P> The ideas presented in this article was in response to a very horrible lesson manuals third year that forced my wife when she was taught this student. The lecture was intended (I assume) an introduction to algebraic concepts.
Without motivation or other reasons
made, the text is a problem: p> If
3 × X = 12, what is X?
It is followed by a description of standard algebraic methods to solve the problem: p> Divide both sides of the equation by three, “Cancel”, “3″ s on left side; Divide 12 by 3; conclude that x = 4
This procedure was a series of diagrams, each of which describes the next step in the transformation. The sponsors were clearly not out of the illusion that a typical graders, or even a third is being developed with the meaning and rationale behind an algebraic derivation to understand. The level of abstraction required to understand the concept of variable, the meaning of an equation, the idea is that the allegation in logically equivalent equations can be transformed and the strategy for developing a solution for years light of any third normal class students met or who might understand. The goal was simply to teach students in a mechanical process, how a computer program. P> In attempting this lesson in a form that could be useful and valuable for students in third year, the challenge is how to make the concepts involved concrete consolidation. If a child thinks of a number (integer) Can you imagine a basket of apples or a pile of coins. If a child may think that the child thinks more, combined with baskets of apples and piles of coins. But what interpretation can give a child an equation or a variable? P> We
start considering how we might imagine a variable that would lead to an equation with one variable in terms of gender, more or less tangible object of meaning to eight years. A variable is a type of object that can be assigned different values. An equation with one variable is a statement based on the value we attribute to the variable in the dependency can be true or false. Thus, we can think of an equation as a kind of answer the question. This machine accepts a number to be called the “value” variables, and the machine responds to the question: “is the equation, the value of the variable is set? One can imagine such a machine running time as in the following diagrams. < P> p> p> p> The machine has a section where we define the input values and are a part called where responses Release labeled product have. We call on machines that an achievement if they produce an entry, entry machines and output are presented. When presented with an input value, shown on the machine, we scored ” X = 3 × 12? Substitutes the value of variable X in the equation X = 3 × 12 assesses whether the resulting equation is true or false, and returns the result. So when we enter the entry “3″, the replacement machine “3″ to “X” in the equation “X = 3 × 12″, which are in the equation “3 × 3 = 12″ which, if we place the value is 3 × 3 “3 × 3″, the equation “9 = 12″, “false”, which is then evaluated production. With this machine, we can restate the original problem as follows: “Find an entry for the” X = 3 × 12? machinery of production “which causes the real machine. P> Although this machine can help students seem to mind an equation like a machine, the True or False visualize the results, it is doubtful that young students would be manipulation of the value of a proxy or understand the values of expressions changed when we participated replace ” 9 “to” 3 × 3 “. Furthermore, it seems likely that the interpretation of an equation not as an affirmation but as a predicate, ie an expression that can be” true “or” bad ” would be confusing for students. We can simplify the problem in two ways. First, get rid of the equation can, in return the next entry / exit of the machine. P>, < , p>
p> p> Here, we replaced the equation, “3 × 12 = X? “To” 3 × X “. Since the equation of the machine when presented with an input value, the machine replaces the value of the variable and evaluates the expression and returns the value. Difference is only in this case The expression value is a number and not a “true” or “poor.” For example, at the entrance with “3″, this machine produces the value “9″. The purpose of this We can machine our problem as “Find an input value of the machine’s” 3 × X ‘to confirm output 12 times. “
We need to interpret an equation as a predicate eliminated, it still requires the student a sense beyond the concept of a variable to simplify and the substitution process in a symbolic expression. So, our last step in the reduction of the problem is the use of variables to eliminate. After all, what does that mean “to 3 × X machine? It Always you enter this and it is multiplied by 3, we can describe this without using a variable. We call it the “Times” 3 “machine (or, if the” Times, 3 prefer “or perhaps be “triple” machine).
p> p>;
< , br />
However, our problem is in the spirit of the machine three times: Find a number produced with input from Times 3 engine start 12 P> No doubt some argue that in this formulation, we “dilute” the lesson that we have to eliminate the use of a variable, the notion of an equation as a predicate, and the symbolic manipulation of expressions. Our counter which has two elements. The first is that even if a third grader typical programmed to perform manipulations of meaning, justification and value are completely beyond its capabilities. In other words, if a student can give meaning to variables, equations, predicates, and symbolic manipulation that they are truly ready to learn algebra, “for real” and as far as we know, no one is seriously proposing algebra as a standard for the third program year. Our second point is that, as shown clearly in the discussion that follows, hope to terms of nature outside the scope of a third grader with the concepts and principles to understand the student and who were replaced, are finally more fundamental and important for the development of students in mathematics. P> These concepts include, based on a mathematical function and the relations and operations functions, as represented here by the input devices ‘output.’ The meaning of these terms can not be stressed enough and get their sense of time developing the student’s perception about them is time well invested. To this end, it should not be limited to digital or even mathematical machines. P> We have the experience
behave with many examples of objects or systems, such as input devices and output. An automaton is a good example. The entrance of the machine is money and the output is candy or whatever product vending machines. Real machines are a bit more complicated than what is natural, as is generally the entry money plus a selection of items that we can do by pressing a button or pulling a button. We could picture such a machine in this way. P>
; < br /> There is no problem concerning the extension of the concept of machine input / output to allow multiple entries. There is also no problem with the machines that have the input / output by more than one exit. For example, we could have an additional output for the change. P>
, a plant Another example is of a type of computer input / output. The inputs and outputs are the raw materials for finished products. The plant takes in cocoa and sugar as the inputs and outputs chocolate bar. P>
Of course, the machine input / output should not stone or metal to produce. A person can bake cupcakes designed as a machine input / output. The entries are the ingredients (flour, etc.) and the result is a cupcake. A leaf can be used as a machine input / output, which takes sunlight, water, carbon dioxide, and thinking and output of sugar and oxygen. A pet can enter / exit the machine as oxygen and sugar as an input and output of carbon dioxide and water takes thought. You can even think to put on your boots and socks machine input / output. Admission is barefoot, a pair of socks and a pair of shoes and the output is that you in your socks and shoes.
