09 Jul 2005 - 16:03

Visual teaching & learning K - 2

The comments thread of this post will be the page Becky C requested on the use of direct, explicit visual aids in teaching K-2 math.

Everyone can comment, edit & revise, so please share your experience & thoughts.

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page requested by Becky C

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Just testing to see how comments look--

-- CatherineJohnson - 09 Jul 2005

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I think one of the most important parts of using manipulatives is to connect that work immediately to the symbols of math. My children call it the "secret code of math".. 2 grapes and 3 grapes combined make 5. Kids need to also see that work in written form 2+3=5, 3+2=5, 5=3+2 etc. and written vertically. The problem with the constuctivist teaching in our local school is the inability of the teachers to include this step!

-- LoneRanger - 09 Jul 2005

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-- LoneRanger - 09 Jul 2005

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My kids enjoyed making popsicle sticks into sticks of ten. We glued 10 macaroni pieces onto each stick. These sticks were very helpul in explaing place value (15=1 ten stick and 5)and in teaching regrouping.

-- LoneRanger - 09 Jul 2005

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One of our teacher commenters swears by those sticks.

I'm going to get some for Andrew.

-- CatherineJohnson - 10 Jul 2005

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Hello Catherine, and thanks for the opportunity to consider this topic here. Lone Ranger, I agree completely with connecting number sentences to the number story to the picture, early and often. But first,

To make a long story short, it was a shock to me when my twin boys came home halfway through third grade and they were still using the Touchmath method to calculate answers to addition and subtraction problems with very small numbers. Like 3 + 4. Which was not the right place for them to be at when it was time to learn multiplication. So we had little choice except to have them memorize the times tables by rote, to qualify for the ice cream sundae party at the end of the school year. Luckily, they only needed to test on one factor at a time.

To make a short story long, their first grade teacher taught them Touchmath, which the boys called "touchpoints" (Thanks Catherine). My husband recognized it from when he was a kid, and I didn't complain because I thought it would turn out to be a harmless way station, that with lots of practice in addition and subtraction the boys would naturally migrate to more efficient methods, e.g. counting on from the higher number. Or noticing, for the umpteenth time that they had to calculate 3 + 4, that it is 7. With Touchmath, the children are taught to visualize each numeral as having a specific number of points that are touched, one by one, as the child counts up the total, going from one numeral to the next in an addition sentence. The number three is the most obvious numeral to "point", but the rest of the numerals 1 through 9 are pointed in a way that is not ... robust? Touchmath is like having counters embedded in the printed numeral on the page. My boys mastered this method of seeing numerals as collections of points very quickly, but they didn't move on from it. It's a very quiet method of counting by ones, much less obvious than if your child is still counting on their fingers. But every addition and subtraction sentence they ever saw for the last three years looked just like a workmat with counters on it. These counters, arranged as they are along the stems of numerals, don't lend themselves to any sense of distance or length or dimension. They are arbitrary, yet regular, piles of counters.

Last December, I quizzed my boys really carefully. I asked them, "Do you ever see 3 + 4 as a length? As stacking 4 blocks on top of 3 blocks? As climbing 3 steps up a ladder and then climbing 4 more steps? As moving 4 inches beyond 3 inches?" And the answers were no; no; no; and never. In kindergarten, first, second, and third grade, they successfully participated in mathematics units on data represention, and on measurement. They had successfully created bar graphs with data and answered questions correctly using the graphs. They had successfully measured objects using many measures -- paper clips, crayons, a ruler. But these and many other experiences of dimension were not informing their practice of addition and subtraction. They had never been assessed for speed in adding and subracting, before third grade. I thought speed would just happen. But they were still touching points in third grade.

So this summer, our math boot camp is all about dimension in whole number operations. Visualizing linear distances. I am preaching the gospel of Part-Part-Whole, and we're studying one whole number at a time. I want them to start fourth grade knowing by heart that 3 + 4 belongs to 7, and it will never belong to 6, and it will never belong to 8. They are learning that 7 - 4 gives the same information as 3 + 4 and 4 + 3 and 7 - 3. We are playing Go Fish with whole-number fact families.

