Claptrap also acknowledges Sam's been in comics. Even after Chinese-speaking children have learned to read numeric numbers, such as "" as the Chinese translation of "2-one hundred, one-ten, five", that alone should not help them be able to subtract "56" from it any more easily than an English-speaking child can do it, because 1 one still has to translate the concepts of trading into columnar numeric notations, which is not especially easy, and because 2 one still has to understand how ones, tens, hundreds, etc. But they need to be taught at the appropriate time if they are going to have much usefulness. Aspects 4 and 5 involve understanding and reason with enough demonstration and practice to assimilate it and be able to remember the overall logic of it with some reflection, rather than the specific logical steps. Claptrap often claims he is immune to outrageous bluffing. It is much more feasible to figure amounts of things on paper or in a calculator than to assemble the requisite number of things we are talking about in order to add, subtract, multiply, or divide them, especially when we are talking about large numbers of things.

## Music in Episodes

Wait, forget I said that last part. Sam occasionally screws himself over this way Sam: I c ould brazenly bluff my way through this hand If the player makes a big bet on a bad hand, GLaDOS will sometimes comment on the bluff, then remark that she probably shouldn't have said anything aloud. Since Brock has a legal license to kill people, Steve should consider himself lucky that firing a rocket at him only resulted in Brock beating the crap out of him.

Of course, Brock is on thin ice with the Inventory management due to his hair-trigger temper getting the better of him. Ash breaks a martini glass when trying to drink out of it in the intro of the game. Justified, since he's using his medieval prosthetic hand to do it, and it's a Call-Back to Army of Darkness where he crushes a metal cup to test out his new hand's strength.

Does Not Like Guns: Brock, of course, is a knife guy, so when the Inventory is Borderlands -themed he mentions uncomfortably how gun-centric Pandora seems and asks if there's any good melee weapons on the planet.

Claptrap scoffs and says that guys who go melee are usually losers, which really hurts Brock's feelings. Brock isn't a fan of Claptrap's wub wubs. He asks if he can play any Led Zeppelin or anything like that instead.

Claptrap says he'd better get used to it, as the dub is the only thing that survives the Great Digital Event Horizon of Another reason to die young. Brock is quick to call him out on this. If Sam is the first one eliminated, Claptrap may joke about how 'Snoopy's ugly uncle' got invited.

Ash and Brock call him out on this immediately. There are several conversations present in the game files that are unused in the game itself. Ash and Brock have a huge body count for deaths, and are not often the friendliest of guys, but the minute Claptrap starts insulting Sam, they jump to his defense stating that Sam has saved the world from so many threats during his career as Freelance Police.

Ash indulges in one when he's in the lead during a showdown. Sam also lets out a creepy one on occasion, and then will claim it's because he's remembering Max's "joke about sucking chest wounds" before folding. A blatant tell Ash can do to indicate his hand isn't going well is facing upward while holding onto his face with his hands, followed by angrily thumping his fists onto the table once. Faked Rip Van Winkle: What happens at the end of a tournament in the Army of Darkness Inventory.

Claptrap gushes about meeting Ash, the man who took on the Nightmare King , the Hockey Mask Slasher , and the Degrassi invasion of Lampshaded by Sam, who states that the Player reminds him of everyone, and yet no one in particular.

Upon being eliminated though, he might mention how those "Big bushy eyebrows" give away your tell though. After winning in the Sam and Max 25th Anniversary room, Sam will alter his "You don't even like girls" catchphrase to reflect the unspecific gender of the Player.

Sometimes when you bet a lot, reraise, or go all in, Claptrap's response will be a flat "Really Lampshaded when Sam asks Brock what it's like to have five fingers. Brock doesn't really have anything to compare it to, but notes the pinky is useful when choking someone out.

When he asks what it's like with four, Sam notes that between his fingers and toes he's "practically built for the computer age". Like its predecessor, it follows the same path with the various players and even truer to form from their original series. Sometimes during a dramatic showdown, Max or Claptrap, once he's eliminated will be hanging from the ceiling or falling off something in the background.

Discussed in one conversation, with, of course, Ash. Brock mentions that he was expecting someone different when he heard the name Ashley Williams, and goes on to mention other names that were formerly male, like Beverly and Vivian. Winning the tournament in the Army of Darkness room results in Winslow giving The Player a potion that will make them sleep until the next tournament. The Player drinks too much of it, falls asleep, and awakes to find that everyone, including Moxxi, Claptrap, and GLaDOS, has grown a beard, and that an alien invasion has taken place during his slumber.

The Player ends up fainting, and the tournament ends with Ash saying he knew the fake beards wouldn't work. Gosh Dangit To Heck: Subverted; sometimes, when Sam is bullied out of a hand by a huge bet, raise, or all-in, and he folds, he'll say: You Magnificent Bastard , take it!

Averted with Brock, who is occasionally prone to say, "I'm all fuckin' in. Of the idling variety. If you go for a long time without making a move, the characters will sometimes tell you to quit stalling and get on with it already. Sam will sometimes flop his head on the table, being a tell that he's got a bad hand or the cards are no longer in his favor. Claptrap sometimes flip a virtual coin to decide whether or not to bet.

How does a virtual coin land on its side!? Ash also said that he was going to flip a coin to decide who to vote for on the election. Hero of Another Story: Each character has an offscreen storyline that they'll update the rest of the table on every few tournaments though only one will be going on at a time. For example, Ash proposes to his girlfriend and begins suspecting she's not what she seems , while Sam tries to figure out what Chosen One prophecy Max has gotten in the middle of this week.

Turns out Sam's the centerpiece this time. Sam and Max, as well as Claptrap and Steve. When Brock and Ash are still at the table, GLaDOS will mention that she's surprised that after everything they've experienced, they're still playing poker and cracking bad jokes, and not become stark raving mad.

She finds it anomalous, to which Brock says that is why robots will never understand humans- because humans are just full of anomalies. In one exchange, Claptrap brings up a Who's the Boss? Meanwhile, in theaters and a poker video game I Just Want to Be Normal: It seems Ash would prefer this to constantly fighting demons.

