There are reasons for why this happens and sometimes the reasons are better understood than others. In learning geometry there is a model that helps explain the learning process more completely than in any other area of mathematics. This model, referred to as the van Hiele model, is not complete nor is it absolutely correct. It will not solve all of your classroom problems, but it can give you a way to think about teaching and learning geometry which will help your students understand and enjoy geometry.

The van Hiele Model of Geometric Thought is the outgrowth of the doctoral work done in the 1950s by a Dutch couple, Dina van Hiele-Geldorf and Pierre van Hiele, who were mathematics teachers and graduate students of a well-known Dutch mathematician and mathematics educator Hans Freudenthal. Their dissertations grew directly out of classroom experiences like that described above and they reached the conclusion that learning geometry involves a developmental sequence and that this sequence is essential. If our students try to skip levels they are doomed to fail. It is possible that Janet and Johnny can run to the top of the ladder taking two steps at a time and not fall, but you know that if you teach ladder climbing by insisting they run up a ladder that is missing half its steps, a majority of your students would fall, fail, and be injured. A majority of our students are failing on the geometry ladder and the van Hiele model says it is because the ladder is missing some steps.

Their work was incorporated into the curriculum of the Soviet Union in the 1960s but it was not until the late '70s that it began to receive international attention. It has received more intense scrutiny from North American mathematicians in the last decade and was a particular influence in the curriculum work of Hoffer (Geometry; A model of the universe ), Usiskin, Coxford and Hirshorn (Geometry, The University of Chicago School Mathematics Project) and Serra (Discovering Geometry: an inductive approach ). Although not universally accepted by all North American educators, the van Hiele model is mathematically elegant and worthy of investigation.

From their research the van Hieles constructed a hierarchy of reasoning (shown below). Be aware when reading research or articles about the van Hiele model that many of the researchers-for better or worse-have added to the levels or have changed the numeration.

van Hiele Levels of Understanding

Level 0 vH 0 Recognition

Level 1 vH 1 Analysis

Level 2 vH 2 Order or Abstract or Informal Deduction

Level 3 vH 3 Deduction or Formal Deduction

Level 4 vH 4 Rigor

van Hiele Level 0 Recognition

Students can learn names of figures and recognize shapes as a whole.

Students at this level are aware of space only as something that exists around them. Geometric concepts are viewed as total entities rather than as having components or attributes. Geometric figures, for instance, are usually recognized only by their shapes as a whole, not by their properties. Students frequently refer to visual prototypes-a rectangle looks like a door or a ball is like a circle. If students do consider properties, they may use imprecise or improper ones to compare drawings and/or identify shapes-using orientation on the page so that a triangle in this position, V, may not be considered a triangle; a canted square may be called a diamond but not accepted as a square. They may also exclude necessary attributes-triangles with curved sides, or rectangles where one side is not quite closed. Predictably they use a wide mix of informal and formal vocabulary.

Learners interpret and react to space and geometrical objects without actual analysis. Properties, components and attributes of figures, if explicitly recognized, will be incidental to the shape. Triangles, squares, and rectangles may be identified and their names learned because "these all look alike." At this level a person will be able to put the square peg in the square hole, but will not recognize that rectangles have all right angles. They may not recognize that a square that has been turned 45° is still a square.

van Hiele Level 1 Analysis

Students can investigate properties of figures.

Students can learn appropriate vocabulary. and symbols.

Students can now begin to delineate properties of figures through investigation and observation. They can identify and describe a figure by its properties and use properties to form classes of shapes. There is, however, limited understanding of definitions and interrelationships between figures. Furthermore, relationships between properties are not yet understood. Thus, they may reject some class inclusions-parallelograms may not have right angles, excluding rectangles as a subclass. They rely almost exclusively on a very limited number of drawings and observations to make conclusions. Mathematical proof may be explicitly misunderstood and certainly unappreciated.

Analysis is the stage when formal geometric concepts begin to emerge. The properties, components, and attributes of shapes and figures that were not explicitly recognized before now become explicit and important. A person at this level is able to recognize rectangles by their properties - right angles and parallel sides. He can identify equal angles in a variety of figures, but he may not know that squares are also rectangles. Interrelationships between figures will generally not be seen. Definitions and deductions are not generally understood.

van Hiele Level 2 Abstract or Informal Deduction

Students can develop a network of figures and relationships.

