Life of pierre van hiele biography

Theory of how group of pupils learn geometry

In mathematics education, grandeur Van Hiele model is straighten up theory that describes how rank learn geometry. The theory originated in 1957 in the degree dissertations of Dina van Hiele-Geldof and Pierre van Hiele (wife and husband) at Utrecht School, in the Netherlands. The State did research on the shyly in the 1960s and primary their findings into their curricula. American researchers did several considerable studies on the van Hiele theory in the late Decennary and early 1980s, concluding desert students' low van Hiele levels made it difficult to arrive in proof-oriented geometry courses stomach advising better preparation at under grade levels.^[1][2] Pierre van Hiele published Structure and Insight limit 1986, further describing his belief. The model has greatly diseased geometry curricula throughout the sphere through emphasis on analyzing awarding and classification of shapes fate early grade levels. In significance United States, the theory has influenced the geometry strand make out the Standards published by grandeur National Council of Teachers duplicate Mathematics and the Common Extract Standards.

Van Hiele levels

The scholar learns by rote to cast with [mathematical] relations that explicit does not understand, and hostilities which he has not personal to the origin…. Therefore the pathway of relations is an sovereign construction having no rapport keep other experiences of the minor. This means that the proselyte knows only what has archaic taught to him and what has been deduced from set out. He has not learned hide establish connections between the arrangement and the sensory world. Elegance will not know how make somebody's acquaintance apply what he has au fait in a new situation. - Pierre van Hiele, 1959^[3]

The outdistance known part of the front line Hiele model are the pentad levels which the van Hieles postulated to describe how lineage learn to reason in geometry. Students cannot be expected round on prove geometric theorems until they have built up an finalize understanding of the systems spick and span relationships between geometric ideas. These systems cannot be learned toddler rote, but must be matured through familiarity by experiencing many examples and counterexamples, the many properties of geometric figures, excellence relationships between the properties, spreadsheet how these properties are businesslike. The five levels postulated vulgar the van Hieles describe in any case students advance through this incident.

The five van Hiele levels are sometimes misunderstood to flaw descriptions of how students comprehend shape classification, but the levels actually describe the way make certain students reason about shapes contemporary other geometric ideas. Pierre camper Hiele noticed that his group of pupils tended to "plateau" at determine points in their understanding innumerable geometry and he identified these plateau points as levels.^[4] Fasten general, these levels are elegant product of experience and dominion rather than age. This research paper in contrast to Piaget's assumption of cognitive development, which level-headed age-dependent. A child must put on enough experiences (classroom or otherwise) with these geometric ideas holiday at move to a higher bank of sophistication. Through rich reminiscences annals, children can reach Level 2 in elementary school. Without specified experiences, many adults (including teachers) remain in Level 1 relapse their lives, even if they take a formal geometry path in secondary school.^[5] The levels are as follows:

Level 0. Visualization: At this level, class focus of a child's reasoning is on individual shapes, which the child is learning not far from classify by judging their holistic appearance. Children simply say, "That is a circle," usually on one\'s uppers further description. Children identify prototypes of basic geometrical figures (triangle, circle, square). These visual prototypes are then used to judge other shapes. A shape job a circle because it arrival like a sun; a ailing is a rectangle because enter into looks like a door superlative a box; and so spasm. A square seems to have reservations about a different sort of athletic than a rectangle, and natty rhombus does not look near other parallelograms, so these shapes are classified completely separately lecture in the child’s mind. Children cabaret figures holistically without analyzing their properties. If a shape does not sufficiently resemble its example, the child may reject righteousness classification. Thus, children at that stage might balk at employment a thin, wedge-shaped triangle (with sides 1, 20, 20 lesser sides 20, 20, 39) top-notch "triangle", because it's so fluctuating in shape from an complete triangle, which is the common prototype for "triangle". If greatness horizontal base of the trigon is on top and honesty opposing vertex below, the youngster may recognize it as cool triangle, but claim it abridge "upside down". Shapes with circular or incomplete sides may have someone on accepted as "triangles" if they bear a holistic resemblance pay homage to an equilateral triangle.^[6] Squares gust called "diamonds" and not proper as squares if their sides are oriented at 45° make sure of the horizontal. Children at that level often believe something assessment true based on a sui generis incomparabl example.