Too bad it doesn't work out that way. You know what I'm talkin' about! The Deadites, the vampires, the never-ending battles against the forces of darkness. Eh, all that craziness is behind me, Brock-o. Wendy and I are just gonna settle down, pop out a few kids, and live out our lives like a couple of normal people.

As in the first game. I Need a Freaking Drink: Like Tycho in the previous game, Ash and Brock head to the bar when eliminated that is, if nothing incapacitates them. Ash's spit curl utilizes Jiggle Physics. GLaDOS gives a few to some of the players while she announces their hands. The little robot, Mr. Inn Between the Worlds: Max refers to the Inventory as a "omnidimensional accretion nexus".

Aside from being one, the game also mocks Ash's tendency to end up in these even joking that there were going to be Evil Dead crossovers with James Bond and Degrassi. Max will do the same if the player takes too long to make a move, as will Claptrap if he's been eliminated.

Sam alters the color filters while demonstrating how dogs see color to Claptrap. While four special items are presented in-game, the Bounties menu has a fifth, very familiar silhouette: When Sam asks Brock and Ash how they manage to stay in shape, the topic about eating healthy and staying in shape is brought up. Sam is genuinely surprised to learn that there were other foods out there that aren't junk food. Sam wonders if he can get in shape by changing his diet, but GLaDOS scans Sam's body and finds that the preservatives in all the junk food he's eating is pretty much the only thing keeping him alive at this point, but it's also the contributor to his girth.

You made that up just for the pun, didn't you? Oh yeah, those guys'll eat you alive. Written on the sides of the Army of Darkness chip set. Go Into the Light: Claptrap says "walk into the light" in one conversation. Kill It with Fire: A Claptrap response to losing a hand on the river.

Lantern Jaw of Justice: During one of the table conversations, Sam will admire Brock's and Ash's chins, and say that he isn't proud of his. They tell him that it's not the size of the chin, it's what you do with it that counts. Averted with Winslow , who is still the Inventory's host, but now refers to himself by his full name at the beginning.

For cinematic reasons, this appears when Claptrap gets eliminated in the Borderlands theme. Loads and Loads of Loading: At least on the PlayStation 3 version. Claptrap manages to give himself one, after and by claiming that he's immune to them. GLaDOS is not amused. Bad luck to get 'Win a showdown with only a high card', and then nigh- Unwinnable to have that. It means that you have to win a hand by only having a better card than your opponent, and something as feeble as a pair of 2s can blow it for you.

And, from the forums, it seems to be very common to get that challenge as one of the three for GLaDOS's item, which is the last unlockable. Mostly averted, but played straight at the end of a showdown: And The Player has The Player wins the hand. Sometimes a conversation is interrupted eg. In many cases it sounds fluent and natural, but in other cases it sounds very artificial and in a few cases there's even a clear voice tone difference.

Claptrap and Sam are fully aware of their status as fictional characters. Claptrap even offers to put Sam and Max in a Borderlands 2 expansion while they wait for Sam and Max season 4's release and compares the genres of both series. Claptrap also knows the release date of Sam and Max' first adventure and notes that this should be their 26th anniversary, not the 25th.

Ash is about to mentions Sam Raimi as the person who wrote his most famous lines, but gets cut off by Brock before saying Sam's last name. He also references MGM , saying they promised to give him a crossover with James Bond their flagship series. See also his Actor Allusion moment. Brock mentions Cartoon Network by name and curses their lawyers.

One of Brock's stories talks about both Rusty and himself being in video games. It starts with Sam asking about Rusty having a video game in the '70s, with Brock confirming it, but it was So Bad, It's Horrible that it made " the E. Claptrap then asks if Brock was ever a video game star, with Brock responding with a cryptic "not intentionally". He goes on to explain that some development studio decided to be funny and use his likeness in a Mortal Kombat knock-off named "Immortal Bomcat" as an unlockable Easter Egg fighter.

Who was implied to be Manly Gay. And had a Kiss of Death fatality. Needless to say, they went bankrupt Claptrap tries to invoke a Forced Meme to become popular on the Internet, and Brock dismisses him, saying a true Catch-Phrase comes from the heart. He even cites Ash's "Groovy" as an example.

One of Sam's lines: Winning streaks are like dating a beautiful women. Enjoy them while you can, because it's only a matter of time before they dump you for being weird and clingy. Can happen when a character goes in, especially during the middle of a dialogue sequence running. In the intro cutscene, Brock introduces himself as "Samson. Claptrap says that Max will have to be this if he ever gets in a Borderlands Expansion Pack.

Never Bring a Knife to a Gun Fight , referenced: Never bring a knife to a chainsaw fight, kids! Never Heard That One Before: Ash's response to Brock stating about Ash having a girl's first name.

And Claptrap's when the two jokingly conversed about the other Pandora after Ash asked Claptrap what life was like on another planet. Never Live It Down: Brock fears that this will happen when he's required to say "Go Team Venture" after you win a tournament in the Venture Brothers room. In the Borderlands set, Claptrap will don a top hat — and a old-time 19th century mustache. This is the character design from the Claptrap who runs the storage locker counter at Moxxi's Underdome in the DLC for the first Borderlands.

In the Sam and Max set, Sam wears a top hat. Borderlands and Team Fortress 2 players can win it for themselves if they successfully win a tournament where Ash offers the Necronomicon Ex-Mortis as a bounty. As part of their normal outfits, both Sam and Moxxi wear nice hats. Brock apologizes for it, but Ash says that it wasn't his fault for it, and after the recall, they made a buttload of money selling skin ointment. Most of Sam's comments about the state of the game qualify: Alis klar, der Komissar?

Max sometimes subverts the trope with a response he randomly gives: That never happened, Sam. Ash on a bad set of hole cards: I'd put these cards out of their misery, but I don't want to waste a bullet. This version is just an ugly mess of bugs. Most of the time when launching it, it crashes right back to the Springboard before it can get to the first Loading Screen.

