This is designed to be a guide to the concepts of Al'Mat and Al'Jabr for an instructor. Implementation would have to be designed by the instructor. Page renders best at 800 by 600 resolution. Photographic images are 24 bit JPEGs.

Al'Mat is a concrete model of arithmetic. Al'Mat uses yarn to group marbles and pompoms in arithmetic expressions. Al'Mat builds from concrete set notations into abstract integer arithmetic. The yarn, marbles, and pompoms can be substituted for any locally available like materials such as cowry shells and basalt chips for the marbles and pompoms. Although commercially made bicolor chips are made expressly for Al'Jabr type models, their use is not desirable here as it presents a model that requires a materials budget to order from abroad. Students who later go on to become teachers are more likely to use Al'Mat if it is done with locally available materials.

Al'Mat may be preceded by set theory discussions. In Al'Mat, any yarn circle is a set boundary. Al'Mat can be preceded by a full treatment of sets using yarn circles and a variety of objects to examine subsets, union, intersection, and other set properties. This is the approach in the cognitive math sequence.

Al'Jabr is the algebraic counterpart to Al'Mat.

Al'Mat consists of sets of marbles and sets of pompoms. There are various rules called
operations that govern Al'Mat. An operation is __closed __if the resulting sets are the
same type of item the starting sets. An __identity__ for an operation is that set which
has no effect on the starting set under that operation.

The empty set, represented by an empty yarn circle, contains neither marbles nor pompoms. The empty set is neither marble nor pompom.

This document, for brevity, only provides singular illustrations of some of the key concepts. Note that the use of non-traditional terminology is intentional. One of the goals of Al'Mat and Al'Jabr is to avoid words that might trigger old habits or old phobias. The sulp operation is done concretely and is based on Union, a concept that is used in the cognitive approach as the term is fairly unfamiliar to our students and was previously introduced to them during work on sets. The cognitive approach course began with set theory and enumerating subsets.

A singular example of sulp:

sulp is

Does sulp ever produce anything other than marbles? If no, then it is closed. If yes, then it is not closed.

What is the identity for sulp? That is, make the following statement come true:

sulp _____ is

How many marbles do you sulp to four marbles to get four marbles? [No marbles are added, thus the identity is zero marbles. Obviously there are subtleties being passed by here: is the absence of marbles numerically zero marbles or a state of non-existence of marbles, a marbular null entity.]

The order in which sulp is done does not matter. This can be used to explore commutivity.

sulp sulp is

Let this be writable as sulpsulp 3 is

That is, let the number of repeats be written as an Arabic numeral.

Is sulpsulp closed? That is, does sulpsulp always produce marbles?

What is the identity for sulpsulp? That is, make the following statement come true:

sulpsulp _____ is

This is a little more difficult. Think of this in terms of how many time the first group must be repeated to get the result? [The result is one repeat, thus one is the identity]

Commutivity with sulpsulp is less clear: the marbles are objects while the repeat count is a symbol. This distinction is subtle but real. In one term the class had previously sulpsulped sets:

{golf ball, pingpong ball, tennis ball} sulpsulp 2 = {golf, golf, pingpong, pingpong, tennis, tennis }

While the above makes sense, the following does not:

2 sulpsulp {golf, pingpong, tennis}. Marbles are still marbles at this stage in Al'Mat and sulpsulp is not commutative. Only if and when the marbles are equatable to their numerical equivalent does sulpsulp become commutative. This subtlety is important in both Al'Jabr and algebra: when adding to both sides of an equals sign the constant adds to only the constant term, variable terms only to their respective like variable term. But multiplication is fundamentally different: all terms get multiplied against. Students here often err by not distributing to the second term. Sulpsulp is fundamentally quite different from sulp.

sunim is

The above can also be read: five remove two is three. Here too is a subtlety: where sulp is merely a simple union of all the marbles lying around (so to speak), sunim involves counting the second set and removing the count number from the first set. This is a radically different and much more complex operation. Al'Mat will be extended further below to return simplicity to the sunim operation.

In concrete object set theory sunim (removal or disunion) involved calculating what was not in the intersection of the sets.

{golf, marble, marble, pingpong, pompom } sunim {marble, pingpong} involved the following diagram:

In the intersection of the two sets diagrammed is {marble, pingpong}. The result of the sunim are those elements that are not in the intersection. A sunim symbol was used: a slash bar through an intersection symbol when set theory sunim was done. Note that sunim is sufficiently complex that no standard font set even includes this bizarre beast (the elements that are not in the intersection).