No more pointile counters distributed arbitrarily in space. I made up numeral-free, mathematically-true, base10-friendly distance flashcards that illustrate addition and subtraction. The boys can illustrate their own addition and subtraction sentences, and it looks a lot like... Singapore bar models. We're focusing on the x-axis (though I don't call it that, yet). Piaget is rolling in his grave.

But, it's working. Just like when they learned to read at home with a systematic, explicit phonics-based reading program. The boys are even helping me figure out better ways to teach arithmetic to their little brother, who will start kindergarten in the fall. More about that next. And about this funky old book I found at Powell's, on how to teach arithmetic. I want to find more like it. There's so much wisdom out there, and it's so hard to find (for free).

-- BeckyC - 10 Jul 2005

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Becky--I'm going to sound like a dunce, but how do you play Go Fish with fact families?

(I've sort of forgotten how to play Go Fish in the first place...)

-- CatherineJohnson - 11 Jul 2005

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And about this funky old book I found at Powell's, on how to teach arithmetic.

I can't wait to hear about this book!

-- CatherineJohnson - 11 Jul 2005

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Becky--What is the Touchpoints method?

-- CatherineJohnson - 11 Jul 2005

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number lines are critically important, I think.

I just learned today that Andrew, who is severely autistic and just barely verbal (but probably very bright as well) is learning math almost entirely through a large, desk-size number line used with money.

The teacher puts one nickel on each of the 5s--5, 10, 15, 20--then has Andrew hand her the sum (20 cents, or whatever it is).

At least, I think that's the way she does it.

In any case, Andrew has incredible powers of visual learning and visual memory, and my sense is that the teacher is using the number line to prevent him simply memorizing each individual sum of money the way a Chinese child must memorize each ideogram.

I'm getting a number line for Christopher & Andrew both to use.

-- CatherineJohnson - 11 Jul 2005

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Hi Becky,

Boy, you brought back memories. One of my sons is LD/ADHD with developmental delays, but he is very visual and has almost weird visual memories at times. He was in a self-contained classroom for the first three years and then in another one during 4th and 5th. In the first three, I remember the teacher sending home the Touchpoints method, which seemed fine, like you said, until you needed to remember where the points were. This seemed like an added layer to me and we had enough problems with short-term memory as it was without some arbitrary way of looking at a number. I could see a regular kid having fun memorizing the points on 8 or 9, but to your average LD kid, it seemed like another obstacle.

I remember the 4th/5th grade teacher sending home the same stuff. I had the same reaction. I knew that he'd have no trouble until about 6 (or 8 or one of the ones past 5 where the little points aren't so obvious.) It made no sense to me to start this visual idea with numbers that didn't connect to anything else. The teacher tried hard that year to teach it, though.

It's interesting to me that you mention that your sons didn't seem to be moving on and connecting the numbers to other math forms. That was the thought I had but I didn't know how to express it. One thing about LD kids--things really have to be efficient and to the point. Lots of distracting ideas is not helpful in my experience.

I will concur with Catherine (whom I tend to always concur with) that the number line was the key for my son. I was extremely tense about the fact that he needed his number line so much more and longer than anyone else. I didn't understand what the heck was going on. His LD teacher just kept telling me to pull it out and use it as long as he needed. I swear I never thought that numbers would ever go "into his head." He used this big, yellow laminated number line all the way to fourth grade. I often had to keep physically counting with him on the line, he just couldn't ever remember how it worked. Many tears during those years (mostly mine.) Then, it just happened. Almost overnight.

I think that's the developmental part that is very difficult for parents to wrap their brains around. Sometimes it's just gonna happen when they're good and ready and not one minute earlier.

I know your situation is a little different, but I just remember wondering how the touchpoints aided in developing true number sense.

My son is a few years behind in math, but I'm pleased to see him plugging along in the Saxon 6/5 homeschooling course. I am still surprised when he skip counts and does the mental math at the beginning of the chapters. Actually, I'm shocked. I truly thought it would never happen.