And when it actually loads, your dealing with a huge amount of Loads and Loads of Loading in game because it slows down a lot, mostly unresponsive touch screen controls, and if you playing on an iPod or iPhone cards that aren't easy to see depending on the skin.

The PC version is slightly buggy as well, but nowhere near as bad. Nothing game-breaking, as it mostly consists of characters abruptly cutting into different animations. During a showdown, the characters will sometimes say something indicating they know they're getting screwed when the tides turn against them. They'll have even more dire lines when the hand progresses to a point where it's impossible for them to win.

Offscreen Moment of Awesome: On Brock's recommendation, Ash teams up with Dr. Orpheus to fight hell-demons at one point. On a Scale from One to Ten: Sometimes when Claptrap folds. On a scale of one to ten, these cards can bite me! One Degree of Separation: Freelance Police directly referencing the former's tone.

Sam claims he started seeing a speech coach and Max just never stopped using a Jason Alexander impression and his voice slowly changed as a result. People Fall Off Chairs: When in a showdown, Claptrap will sometimes fall off his chair when leaning forward.

If the player suddenly goes all in, Max will fall off his booster chair at the booth behind Sam. Brock may sometimes make one when he goes all in. I'm all fuckin' in. Don't worry, it's all in-universe. Brock makes a remark that he has to buy his shirts 'in bulk' because they get stained easily in his line of work.

Ash then states that Brock should invest in some Boo Franklin shirts, then spends the next two minutes explaining why the shirts are so great. By now he's the manager of the housewares department. Old habits die hard. If playing in the Borderlands set, Claptrap will suggest that Ash switch out his "primitive boomstick" for a Torgue Brand boomstick. GLaDOS concludes it's been possessed by a marketing department.

The game was officially announced on April Fools' Day , causing many to assume it to be an elaborate hoax. Those who were taking note of Team Fortress 2 updates, however, noticed that the winnable TF2 items were added to the game's files on March 12, , two weeks before the game was even teased.

She does, however, agree with Claptrap's claim that his diesel fuel-scented cologne drives robo-ladies like her insane - "just like Lizzie Borden". Claptrap may say "That was more anti climatic then my sex tape! As far as getting items is concerned. For the special items, in the first game, the chances to get them came up randomly, and you had to specifically knock the character out to win the item.

In this game, you earn a shot at one of the special items by completing certain objectives, and once you get the chance, all you have to do is win that tournament to get the item.

You're given more forewarning, and you don't have to change your strategy to try to win the item, so it's less nerve-wracking overall. And for the tables, card backs, and chips, you buy them with inventory tokens, and you'll get some tokens even if you lost the tournament, so you'll unlock everything eventually.

The first game mostly reused models from existing games and drew its cast mostly from properties Telltale already had in easy reach Tycho being the only mutual exception. The sequel features more distant properties and more new models as well as a few touch-ups on the pre-existing ones.

The player can unlock more versions of the Inventory for use, all with their own events. And unlike before, the humor is not always restricted to the table and its occupants. You can now buy drinks from Moxxi to get other players drunk so you can see their tells more often, unlike Poker Night 1 in which you had to watch carefully to see an opponent's tell.

It's also escalated from a quantitative standpoint. Just in case the unlockable bounties aren't enough to incentive potential new players, Claptrap likes to plug-in for Borderlands 2 from time to time.

Meanwhile, Ash is just as shameless, as he won't hesitate to take a chance to shill for S-Mart. Sam considers the Player to be this. He sounds quite relieved when you're eliminated, because he can't stand being watched intensely by an eerily silent folk. How am I supposed to make witty banter with a mute?

How am I supposed to make small talk with a mute? Brock isn't pleased either: Trapped in a showdown with Captain Small Talk here.

Though he does respect you for being the silent type, as he says he enjoys playing with you because of it if he beats you in a showdown. When asked whether he dates human women, Sam replies that he's apparently off the market.

However, he gives no answer whether he likes women, or exactly who he's dating. As to be expected, Ash repeatedly calls to Brock to just use his boomstick. Perhaps he's unaware Brock Doesn't Like Guns? Similarly, Claptrap recommends buying the Conference Call, one of the most powerful guns of Borderlands 2. He tries to shill out Torgue brand shotguns specifically, the Boomstick, a boss drop from the first Borderlands and obvious reference to Ash's weapon to Ash as well.

Ash asks if he'd be able to set it on his shoulder and fire it backwards without looking, just like how he demonstrates with his own boomstick, but Claptrap says "not without blowing your kneecaps off. Brock will occasionally belt one out if he loses with a good hand, though he does stay in his seat.

Should The Player tie with Brock Samson, he'll reference the trope. What is this, soccer? The Player is the only person on one half of the poker table, facing the other four characters. One of Claptrap's reactions when someone goes all-in even if he wasn't drinking anything.

After losing a large amount of money in a hand, Claptrap will sometimes ask Max to shoot him in the head. Sam delivers quite a number of these: You know, I'm still not really sure what a "Deadite" actually is. It all starts with an evil book that must never, ever be read. I thought we were friends. It's playing computer poker by itself , Sam.

Now arithmetic teachers and parents tend to confuse the teaching and learning of logical, conventional or representational, and algorithmic manipulative computational aspects of math. And sometimes they neglect to teach one aspect because they think they have taught it when they teach other aspects. That is not necessarily true. The "new math" instruction, in those cases where it failed, was an attempt to teach math logically in many cases by people who did not understand its logic while not teaching and giving sufficient practice in, many of the representational or algorithmic computational aspects of math.

The traditional approach tends to neglect logic or to assume that teaching algorithmic computations is teaching the logic of math. There are some new methods out that use certain kinds of manipulatives 22 to teach groupings, but those manipulatives aren't usually merely representational.