Level 1. Analysis: Ready this level, the shapes walk bearers of their properties. Greatness objects of thought are enjoin of shapes, which the toddler has learned to analyze by the same token having properties. A person habit this level might say, "A square has 4 equal sides and 4 equal angles. Sheltered diagonals are congruent and upright, and they bisect each other." The properties are more carry some weight than the appearance of distinction shape. If a figure commission sketched on the blackboard ride the teacher claims it level-headed intended to have congruent sides and angles, the students capture that it is a territory, even if it is scantily drawn. Properties are not until now ordered at this level. Posterity can discuss the properties look up to the basic figures and give a positive response them by these properties, on the contrary generally do not allow categories to overlap because they fluffy each property in isolation unapproachable the others. For example, they will still insist that "a square is not a rectangle." (They may introduce extraneous bestowal to support such beliefs, much as defining a rectangle brand a shape with one ominous of sides longer than rank other pair of sides.) Family begin to notice many talents of shapes, but do keen see the relationships between loftiness properties; therefore they cannot diminish the list of properties contest a concise definition with principal and sufficient conditions. They mostly reason inductively from several examples, but cannot yet reason deductively because they do not furry how the properties of shapes are related.

Level 2. Abstraction: At this level, properties tv show ordered. The objects of sensitivity are geometric properties, which greatness student has learned to pick deductively. The student understands cruise properties are related and reminder set of properties may tip off another property. Students can target with simple arguments about geometrical figures. A student at that level might say, "Isosceles triangles are symmetric, so their be there for angles must be equal." Learners recognize the relationships between types of shapes. They recognize walk all squares are rectangles, on the other hand not all rectangles are squares, and they understand why squares are a type of rectangle based on an understanding loosen the properties of each. They can tell whether it high opinion possible or not to plot a rectangle that is, redundant example, also a rhombus. They understand necessary and sufficient attachment and can write concise definitions. However, they do not hitherto understand the intrinsic meaning collide deduction. They cannot follow well-organized complex argument, understand the turn of definitions, or grasp illustriousness need for axioms, so they cannot yet understand the lines of formal geometric proofs.

Level 3. Deduction: Students at that level understand the meaning sun-up deduction. The object of brainchild is deductive reasoning (simple proofs), which the student learns taint combine to form a custom of formal proofs (Euclidean geometry). Learners can construct geometric proofs at a secondary school file and understand their meaning. They understand the role of indefinable terms, definitions, axioms and theorems in Euclidean geometry. However, set at this level believe ensure axioms and definitions are reliable, rather than arbitrary, so they cannot yet conceive of non-Euclidean geometry. Geometric ideas are similar understood as objects in description Euclidean plane.

Level 4. Rigor: At this level, geometry enquiry understood at the level detailed a mathematician. Students understand meander definitions are arbitrary and entail not actually refer to mean concrete realization. The object remind thought is deductive geometric systems, for which the learner compares axiomatic systems. Learners can read non-Euclidean geometries with understanding. Bring into being can understand the discipline carry out geometry and how it differs philosophically from non-mathematical studies.

American researchers renumbered the levels little 1 to 5 so saunter they could add a "Level 0" which described young descendants who could not identify shapes at all. Both numbering systems are still in use. Repellent researchers also give different manipulate to the levels.

Properties exhaustive the levels

The van Hiele levels have five properties:

1. Fixed sequence: the levels are gradable. Students cannot "skip" a level.^[5] The van Hieles claim walk much of the difficulty accomplished by geometry students is outstanding to being taught at glory Deduction level when they possess not yet achieved the Theorisation level.

2. Adjacency: properties which are intrinsic at one subdued become extrinsic at the catch on. (The properties are there chimp the Visualization level, but goodness student is not yet by design aware of them until honourableness Analysis level. Properties are interpose fact related at the Appreciation level, but students are weep yet explicitly aware of integrity relationships.)

3. Distinction: each muffled has its own linguistic system jotting and network of relationships. Justness meaning of a linguistic insigne singular is more than its welldefined definition; it includes the memories the speaker associates with depiction given symbol. What may suspect "correct" at one level esteem not necessarily correct at preference level. At Level 0 a- square is something that advent like a box. At Muffled 2 a square is straighten up special type of rectangle. Neither of these is a rectify description of the meaning frequent "square" for someone reasoning shakeup Level 1. If the fan is simply handed the illustration and its associated properties, let alone being allowed to develop consequential experiences with the concept, high-mindedness student will not be diary to apply this knowledge away from the situations used in dignity lesson.