A very interesting and important thing about the machine input / output, is that sometimes you can take two or more machine input / output and connecting them to new type of input to / output machines. As an example, I take on machines for sale, and I have two types of machines. A cupcake is a decision machine which mixed the cupcakes. The other is a chocolate coating machine, which you give, (dogs, cats, children, clubs, balls, cookies, fruit, whatever!) Who is on it, and chocolate icing. P>
p>
One day, I received a call from someone a special machine that makes Chocolate Cupcakes wants Frosted . I say: “Now we have a machine that makes cakes and we have a machine that lays chocolate icing on things, but I’m not a machine, a chocolate cupcake are dull. Maybe the boss machine (CMM) can find a way to do? “The CMM wrote:” No problem, we took a small chocolate cake and a freezer and connect the output of cupcakes with chocolate coming from the freezer and we put it all in a box and call it the Chocolate Frosted Cupcake Maker. “ P>
< br /> So, now I sell three machines: my machine cake, my freezer chocolate, chocolate shortcake and my creator. After a while, add a “Cherry Topping ‘machine for my inventory. Cherry Topper takes what you put in and put a cherry on the cake. P>
Things are going along and then one day I get a call from a customer who has a machine that wants to make a chocolate frosted cake with a cherry on the cake get. I can not I go to my chief engineer of the machine to make White. He said: “No problem” and designed a machine from a chocolate cupcake-making mat machine connected to a machine head with a cherry frame around the set. Thus, the gear it seems. P>
; < br />
The operation of combining two machines I / O, the composition is known, as with the formation of a composite “or” connection. “As any mathematician will agree is the composition of basic operations in mathematics. Composition is what we can build arbitrarily complex, concepts and theories of simple components. It is also the basis of inference logic: if A can be derived from B and C can be deduced from B will be derived from A. It is not surprising since the basic principle of mathematical composition concerns. We can use this principle with a dishwasher. P> as shown below, our dishwasher is built from three components to the machine: washing machine, rinser and dryer. P> p> p> p> p> p> p> The dishwasher is a machine, dirty dishes and clean incoming and dry patches outputted. Research in the dishwasher, we see that the plate dirty first enters a washer that produces a clean, soapy plate, but the output. The soap then enters the “Rinse” are unique, but the wet-plate produced, and the wet plate is then placed in the kiln that the last edition produced a clean, dry plate. P> It has a number of different forms
organizational components of the dishwasher. One way is to simply see them as three distinct elements the dishwasher. Another, shown below, rinser is associated with the washing machine and see the pair to form a cleaning machine rinser, which is then connected to a dryer. “ P> ; p> p> < p> p> p> Another possibility is to take into account elements related to rinser rinser-dryer into a dry form and connect the washer to the dryer rinser. < P> p> p> p> p> p> Now, we ask : Is this the way we look at organizational elements: p> A disk connected to a rinsing machine connected with a hair dryer or cleaning and rinsing VerbindungEin a dryer or a washing VerbindungEine < p> rinser-dryer makes no difference on the behavior of the dishwasher? To answer this question, consider the behavior of the dishwasher in the corner of the plate. No matter how we look at the organization components that provides all the record “is that it is first washed by the window, then by the rinser rinsed then dried by the dryer. Thus, the results of the same. This is the basic principle of mathematics. It is sometimes called the associative law of composition. He said that no matter how we, the elements in a composition of input / output assignment of machines, for example, participation of the washer rinser with the participation of the rinser dryer with the behavior of the composite machine is the same. P> The associative principle could lead us to wonder whether the order in which machines appear in a composition issues. The answer is a resounding yes! Consider a composite consisting of two garments machine: take a person into and put in his underwear and the second makes a person focuses on the sweat. If we take the machine in an order, we log into a machine correctly a person’s clothing with underwear and outerwear under the exterior. If we connect the machines in reverse order, then we get our machine is on our outerwear to underwear. These machines are definitely not equivalent. So basically, if the machines are connected, order that the P> The key concept for solving problems like our lesson, the concept of inversion that we get from a constitute “a machine.” Consider an example.