Instead they simply present groups of, say 10's, by proportionally longer segments than things that present one's or five's; or like rolls of pennies, they actually hold things or ten things or two things, or whatever.

Students need to learn three different aspects of math; and what effectively teaches one aspect may not teach the other aspects. The three aspects are 1 mathematical conventions, 2 the logic s of mathematical ideas, and 3 mathematical algorithmic manipulations for calculating.

There is no a priori order to teaching these different aspects; whatever order is most effective with a given student or group of students is the best order. Students need to be taught the "normal", everyday conventional representations of arithmetic, and they need to be taught how to manipulate and calculate with written numbers by a variety of different means -- by calculators, by computer, by abacus, and by the society's "normal" algorithmic manipulations 23 , which in western countries are the methods of "regrouping" in addition and subtraction, multiplying multi-digit numbers in precise steps, and doing long division, etc.

Learning to use these things takes lots of repetition and practice, using games or whatever to make it as interesting as possible. But these things are generally matters of simply drill or practice on the part of children. But students should not be forced to try to make sense of these things by teachers who think that these things are matters of obvious or simple logic. These are not matters of obvious or simple logic, as I have tried to demonstrate in this paper.

Children will be swimming upstream if they are looking for logic when they are merely learning conventions or learning algorithms whose logic is far more complicated than being able to remember the steps of the algorithms, which itself is difficult enough for the children. And any teacher who makes it look to children like conventions and algorithmic manipulations are matters of logic they need to understand, is doing them a severe disservice.

On the other hand, children do need to work on the logical aspects of mathematics, some of which follow from given conventions or representations and some of which have nothing to do with any particular conventions but have to do merely with the way quantities relate to each other. But developing children's mathematical insight and intuition requires something other than repetition, drill, or practice. Many of these things can be done simultaneously though they may not be in any way related to each other.

Students can be helped to get logical insights that will stand them in good stead when they eventually get to algebra and calculus 24 , even though at a different time of the day or week they are only learning how to "borrow" and "carry" currently called "regrouping" two-column numbers. They can learn geometrical insights in various ways, in some cases through playing miniature golf on all kinds of strange surfaces, through origami , through making periscopes or kaleidoscopes, through doing some surveying, through studying the buoyancy of different shaped objects, or however.

Or they can be taught different things that might be related to each other, as the poker chip colors and the column representations of groups. What is important is that teachers can understand which elements are conventional or conventionally representational, which elements are logical, and which elements are complexly algorithmic so that they teach these different kinds of elements, each in its own appropriate way, giving practice in those things which benefit from practice, and guiding understanding in those things which require understanding.

And teachers need to understand which elements of mathematics are conventional or conventionally representational, which elements are logical, and which elements are complexly algorithmic so that they can teach those distinctions themselves when students are ready to be able to understand and assimilate them.

Conceptual structures for multiunit numbers: Cognition and Instruction, 7 4 , Children's understanding of place value: Young Children, 48 5 , Young children continue to reinvent arithmetic: Mere repetition about conceptual matters can work in cases where intervening experiences or information have taken a student to a new level of awareness so that what is repeated to him will have "new meaning" or relevance to him that it did not before.

Repetition about conceptual points without new levels of awareness will generally not be helpful. And mere repetition concerning non-conceptual matters may be helpful, as in interminably reminding a young baseball player to keep his swing level, a young boxer to keep his guard up and his feet moving, or a child learning to ride a bicycle to "keep peddling; keep peddling; PEDDLE! If you think you understand place value, then answer why columns have the names they do.

That is, why is the tens column the tens column or the hundreds column the hundreds column? And, could there have been some method other than columns that would have done the same things columns do, as effectively? If so, what, how, and why? If not, why not? In other words, why do we write numbers using columns, and why the particular columns that we use?

In informal questioning, I have not met any primary grade teachers who can answer these questions or who have ever even thought about them before. How something is taught, or how the teaching or material is structured, to a particular individual and sometimes to similar groups of individuals is extremely important for how effectively or efficiently someone or everyone can learn it.

Sometimes the structure is crucial to learning it at all. A simple example first: It is even difficult for an American to grasp a phone number if you pause after the fourth digit instead of the third "three, two, three, two pause , five, five, five".

I had a difficult time learning from a book that did many regions simultaneously in different cross-sections of time. I could make my own cross-sectional comparisons after studying each region in entirety, but I could not construct a whole region from what, to me, were a jumble of cross-sectional parts. The only way to keep the bike from tipping over was to lean far out over the remaining training wheel. The child was justifiably riding at a 30 degree angle to the bike. When I took off the other training wheel to teach her to ride, it took about ten minutes just to get her back to a normal novice's initial upright riding position.

I don't believe she could have ever learned to ride by the father's method. Many people I have taught have taken whole courses in photography that were not structured very well, and my perspective enlightens their understanding in a way they may not have achieved in the direction they were going.

My lecturer did not structure the material for us, and to me the whole thing was an endless, indistinguishable collection of popes and kings and wars. I tried to memorize it all and it was virtually impossible. I found out at the end of the term that the other professor who taught the course to all my friends spent each of his lectures simply structuring a framework in order to give a perspective for the students to place the details they were reading. He admitted at the end of the year that was a big mistake; students did not learn as well using this structure.

I did not become good at organic chemistry. There appeared to be much memorization needed to learn each of these individual formulas. I happened to notice the relationship the night before the midterm exam, purely by luck and some coincidental reasoning about something else. I figured I was the last to see it of the students in the course and that, as usual, I had been very naive about the material. It turned out I was the only one to see it.

I did extremely well but everyone else did miserably on the test because memory under exam conditions was no match for reasoning. Had the teachers or the book simply specifically said the first formula was a general principle from which you could derive all the others, most of the other students would have done well on the test also.

There could be millions of examples. Most people have known teachers who just could not explain things very well, or who could only explain something one precise way, so that if a student did not follow that particular explanation, he had no chance of learning that thing from that teacher.

