cells. "

Look Tartakovsky, I hate to be the one to rain on your parade, what with publication in Forbes and all, but I think that even with two calculators and a laptop you wouldn't be able to prove Ramanujan's thesis.

He goes on and completes his proof of the uselessness of math by showing that both Benjamin Franklin and Winston Churchill were bad at math but went on to illustrious careers in spite of it.

Other than the fact that Tartakovsky has no use for math, I don't know what point he is trying to make. I only hope that as our recession worsens, he doesn't write an essay scolding big business for hiring engineers from China and India when so many people in the U.S. need jobs.