Students can establish the interrelationships of properties within and among figures. They can also form complete definitions that are applied explicitly. Thus they can deduce properties of a figure and recognize classes of figures, now understanding class inclusions. Informal arguments can be followed. They can also formulate deductions of a few steps, using properties of figures and some logic. They can follow a formal proof but cannot construct one nor can they find a different path of reasoning.

Here students begin to understand interrelationships between figures and recognize common properties. The students would know that triangles and rectangles were both polygons and they would know that a square is also a rectangle because it has all of the properties of a rectangle. Students can understand and begin to formulate informal deductive arguments. Definitions can be understood, but the role of deduction is not understood. Students will mix empirical reasoning with deductive reasoning. They cannot generally extend arguments to new situations and they do not understand the role that axioms play in the formal system of high school geometry.

van Hiele Level 3 Formal Deduction

Students understand proof.

Students at this level understand the interrelationships and roles of the system components - undefined terms, axioms, postulates, definitions, theorems, etc. They can construct a proof and understand when it is presented in a different way. They can understand the difference between necessary and sufficient conditions and distinguish between a statement and its converse. Students may see the significance of deduction as a way of establishing geometric theory within an axiomatic system, however, they do not realize that different axioms would produce a different system and different theorems. This is the traditional level of high school geometry. It is not necessarily what actually happens in the classroom, but it is what we traditionally have hoped would happen. Students at this level will understand the process and role of deduction, the place of axioms, and the need for definitions. Students in Levels 2 and 3 will see that the diagonals of a square are perpendicular bisectors of each other; only the Level 3 students will recognize the need for a formal proof. They can construct proofs and understand that many deductive arguments or proofs may lead to the same conclusion.

van Hiele Level 4 Rigor

Students can develop a network of relationships.

Students at this level can study and compare a variety of axiomatic systems. Geometry can be studied and understood completely abstractly. Few students reach or work at this level. This is the level of much advanced college level mathematics where many of the activities are dominated with formal definitions, constructs, and different axiom systems. Level 4 has received little formal study.

A large percentage of the population never rises above the first two levels. This is related to experiences, not maturity. It is difficult to communicate across levels. If you talk at level 2 with a student thinking at level 0 very little good will come of it. More than likely, this is what has been happening when our students tell us the angle has 135°. Students who enter a high school geometry class functioning at levels 2 and 3 are very successful. Students functioning at levels 0 and 1 have very low success rates.

For the classroom teacher the first four levels are of primary concern. How do they relate to your students, the SOL that they are responsible for, and your student's success in more advanced mathematics? We do not need to know if the model is exactly right, or what precisely separates levels 1 and 2. The essential issue is that our students are functioning at levels 0 and 1 and by the time they get to high school they need, if they are going to succeed, to function at levels 2 and 3. We can look at this and see that there is a world of difference between these levels, and that a student working at level one does not have a fair chance for success in a high school geometry course operating at level three.

There is an lot that can be said about the process of moving a child from level 0 to 3. This model can give you a useful framework for thinking about teaching geometry. If used with common sense and hard work it can help your students succeed in their geometry studies.

Assumptions of the van Hiele Model

1. The reasoning levels are in a fixed sequence and a student cannot be at vH n without having gone through vH n-1.

2. At each level of thought, what was intrinsic in the preceding level becomes extrinsic in the current level. For instance, at vH 0, a student will see only a rectangle; it is not until vH I that the rectangle is analyzed and its components and properties are discovered.

3. Each level has its own linguistic symbols and its own network of relationships connecting those symbols. A relationship that is correct at one level may be modified at another level. A student will recognize squares, rectangles and parallelograms at vH 1 but does not understand until vH 2 that figures may have more than one name, i.e., a square is also a rectangle and a parallelogram.

4. If students are at one level and instruction is at another level, then understanding and progress may not take place. Students will simply not be able to follow the thought processes being used in instruction.

5. Progress and understanding depend more on the content and methods of instruction than on age. Thus, instruction, not maturation, is a key to success. No method of instruction allows a student to skip a level; some methods enhance progress; other methods retard or prevent progress. A student who memorizes the formula for the surface area of a figure without knowing the reasoning behind the formula generally cannot apply that knowledge in a problem solving situation or to a proof.

The van Hiele model provides a hierarchical framework for assessing student backgrounds and for sequencing instruction in order to move students gently from one level to the next. A note of caution-however accurate the characteristics of each van Hiele level might be, they are not necessarily absolute for all students or for all concepts. The framework should be used as a guide to plan instruction, not to label individual students.