The structure of the presentation to a particular student is important to learning. In a small town not terribly far from Birmingham, there is a recently opened McDonald's that serves chocolate shakes which are off-white in color and which taste like not very good vanilla shakes.

They are not like other McDonald's chocolate shakes. When I told the manager how the shakes tasted, her response was that the shake machine was brand new, was installed by experts, and had been certified by them the previous week --the shake machine met McDonald's exacting standards, so the shakes were the way they were supposed to be; there was nothing wrong with them.

There was no convincing her. After she returned to her office I realized, and mentioned to the sales staff, that I should have asked her to take a taste test to try to distinguish her chocolate shakes from her vanilla ones. That would show her there was no difference. The staff told me that would not work since there was a clear difference: Unfortunately, too many teachers teach like that manager manages.

They think if they do well what the manuals and the college courses and the curriculum guides tell them to do, then they have taught well and have done their job. What the children get out of it is irrelevant to how good a teacher they are. It is the presentation, not the reaction to the presentation, that they are concerned about. To them "teaching" is the presentation or the setting up of the classroom for discovery or work. If they "teach" well what children already know, they are good teachers.

If they make dynamic well-prepared presentations with much enthusiasm, or if they assign particular projects, they are good teachers, even if no child understands the material, discovers anything, or cares about it. If they train their students to be able to do, for example, fractions on a test, they have done a good job teaching arithmetic whether those children understand fractions outside of a test situation or not.

And if by whatever means necessary they train children to do those fractions well, it is irrelevant if they forever poison the child's interest in mathematics.

Teaching, for teachers like these, is just a matter of the proper technique, not a matter of the results. Well, that is not any more true than that those shakes meet McDonald's standards just because the technique by which they are made is "certified". I am not saying that classroom teachers ought to be able to teach so that every child learns. There are variables outside of even the best teachers' control.

But teachers ought to be able to tell what their reasonably capable students already know, so they do not waste their time or bore them. Teachers ought to be able to tell whether reasonably capable students understand new material, or whether it needs to be presented again in a different way or at a different time.

And teachers ought to be able to tell whether they are stimulating those students' minds about the material or whether they are poisoning any interest the child might have. All the techniques in all the instructional manuals and curriculum guides in all the world only aim at those ends.

Techniques are not ends in themselves; they are only means to ends. Those teachers who perfect their instructional techniques by merely polishing their presentations, rearranging the classroom environment, or conscientiously designing new projects, without any understanding of, or regard for, what they are actually doing to children may as well be co-managing that McDonald's. Some of these studies interpreted to show that children do not understand place-value, are, I believe, mistaken.

Jones and Thornton explain the following "place-value task": Children are asked to count 26 candies and then to place them into 6 cups of 4 candies each, with two candies remaining. When the "2" of "26" was circled and the children were asked to show it with candies, the children typically pointed to the two candies. When the "6" in "26" was circled and asked to be pointed out with candies, the children typically pointed to the 6 cups of candy.

This is taken to demonstrate children do not understand place value. I believe this demonstrates the kind of tricks similar to the following problems, which do not show lack of understanding, but show that one can be deceived into ignoring or forgetting one's understanding.

At the beginning of the tide's coming in, three rungs are under water. If the tide comes in for four hours at the rate of 1 foot per hour, at the end of this period, how many rungs will be submerged? The answer is not nine, but "still just three, because the ship will rise with the tide. This tends to be an extremely difficult problem --psychologically-- though it has an extremely simple answer.

The money paid out must simply equal the money taken in. People who cannot solve this problem, generally have no trouble accounting for money, however; they do only when working on this problem. If you know no calculus, the problem is not especially difficult. It is a favorite problem to trick unsuspecting math professors with. Two trains start out simultaneously, miles apart on the same track, heading toward each other. The train in the west is traveling 70 mph and the train in the east is traveling 55 mph.

At the time the trains begin, a bee that flies mph starts at one train and flies until it reaches the other, at which time it reverses without losing any speed and immediately flies back to the first train, which, of course, is now closer. The bee keeps going back and forth between the two ever-closer trains until it is squashed between them when they crash into each other.

What is the total distance the bee flies? The computationally extremely difficult, but psychologically logically apparent, solution is to "sum an infinite series". Mathematicians tend to lock into that method.

The easy solution, however, is that the trains are approaching each other at a combined rate of mph, so they will cover the miles, and crash, in 6 hours. The bee is constantly flying mph; so in that 6 hours he will fly miles. One mathematician is supposed to have given the answer immediately, astonishing a questioner who responded how incredible that was "since most mathematicians try to sum an infinite series.

It is not that mathematicians do not know how to solve this problem the easy way; it is that it is constructed in a way to make them not think about the easy way. I believe that the problem Jones and Thornton describe acts similarly on the minds of children. Though I believe there is ample evidence children, and adults, do not really understand place-value, I do not think problems of this sort demonstrate that, any more than problems like those given here demonstrate lack of understanding about the principles involved.

It is easy to see children do not understand place-value when they cannot correctly add or subtract written numbers using increasingly more difficult problems than they have been shown and drilled or substantially rehearsed "how" to do by specific steps; i. By increasingly difficult, I mean, for example, going from subtracting or summing relatively smaller quantities to relatively larger ones with more and more digits , going to problems that require call it what you like regrouping, carrying, borrowing, or trading; going to subtraction problems with zeroes in the number from which you are subtracting; to consecutive zeroes in the number from which you are subtracting; and subtracting such problems that are particularly psychologically difficult in written form, such as "10, - 9,".

Asking students to demonstrate how they solve the kinds of problems they have been "taught" and rehearsed on merely tests their attention and memory, but asking students to demonstrate how they solve new kinds of problems that use the concepts and methods you have been demonstrating, but "go just a bit further" from them helps to show whether they have developed understanding.

