# Me Too movement

The count starts at 0 for a freshly shuffled deck for "balanced" counting systems. Contact us for more information on what puppies are available. While I was able to win with it, I do not believe that my test was long enough to be scientifically valid, and therefore I am unwilling to express an opinion at this time. You could also use a blackjack simulation software program e. I reached in my wallet and took out five one-hundred dollar bills.

## Recently Played Games

Your chips should be in one stack. If you are betting multiple denominations of chips, place the larger valued chips on the bottom of the stack, and the smaller value chips on top. Once the cards have been dealt, you are not allowed to touch the bet in the circle.

If you need to know how much you have bet for doubling or splitting explained later , the dealer will count down the chips for you. Once the hand is over, the dealer will move around the table to each position in turn, paying winning hands and collecting the chips from losing hands. After the dealer has paid you, you can remove your chips from the circle, and place your next bet.

If you want to let your winnings ride, you will need to form one stack of chips from the two or more stacks on the table after the dealer pays you. Remember, higher value chips should be placed on the bottom of the stack. When you are ready to leave the table, you do not cash in your chips the same way you bought them. The dealer cannot give you cash for the chips at the table. To do that, you must take the chips to the casino cashier.

If you have a lot of low denomination chips in front of you at the table, you should trade them for the equivalent higher value chips instead. In between hands, just tell the dealer you want to "color up", and he will have you push your chips into the middle of the table. He will count them down, and give you a smaller stack of chips that amount to the same value.

This makes them easier to carry for you, and for the dealer it maintains his supply of smaller chips. Now you can take those chips to another table for more play, or head to the casino cashier where you can exchange them for cash.

So, if you have made it this far, congratulations. You should have a good idea of what to expect when you sit down at a blackjack table in the casino.

What we have not talked about is how to actually make the best decisions while playing the game. That is a whole subject all its own. To have the best chance of winning, you should learn and practice "basic strategy", which is the mathematically best way to play each hand against each possible dealer upcard. For a free chart that shows the right play in every case, visit our Blackjack Basic Strategy Engine. If you are looking to play from the comfort of your home, you can visit our online blackjack or live dealer blackjack sections for further resources.

Hopefully I've covered just about everything you need. But if you have other questions, feel free to post a reply at the bottom of the page. Our free blackjack game lets you play at your pace, and the Strategy Coach provides instant feedback on the best strategy. To find the best strategy, use our most popular resource: The Blackjack Strategy Engine provides free strategy charts that are optimized for your exact rules.

If you prefer a plastic card that you can take to the table with you, we have those too: Blackjack Basic Strategy Cards. The original version of this explanation of the rules of blackjack has a very long history here at BlackjackInfo. I created and published it here sometime in It was widely copied by other sites, and it has appeared without my permission on literally hundreds of sites over the years.

When I relaunched BlackjackInfo with a new mobile-friendly design in , I took the opportunity to write this all-new version. Hmmm seems I have been making some bad calls for years now, I thought splitting tens against anything but an ace or 10 was a good move: If a player decides to stand on 15 for whatever reason and the dealer has 16, must the dealer still draw another card since it is less than 17?

Is there any significance in blackjack when you have a black jack paired with a black ace, same suit? The question came up on the multiple choice question on Millionaire. I guessed 16 but the answer was 32??? In playing 21 with one deck off cards aND two people playing, in playing Blackjack with one deck of cards and two people playing what is the most black jack show up.

Ken, This may not be the most appropriate page to post this, but let me explain the situation. I aspire to hopefully gather a group of trustworthy guys together to form a blackjack team. Team play is complicated and far more involved than a group of friends pooling resources.

There is not much published on team play. The following book may be helpful. The strategy does not change, but the player is worse off by around 0. As the dealer I get up to Can the dealer chose to stay and take the chips bet from player on the left. But pay the player on the right?

His rules are fixed. He must hit until he has 17 or higher, and then he must stand. Even if all the players at the table have 18, the dealer must stand if he ends up with a Most casinos now deal games with an extra rule about soft This is covered in detail in the article above. I have a question. I signed up on an online casino and I was getting ready to play blackjack for real money and I asked the live chat help person how many decks were being used and she said 24 decks.

Casinos, both online and land-based, can deal the game pretty much any way they like, including increasing the number of decks to a ridiculous 24! Fortunately, once you get to 8 decks, the game does not get much worse for the player by adding even more decks.

