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The Great Fire Wall of China

As my regular readers know, I am writing from China these days, and have been doing so four years so far. Sometimes the blog becomes inaccessible to me, making it impossible to post regularly. In fact, starting in late September 2014, China began interfering with many Google-owned entities of which Blogspot is one. If the blog seems to go dark for a while, please know I will be back as soon as I can get in again. I am sometimes blocked for many weeks at a time. I hope to have a new post up soon if I can gain access. Thank you for your understanding and loyalty.


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Monday, September 26, 2016

California Proposition 58: A Solution Looking for a Non-existent Problem

In 1998, it was Hispanic parents who clamored to get rid of bilingual education. Bilingual education was not a bridge but ajail. Hispanic children languished in bilingual classrooms for years and years, and never attained the proficiency that would allow them to go to college. The parents were successful in getting Prop 227 passed, not so much because of the force of their own arguments, but largely through the ideological effect of English-only whites who maintain that America is an English-speaking country, and so students should be taught in English.

The California Department of Education says they are already ensuring that English learners:

...acquire full proficiency in English as rapidly and effectively as possible and attain parity with native speakers of English. AND ...within a reasonable period of time, achieve the same rigorous grade-level academic standards that are expected of all students.

Bilingual education is already readily available. Previously, the school decided whether a child would be placed in the bilingual program. Currently, it is the parents who decide. Most parents choose English immersion.

Since the main effect of Prop 58 is to undo Prop 227, we must next investigate whether removing the decision-making power from parents is actually good for kids.

The proponents of Prop 58 claim that the American Institutes for Research (AIR) concluded “there is no conclusive evidence to support the opponents claims that Prop 227 has been successful. AIR is a credible source, so I took a closer look. The first thing to note is the cited report was published in 2006, and looked at the previous 5 years. It specifically said that because of the many variables involved, “There is no conclusive evidence that one instructional model for educating English learners, such as full English immersion or a bilingual approach, is more effective for California’s English learners than another(method)...”

In other words, a dichotomous approach does not work. AIR was unable to isolate pre-Prop 227 or post-Prop 227 as the independent variable. That is a little different from what the proponents of Prop 58 claim the AIR report said. Furthermore, AIR was unable to control for the numerous other variables that impact Hispanic achievement. So what kind of evidence is there? The AIR report itself observed,

During this time, the performance gap between English learners and native English speakers has remained virtually constant in most subject areas for most grades. That these gaps have not widened is noteworthy given the substantial increase in the percentage of English learners participating in statewide tests, as required by federal and state accountability provisions.

So even though many more English learners took the statewide tests, they did not bring down test scores as was expected. Ten years have passed since that AIR report was published. Who knows, but what a new report might find conclusive evidence, or at least a greater quantity of circumstantial evidence.

Lacking an experimental methodology, the AIR evaluation often relied on case studies, which is simply a systematic look at an anecdote. I have a few anecdotes/case studies of my own. I have not worked much with Hispanic students because most of my 40 years of teaching took place in Department of Defense Dependent Schools in Japan or in private schools located in Japan and Shanghai. I was also an in-service training provider to public secondary schools located on the Navajo reservations in Arizona. One time I was also assigned to a 6-year-old Hispanic boy who had been in horrible car accident to be his at-home teacher. I will summarize each experience.

1. The Hispanic kindergarten student. I began teaching this boy the last three months of school. I met with his classroom teacher who gave me a packet of papers to color and a copy of his third-quarter report card. His grades were very bad, and his progress was below grade level on almost every measure. His teacher referred to him as “one of those typically dumb Hispanics.” Back home, I threw away the papers she gave me, and spent the weekend creating a kindergarten program for this boy. Within three months, he was reading English and performing at grade level on every measure. His parents were thrilled.

2. The Navajos. I did not work with the Navajo students directly. At my in-service presentations, their secondary teachers complained that they did not need the information I had been commissioned by the administrators to present. They asked me to tell them how they could use their subject area textbooks to teach their students to read. I chucked my carefully planned presentation (including hands-on activities) and immediately improvised a seminar on phonics and reading comprehension using science and history books. The teachers loved it (but the administration was peeved at me. Whatever).