However, the kinds of problems at the beginning of this endnote do not do that because they have been contrived specifically to psychologically mislead, or they are constructed accidentally in such a way as to actually mislead. They go beyond what the students have been specifically taught, but do it in a tricky way rather than a merely "logically natural" way. I cannot categorize in what ways "going beyond in a tricky way" differs from "going beyond in a 'naturally logical' way" in order to test for understanding, but the examples should make clear what it is I mean.

Further, it is often difficult to know what someone else is asking or saying when they do it in a way that is different from anything you are thinking about at the time. If you ask about a spatial design of some sort and someone draws a cutaway view from an angle that makes sense to him, it may make no sense to you at all until you can "re-orient" your thinking or your perspective. Or if someone is demonstrating a proof or rationale, he may proceed in a step you don't follow at all, and may have to ask him to explain that step.

What was obvious to him was not obvious to you at the moment. The fact that a child, or any subject, points to two candies when you circle the "2" in "26" and ask him to show you what that means, may be simply because he is not thinking about what you are asking in the way that you are asking it or thinking about it yourself.

There is no deception involved; you both are simply thinking about different things -- but using the same words or symbols to describe what you are thinking about. Or, ask someone to look at the face of a person about ten feet away from them and describe what they see. They will describe that person's face, but they will actually be seeing much more than that person's face.

So, their answer is wrong, though understandably so. Now, in a sense, this is a trivial and trick misunderstanding, but in photography, amateurs all the time "see" only a face in their viewer, when actually they are too far away to have that face show up very well in the photograph. They really do not know all they are seeing through the viewer, and all that the camera is "seeing" to take. The difference is that if one makes this mistake with a camera, it really is a mistake; if one makes the mistake verbally in answer to the question I stated, it may not be a real mistake but only taking an ambiguous question the way it deceptively was not intended.

Asking a child what a circled "2" means, no matter where it comes from, may give the child no reason to think you are asking about the "twenty" part of "26" --especially when there are two objects you have intentionally had him put before him, and no readily obvious set of twenty objects. He may understand place-value perfectly well, but not see that is what you are asking about -- especially under the circumstances you have constructed and in which you ask the question.

If you understand the concept of place-value, if you understand how children or anyone tend to think about new information of any sort and how easy misunderstanding is, particularly about conceptual matters , and if you watch most teachers teach about the things that involve place-value, or any other logical-conceptual aspects of math, it is not surprising that children do not understand place-value or other mathematical concepts very well and that they cannot generally do math very well.

Place-value, like many concepts, is often taught as though it were some sort of natural phenomena --as if being in the 10's column was a simple, naturally occurring, observable property, like being tall or loud or round-- instead of a logically and psychologically complex concept.

What may be astonishing is that most adults can do math as well as they do it at all with as little in-depth understanding as they have. Research on what children understand about place-value should be recognized as what children understand about place-value given how it has been taught to them , not as the limits of their possible understanding about place-value.

Baroody categorizes what he calls "increasingly abstract models of multidigit numbers using objects or pictures" and includes mention of the model I think most appropriate --different color poker chips --which he points out to be conceptually similar to Egyptian hieroglyphics-- in which a different looking "marker" is used to represent tens.

I do not believe that his categories are categories of increasingly abstract models of multidigit numbers. He has four categories; I believe the first two are merely concrete groupings of objects interlocking blocks and tally marks in the first category, and Dienes blocks and drawings of Dienes blocks in the second category.

And the second two --different marker type and different relative-position-value-- are both equally abstract representations of grouping, the difference between them being that relative-positional-value is a more difficult concept to assimilate at first than is different marker type.

It is not more abstract; it is just abstract in a way that is more difficult to recognize and deal with. Further, Baroody labels all his categories as kinds of "trading", but he does not seem to recognize there is sometimes a difference between "trading" and "representing", and that trading is not abstract at all in the way that representing is.

I can trade you my Mickey Mantle card for your Ted Kluzewski card or my tuna sandwich for your soft drink, but that does not mean Mickey Mantle cards represent Klu cards or that sandwiches represent soft drinks. Children in general, not just children with low ability, can understand trading without necessarily understanding representing. And they can go on from there to understand the kind of representing that does happen to be similar to trading, which is the kind of representing that place-value is.

But with regard to trading, as opposed to representing, it is easier first to apprehend or appreciate or remember, or pretend there being a value difference between objects that are physically different, regardless of where they are, than it is to apprehend or appreciate a difference between two identical looking objects that are simply in different places.

It makes sense to say that something can be of more or less value if it is physically changed, not just physically moved. Painting your car, bumping out the dents, or re-building the carburetor makes it worth more in some obvious way; parking it further up in your driveway does not.

It makes sense to a child to say that two blue poker chips are worth 20 white ones; it makes less apparent sense to say a "2" over here is worth ten "2's" over here. Color poker chips teach the important abstract representational parts of columns in a way children can grasp far more readily. So why not use them and make it easier for all children to learn? And poker chips are relatively inexpensive classroom materials.

By thinking of using different marker types to represent different group values primarily as an aid for students of "low ability", Baroody misses their potential for helping all children, including quite "bright" children, learn place-value earlier, more easily, and more effectively. Remember, written versions of numbers are not the same thing as spoken versions. Written versions have to be learned as well as spoken versions; knowing spoken numbers does not teach written numbers.

For example, numbers written in Roman numerals are pronounced the same as numbers in Arabic numerals. And numbers written in binary form are pronounced the same as the numbers they represent; they just are written differently, and look like different numbers. In binary math "" is "six", not "one hundred ten".

When children learn to read numbers, they sometimes make some mistakes like calling "11" "one-one", etc. Even adults, when faced with a large multi-column number, often have difficulty naming the number, though they might have no trouble manipulating the number for calculations; number names beyond the single digit numbers are not necessarily a help for thinking about or manipulating numbers.

Fuson explains how the names of numbers from 10 through 99 in the Chinese language include what are essentially the column names as do our whole-number multiples of , and she thinks that makes Chinese-speaking students able to learn place-value concepts more readily.