Eight decks is the most typically seen in brick-and-mortar casinos, but in most jurisdictions, land-based or online, there is no legal requirement for any specific number of decks. Is there any standard in the way a dealer deals from the deck? You are describing a CSM continuous shuffle machine , where after each hand the dealer immediately puts the used cards back into the shuffler.

As you note, this eliminates the ability to count cards, or to even observe a useful bias. Since used cards can return into play immediately, the penetration is effectively zero. These machines have spread widely. The only recourse is to see if your casino also still offers regular games either hand-shuffled, or machine-shuffled but with a normal discard tray. If not, voice your discontent and look elsewhere. Should I be able to ask a casino to provide a copy of their rules?

I assume you are talking about online casinos. I didnt catch it at first but sure enough when I drew another card I busted. This particular game would not count both my Aces as one.

Is this ever done at a casino? Who knows what else they are doing wrong! To answer your question, no, this is never done at any casino I have visited. Is this a good bet? What is the house percentage? Yes, there are outstanding opportunities afforded by some sidebets under some circumstances, but if you are able to figure out how to beat them, you no longer need to ask about them. So, the answer to that question is always no. I read the following in the Casino Verite Software: Can you help me with this?

There you can find: This works because of the unbalanced nature of the count. Or do I need to count them equally? Treat them independently, and reassess after each drawn card changes the hand.

Here are some examples: You will draw each of those hands about once every hands or so. As to whether or not to double, 9,3 is a hard 12 and should certainly never be doubled. Also against a dealer 4 in some 1 and 2 deck games. See the Strategy Engine for accurate advice for any game. I assume you are asking about accessing the charts on your smartphone in the casino.

I do not recommend that. In the US, there are laws prohibiting the use of devices in gambling. Most of these laws are based on the Nevada version, which is written in a way that it could be interpreted to mean even something as harmless as looking up a strategy.

If you want access to the strategy in the casino, get a plastic card instead. The shoe game will be easier for a visually impaired player, because both the other players and the dealer have full access to the needed information about the hands. I have played at the table on many occasions with blind players, and the dealer has always assisted by verbally providing the details of the hand. Awesome page, One question.. Is this a standard rule?

Need to be sure which is the standard and which is the exceptionâ€¦ Thanks. I have seen the behavior you describe in online casinos, but I have never seen it dealt that way in any land-based casino.

There, the dealer always deals a card to the first split hand, finishes that hand, and then deals a card to the second split hand and begins it.

I was in Cancun in a casino and in one hand I had 2 Aces against dealer King. I split the 2 Aces and I,ve got Queens on both aces but the dealer draw a card that was an Ace and said that he won since he had blackjack and I only had Blackjack is 21 in the first two cards, and it beats a total of 21, even the case of splitting and getting two hands that look like blackjack.

They are just 21s. But when the dealer does not take a hole card and all bets are subject to losing to a potential dealer blackjack , you should not double or split against a dealer ten or ace because of this. Hi, can please anyone help me with an answer. Do casinos now change the deck of cards after every game or not?

Because I heard lot of people saying that now it is impossible to memorise cards that are still in the game. But I am not sure if this is due to change or because casino use so many decks at once I read above that there are 8 decks in the game. I think what you are trying to describe are continuous shuffle machines CSMs , which allow the dealer to put the discards back into the machine immediately instead of waiting until the shuffled shoe is completed.

The use of a CSM does eliminate the ability to count cards. Fortunately, there are still plenty of games available in most areas that do not use CSMs. The difference is that you must draw a card to your total of 9, whether you double or not. Standing is not an option. With A8 instead, you already have a completed solid hand of Basic strategy will reduce the house edge, but not eliminate it. In typical games, your long-run expectation will be to lose around half a percent of your total action.

Yes, tipping the dealer is common. If a dealer is pleasant, I like to tip a small amount on the hand after I get blackjack, but even then I skip some hands. It was a cool list. Despite the title, it covered many unusual rule variations. Understand the rules here, and then visit the Strategy Engine to see the appropriate plays. Practice with the Trainer. Are you asking about a rule that pays you for having 5 cards and not busting? The free Windows software at this site will analyze these rules for you, including optimal strategy and the effect on the game.

Free BJ Combinatorial Software. Tell me about the best place to sit at a table in Vegas. I have heard that you do not want to be on the end. If you are a basic strategy player, it makes no difference where you sit. If you are a card counter, you can see an extra few cards before you play your hand by sitting at the end, which helps a small bit. For that reason, many players avoid it. When the dealer has a 2 through 6 up, they must draw another card except A6 depending on the table rules.