3. Japan. One fall, a large group of parents suddenly enrolled their children in the junior high where I was actually teaching science. The parents took this drastic measure because their children were refusing to go to school due the extreme bullying that sometimes occurs in Japanese schools. The principal pulled me out of my morning classes, and asked me to create a half-day transitional program for these kids. They studied art, music and PE in the afternoon in the mainstream class. After three months, I put them in the mainstream math classes. After the second term, I put them in my mainstream science class. After the third term, they were fully mainstreamed, including English and social studies.

4. China. I have spent 4 academic years teaching in China using English only with great success, even with first graders who speak zero English when they start. Within one year, all but just a handful were reading and comprehending at American second-grade level. Their English speech still retains errors attributable to Chinese syntax, but those errors will fix themselves eventually.

42.8% of community college students are Hispanic in 2015. Overall, the number of Hispanic students in college has been increasing dramatically year over year, while the number of white students in college has been falling over the last five years. According to Pew, a record number of Hispanic students have enrolled in college, and the high school drop-out rate is the lowest it has ever been. The numbers on both measures have been positive since 2000. English Only as a factor contributing to these results did not occur to Pew, but it is as likely a factor as any of the others that Pew did suggest. According to the National Conference of State Legislatures, Latino college completion is on the rise and in the past decade the number of Latinos with bachelor’s degrees or higher increased 80 percent. Of course, this achievement is not due solely to Prop 227. Programs such as AVID, TRIO, Gear Up, and others also contribute to positive outcomes.

As far as the proponents' claim that Prop 58 would expand second language opportunities for native or fluent English-speaking students, a proposition is unnecessary. Schools are already free to add foreign languages to their curriculum, or create foreign language immersion programs.

It is unfortunate that a majority of the California legislature supports Prop 58. They seem unaware of the history since the legislature in place when Prop 227 was approved has either retired or termed out. The legislature also seems unduly impressed by the articulate but empty arguments of the proponents when compared compared to the emotional tenor of the opponent's arguments. In fact, the proponents' statements in the Voter's Guide read like one of those long-winded sales pitches with a lot of beautiful words that actually say nothing. For example, the proponents introduce a paragraph in the Voter's Guide by saying, "Here's what Prop. 58 actually says:," and then proceeds to quote, not Prop 58, but the already existing California Education Code. In this way, proponents mislead voters into thinking that Prop 58 will do something that is not already mandated, when in fact the law already mandates it, and Prop 58 is unnecessary. No wonder the less sophisticated opponents got emotional.

There is no need to fix a non-existent problem. In short, the stakeholders with the most compelling interest, that is, parents of Hispanic students, do not want Prop 58. That should be good enough for the rest of us. Vote NO on Prop 58.

Thursday, August 18, 2016

This is What's Wrong with Tech Articles

GreatSchools has an article about evaluating the effectiveness of technology in your child's school. Just like most such articles, it does not even question the assumption that technology should be used. The unexamined assumption is of course technology should be used. It is only a matter of whether it is being effectively used.

The assumption ignores two considerations. One, technology has always been used in schools. There are people still alive who remember the old mimeograph machines that produced odorous purple worksheets.

Language labs once used huge reel-to-reel tape players.

There are people who remember helping their teacher carefully thread the filmstrip projector.

Eventually, the the projector gave way to VHS tapes which finally gave way to You-tube videos projected from flash drives.

The point is there is no stopping technology. Which brings us to the second consideration. Back in those days, there were no articles discussing whether technology was effective or not. Technology was a tool, but not a panecea. We had not yet mentally endowed technology with mythological superpowers. Technology was not "a thing." Today, technology is a bandwagon to jump on merely for technology's sake. Tech for tech's sake is expensive and unnecessary.

"Research shows these (smart) boards can increase both student interest and participation," (but this does not necessarily translate to increased understanding or achievement, especially if it doesn't) "change the dynamic of the classroom...Because it’s the teaching practices associated with technology use that matter most.”