Billy and Sally want to send love to each other notes in class. Since they do not want other people to read, they use a PIN. The way the code works is that each letter of the message after entering the letter in the alphabet will be replaced. Thus, ‘A’ is replaced by “B”, “B” is replaced by “C” and so on. Of course, “Z” is the next letter, so we replace them with “A” to decode the message that we do exactly the opposite: replace “B” with “A”, “C” to “B”. .. ‘Z’ “Y” and then “A” with “Z”. We can imagine this process in terms of “Alphabet Circle”, as shown below. The encoding process replaces next letter in the direction schedule and replaced by decoding the next letter in the counterclockwise ..
;
p> The Billy the message it sends to the encoding code, and Sally that obtained, it decrypts it using the decoding. We can visualize this on the basis of machines / O, as in the table below. P> />
So, what is done by encoding decoding machine from the machine reversed, so if I feed the output of the encoding of the machine in the decoding machine, I wrote back. In other words, when I log into the machine code to decode the computer I get a machine that is exactly what you put out of action. For example, we “I love you” and we “I love you” to. A machine whose output is always identical to the input into a machine called identity. A telephone is Another example of an identity of the machine. P> “/> p ‘/ p>” /> p’ / p> p> If you talk on the phone, a part of the telephone as the microphone sound as input and generates an electrical signal output. The electrical signals are then sent to the mobile phone of the person on the other computer is the name of a speaker takes the electrical signal input and produced the same sound speak for the other person has sent. Again we have an example of a machine in this case, the President, what other machines, the microphone that if both are connected, we obtain a machine identity raises the sound. We are a machine, a machine that different from the first machine unmachine are not reversed. Thus, the decoder unmachine a call for coders and high- unmachine speaker is for the microphone. p> If we unmachine one machine, then we can answer questions of the form: “What was the input that produces that result?” by the power output in unmachine. Consider the “Add 2″ of the machine. located in a row and is the number +2 So if you have received 5, it will exit 7, and if you want it to be coming from 12 14, the output is 23 and we want to find the entrance? We can solve this problem if we can find a non-Add 2 machine. How can we do to reverse with 2? look at some examples in which we know that entry. “ P> p> p> p> p> So, if we add 3 to A-2, we get when we entered 1 and 4, we 2 and increase when we are 11 we have received nine entries. After we see if there are enough examples or perhaps we will see it immediately, because our teacher said subtraction this way, we know how to avoid canceling the addition of 2 to 2. So, the UN-Add 2 Machine is the machine and subtract 2, you subtract 23 supplying the machine 2 gives us the answer 21st After a bit more problems than we recognize that the way to overturn it by any number is calculated by subtracting the number. In addition to mathematical terms a number and subtract this number are opposite operations. Similarly, we discover the division by a number unmachine is for the multiplication we can solve the problems we have started. p> p> To make things even more interesting we will combine the addition and multiplication guess. the result of multiplying a number by 6 and then added 14 56th How many? P> P> So our problem is how to make irreversible the Times of 6 then adds 14 “. After computing the value of time 6 and then inserts identified 14 different entries for some of us that both 6 and 14 machines by connecting a “6 times” can be constructed at an “Add 14″ machine . P> p> We now know that “decided by 6″ Split Times 6 “and” subtract 14 “triggers” Add 14. “Is it possible that we can use this unmachines unmachine for the construction of a composite
Consider a familiar example: two-step” dressing machine. sets “As the first illustration below shows that the element on my underwear first and second component is on my pants and shirt. The second table shows the A-dressing machine. It consists of two components: which triggered a shirt and pants and reverses others set dressing underwear. But note that the order of unmachines reverse the order of machines. If the last thing you did was dress, pants and shirt, then pulled the first thing you do not undress in your pants and shirt is removable. P> P> P> P> P> It This is a general principle that works for all input / output machines, arithmetic, or otherwise. unmachine If A is a United Nations A and B is a B-a UN-PC -B, then the UN to the UN-A is connected to the unmachine from A to B. This compound is table below. An entry “x” in the machine, AB where he is first passed through A generates an output “a” then “A” is passed through an output device B “B.” If we feed then “b” in the production A-BA-A, “b” is passed only by A-B, we need “a” and “a” is passed by the UN-A, which we x to ‘” p> p> How
irreversible six times, then 14, we first subtract 14, to be reversed by adding 14 and then divided by 6 to cancel multiplying by 56 to the sixth power unmachine we
< ; / p> In fact, 7 × 6 + 42 + 14 = 14 = 56 p> This article We have just the surface of what it is based on a basic level through the concepts in the concept machine are explained scratched. In addition to the use of these terms and students for more advanced fields such as algebra, they produce can be used for a deeper understanding of the meaning and use of numbers and operations to provide figures. We will explore these ideas in a forthcoming paper. “/ P>