As a result, the dealer busts more often with those upcards. This is why standing on player totals of is the best play against the small dealer upcards. As for 12v2 and 12v3, those are just exceptions you have to remember.

No, pairs and soft hands are completely different. Use the Strategy Engine to see how to play any situation. Any thoughts on this difference? It is likely that you saw players doubling A7v2, not just hitting it. Hitting would simply be a mistake, as it is the third best way to play the hand. 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. This can be at a young age, if children are given useful kinds of number and quantity experiences.

Age alone is not the factor. Thinking or remembering to count large quantities by groups, instead of tediously one at a time, is generally a learned skill, though a quickly learned one if one is told about it. Similarly, manipulating groups for arithmetical operations such as addition, subtraction, multiplication and division, instead of manipulating single objects.

The fact that English-speaking children often count even large quantities by individual items rather than by groups Kamii , or that they have difficulty adding and subtracting by multi-unit groups Fuson may be more a lack of simply having been told about its efficacies and given practice in it, than a lack of "understanding" or reasoning ability.

I do not think this is a reflection on children's understanding, or their ability to understand. There are many subject areas where simple insights are elusive until one is told them, and given a little practice to "bind" the idea into memory or reflex. Sometimes one only needs to be told once, sees it immediately, and feels foolish for not having realized it oneself.

Many people who take pictures with a rectangular format camera never think on their own to turn the camera vertically in order to better frame and be able to get much closer to a vertical subject.

Most children try to balance a bicycle by shifting their shoulders though most of their weight and balance then is in their hips, and the hips tend to go the opposite direction of the shoulders; so that correcting a lean by a shoulder lean in the opposite direction usually actually hastens the fall. The idea of contour plowing in order to prevent erosion, once it is pointed out, seems obvious, yet it was never obvious to people who did not do it.

Counting back "change" by "counting forward" from the amount charged to the amount given, is a simple, effective way to figure change, but it is a way most students are not taught to "subtract", so store managers need to teach it to student employees.

It is not because students do not know how to subtract or cannot understand subtraction, but because they may have not been shown this simple device or thought of it themselves. I believe that counting or calculating by groups, rather than by one's or units, is one of these simple kinds of things one generally needs to be told about when one is young and given practice in, to make it automatic or one will not think about it.

I do not believe having to be told these simple things necessarily shows one did not have any understanding of the principles they involve. As in the trick problems given earlier, sometimes our "understanding" simply gets a kind of blind spot or a focus in a different direction that blocks a particular piece of knowledge. Since understanding is so immediate upon simply being told the insight, it seems a different kind of thing from teaching someone a whole new idea they did not understand before, were not ready to understand, or could not understand.

I suspect that often even when children are taught to recognize groups by patterns or are taught to recite successive numbers by groups i. And they are not given practice counting objects that way. So they don't make the connection; and when asked to count large quantities, do it one at a time. Different color poker chips alone, as Fuson notes p. Children can be confused about the representational aspects of poker chip colors if they are not introduced to them correctly.

And if not wisely guided into using them effectively, children can learn "face-value superficial grouping " facility with poker chips that are not dissimilar to the face value, superficial ability to read and write numbers numerically. The point, however, is not to let them just use poker chips to represent "face-values" alone, but to guide them into using them for both face-value representation and as grouped physical quantities.

What I wrote here about the use of poker chips to teach place-value involves introducing them in a particular but flexible way at a particular time, for a particular reason. I give examples of the way they need to be used to teach place-value in the text.

The time they need to be introduced this way is after children understand about grouping quantities and counting quantities "by groups". And I explain in this article precisely why different color poker chips, when used correctly, can better teach children about place-value than can base-ten blocks alone. Poker chips, used and demonstrated correctly, can serve as an effective practical and conceptual bridge between physical groups and columnar representation, because they are both physical and representational in ways that make sense to children --with minimal demonstration and with monitored, guided, practice.

And since poker chips stack fairly conveniently, they can be used at earlier stages for children to count individually and by groups, and to manipulate by groups. Columns of poker chips can also be used effectively to teach understanding about many of the more difficult conceptual and representational aspects of fractions, which is another matter about teaching that I only mention here to point out the usefulness of having a large supply of poker chips in classrooms for a number of different mathematics educational purposes.