Sunday, June 26, 2016

Wrongness

The topic of being wrong pops up more and more frequently in public discourse these days. Author Chuck Klosterman, maintains we are probably wrong about everything we think we know, including and maybe especially gravity. Meanwhile, we are chided for being “ant-science” if we disagree with the consensus of scientists. In a famous Last Week Tonight spot, Bill Nye (the Science Guy) leads a climate change “debate” that was no more than Bill with 96 white-coated people representing the 97% of the scientific consensus against 3 other people representing the 3% of the science community refusing to join the bandwagon. Case closed, apparently.

We all “know” that Republicans are the anti-science party, right? Except, according to Neil Degrasse Tyson, there is plenty of anti-science on the Liberal side of the aisle as well. Steven Novella, MD, a contributor to Neuroligica Blog, supports Dr. Tyson’s assertions with some survey results, concluding, “My synthesis of all this information, which is admittedly incomplete, is that people tend to be anti-science whenever science confronts their ideology.”

Dr. Novella elaborates,

I think it is more meaningful to understand these issues by breaking them down to specific ideologies and how they influence acceptance or rejection of science. Conservatives tend to value freedom, the sanctity of life, and the free market and they distrust government. Liberals value nature and the environment and distrust corporations. Individual issues are complicated because they can cut across multiple ideologies. In terms of the question of who is more anti-science, my approach is this – you don’t get credit for being pro science for accepting an issue that is compatible with your ideology (bold added). Liberals acceptance of manmade global warming does not mean they are necessarily pro science, because this issue is right in line with their ideology (pro nature, anti corporate). Conservatives don’t get credit for being pro nuclear for the same reason. Evidence for being pro science is when you accept a scientific consensus that conflicts with your ideology. You have to demonstrate that science comes before your ideology, (bold added).

The thing is the 3% of scientists who disagree with the 97% are not wrong simply because they are outnumbered, as Bill Nye implied. Science is not a majority-rules proposition. Throughout history, there have been scientists who have disagreed with mainstream science. Some suffered, at worst, outright scorn and ridicule, or at best, indifference, only to be found to have been right all along. One big reason why accusations of being “anti-science” carry no weight with either camp is because everybody knows that settled science is settled only until a scientist unsettles it.

“Anti-science” is the new heresy. There is nothing wrong with disagreeing with settled science. The problem is when disagree-ers (of any stripe) have no basis for the disagreement except ideology. That’s a problem that seriously impedes useful discourse on any issue.

Sunday, April 24, 2016

Missing Key to Understanding Place Value

I write a lot about place value. Place value (along with zero) may arguably be the most important math concept because it underlies every single calculation we do. Yet teachers often do not teach place value well. Teachers (and most curriculum) are satisfied with a very superficial understanding of this essential concept. If a child can identify the place name of a given digit or put a digit in a given place, most teachers deem the child to have a good understanding of place value. Place value is so much more.

Groups of Ten

Place value is all about making groups of ten. Well, yeah, the reader might say. Tell me something I don’t know. The key to understanding place value is the realization that each succeeding place represents a group of ten of the preceding place. Duh. Stay with me here. The curriculum and instruction alludes to this key, but rarely makes it explicit. Most textbooks have replaced “borrowing and carrying” with “regrouping,” and this was a positive step, but students still take a mechanical view. They still borrow and carry as they move leftwards through an addition or subtraction problem without realizing that they are actually making or breaking a group of ten at each successive place. For example, if they carry a one from the tens place to the hundreds place, they mechanically add that one to the other digits in the place without realizing that the carried one represents making a group of ten. In fact, most students will say, (correctly on a superficial level), that they made a group of 100 because they put the “1” at the top of the column named “100s place.”

Place value is all about making groups of ten. Subtraction is all about breaking groups of ten into loose ones and dumping them with the other loose ones. Every place except the loose ones is a group of ten something. Teachers tell students that each succeeding place is larger by a magnitude of ten, but somehow children fail to grasp the significance of this fact. The reason the standard addition algorithm works is because you are gathering up groups of ten at every place. Likewise, the reason the standard subtraction algorithm works is because you are breaking a group of ten at every place.