Implications for Instruction

The critical entry level for high school geometry appears to be vH I (analysis). All the research shows that a beginning student at vH I has a 50-50 chance of mastering proof writing by the end of the year. Students at vH 2 (abstract) or vH 3 (deduction) have a significantly greater chance of mastering proof writing. There will be, however, some students who enter geometry operating at the recognition level (vH 0). It is impossible for teachers to provide total remediation for those at vH 0 or vH I. However, by assessing the knowledge, reasoning and attitude of the class, and by providing appropriate quality instruction, teachers can improve the chances of their students to understand and pass geometry.

The van Hieles developed a sequencing pattern for the classroom which they assert will take students through a reasoning level. The sequence is shown in outline form below.

1. Information: New topics are introduced through guided dialog.

A. Teacher assesses student's vocabulary, interpretations, and prior knowledge.

B. Students learn what direction further study will take.

2. Directed Orientation: Students explore topic through activities and materials sequenced by teacher.

A. Much of the material will be short tasks designed to elicit specific responses.

B. Teacher provides clarification where needed.

3. Explication: Students refine their conceptualizations and vocabulary.

A. Students express opinions about the structures observed.

B. Students observe relations within the structure.

C. Teacher moderates student discussion and helps them reach consensus.

4. Free Orientation: Students are challenged with more complex tasks of problem solving nature.

A. Students discover new relationships and resolve problems on their own.

B. Teacher acts as guide for discovery.

5. Integration

A. Students review and summarize their observations forming a synthesis of new concepts and relationships.

B. Teacher provides direct explanations to assist students in refining and internalizing concepts and procedures.

Activities for the Reasoning Levels

van Hiele Level 0 Recognition

1. Students should identify figures

a. from a single drawing,

b. within a given set,

c. embedded within other shapes, and

d. in a variety of orientations.

2. Students should relate figures to real world objects within their experience.

3. Students should create figures by copying on dot paper, grid paper, etc., drawing, constructing on geoboards, constructing with sticks, straws, or pipe cleaners, tiling with manipulatives.

4. Students should describe geometric shapes verbally using appropriate standard and nonstandard language.

5. Students should solve problems by managing shapes, measuring, and counting, i.e., using two triangles to make a rectangle or finding the area by tiling and counting.

van Hiele Level 1 Analysis

1. Students should investigate properties of figures and geometrical relationships by

a. measuring, physically or via computer,

b. folding,

c. modeling, and

d. tiling.

2. Students should describe a class of figures by its properties

a. verbally, speaking or writing,

b. using charts, and

c. with property cards.

3. Students should compare shapes according to characterizing properties.

4. Students should sort shapes by a given attribute, i.e., cutouts of quadrilaterals by number of parallel sides.

5. Students should identify and/or draw figures, given

a. an oral or written description, or

b. a logic puzzle with a series of clues, or

c. a series of visual clues.

6. Students should empirically develop rules, definitions, and formulas.

7. Students should investigate and classify properties used to characterize classes of figures.

8. Students should investigate to discover properties of unfamiliar classes of objects, given examples and non-examples of the class.

9. Students should solve problems using properties of figures and geometric relationships.

van Hiele Level 2 Informal Deduction

1. Students should study relationships developed at vH 1, looking for inclusions and implications by

a. using property cards, or

b. using geoboards to change figures, i.e., quadrilateral to trapezoid, observing the transformation.

2. Students should identify minimum sets of properties that describe a figure.

3. Students should develop and use definitions.

4. Students should follow and present, informal arguments.

5. Students should follow deductive arguments and supply some reasons.

6. Students should provide more than one approach or explanation i.e., define a parallelogram in two ways.

7. Students should be able to work with a statement and its converse.

8. Students should solve problems using properties of figures and interrelationships.

van Hiele Level 3 Formal Deduction

1. Students should identify what is given and what is to be proved in a problem.

2. Students should identify information implied by a figure or by given information by analyzing a given figure and/or Writing an if .... then.... statement from given information.