What is the identity for sunim?

Is sunim closed? This is best explored by examining systems such as:

sunim is ??

In other words, three marbles take away five marbles. Three of the five can be removed, and then there are no more marbles to be removed. This is similar to the problem in set theory of "What is the result of removing a tennis ball from a set composed of a marble, a pingpong ball, and a pompom." The sentence does not make any mathematical sense. There is no tennis ball to remove.

One option, one that was suggested was to ban this operation as impossible. Around here, however, one can sometimes buy ten dollars worth of groceries even if one only has three dollars. Three take away ten is possible if you buy on credit. This led to the following concept:

Let the set "owe" the world one tennis ball or, in the case further above, three marbles.

In Al'Mat the "owed marbles" or "debt" is marked with pompoms.

sunim is

Where the pompoms signify that the "table" is owed two marbles.

Sunim with marbles does not always produce marbles. Sunim produces other objects. Closure should be viewed as a norm: dogs produce dogs and cats produce cats. A lack of closure in biology would mean dogs could produce cats. Lack of closure is an aberration. Fascinating and obviously inherently complex.

Now Al'Mat can be explored with pompoms, where pompoms are marbles that are owed. Debts can be sulped:

sulp is

'Two owed marbles sulp one owed marbles results in the owing of three marbles."

Note that:

sulp is (no marbles).

pays off the debt. The debt is cleared by the paying in of the three marbles. The result is no marbles and no debt. Thus pompoms are anti-marbles and marbles are anti-pompoms. When they meet they destroy each other one for one.

This leads to a whole new set of rules. Pompoms sulp marbles can be pompoms, marbles, or the empty set (no marbles). If the marbles outnumber the pompoms, then sulp produces marbles. If the pompoms outnumber the marbles, then sulp produces pompoms.

Note that the pompoms provide a way to eliminate sunim altogether:

sunim is can be rewritten as a sulp statement with the same result:

sulp is

Note that the first of the two equations CANNOT be order reversed and yield the same result:

sunim is

But the second of the two equations can be order reversed and attain the same result:

sulp is

Sulp can be order reversed, sunim cannot. In math this means sulp can commute and sunim cannot.

Pompoms can be also be sulpsulped:

sulpsulp 2 is

Note that the repeat count is always, in modern terms, a positive integer. Objects can only be repeated in a positive sense. Repeat twice, thrice, and so forth. If an equivalency is asserted between marbles and positive integers, then sulpsulp is only done with marbles as the number of repeats.

Note the following:

[ sulp ]
sulpsulp 2

is

[ sulp ][ sulp ]

which is the same number of marbles as:

This same result could have been attained by "dealing" the sulpsulp operation into the brackets;

[ sulpsulp 2
sulp sulpsulp 2]

is

[ sulp ]

which is the same number of marbles as:

This "dealing" into the parentheses of the repeat count is called distribution. This can be used to get at a fundamental mystery: what happens if sulpsulping is done with pompoms as the repeat element.

The outline of the argument goes as follows:

sulp is (no marbles)

If marbles are asserted to be equivalent to positive integers, then:

[ sulp ] sulpsulp 1 (is still no marbles)

can be rewritten

[ sulp ] sulpsulp

Distributing the marble results in the following:

[ sulpsulp sulp sulpsulp
] (must still be no marbles for
consistency)

bear in mind that marble sulpsulp marble is marble and pompom sulpsulp marble (pompom
repeated once) is pompom (in order to generate the consistent result of no marbles).
This is still obvious but important: repeating pompoms yields pompoms. Bear
in the back of one's mind that what we are saying is that a negative times a positive is
negative, but do not yet reveal this to the class.

This now provides a way to do:

[ sulp ] sulpsulp
is no marbles

is

[ sulpsulp sulp
sulpsulp ] must still be no marbles

but the front is just:

[ sulp sulpsulp ]
must somehow result in no marbles

therefore we had better choose to let
sulpsulp be equal to !

Weird but true, pompoms sulpsulped with pompoms are marbles.