Catherine, I imagine you could google it and you'd get some info on it.

-- SusanS - 11 Jul 2005

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I will concur with Catherine (whom I tend to always concur with) that the number line was the key for my son.

Boy am I glad to hear that.

Because I am gearing up to teach math to Andrew, and teaching math to Andrew is going to make the past year with Christopher look like child's play.

-- CatherineJohnson - 11 Jul 2005

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I think that's the developmental part that is very difficult for parents to wrap their brains around. Sometimes it's just gonna happen when they're good and ready and not one minute earlier.

I've gotten so much calmer on all this (not that that should necessarily translate to anyone else's experience...)

Having raised two autistic kids, one of whom was considered to be really hopeless, I've seen that they absolutely do change, develop, and learn.

We now are less panicked, often, than the teachers, simply because we've seen Jimmy grow up.

This year Andrew's teacher started freaking out because Andrew wasn't pulling his pants up when before he went to the bathroom. He'd just walk right out into the hall or the class while hauling them up.

We were completely unflapped by that, because Jimmy, when he was young, would refuse to wear any clothes at all, ever and would leave the house stark naked if someone forgot to lock it.

At some point, Jimmy decided he needed to wear clothes. He didn't get taught to wear clothes, even though we constantly harrassed him about it.

He just decided to wear clothes, or came to feel he ought to wear clothes.

Of course, otoh, lately he has taken to pulling down his pants and quickly touching his silverware, food, or cup to his penis before he feels he can eat, drink, or deposit whatever he's holding in the dishwasher.

So...uh...maybe I shouldn't be quite so confident about Andrew's social future....

-- CatherineJohnson - 11 Jul 2005

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The interesting thing about the new peen obsession ('peen' has always been our family word) is that I'm pretty sure Jimmy does not do this in public. (I need to ask his teachers and program directors, but I'm positive I would have heard about it if he were doing it.)

So as odd as he is, he has some kind of mature sense of social rules.

-- CatherineJohnson - 11 Jul 2005

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btw, another thing I have to blog: PRINCIPAL'S GUIDE says that one of the math neuroscience researchers believes humans have an internal number line, inside our brains.

I'd be willing to bet we do not have an internal Touchpoints.

-- CatherineJohnson - 11 Jul 2005

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At this point in my limited experience with remediation... I am going with a number line as the better visual support for a child who counts by ones up and down. Thanks, Susan, for the background on your son's experience. I am so encouraged that my boys will have the numbers go in, someday. Sooner if we're lucky.

Until or unless a parent or teacher further up the food chain at kitchentablemath intervenes in this thread or elsewhere with dire warnings of unintended consequences of number lines, it's got to be better to have a child constantly experience equally spaced segments on a number line. Each unit on a number line shares that quality of equality of length, whereas in Touchmath, the points are spatially arbitrary. Perhaps a child hears or feels nearly equally spaced taps in time, and this contributes to their developing sense of proportions... but I doubt it. I do recognize it's important for children to be able to count the members of a set without requiring the members of the set to be lined up shoulder to shoulder. If I'm asking a child to count a set of flowers, they need to recognize that the flowers share the quality of flowerness, whether they are of different colors or sizes, and that's all. We don't need to pick the flowers and line them up.

I chatted with one of my boys today about why we're not studying more than pairs of parts of whole numbers, i.e. we're not studying the fact 1 + 1 + 2 + 1 = 5. I told him that there was no use in studying how to get to 5 by one step and one step and one step and one step and one step. And he thought about it and said, "we're learning to jump by 2 then leap by 3!" Very sweet.

On the subject of developmental readiness in children who are not diagnosed with a learning disorder, I believe there's just too much temptation to wait. To leave kids where they are at. A first grade teacher can claim a child isn't ready this year, and if the kid doesn't get it next year, well, nobody brings that child back to the first grade teacher to fix. WAIT FOR THE CHILD is the siren song of constructivism. I mean, it's rather dog-eat-dog. When Steven Pinker says "Old-fashioned practice at connecting letters to sounds is replaced by immersion in a text-rich social environment, and the children don't learn to read", he's not suggesting it's a great thing that only the strongest (will teach themselves to read and) survive. How about, "old-fashioned practice at connecting parts to wholes is replaced by immersion in a strand-rich social environment, and the children don't learn any number sense"?