Students betray this lack of deeper understanding when they express surprise that given the number 437, that an equally correct answer to the question “How many tens?” is 43. They are also surprised to learn that when we say 2 tens and 5 ones equals 25, what we really mean is 2 tens and 5 ones equals 25 ones.

A better way to express it is “2 groups of ten and 5 loose (not in a group) ones equals 25 loose ones.” Therefore, I spend a lot of time having students expand large numbers in a variety of ways.

Methods of Expansion

Expansion basically means counting numbers of groups. There are several ways to express this accounting. Given the number 47,396:

Standard Methods:

Place Value Names: 4 ten thousands, 7 thousands, 3 hundreds, 9 tens, 6 loose ones

Multiplication: (4 x 10,000) + (7 x 1000) + (3 x 100) + (9 x 10) + (6 x 1)

Exponents: (4 x 10^4) + (7 x 10^3) + (3 x 10^2) + (9 x 10^1) + (6 x 10^0)

Notice that using exponents displays the idea that each succeeding place is a group of ten, however, most teachers do not make this understanding explicit. Most students just view, for example, the number 10000 or 104 as merely another way of expressing the place value name “ten thousands.”

I give my students practice with alternative expansions.

Alternative Expansion

47, 396 = _______ thousands, ________tens, _____ ones

47, 396 = _______ ten thousands, ________hundreds, _____ ones

47, 396 = _______ tens, _____ ones

And of course, we can repeat this exercise with multiplicative expansion and exponential expansion. This sort of practice has the side effect of helping students later understand rounding to a given place. I am also very picky about counting and zeroes. 0 is a real counting number, and I expect students to show that they know that 102 has 0 tens, or (0 x 10) or (0 x 10^1).

Place Value in Later Mathematics

This sort of foundational learning of place value pays dividends in later mathematics. To give just a couple examples:

Bases: Each succeeding place is a group of the given base. This understanding gives logic to “borrowing and carrying” in other bases besides base ten.

Polynomial expressions: Quadratic and other equations of the form Ax^n + Bx^n-1 + …Gx^1 + Hx^0 are essentially equations expressed in base x. Students will find that working in other bases is greatly simplified if they exponentially expand the number and replace the base with x.

Polynomial (and by extension, synthetic division: When students learn to divide equations such as Ax^3 + Cx^1 + D by say, x + 1, they must remember to insert the missing term, 0x^2. Students do learn to replace the missing term in a mechanical way. However, if they have regularly understood zero as a real counting number and included the zero term in their elementary expansions, it seems obvious to them that of course they must have the zero term if they expect to successfully complete the division.

More attention to a deep understanding of place value in the early years would make much of later mathematics less mechanical and more intuitively comprehensible, thus actually saving instruction time and allowing teachers to teach more math.

Saturday, March 12, 2016

Zero is a Real Number

Zero is a real number. Could such a headline possibly be click bait? If so, it is pretty lame. Of course everyone knows zero belongs to the set of real numbers. The problem is the word “real.” A sentence such as “zero is a real number” immediately puts people into mathematics mode wherein they consider the word “real” in only its mathematical sense. Sometimes people recall set theory theory and the curious case of a set containing only one member, zero, as opposed to an empty set with no members. The problem here is that set theory leads people to objectify zero. They think of zero as an object rather than a number.

Zero is a real number. When the truth of this statement dawns, the world changes forever. If you are thinking, “Well, of course zero is a real number. What a stupid waste of time to write about it,” you may be one of those people for whom the realization of this truth in all its depth and beauty has not yet occurred.