Those are the rule of Al'Mat. In modern terms:

- marbles sulp marbles are marbles: pos + pos= pos
- marbles sulp pompom are whichever is the greater numerically: pos + neg= pos or neg
- pompoms sulp pompoms are pompoms: neg + neg = neg
- marbles sulpsulp marbles are marbles: pos * pos = pos
- pompoms sulpsulp marbles are pompoms: neg * pos = neg
- pompoms sulpsulp pompoms are marbles: neg * neg = pos

Now introduce red yarn circles as yarn circles containing an unknown, unspecified number of marbles or pompoms. Each red circle must contain the same number of marbles or pompoms.

If are the same as

How many marbles should be in each red yarn circle given that each red yard circle should have the same number of marbles? A series of problems with varying numbers of yarn circles and marbles can be done. Bear in mind that in class this is done with groups of two to four students having actual yarn circles and marbles. Marbles are dealt (like cards are dealt around a table to players) into the yarn circles. At this point some students have asked, "When are we going to do math?" I take such comments to mean I am being successful: the students are busy doing 2x = 6 and they don't even know it. Don't tell them either.

To find the answer to an Al'Jabr problem with red yarn circles on one side and marbles on pompoms on the other, "deal" the marbles or pompoms into the red yarn circles.

In Al'Jabr, since all red yarn circles contain the same number of marbles or pompoms, the answer can be abbreviated by giving the number of marbles in one red yarn circle.

[Instructor should set five or so problems of the above type and then walk the room checking for correct "solutions".]

Another type of Al'Jabr involves having marbles or pompoms on the same side as the red yarn circles. Below, for brevity, "is" replaces "are the same as," although I am not certain this is a wise replacement to make in class.

sulp is

By looking, one should deduce that one marble should be placed in each red circle. One way to work towards this would be to find a way to "wipe out" the marbles on the left side. But what are pompoms for anyway? Toss five pompoms into the left side [boom!], toss five into the right side [boom!] and when the smoke clears one is left with:

is

Deal out the marbles and the result is the answer: a red yarn circle has one marble. Now we have a system for solving these Al'Jabr problems. Note that the system demands that what we do on one side of the "is" we must also do on the other side of the "is." We must be fair. If the left side gets five pompoms, then the right side must get five pompoms. Again, bear in mind that this is being done with real yarn and marbles in the classroom. A pen, pencil, or other object "separates" the two sides.

[Teacher's note: We have just solved 3x + 5 = 8 by adding -5 to both sides and dividing the remaining equation by 3. Do not, however, tell the students this, it is premature and would drag all sorts of prior "mental baggage" into our college student's minds].

At this point various Al'Jabr problems are done, including ones with pompoms:

sulp is

The solution, and let the students work on this, can be arrived at by either taking three pompoms from both sides or by "tossing in" (sulping three marbles on both sides and then dealing the pompoms out. As you create problems, be sure they have an integer solution. Fractional solutions introduce complexities that are probably better avoided at this time. The solution above is each circle winds up with two pompoms.

Consider the Al'Jabr:

sulp

is

Tossing in four pompoms to both sides and dealing marbles is one way to find the answer. There is another for this problems: breaking apart. [Do this by physically sliding the elements around on a table - anyway in which stuff can moved except across the "is" is legal because sulp was commutative!]. The problem can be broken into three problems:

sulp is

sulp is

sulp is

Now just solve one of the Al'Jabrs.

Note that in the break apart, the red yarn circles, the left side marbles, and the right side marbles are all broken apart. Later this "break apart" process will used to explain why, when we multiply both sides by 1/3, we must "distribute" the one third, but when we add three to both sides we do not need to "distribute." Adding is simply tossing in three marbles.

Al Jabrs with red yarn circles on both sides:

sulp

is

sulp

Just as marbles can be removed from both sides, yarn circles can also be removed from both sides. This leads to "yarn circles are removed from yarn circle and marbles/pompoms are removed from marbles/pompoms." Without actually saying so, the concept of "like terms" has arisen naturally from the system. Without realizing it, students are solving 3x + 5 = x + 8.

At this point I usually go ahead and transition into symbolic algebra, spending a class showing the equivalencies between Al'Mat, Al'Jabr, arithmetic, and algebra.

Note from 2017: "Negative" yarn circles could be deployed to handle negative x values. A negative yarn circle with marbles would be equivalent to the positive yarn circle with pompoms, a positive yarn circles with pompoms would be a negative circle with marbles. This would potentially prove useful for solving x²-8x+15=0 in the next section. Perhaps.