My boys and I have experienced the reality of Pinker's assertion that mathematics is "ruthlessly cumulative".

One last little third grade flashback for tonight, to illustrate. The boys were given multidigit multiplication problems. Encouraged at school to think in terms of arrays, they dutifully created arrays of dots, and dutifully counted those dots one by one. If there were a lot of dots, more chance that they'd miss one though they were careful. Classwork began coming home with half the problems wrong, half the problems left undone, and NO comments by the teachers. Nothing, not a drop or a shred to help me figure out where they were at and where they were going. But if you think that huge arrays of 56 points would be enough to motivate the boys to memorize their times table... Um, no. When the horrible unit on multiplication was done, it was time to study the times table at home. The third grade team was explicit about this -- that this was a job for parents to do. There WERE NO HELPFUL SUGGESTIONS for this homework. No likely strategies shared with all parents, and certainly nothing specific shared with me for my boys to use. The boys had no problem sitting down at the kitchen table and working through 1X and 2X. But as we began to talk about 3X... that's when I found out they were unable to add 12 + 3 = 15, 15 + 3 = 18, etc. without Touchmath. That's when I started to ask them more questions about their basic facts, and what, if anything, they could visualize.

More on suggestions from the arithmetic activities textbook tomorrow or the next day.

-- BeckyC - 11 July 2005

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Math is not a language, but let me draw this analogy between developing number sense, and teaching phonics, in K-2:

When they were 6yo, my boys responded beautifully to a systematic explicit phonics program -- it was a series of 72 booklets, and each booklet focused on connecting one sound with its constituent letters. There are many such phonics programs available at bookstores and on the internet for parents to see and choose from, e.g. Bob Books (no endorsement -- not the one we used anyway -- but knowing what I know now, I think it would work and so I'm mentioning it here). And parents can walk into any public library and find many competing series of such booklets published. Even if we hadn't lucked into the program we used, we could have found many programs to embark upon, leading to a similar happy result.

So, why can't I walk into the library, or a bookstore, or search the internet and find a series of math booklets, each focused on a number from 0 to 20? And see, for instance, the "Make 7" booklet that tells compelling (perhaps humorous?) addition and subtraction number stories with great pictures and number lines and the number sentences? Under the organizing principle that: it's all about 7, in the "Make 7" booklet? Perhaps there would even need to be separate 7-0-0, 7-6-1, 7-5-2, and 7-4-3 booklets. I mean, I don't pick up a phonics booklet about the short vowel sound "ow" and expect to find "The quick brown fox jumped over the lazy dog".

The shelves are groaning with humorous and gorgeous and themed counting books, but one is hard pressed to find a book that shows simple number sentences with corresponding pictures. I found two with sums that stayed under 10 and that weren't simply adding by ones (counting). Whereas there must be 200 counting books.

Are systematic explicit arithmetic booklets found at Kumon and Mathnasium and Sylvan?

-- BeckyC - 12 July 2005

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Becky,

Maybe you have found a void you can fill! I often find some area lacking at the library. I agree about the beautiful counting books. It all sort of stops there.

The whole language vs. phonics debate drives me nuts, too. It took me a while to learn more about what was really going on with all of that. Some kids do read by "pulling words out of the air." Some don't. I imagine most probably don't. Even the ones who just wake up one day and get it due to all of the "exposure" they're receiving still have to deal with spelling down the road. The Look-say crowd (with little to no phonics training) does well until they get to a word they've never seen. The phonics-based kid is more likely to be able to decode it. I understand the fear of the old kill-and-drill returning, but once again, it seems they've taken it too far.

I wonder, too, if the kids who were finally recognized as having a reading problem by 2nd grade might have been better off had they been drilled more in phonics from Pre-K on up. Most schools still seem to view it as a tool, but they don't really teach it any systematic way from what I've read, but as an embellishment almost.