My student teacher this year was one of those people in September. 27 years old and she never knew zero was a real number. She thought she knew it, but she betrayed herself when she began teaching first graders to answer the question “how many?.” Although she never explicitly said so, she gave her charges to understand that the minimum answer to the question was “one.” I surprised her by reminded her that “zero” is a legitimate answer to the question, “how many?” She did not quite believe me. “Think,” I said, “Of a time when you may have looked for eggs in the refrigerator and found there were zero eggs.” Her eyes widened. “Oh...yeah!” she said, “I hadn’t really thought about it.” I reminded her that when she set up her counting situations, to let zero often be the answer. Children come to school already preconditioned to disregard zero. Their parents and preschool teachers have given them 6 years of experience ignoring zero. One of the first math tasks at school is to undo that misconception.

Zero is a real number. Tax season provides a perfect example. Consider two taxpayers. One person may complete a tax return and find that his tax liability is zero. Therefore when he pays his taxes, he pays zero dollars. Another person does not even complete the form. One person paid no taxes. However, the other one did pay his taxes, and he paid zero dollars. “Zero” and “nothing” are not the same thing. Set theory was supposed to make this distinction clear, but too often we go into math mode and miss the point.

Zero is a real number. I will never forget the day in November when this realization struck my student teacher. She was in the middle of teaching first grade math when she looked at me sitting in the back of the room and said incredulously, “Zero is a real number,” as if it were her own discovery and not something I had said again and again for more than two months.

Tuesday, February 2, 2016

Is Division Repeated Addition?

One of Stanford mathematician Keith Devlin's pet peeves is the common “division is repeated addition” meme . He despises it so much he has something like a mantra, “Repeat after me. Division is NOT repeated addition.” Naturally, math teachers give him a lot of pushback because division is indeed repeated addition (except when it's not).

It seems there are actually two topics in play here. 1) Multiplication as repeated addition and 2) the skill of elementary math teachers. I completely understand his frustration with prospect of undoing the poor math instruction college students typically receive during their elementary school years. I also experience the same frustration as a secondary and college level instructor.

Multiplication as repeated addition is not a definition of multiplication, even though many elementary math teachers erroneously think the definition of multiplication is precisely repeated addition. Repeated addition is merely another name for the group model of multiplication. There are other models, such as the array model, the area model and the number line model, to name the ones most commonly presented to elementary students. Devlin rightly maintains that it is inaccurate to say that multiplication is repeated addition, period. As a misleading misstatement, it is right up there with “you cannot subtract a bigger number from a smaller number.”

However, as properly taught (a giant qualifier, I know), the group model is merely the first element in a teaching sequence which eventually progresses to the area model, then to the use of the area model to multiply fractions, and beyond. For example, you can definitely model a positive whole number times a negative rational number on the number line where it very much looks like repeated addition of the given negative number. Turn that number line vertically, and it makes even more sense to students because it reminds them of another number line they are very familiar with, the thermometer.

Devlin writes, “Addition and multiplication are different operations on numbers. There are, to be sure, connections. One such is that multiplication does provide a quick way of finding the answer to a repeated addition sum.” Exactly, and this is precisely the way a good teacher presents the group model. Children sometimes ask questions like, “Instead of saying 8 + 8 + 8 + 8 + 8, and then saying the answer, can't we just say “8, five times” and then say the answer?” Of course we can, and that is what we do when we say 5 x 8 = 40. The group model is meant to express this particular connection between addition and multiplication. The group model is not meant to be a definition of multiplication. Nevertheless, I agree there are too many elementary math teachers who fail to make the distinction, or properly progress through the models.

An umbrella idea I like involves the word “of” as an English language expression of multiplication. We can say “5 groups of 3,” or “5 groups of -3,” or “1/2 of 3,” or “1/3 of 4/5,” or “16/100 of 40” or “75% of 200,” etc and neatly cover most examples of multiplication that children are likely to encounter before junior high. Devlin prefers scaling as the dominant meme and argues that children should readily understand scaling because examples of scaling surround them. The problem is most eight-year-olds have difficulty comprehending scaling as a model and effect of multiplication. Even though they can readily see that a scale model is a perfect replica of the original, they do not understand how it is possible that doubling the dimensions of a garden (to take a simple example) results in a garden four times larger. Most of the scaling children see is usually on maps where the scale is for them an unimaginably large (or small, depending on viewpoint) number.