3. Students should demonstrate an understanding of the meaning of undefined term, postulate, theorem, definition, etc.

4. Students should demonstrate an understanding of necessary and sufficient conditions.

5. Students should prove formally the relationships developed in vH 2.

6. Students should prove unfamiliar relationships.

7. Students should compare different proofs of a theorem, i.e., the Pythagorean Theorem.

8. Students should use a variety of proof techniques, such as, synthetic, transformations, coordinates, and vectors.

9. Students should develop strategies for proof.

10. Students should solve problems using both deductive and inductive thinking, and basic logic.

In order to be successful in geometry, students must be able to do-as defined below-critical thinking, mathematical reasoning, deductive reasoning, and inductive reasoning. All this and logic too from 15, 16, and 17 year olds!

Research in geometry reasoning has shown that growth in the ability to recognize the invalid conditional inference patterns is very slow until about age 16 or grade 10. Obviously, since adolescents mature at different rates some students in geometry classes will not have this ability at all. Although some students find it hard to acknowledge the validity of the contrapositive, it is, in general the easiest logic inference pattern for them to use correctly. The inverse is the next easiest and the converse is the most difficult. Many have problems with If- then statements because they interpret them as being the same as If and only if. Almost all have trouble dealing with negations in conditional statements. For a variety of reasons, students frequently have a hard time accepting the absoluteness of a theorem or a definition.

Inductive errors stem from making inferences from a very limited number of samples and not subjecting the inductive hypothesis to sufficient tests for invalidity. Students are often reluctant to let an inductive hypothesis go once it is formed, even when they can find no supporting evidence. They frequently use intuitive rules that resemble formal procedures. These may be appropriate but are often insufficient for proof.

Errors in reasoning are often caused by:

1. altering, adding to, or ignoring items from the premise,

2. letting factual content override inference patterns, o letting traditional patterns of everyday language override logic,

3. language difficulties, number and location of negations, sentence and word length, cognitive overload, and

4. an inability to accept the hypothetical.

Teachers can help students avoid some of these problems by:

1. regular use of If-then statements in class discussion, assignments and on tests,

2. paying particular attention to vocabulary, especially those words that may have very different meanings in everyday discourse,

3. emphasizing the everyday application as well as the mathematical meaning of words and phrases such as all, no, some, at least, not more than, and

4. stressing that definitions mean exactly what they say, no more, no less.

A 1983 CBSM report recommends, in part, "We recommend that classes work through short sequences of rigorously developed material, playing down column proofs, which mathematicians do not use. These proof sequences should be preceded by some study of logic itself. Important theorems not proved can still be explained and given plausibility arguments, and problems involving them can be assigned. The time which becomes available because proofs are de-emphasized can be devoted to study of algebraic methods in geometry, analytical geometry and vector algebra, especially in three dimensions. Work in three dimensions is essential if one is to develop any pictorial sense of relations between many variables, and handling many variables is essential if one is to model phenomena realistically.

There is much room for using computers in geometry. The power of graphics packages makes it much easier for students to get a visual sense of geometric concepts and transformations. . . ."

The last statement is a reminder that visualization paired with reasoning can be a powerful mathematical tool.

Reasoning Definitions:

Critical thinking is a process of effectively using thinking skills to help one make, evaluate, and apply decisions about what to believe or do.

Mathematical reasoning is a part of mathematical thinking that involves forming generalizations and drawing valid conclusions about ideas and how they are related.

Deductive reasoning is a mathematical reasoning process in which valid inference patterns are used to draw conclusions from premises.

Inductive reasoning is a mathematical reasoning process in which information about some members of a set is used to form a generalization about other or all members of the set.

References

Tommy Dreyfus, Nubit Hodas, "Euclid May Stay-and Even Be Taught" Learning and Teaching Geometry K-12, Mary Lindquist & Albert Shulte, editors, NCTM 1987.

Mary C. Crowley, "The van Hiele Model of the Development of Geometric Thought" Learning and Teaching Geometry K-12, Mary Lindquist & Albert Shulte, editors, NCTM 1987.

Philip Oppenheimer, "Assessing Readiness for Tenth Grade Geometry" paper for Woodrow Wilson Geometry Institute 1993

Phares G. O'Daffer and Bruce A. Thornquist, "Critical Thinking, Mathematical Reasoning, and Proof" Research Ideas for the Classroom: High School, Patricia S. Wilson, editor, MacMillan Publishing Co., 1993.

Phares G. O'Daffer and Bruce A. Thornquist, "Critical Thinking, Mathematical Reasoning, and Proof" Research Ideas for the Classroom: High School, Patricia S. Wilson, editor, MacMillian Publishing Co., 1993.