I've often wondered if this whole math debate isn't just an extension of the old parts-to-whole, whole-to-parts argument that has been raging since the whole reading days.

Parts-to-whole teaching, according to its advocates, gives students the facts--the building blocks--and then they build them into some meaningful structure. Whole-to-parts reveals the entire structure and then pulls various parts out and explains them one at time whenever the child encounters them. Whole-to-parts instructions appears to me to require analytical thought which really doesn't appear in the child till late grade school or middle school. This could be what's behind the frustration of many parents when they run across these oddly inappropriate problems (and/or projects) that pop up throughout the year.

One professor and fan of whole language once said, "Accuracy is not an essential goal of reading." This same logic seems to apply to the Standards Math curriculums.

-- SusanS - 12 Jul 2005

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"Accuracy is not an essential goal of reading." That statement reminded me of the following, posted by the blogger, "Instructivist"

"Prof. Plum's latest offering reminds me of a wonderful piece of satire by Kerry Hempenstall I read a while back. The virtue of this satirical piece is that it nicely and perfectly encapsulates the progressive/constructivist ideology -- a fantastic ideology that would be driven out of town in any other profession.

Mr. Hempenstall, one of the foremost reading experts, tells how the founders of whole language would teach how to play golf. I wonder how they would train pilots and physicians?

Hempenstall, K. (1996). Whole language takes on golf. Effective School Practices, 15(2), 32-33.

Well folks here we are at the WL School of Golf with our two founders - Smith and Goodman. What can you tell us about your method of teaching beginning golfers?

"Yes, well, our approach to teaching golf is more of a philosophy than a method. We consider that golf is an holistic experience which comprises more than the sum of its parts. Golf, to us, is an irreducible experience best learned by doing, so we enter all our novices in the Australian Open because that's authentic golf. Our role is that of motivator/facilitator, we empower our students to grow in golf. We do not teach skills of course; even though some students request help with their swing, we explain that swing is only a sub-skill of golf, and to emphasise it out of the context of authentic golf is time-wasting or even harmful.

We do like to see our learners practise their invented swing during the Open itself of course; the principles of the swing are eventually induced by the learner who is highly motivated during an Open, but probably bored to tears and disheartened by artificially timetabled swing practice. Thus we (along with another former champion, "Jocular" Johnny Rousseau) consider that the swing will evolve naturally, that feedback is pointless and it may even damage the essential confidence that learners need if they are to take risks with their golf. Since golf is as natural as learning to speak, we allow it to develop, rather than forcing it - just as speech developed.

Golf being such a natural pursuit, there is no need to demonstrate grip, stance, or even which end of the club is best to hold - gradually, through playing in authentic tournaments, the efforts of the novice will more and more closely approximate that of Greg Norman. If for any reason development is slow, probably caused by earlier misguided attempts at skill instruction, we provide entry into other golfing majors, such as Augusta, or St Andrews - more immersion in real golf is the answer. Golf improvement depends largely on the learner's establishment of a self-regulating and self-improving system, not on anything an instructor provides.

We also ensure that our students don't practise their chipping or bunker shots as that involves fractionating the great game. Similarly, we consider driving ranges and putting greens are merely mind numbing traps only used by old-fashioned, ignorant instructors who fail to understand the implications of the new research literature on preferred golfing styles. Golfing-for-meaning is our mantra, because of course golf is a very personal activity. Only by considering the golf experience from a developmentalist-constructivist-relativist perspective can we move away from the notion of goals prescribed autocratically from above.

We believe that players can progress far beyond the shallow objectives of the ball-in-hole-in-minimum-strokes model which dominates in certain quarters. Our players are encouraged to achieve satisfaction of their own diverse needs, which may be markedly different from those of course-designers, or self-appointed traditionalists. The golfers transact with the course, bringing their own unique understandings and experiences to the event; they should not feel tied down by conventional notions of what the process should mean to the player.