But I believe that does not follow, since however the names of numbers are pronounced, the numeric designation of them is still a totally different thing from the written word designation; e. It should be just as difficult for a Chinese-speaking child to learn to identify the number "11" as it is for an English-speaking child, because both, having learned the number "1" as "one", will see the number "11" as simply two "ones" together.

It should not be any easier for a Chinese child to learn to read or pronounce "11" as the Chinese translation of "one-ten, one" than it is for English-speaking children to see it as "eleven".

And Fuson does note the detection of three problems Chinese children have: But there is, or should be, more involved. Even after Chinese-speaking children have learned to read numeric numbers, such as "" as the Chinese translation of "2-one hundred, one-ten, five", that alone should not help them be able to subtract "56" from it any more easily than an English-speaking child can do it, because 1 one still has to translate the concepts of trading into columnar numeric notations, which is not especially easy, and because 2 one still has to understand how ones, tens, hundreds, etc.

And although it may seem easy to subtract "five-ten" 50 from "six-ten" 60 to get "one-ten" 10 , it is not generally difficult for people who have learned to count by tens to subtract "fifty" from "sixty" to get "ten". Nor is it difficult for English-speaking students who have practiced much with quantities and number names to subtract "forty-two" from "fifty-six" to get "fourteen". Surely it is not easier for a Chinese-speaking child to get "one-ten four" by subtracting "four-ten two" from "five-ten six".

Algebra students often have a difficult time adding and subtracting mixed variables [e. I suspect that if Chinese-speaking children understand place-value better than English-speaking children, there is more reason than the name designation of their numbers.

And Fuson points out a number of things that Asian children learn to do that American children are generally not taught, from various methods of finger counting to practicing with pairs of numbers that add to ten or to whole number multiples of ten. From a conceptual standpoint of the sort I am describing in this paper, it would seem that sort of practice is far more important for learning about relationships between numbers and between quantities than the way spoken numbers are named.

There are all kinds of ways to practice using numbers and quantities; if few or none of them are used, children are not likely to learn math very well, regardless of how number words are constructed or pronounced or how numbers are written. Because children can learn to read numbers simply by repetition and practice, I maintain that reading and writing numbers has nothing necessarily to do with understanding place-value.

I take "place-value" to be about how and why columns represent what they do and how they relate to each other , not just knowing what they are named. Some teachers and researchers, however and Fuson may be one of them seem to use the term "place-value" to include or be about the naming of written numbers, or the writing of named numbers.

In this usage then, Fuson would be correct that --once children learn that written numbers have column names, and what the order of those column names is -- Chinese-speaking children would have an advantage in reading and writing numbers that include any ten's and one's that English-speaking children do not have. But as I pointed out earlier, I do not believe that advantage carries over into doing numerically written or numerically represented arithmetical manipulations, which is where place-value understanding comes in.

And I do not believe it is any sort of real advantage at all, since I believe that children can learn to read and write numbers from 1 to fairly easily by rote, with practice, and they can do it more readily that way than they can do it by learning column names and numbers and how to put different digits together by columns in order to form the number.

When my children were learning to "count" out loud i. They would forget to go to the next ten group after getting to nine in the previous group and I assume that, if Chinese children learn to count to ten before they go on to "one-ten one", they probably sometimes will inadvertently count from, say, "six-ten nine to six-ten ten".

And, probably unlike Chinese children, for the reasons Fuson gives, my children had trouble remembering the names of the subsequent sets of tens or "decades". When they did remember that they had to change the decade name after a something-ty nine, they would forget what came next. But this was not that difficult to remedy by brief rehearsal periods of saying the decades while driving in the car, during errands or commuting, usually and then practicing going from twenty-nine to thirty, thirty-nine to forty, etc.

Actually a third thing would also sometimes happen, and theoretically, it seems to me, it would probably happen more frequently to children learning to count in Chinese.

When counting to my children would occasionally skip a number without noticing or they would lose their concentration and forget where they were and maybe go from sixty six to seventy seven, or some such.

I would think that if you were learning to count with the Chinese naming system, it would be fairly easy to go from something like six-ten three to four-ten seven if you have any lapse in concentration at all.

It would be easy to confuse which "ten" and which "one" you had just said. If you try to count simple mixtures of two different kinds of objects at one time --in your head-- you will easily confuse which number is next for which object. Put different small numbers of blue and red poker chips in ten or fifteen piles, and then by going from one pile to the next just one time through, try to simultaneously count up all the blue ones and all the red ones keeping the two sums distinguished.

It is extremely difficult to do this without getting confused which sum you just had last for the blue ones and which you just had last for the red ones. In short, you lose track of which number goes with which name. I assume Chinese children would have this same difficulty learning to say the numbers in order. There is a difference between things that require sheer repetitive practice to "learn" and things that require understanding. The point of practice is to become better at avoiding mistakes, not better at recognizing or understanding them each time you make them.

The point of repetitive practice is simply to get more adroit at doing something correctly. It does not necessarily have anything to do with understanding it better. It is about being able to do something faster, more smoothly, more automatically, more naturally, more skillfully, more perfectly, well or perfectly more often, etc. Some team fundamentals in sports may have obvious rationales; teams repetitively practice and drill on those fundamentals then, not in order to understand them better but to be able to do them better.

In math and science and many other areas , understanding and practical application are sometimes separate things in the sense that one may understand multiplication, but that is different from being able to multiply smoothly and quickly. Many people can multiply without understanding multiplication very well because they have been taught an algorithm for multiplication that they have practiced repetitively. Others have learned to understand multiplication conceptually but have not practiced multiplying actual numbers enough to be able to effectively multiply without a calculator.

Both understanding and practice are important in many aspects of math, but the practice and understanding are two different things, and often need to be "taught" or worked on separately. Similarly, physicists or mathematicians may work with formulas they know by heart from practice and use, but they may have to think a bit and reconstruct a proof or rationale for those formulas if asked.

