These facts must be recalled accurately, with little mental effort. Procedural memory is used to recall how to do things -- such as the steps to reduce a fraction or perform long division.

## Development of the Mathematical Mind

Experience a problem with basic facts. Active working memory is the ability to remember what you're doing while you are doing it, so that once you've completed a step, you can use this information to move on to the next step. In a way, active working memory allows children to hold together the parts of math problems in their heads. For example, to perform the mental computation 11 x 25, a child could say, "10 times 25 is and 1 times 25 is 25, so adding with 25 gives me Pattern recognition also is a key part of math.

Children must identify broad themes and patterns in mathematics and transfer them within and across situations. When children are presented with a math word problem, for example, they must identify the overarching pattern, and link it to similar problems in their previous experience. Finally, memory for rules is also critical for success in math. When children encounter a new problem, they must recall from long-term memory the appropriate rules for solving the problem. For example, when a child reduces a fraction, he or she divides the numerator and the denominator by the greatest common factor -- a mathematical rule.

Memory skills help children store concepts and skills and retrieve them for use in relevant applications. In turn, this kind of work relating new concepts to real-life contexts enhances conceptual and problem-solving skills. To solve the problem, "If there are six children, each with one pair of shoes, how many shoes in total? Math and Language The language demands of mathematics are extensive. Children's ability to understand the language found in word problems greatly influences their proficiency at solving them.

In addition to understanding the meaning of specific words and sentences, children are expected to understand textbook explanations and teacher instructions.

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Math vocabulary also can pose problems for children. They may find it confusing to use several different words, such as "add," "plus," and "combine," that have the same meaning.

Other terms, such as "hypotenuse" and "to factor," do not occur in everyday conversations and must be learned specifically for mathematics. Sometimes a student understands the underlying concept clearly but does not recall a specific term correctly. Math and Attention Attention abilities help children maintain a steady focus on the details of mathematics.

For example, children must be able to distinguish between a minus and plus sign -- sometimes on the same page, or even in the same problem. In addition, children must be able to discriminate between the important information and the unnecessary information in word problems. Attention also plays an important role by allowing children to monitor their efforts; for instance, to slow down and pace themselves while doing math, if needed.

Temporal-Sequential Ordering and Spatial Ordering While temporal-sequential ordering involves appreciating and producing information in a particular sequential order, spatial ordering involves appreciating and producing information in an appropriate form. Each plays an important role in mathematical abilities. Levine points out that "Math is full of sequences. Sequencing ability allows children to put things, do things, or keep things in the right order. For example, to count from one to ten requires presenting the numbers in a definite order.

When solving math problems, children usually are expected to do the right steps in a specific order to achieve the correct answer. Recognizing symbols such as numbers and operation signs, being able to visualize -- or form mental images -- are aspects of spatial perception that are important to succeeding in math. The ability to visualize as a teacher talks about geometric forms or proportion, for example, can help children store information in long-term memory and can help them anchor abstract concepts. In a similar fashion, visualizing multiplication may help students understand and retain multiplication rules.

The Developing Math Student Some math skills obviously develop sequentially. A child cannot begin to add numbers until he knows that those numbers represent quantities. Certain skills, on the other hand, seem to exist more or less independently of certain other, even very advanced, skills. A high school student, for example, who regularly makes errors of addition and subtraction, may still be capable of extremely advanced conceptual thinking.

The fact that math skills are not necessarily learned sequentially means that natural development is very difficult to chart and, thus, problems are equally difficult to pin down. Educators do, nevertheless, identify sets of expected milestones for a given age and grade as a means of assessing a child's progress.

Learning specialists, including Dr. Levine, pay close attention to these stages in hopes of better understanding what can go wrong and when. Levine outlines many of these milestones for four age groups, pre-school through grade Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects.

A major force behind intuitionism was L. Brouwer , who rejected the usefulness of formalized logic of any sort for mathematics. His student Arend Heyting postulated an intuitionistic logic , different from the classical Aristotelian logic ; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms.

Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers , first introduced by Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science. Like intuitionism, constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse.

In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols.

Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction. Important work was done by Errett Bishop , who managed to prove versions of the most important theorems in real analysis as constructive analysis in his Foundations of Constructive Analysis. Finitism is an extreme form of constructivism , according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps.

In her book Philosophy of Set Theory , Mary Tiles characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists. The most famous proponent of finitism was Leopold Kronecker , [18] who said:. Ultrafinitism is an even more extreme version of finitism, which rejects not only infinities but finite quantities that cannot feasibly be constructed with available resources. Structuralism is a position holding that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their places in such structures, consequently having no intrinsic properties.

For instance, it would maintain that all that needs to be known about the number 1 is that it is the first whole number after 0. Likewise all the other whole numbers are defined by their places in a structure, the number line. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra. Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value.

However, its central claim only relates to what kind of entity a mathematical object is, not to what kind of existence mathematical objects or structures have not, in other words, to their ontology. The kind of existence mathematical objects have would clearly be dependent on that of the structures in which they are embedded; different sub-varieties of structuralism make different ontological claims in this regard.

The ante rem structuralism "before the thing" has a similar ontology to Platonism. Structures are held to have a real but abstract and immaterial existence. As such, it faces the standard epistemological problem of explaining the interaction between such abstract structures and flesh-and-blood mathematicians see Benacerraf's identification problem. The in re structuralism "in the thing" is the equivalent of Aristotelean realism.

Structures are held to exist inasmuch as some concrete system exemplifies them. This incurs the usual issues that some perfectly legitimate structures might accidentally happen not to exist, and that a finite physical world might not be "big" enough to accommodate some otherwise legitimate structures. The post rem structuralism "after the thing" is anti-realist about structures in a way that parallels nominalism. Like nominalism, the post rem approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure.

According to this view mathematical systems exist, and have structural features in common. If something is true of a structure, it will be true of all systems exemplifying the structure. However, it is merely instrumental to talk of structures being "held in common" between systems: they in fact have no independent existence.

Embodied mind theories hold that mathematical thought is a natural outgrowth of the human cognitive apparatus which finds itself in our physical universe. For example, the abstract concept of number springs from the experience of counting discrete objects. It is held that mathematics is not universal and does not exist in any real sense, other than in human brains.

Humans construct, but do not discover, mathematics.

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With this view, the physical universe can thus be seen as the ultimate foundation of mathematics: it guided the evolution of the brain and later determined which questions this brain would find worthy of investigation. However, the human mind has no special claim on reality or approaches to it built out of math.

If such constructs as Euler's identity are true then they are true as a map of the human mind and cognition. Embodied mind theorists thus explain the effectiveness of mathematics—mathematics was constructed by the brain in order to be effective in this universe. For more on the philosophical ideas that inspired this perspective, see cognitive science of mathematics. Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be literally realized in the physical world or in any other world there might be.

It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized. For example, the number 4 is realized in the relation between a heap of parrots and the universal "being a parrot" that divides the heap into so many parrots.

## Montessori - Mathematics - Introduction

Mayberry, following Euclid, considers numbers to be simply "definite multitudes of units" realized in nature—such as "the members of the London Symphony Orchestra" or "the trees in Birnam wood". Whether or not there are definite multitudes of units for which Euclid's Common Notion 5 the Whole is greater than the Part fails and which would consequently be reckoned as infinite is for Mayberry essentially a question about Nature and does not entail any transcendental suppositions.

John Stuart Mill seems to have been an advocate of a type of logical psychologism, as were many 19th-century German logicians such as Sigwart and Erdmann as well as a number of psychologists , past and present: for example, Gustave Le Bon. Psychologism was famously criticized by Frege in his The Foundations of Arithmetic , and many of his works and essays, including his review of Husserl 's Philosophy of Arithmetic. Edmund Husserl, in the first volume of his Logical Investigations , called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it.

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The "Prolegomena" is considered a more concise, fair, and thorough refutation of psychologism than the criticisms made by Frege, and also it is considered today by many as being a memorable refutation for its decisive blow to psychologism. Mathematical empiricism is a form of realism that denies that mathematics can be known a priori at all.