There is a difference between regrouping poker chips between 10 and 18, and regrouping written numbers between 10 and 18, since when you regroup with poker chips, you change ten of the white ones into a blue one, or vice versa but when you regroup 18 in written form you merely end up with a number that looks like what you started with. When you regroup and borrow in order to subtract, say in the problem 35 - 9, you regroup the 35 into "20 and 15" or, as I say pointedly to students "twentyfifteen".

Then you write the "15" in the one's column where the digit "5" was and you have a "2" in the column where the "3" was, so it even kind of looks like "twentyfifteen". However, in numerical written form, when you start with a number from 10 through 18, if you "scratch out" the "1" and then add ten to the "8" in the one's column, you end up with "18" in the one's column, which is essentially the same in appearance as what you started with.

There is a perceptual point in changing 35 into 2[ 15 ]; there is not a perceptual point in changing 18 into [ 18]. With poker chips there is a perceptual difference between "one blue ten and eight white ones" and "18 white ones". That is part of how poker chips help children conceptually understand representational regrouping. Instead of teaching them to construct numbers by numerals and columns, you have previously taught them simply to write numbers.

By using the poker chips, you have helped them group quantities representationally in terms of ten's and one's where the ten's are different from the one's in some characteristic. Then you show them that written numbers actually also group quantities that way --that written numbers are not just indivisible monadic symbols but that they have a logical structure and rationale to them.

That gives them a feeling of discovery and it makes more sense to them than does trying to start out teaching them to write numbers in terms of numerals and columns, which will mean nothing to them, or seem of no special significance. In any base math, you simply add another column whenever you "get stuck" because you have run out of numeric symbols and combinations of them.

And you call that column by the name of the first number you need to have a new column in order to write the number. Hence, in binary arithmetic, you have "one's", "two's", "four's", "eight's", "sixteen's", "thirty-two's" columns, etc. Then you can write "10" for "two" and "11" for "three", and you again run out of numerals and combinations. To write "four" you need a new column hence, it is the "four's" column and you then can make four different combinations "" for "four", "" [one four, no two's, and one one] for "five", "" [one four, one two, and no one's] for a "six", and "" [a four, a two, and a one] for a "seven".

And this is how we actually do the calculation though in a different order when we multiply, since you multiply five times three and then five times forty and then add it together in the same number and add that to the sum of thirty times three and thirty times forty.

But, of course, we don't think of it this way; and many people who can perfectly well multiply would be unable to think of it this way on their own. You can see that five rows of seven, for example is the same as five rows of four plus five rows of three, because the two sets of five rows are lying right beside each other.

At any rate, the manipulations we learn using pencil and paper, have a rationale, but the rationale is not something we generally learn, and not something that in a sense is as easy as the manipulations. Further, for large numbers, conceptualization and physical representation are difficult or impossible.

So once one learns the rationale or is able to understand or see it, one does not necessarily employ the conceptualization of it for every application. I believe there is a certain irony in calling actual physical quantities of things manipulatives, while considering "pure" numbers not to be manipulatives.

In a sense it seems to me that is just the reverse of the truth. 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. And this is true whether we are talking about billions of dollars of money or thousands of gallons of gasoline.

In liquid measures we often calculate volumes by multiplying dimensions, not by individually scooping out and transferring unit volumes.

In all these cases, we manipulate numbers, not things. Unfortunately in real life, quantities do not conform to simple arithmetic, and so science is empirical rather than a priori. Velocities do not combine with each other by simple addition although at relatively low velocities they seem to ; forces do not combine with each other by simple addition; nor are forces three times the distance acting at one third the strength; working twice as fast may not get you done in half the time because you may wear out before you finish if you work harder than your capacity ; and 10, T-shirts purchased at one time probably won't cost 10, times the price of one T-shirt.

Mixing equal volumes of things that dissolve in one another won't give you twice the volume of either of them. Figuring out the way in which various quantities of things relate to each other is part of what science is about; and it is not always a very easy endeavor that conforms to the arithmetical manipulation of numbers.

In other words, real objects do not always manipulate in the same way that numbers do; and manipulating objects is not the same thing as manipulating numbers. And, it seems to me, the child who is manipulating objects in rows and columns in order to demonstrate or understand multiplication is doing something quite different from the person who is manipulating numbers on paper or in his head. Multiplication is easily seen to be commutative i. In correspondence with me from Peabody, Paul Cobb has said he "would argue that mathematics at the elementary level should not involve mechanical skills even though it is currently often taught this way.

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