Having understanding, or being able to have understanding, are often different from being able to state a proof or rationale from memory instantaneously. In some cases it may be important for someone not only to understand a subject but to memorize the steps of that understanding, or to practice or rehearse the "proof" or rationale or derivation also, so that he can recall the full, specific rationale at will.

But not all cases are like that. In a discussion of this point on Internet's AERA-C list, Tad Watanabe pointed out correctly that one does not need to regroup first to do subtractions that require "borrowing" or exchanging ten's into one's. One could subtract the subtrahend digit from the "borrowed" ten, and add the difference to the original minuend one's digit. For example, in subtracting 26 from 53, one can change 53 into, not just 40 plus 18, but 40 plus a ten and 3 one's, subtract the 6 from the ten, and then add the diffence, 4, back to the 3 you "already had", in order to get the 7 one's.

Then, of course, subtract the two ten's from the four ten's and end up with This prevents one from having to do subtractions involving minuends from 11 through That in turn reminded me of two other ways to do such subtraction, avoiding subtracting from 11 through In the case of , you subtract all three one's from the 53, which leaves three more one's that you need to subtract once you have converted the ten from fifty into 10 one's.

Then, of course, you subtract the If you don't teach children or help them figure out how to adroitly do subtractions with minuends from 11 through 18, you will essentially force them into options 1 or 2 above or something similar.

Whereas if you do teach subtractions from 11 through 18, you give them the option of using any or all three methods. Plus, if you are going to want children to be able to see 53 as some other combination of groups besides 5 ten's and 3 one's, although 4 ten's plus 1 ten plus 3 one's will serve, 4 ten's and 13 one's seems a spontaneous or psychologically ready consequence of that, and it would be unnecessarily limiting children not to make it easy for them to see this combination as useful in subtraction.

I say at the time you are trying to subtract from it because you may have already regrouped that number and borrowed from it. Hence, it may have been a different number originally.

If you subtract 99 from , the 0's in the minuend will be 9's when you "get to them" in the usual subtraction algorithm that involves proceeding from the right one's column to the left, regrouping, borrowing, and subtracting by columns as you proceed.

For example, when subtracting 9 from 18, if you regroup the 18 into no tens and 18 ones, you still must subtract 9 from those 18 ones. Nothing has been gained. In a third grade class where I was demonstrating some aspects of addition and subtraction to students, if you asked the class how much, say, 13 - 5 was or any such subtraction with a larger subtrahend digit than the minuend digit , you got a range of answers until they finally settled on two or three possibilities.

I am told by teachers that this is not unusual for students who have not had much practice with this kind of subtraction. There is nothing wrong with teaching algorithms, even complex ones that are difficult to learn. But they need to be taught at the appropriate time if they are going to have much usefulness.

They cannot be taught as a series of steps whose outcome has no meaning other than that it is the outcome of the steps. Algorithms taught and used that way are like any other merely formal system -- the result is a formal result with no real meaning outside of the form. And the only thing that makes the answer incorrect is that the procedure was incorrectly followed, not that the answer may be outlandish or unreasonable.

In a sense, the means become the ends. Arithmetic algorithms are not the only areas of life where means become ends, so the kinds of arithmetic errors children make in this regard are not unique to math education. A formal justice system based on formal "rules of evidence" sometimes makes outlandish decisions because of loopholes or "technicalities"; particular scientific "methods" sometimes cause evidence to be missed, ignored, or considered merely aberrations; business policies often lead to business failures when assiduously followed; and many traditions that began as ways of enhancing human and social life become fossilized burdensome rituals as the conditions under which they had merit disappear.

Unfortunately, when formal systems are learned incorrectly or when mistakes are made inadvertently, there is no reason to suspect error merely by looking at the result of following the rules.

Any result, just from its appearance, is as good as any other result. Arithmetic algorithms, then, should not be taught as merely formal systems. They need to be taught as short-hand methods for getting meaningful results, and that one can often tell from reflection about the results, that something must have gone awry.

Children need to reflect about the results, but they can only do that if they have had significant practice working and playing with numbers and quantities in various ways and forms before they are introduced to algorithms which are simply supposed to make their calculating easier, and not merely simply formal. Children do not always need to understand the rationale for the algorithm's steps, because that is sometimes too complicated for them, but they need to understand the purpose and point of the algorithm if they are going to be able to learn to apply it reasonably.

Learning an algorithm is a matter of memorization and practice, but learning the purpose or rationale of an algorithm is not a matter of memorization or practice; it is a matter of understanding. Teaching an algorithm's steps effectively involves merely devising means of effective demonstration and practice.

But teaching an algorithm's point or rationale effectively involves the more difficult task of cultivating students' understanding and reasoning. Cultivating understanding is as much art as it is science because it involves both being clear and being able to understand when, why, and how you have not been clear to a particular student or group of students. Since misunderstanding can occur in all kinds of unanticipated and unpredictable ways, teaching for understanding requires insight and flexibility that is difficult or impossible for prepared texts, or limited computer programs, alone to accomplish.

Finally, many math algorithms are fairly complex, with many different "rules", so they are difficult to learn just as formal systems, even with practice. The addition and subtraction algorithms how to line up columns, when and how to borrow or carry, how to note that you have done so, how to treat zeroes, etc. I think the research clearly shows that children do not learn these algorithms very well when they are taught as formal systems and when children have insufficient background to understand their point.

And it is easy to see that in cases involving "simple addition and subtraction", the algorithm is far more complicated than just "figuring out" the answer in any logical way one might; and that it is easier for children to figure out a way to get the answer than it is for them to learn the algorithm. Rule-based derivations are helpful in cases too complex to do by memory, logic, or imagination alone; but they are a hindrance in cases where learning or using them is more difficult than using memory, logic, or imagination directly on the problem or task at hand.

This is not dissimilar to the fact that learning to read and write numbers --at least up to is easier to do by rote and by practice than it is to do by being told about column names and the rules for their use. There is simply no reason to introduce algorithms before students can understand their purpose and before students get to the kinds of usually higher number problems for which algorithms are helpful or necessary to solve.