We also teach a revolutionary strategy in that we encourage our learners to disengage from the tyranny of the ball. The ball is only marginally relevant to the game, and is too often over-emphasised. It is, after all, only one cue to the deeper transacted meaning of the golfing experience. Students are sometimes bemused when we instruct them to pay as little attention as possible to the ball - just a quick glance is all that is needed as they stroll along the fairway (to ensure that their prediction is correct, and it is a ball not a cowpat). Striking any ball that meets the definition of a ball will do, it needn't be your own - in fact such an action is a genuine indicator of the degree to which your comprehension of the true potential of this exciting game is developing.

How much success are we having with our up-to-date, golfer-centered philosophy? We have numerous anecdotes from dedicated teachers who find our approach so much more rewarding - they have no trouble engaging their students; they see the joy on the faces of the students; they are exhilarated to be part of this important redefinition of the essence of the game. Scores? You ask? Unfortunately that question is very revealing of your failure to keep up with modern research. You are still dominated by out-dated reductionist models of golf. One cannot validly and reliably keep scores without interfering in the golfing process; scores do not reflect all that is entailed by golf; they fail to capture more than the most miniscule element of the whole game. Scores are likely to be used to compare golfer to golfer - which is an unconscionable intrusion on the innate developmental trajectory of each individual seeker of golf prowess.

We anticipate our philosophy will sweep the golfing world. It is new, innovative, flexible - everyone's a winner. And we won't stop there either. We already have plans to take on swimming coaching for beginners, using our proven immersion techniques. The sky's the limit - Hey, Kenny G., have you thought about using our approach for beginning skydiver training?" "

-- LoneRanger - 13 Jul 2005

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Good one, LoneRanger^?! I'd laugh even harder if I wasn't choking back tears. I just re-read Steve Leinwand's incoherent piece in Edweek because I wanted to comment, visually speaking, on his "gazinta" dismissal of long division, even though long division is well beyond the scope of this K-2 thread. And while I was searching the site for his incoherent piece, I turned up a letter to the editor from 1994 also taking issue with some yet more ancient insanity from Leinwand that was uncritically given a prominent position at Edweek.

But more on arithmetic soon, some excerpts from the book I got at Powell's. LoneRanger^?, how old are your kids?

-- BeckyC - 13 Jul 2005

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My kids are 10 and 8.

-- LoneRanger - 13 Jul 2005

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I was thinking that perhaps Greg Tang would be the right person to write whole-part-part booklets. He does have a single book out called Math Fables that explores number stories about each number 1-10. I'm not shilling for him because he won't do linear representations, and I haven't seen Math Fables, but here are excerpts from four reviews:

"I had a very hard time finding a counting book that would help me teach 1st graders that numbers are made up of other numbers. This book gives you great poems that are perfect for the exploration of numbers 1-10. The poems are short, descriptive, and beautifully illustrated."

"As he did in Math Appeal, Tang introduces children to the wonders of grouping numbers. Each 'fable' tells a rhyming story in a two- or four-page spread, with each setup more complex than the last. One of the first fables tells of two young birds. One bird takes wing and hits the ground, and the other one falls from the sky and nearly drowns. When the birds practice together, however, they both learn to fly. In another story, 10 beavers leave for work, regrouping and reorganizing their numbers all day."

"Continuing to make arithmetic fun, Math Fables by Greg Tang, illus. by Heather Cahoon, offers 10 rhymes about animals that teach a life lesson while demonstrating basic addition. For the number seven, 'Gone with the Wind' traces the path of monarch butterflies to Mexico, using every possible combination of addends (5+2; 6+1; etc.): 'Their journey would be very far,/ a thousand miles or more./ The monarchs flew both day and night/ in groups of 3 and 4.'"