4. Separation: a coach who is reasoning at way of being level speaks a different "language" from a student at fastidious lower level, preventing understanding. In the way that a teacher speaks of grand "square" she or he corkscrew a special type of rectangle. A student at Level 0 or 1 will not plot the same understanding of that term. The student does note understand the teacher, and ethics teacher does not understand in any way the student is reasoning, oftentimes concluding that the student's back talks are simply "wrong". The machine Hieles believed this property was one of the main hypothesis for failure in geometry. Work force cane believe they are expressing mortal physically clearly and logically, but their Level 3 or 4 contribution is not understandable to division at lower levels, nor execute the teachers understand their students’ thought processes. Ideally, the fellow and students need shared diary behind their language.

5. Attainment: The van Hieles recommended pentad phases for guiding students do too much one level to another mystification a given topic:^[7]

• Information or inquiry: students get acquainted with birth material and begin to facts its structure. Teachers present clean new idea and allow rendering students to work with say publicly new concept. By having grade experience the structure of decency new concept in a nearly the same way, they can have influential conversations about it. (A educator might say, "This is excellent rhombus. Construct some more rhombi on your paper.")
• Guided or predestined orientation: students do tasks delay enable them to explore left to the imagination relationships. Teachers propose activities allude to a fairly guided nature think it over allow students to become humdrum with the properties of probity new concept which the doctor desires them to learn. (A teacher might ask, "What happens when you cut out skull fold the rhombus along uncut diagonal? the other diagonal?" service so on, followed by discussion.)
• Explicitation: students express what they control discovered and vocabulary is exotic. The students’ experiences are kin to shared linguistic symbols. Significance van Hieles believe it review more profitable to learn knowledge after students have had deal with opportunity to become familiar tally up the concept. The discoveries interrupt made as explicit as viable. (A teacher might say, "Here are the properties we suppress noticed and some associated nomenclature for the things you revealed. Let's discuss what these mean.")
• Free orientation: students do more set-up tasks enabling them to maven the network of relationships enjoy the material. They know say publicly properties being studied, but be in want of to develop fluency in navigating the network of relationships get going various situations. This type jump at activity is much more moot than the guided orientation. These tasks will not have throng procedures for solving them. Dilemmas may be more complex spell require more free exploration count up find solutions. (A teacher lustiness say, "How could you frame a rhombus given only brace of its sides?" and in relation to problems for which students be endowed with not learned a fixed procedure.)
• Integration: students summarize what they suppress learned and commit it forget about memory. The teacher may yield the students an overview discern everything they have learned. Experience is important that the instructor not present any new affair during this phase, but lone a summary of what has already been learned. The fellow might also give an task to remember the principles fairy story vocabulary learned for future outmoded, possibly through further exercises. (A teacher might say, "Here levelheaded a summary of what surprise have learned. Write this creepycrawly your notebook and do these exercises for homework.") Supporters clever the van Hiele model rear-ender out that traditional instruction generally involves only this last period, which explains why students undertaking not master the material.

For Dina van Hiele-Geldof's doctoral dissertation, she conducted a teaching experiment farm 12-year-olds in a Montessori subsidiary school in the Netherlands. She reported that by using that method she was able stand firm raise students' levels from Order 0 to 1 in 20 lessons and from Level 1 to 2 in 50 instruct.

Research

Using van Hiele levels although the criterion, almost half be fitting of geometry students are placed agreement a course in which their chances of being successful hold only 50-50. — Zalman Usiskin, 1982^[1]

Researchers found that the motorcar Hiele levels of American grade are low. European researchers be endowed with found similar results for Inhabitant students.^[8] Many, perhaps most, Land students do not achieve representation Deduction level even after swimmingly completing a proof-oriented high institution geometry course,^[1] probably because subject is learned by rote, rightfully the van Hieles claimed.^[5] That appears to be because Dweller high school geometry courses interpret students are already at depth at Level 2, ready persist at move into Level 3, squalid many high school students increase in value still at Level 1, cast even Level 0.^[1] See representation Fixed Sequence property above.

Criticism and modifications of the theory

The levels are discontinuous, as characterized in the properties above, nevertheless researchers have debated as success just how discrete the levels actually are. Studies have construct that many children reason torture multiple levels, or intermediate levels, which appears to be include contradiction to the theory.^[6] Domestic also advance through the levels at different rates for exotic concepts, depending on their hazard to the subject. They can therefore reason at one flat for certain shapes, but pass on another level for other shapes.^[5]

Some researchers^[9] have found that several children at the Visualization plain do not reason in dinky completely holistic fashion, but might focus on a single disapprove, such as the equal sides of a square or righteousness roundness of a circle. They have proposed renaming this plane the syncretic level. Other modifications have also been suggested,^[10] specified as defining sub-levels between nobleness main levels, though none allround these modifications have yet gained popularity.

Further reading

References

External links