Teachers are better off working their way up to the scalar model of multiplication. I have found this is best done by reminding younger students early and often with the idea that we have not yet exhausted the possible models and applications of multiplication. I have found it useful to show some examples of these applications, and say something like, “Later you will learn how you can use multiplication to produce an exact scale model, or use multiplication to produce a real-life-sized object from a scale model.”

Actually, most students get their first solid grip on scaling when they work with similar figures (typically triangles) during high school geometry. Personally, I have found success with older elementary students by giving them basic practice in scaling on the coordinate plane or increasing recipe yield and other types of problems. Students also enjoy the products of their work whether it be art or good eats. The number line model is also a good introduction to scaling because you are scaling only one dimension, as opposed to the two and three dimensions involved in scaling area and volume, respectively.

Devlin also laments the constant push to make math “real.”

No wonder children arrive at college not only having little or no genuine understanding of elementary arithmetic, they have long ago formed the view that math has nothing to do with the world they live in...many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength.

His comments seem contradictory, but they are not. One of the most enjoyable aspects of teaching math is showing students the leap from concrete to abstract. For example, I love showing students a cube and showing them the edge e, the face e x e, then showing them cube e x e x e. I usually let the idea hang, and love when someone asks if I can show e x e x e x e on the cube. No, I answer, I have nothing to show them that. And therein lies the power of math. Math can help us express ideas we can understand, but for which we have no physical representation. I am thrilled when someone asks if e x e x e x e can be time. I ask what would e x e x e x e then express. The children are really products of this century, and one of them is likely to answer, “the coordinates of a time-traveling spaceship.” So much fun. One time, a child said, “Maybe someday we will have a real meaning for more es.”

To the extent that teachers present repeated addition as a property, which in some applications, connects the operations of addition and multiplication, no harm done. However, considering repeated addition as THE definition of multiplication is a serious problem, and Devlin is right to be concerned about it. As Denise Gaskins pointed out, “...if (a particular) model doesn’t work universally, then (the model) certainly cannot be used to define the operation.”

Devlin asks an interesting, and in today's pedagogical environment, nearly taboo question:

The "learn the technique first and understand later" approach is very definitely the only way to learn chess, and millions of children around the world manage that each year, so we know it is a viable approach. Why not accept that math has to be learned the same way?

I would say that technique over understanding has been the preferred approach for centuries, but by all accounts, many adults have never made it to the “understand later” stage. In my experience teaching concept concurrently with technique works the best. The problem I am seeing is that these days too many elementary teachers attempt to teach concepts they barely understand, and then give short shrift to technique because the kids have calculators for that. The result is legions of kids who not only have faulty understanding of the concept, but also lack the ability to perform the technique quickly and accurately.

Additional Discussion:

https://www.quora.com/Why-is-it-incorrect-to-define-multiplication-as-repeated-addition

https://denisegaskins.com/2008/07/01/if-it-aint-repeated-addition/

http://scienceblogs.com/goodmath/2008/07/25/teaching-multiplication-is-it/

https://numberwarrior.wordpress.com/2009/05/22/the-multiplication-is-not-repeated-addition-research/

http://www.quickanddirtytips.com/education/math/is-multiplication-repeated-addition

http://billkerr2.blogspot.com/2009/01/multiplication-is-not-repeated-addition.html

http://rationalmathed.blogspot.com/2008/07/devlin-on-multiplication-or-what-is.html

http://rationalmathed.blogspot.com/2010/02/keith-devlin-extended.html

http://homeschoolmath.blogspot.com/2008/07/isnt-multiplication-repeated-addition.html

http://www.textsavvyblog.net/2008/07/devlins-right-angle-part-i.html

http://mathforum.org/kb/thread.jspa?threadID=2045768

http://www.qedcat.com/archive_cleaned/114.html

Wednesday, December 30, 2015

Nobody Understands Place Value

Parents often ask me to help their children with math homework. My reply is always the same. I am not interested in “helping” with homework. I am very interested in addressing the gaps and misconceptions that give children difficulty with their homework in the first place. Chief among these is a pervasive lack of understanding about place value.