"For example, in 'Going Nuts,' four squirrels frolic in autumn leaves until they realize they need provisions for winter. One begins to explore while three sit on a branch, frightened with worry. Next, '2 squirrels raced to gather nuts' while 'the other 2- buried them in stashes underground.' Finally, 'all 4 slept very well that night,/no longer feeling scared./They learned it's wise to plan ahead/and always be prepared!' Cahoon keeps the different combinations together by enclosing them in ovals, visually emphasizing that although the groupings may look different, they still add up to four."_

-- BeckyC - 14 Jul 2005

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I was wondering how Montessori schools approach developing a child's number sense. I am having the boys work with lego blocks in emphasizing parts to wholes. Today, I found Catherine Stern. Wherever you see the word "materials" and you think "manipulatives", know that the kids are handling these manipulatives with their eyes open -- it's an outstanding visual lesson, always. I particularly like the Number Track.

-- BeckyC - 16 Jul 2005

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With non LD kids, I believe that they hang on to things when they don't have the confidence to make the leap to the next level.

My daughter used a number line in second grade. She could not do any problems without a number line. Once we started the mad minute, she developed confidence in her ability to add and she never asks for the number line any more.

The kids who have confidence in their ability will naturally gravitate to the easiest, most convenient method. Here are two stories to illustrate this:

I was discussing with a neighbor the lack of a long division algorithim in Everyday Math. She stated that her son showed her the method he learned in EM. Then she showed him the traditional algorithim. After that, he started using the traditional algorithim.

When I was a little girl, I showed my mother the method I was taught for subtracting: The traditional borrowing algorithim that we still teach today. She showed me her method which involved adding to the bottom number instead of borrowing from the top number. It looked easier and quicker to me, so I still use that method today.

When a teacher thinks a child is not developmentally ready, she is probably correct. But the answer, in most cases, is not to wait for the child to be ready. The answer is to do what we parents are doing: determine what will give our kids the confidence to make the leap to the easiest, shortest method. Remember, someday they are going to have to be able to teach themselves to understand the math.

-- AnneDwyer - 16 Jul 2005

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Earlier I promised to tell a bit about the funky old book I bought at Powell's...

Edna Sykes published the Arithmetic Activities Handbook in 1975 and it draws on her practical experiences in teaching arithmetic to elementary school children in the 1950s and 1960s. Starting with some excerpts.

"Children are mobile and active creatures, therefore the arithmetic must be mobile, relevant and active." Anybody who ever dared to call children creatures gets a gold star from me. If this offends you, stop reading now!

"In an activity program every child particpates voluntarily, in a cooperative spirit, and actively. This does not mean physical action, nor does it mean that the child always leaves his seat or performs uncontrolled meaningless actions to satisfy the word 'activity.' Learning in itself requires self-discipline, and the small child must learn from you the importance that discipline of the mind and body must assume in arithmetic." Yes, ma'am. "The purpose of this book is to take ordinary lessons and transpose them into active lessons, methods, and devices that not only are enjoyable for all but that also reach the child at his level, encourage mental activity and voluntary spirited involvement, and above all establish memorable points of reference for future recall."

"Unlike the idealist who believes that standards are the sole responsibility of the student, teachers of children know that a student produces that standard of work which is accepted from him. Setting standards, therefore, is a vital part of a successful arithmetic program and major responsibility of the teacher."

"It is recommended that arithmetic time be used each day to teach the small child how to apply the arithmetic skills he has acquired. For the small child much discussion and use of visual aids in required.... Problem solving must be taught. No child is born with an instinctive ability to transfer physical action or the spoken word into written form."

"In the rush to teach the new math and its strange concepts, written skills which help to make up the total picture of arithmetic success for the small child were ignored...and the small child left to his own devices did not develop the working mental discipline which manifests itself in neat and accurate paper work." Hmm. Was Edna thinking of adults asking small children to write down their mathematicallly powerful strategies in a math journal? Probably not! Edna was probably talking about having every child in the classroom simply being able to write their number sentences neatly so that other people could read it.

Edna then outlines her readiness activities:

1. counting orally (naming) 0 – 10 (as we have children sing the alphabet song)

2. connecting the spoken name with the written symbol

3. writing numerals

4. ordering numerals

5. seeing number value by discrete and linear quantities.