Al'Jabr can be, in a limited way, extended to quadratic equations, but it is probably not productive. Here algebra tiles are probably a better option. The key is to use a new type of variable for the square term: a square grid. The filling order of this new variable is very special. See the example below.

Note that an empty grid will not appear in most browers. FrontPage 98 stripped out characters from otherwise empty data cells. When using cascading style sheets, Explorer 5.0 drop colored borders on empty cells. Opera 3.5 doesn't display any borders on the table, whether the cell is empty or otherwise, hence the whole layout is confusing in Opera 3.5. Thus I have put "periods" in as place holders in the square variable.

. | . | . | sulp sulp | is | |

. | . | . | |||

. | . | . |

Remove two marbles from both sides

. | . | . | sulp | is | |

. | . | . | |||

. | . | . |

Start dealing the marbles in the following way. Note that I cannot place the marbles physically inside the yarn circles on this page.

. | . | . | sulp | is | |

. | . | . | |||

. | . |

Next deal:

. | . | . | sulp | is | |

. | |||||

. |

Note that the filling of the square variable must be done such that one always forms a square. The variable must also be considered an infinitely expandable grid. In a classroom one might lash together three egg cartons to create a six by six square variable.

sulp | |||

The answer is the number in each yarn circle or the number of marbles along an edge of the grid. Note that this system is limited to marbular answers.

Students at the College may be able to solve 3x +5 = 17 correctly, but when faced with 5 + 3x = 17 will subtract the 3x from the 17 as their first step. I call this "positional memorization." The student has memorized, "subtract the second element from whatever is on the left side." Al'Jabr will let me say, "Can you remove yarn circles from marbles?" Al'Jabr makes concrete like terms.

Another common difficulty is distribution. In solving 3x + 6 = 18 it is perfectly legal to subtract six from both sides, subtracting that six from only the constant term on the left side of the equation. Once the students learn this, they then become confused when you suddenly insist that multiplying both sides by 1/3 must be applied to both terms on the left side. Why? Adding didn't require adding to both terms, so why should multiplication be different? Bear in mind, some of our students would accept 3x +6 = 9x as a perfectly legal statement. Some would add six to both sides of 3x+ 6 = 18 and get 9x + 12 = 27. Subtracting six from both sides could lead to -3x = 12 in some student's view.

The above adding of six can be explained as tossing in six marbles to both sides. Obviously marbles combine only with marbles. Now multiplying both sides by 1/3 can be equated to "breaking apart." The students now have a concrete, in-their-mind's-eye, view of what the operations are doing.

When I began the journey into Al'Jabr I had no idea whether it would work. The previous term a "transition to algebra" (~40% arithmetic/~60% linear algebra) class performing standard drill and practice from the textbook had experienced a failure rate of 74%. The department chair asked that this not occur again - but course repeat rates were hovering around 50%. Neither number was acceptable to me. I asked for wide latitude to try anything I chose to improve success rates.

While visiting Cordoba, Spain on vacation in 1994, I came to learn of Averröes, Ibn Al-Arabi, Maimonides, King Alfonso X, and Muhammed Al Riquti. The later founded a school at Murcia. It was through these people and in these places, according to what I learned in Cordoba, that the knowledge held in the middle East in the Muslim cultures passed into Europe. I was sitting in the Calahorra Tower when the concept of Al'Mat and Al'Jabr crystallized in my mind.

I used Al'Jabr to begin the first few weeks of PreAlgebra the next term. Al'Jabr acted as an advance organizer, if you will, for the students. Then the class moved into algebra and regular drill, practice, and textbook plus College standard chalk and talk. 60% of the class moved on to the next class, only 40% had to repeat the course. More importantly, I had a whole new language I could use when students made common distributional errors - errors that I saw less frequently than in previous terms.

I also did not see the "rapid extinction" of memorized knowledge so often seen in students in the math courses. The students appeared to be moving from pure memorization of unimportant features such as positionality and moving towards seeing mathematics as a language with a few simple rules.

A year later Al'Jabr was utilized by me in our recently minted lowest level course: MS 095 PreAlgebra. Here Al'Jabr came late in the term, after set theory and other explorations. In this course topics were not organized by mathematical complexity. Thus the world of golf ball algebra had already been done before Al'Jabr.

Notebook material originally created in 1994 by Dana Lee Ling. Edited in 2000 and 2017.