Children's understanding is generally limited to identifying the place value of a given digit or inserting a digit in a given place. I do not blame the children. Most curriculum asks them to do nothing else. Replacing the terms “borrowing” and “carrying” with terms like “exchanging” or “regrouping” represented a tremendous improvement in math education. Even though many elementary math teachers these days play trading games, and the kids appear to know what they are doing, every junior high or high school math teacher has observed that they do not profoundly understand place value, so crucial to understanding the quadratic equation, bases other than base ten, and other topics.

Therefore, I usually start my homework help by first playing simple trading games with the student. The problems I use for the games are ones I know students can calculate correctly, such as 48 + 17. Maybe they can even do it in their heads. No matter.

The first thing I do is dispense with the usual place value names. I use “loose ones” and “packages of ten.” Loose ones is a better term than simply ones. The ones are ones precisely because they are not in a group. They are loose. Children often do not realize that the loose ones' place is fundamentally different from all the other places. Thus they often have trouble with the ones' place in other bases. For example, teachers tell students that when they are working in, say, base five, the largest possible digit in the ones' place is a 4 because “there is a rule that the one's place can be no larger than one less than the base.” Although it is true that the ones' place can be no larger than one less than the base, it is not because of a rule. The reason is much more fundamental than a mere rule. Understanding the ones' place as “loose ones” is key to discovering that fundamental principle.

After we play the trading game for a little while, I have the student do a simple addition problem. When students calculate a problem like 18 + 25, they put a 3 in the ones' place and a 1 above the 1 of 18. Then they add 1 + 1 + 2 and write a 4 in the tens' place, resulting in the answer of 43. Then I ask, “How did you get that answer?”

They usually reply, “I put a 3 in the ones' place and a 1 above the 1 of 18. Then I add 1 + 1 + 2 and write a 4 in the tens' place, so my answer is 43.”

That's fine. Of course they answer in a mechanical, non-mathematical way. They have heard teachers repeatedly explain addition problems to them in much the same way. Then I ask, “Yes, but why did you do that?”

Students invariably reply, “Because that is how the teacher told me to do it.”

Then I ask, “Yes, but why did the teacher tell you to do it that way? Why does it work?” Now they are stymied.

So I show them how to “prove” (not really prove, more like demonstrate) the answer using a picture (similar to this one, but simpler).

I show them how to draw the picture and talk their way through it. “See, you have 15 loose ones. That is enough to make a package of ten. So you gather up a package a ten and put it with all the other packages of ten. You still have 5 loose ones left. Because 5 is not enough to make a package, you leave them loose and show them in the loose ones' column. You add up the packages of ten and put that total in the packages of ten column.”I have them illustrate several problems by drawing the picture.

Very often students realize for the first time that place value is all about making groups of ten. Subtraction is all about breaking groups of ten into loose ones and dumping them with the other loose ones. Every place except the loose ones is a group of ten something. Teachers tell students that each succeeding place is larger by a magnitude of ten, but somehow children fail to grasp the significance of this fact. The reason the standard addition algorithm works is because you are gathering up groups of ten at every place. Likewise, the reason the standard subtraction algorithm works is because you are breaking a group of ten at every place.

Instead of the usual place value mat, I like to use a mat that labels the places a little differently. I start with the loose ones, then packages of 10 loose ones, then cartons of 10 packages, then boxes of 10 cartons, then cases of 10 boxes, then pallets of 10 cases, and so on. I usually stop at cargo ship with 10 shipping containers. Kids love it. Even second graders can easily calculate a multi-digit addition or subtraction problem. In fact, after kids master place value as groups of 10, they often ask me to set them problems with any number of digits. I usually refrain from a problem with more than 13 or 14 digits because even though the kids find the problem easy, they also find it tedious and time-consuming. But hey, tedious and time-consuming is a whole sight better than hard when it comes to doing homework.