6. adding and subtracting with written number sentences

She makes a special note about starting with zero when teaching counting: "... the child who starts with one when counting never realizes that the value of one comes as a result of the step from zero to one, and not from one to two. Even the ruler starts with a zero, but because the child does not see the printed symbol he is not aware of this concept." She is also critical of number lines with printed numerals: "...draw a giant ruler on the board and drill the skill of counting spaces and not numbers. One of the failures of the ‘number line’ approach is that the small child sees the number as an object, and does not comprehend that the value of numbers lies in the addition of spaces as represented in the ruler or number line. Children seem to get more benefit from counting the number [of] dots than they do from the abstract number line." In this area I differ with her slightly. I am working with my kindergartener on both discrete and linear pictures of number value. In her suggested activities, Edna does include activities with quite linear ladders, bridges etc.

The most valuable thing I learned about arithmetic teaching from her book was to focus on one cardinal number at a time. Do not mix mix mix the practice, yet. Have your children focus on one number and all of its fact families. The following four facts belong to 6: 2 + 4 = 6, 4 + 2 = 6, 6 – 2 = 4, and 6 – 4 = 2. For example, 9 – 3 = 6 does not belong to 6.

-- BeckyC - 17 Jul 2005

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Also from the archives,

Catherine Stern published Children Discover Arithmetic in 1971, and her daughter has continued the legacy. I don’t have a copy of Stern’s book yet, but from Structural Arithmetic website, Catherine’s readiness activities would map as follows:

1. seeing number value by their lengths, without direct attention to naming or numerals

2. ordering numbers (lengths) without naming

3. adding numbers (lengths) without naming

4. counting orally (naming) 0 – 10

5. adding and subtracting with names

6. connecting the spoken name with the written symbol

7. writing numerals

8. writing number sentences

Kind of interesting. Whereas Sykes is takes the approach of teaching names, numerals, then number value. Is it true that in German they name the letters by their sounds? Math is not language, but if Stern wrote a book on teaching children to read English, would it go like this?

1. hear and notice all sounds without direct attention to letters or graphemes

2. add sounds to make words without naming

3. given a word, list all sounds

4. hearing all sounds, say the word

5. connect the sounds with the graphemes

6. etc.

Some choice excerpts on teaching arithmetic from Stern:

“What is involved in the formation of concepts? Children seem to reason with mental pictures. Therefore, when teaching children to think, we must develop their ability to form images. Multisensory materials will have fulfilled their purpose when the children can visualize the concepts presented.”

“The teacher should introduce number symbols (numerals) after number concepts have been explored.”

“Children who don’t know math facts (e.g. 4 + 4 = 8) will have had difficulty learning them and will resort to counting them out over and over again because they have become attached to the ‘counting song’.”

About teaching 6 – 4 = 2: “Teachers know that some children, when taking four loose cubes away from six loose cubes, cannot visualize the original amount and wonder just how much they began with. They lose any sense of the relationship of 4 to 6, and of the remaining 2 to the 6 they began with.”

In fact, the Stern folks sell number “frames” from 1 to 10, and so a parent can pull out the “six frame” and have the child discuss number stories and number sentences using number blocks of only those lengths that will fit into the “six frame”.

Not being willing to cough up a lot of money for expensive wooden number blocks and frames at this point, I am having the boys work with Lego blocks, and with graph paper and pencils to illustrate number stories and their correspondent number sentences. With pencil and graph paper, we can represent 6 – 4 = 2 by drawing a line from 0 to 6, and either drawing a return line of length 4 or another line from 0 to 4 just underneath the line that is 6 units long. Then, it really pops out nicely for the boys that we are looking at the negative space, the line of length two that isn’t there but is suggested. We emphasize that 6 – 4 = 2 only gives the same information as 2 + 4 or 4 + 2 (or 6 – 2). It’s not a different animal.

For my younger son, we carefully connect a good number story about car travel to go with the linear distances shown. “I drove out six miles that morning. I came back four miles that afternoon. How far was I from home at the end of the day?” Or, “You drove six miles in your car. I drove four miles in my car. How much farther did you drive?” Works for us. I’m sure other folks could come up with better stories.

-- BeckyC - 17 Jul 2005

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