Levels of Geometric Thinking The van Hiele Levels Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Wednesday, July 20, 2016 8:00 a.m. – 10:00 a.m.

Learning Intention We are learning... one research-based model (van Hiele levels) for the development of students’ understanding of shapes and their attributes. We will be successful when... We can describe the van Hiele levels of students’ geometric thinking about 2-dimensional shapes and their attributes. We can explain some implications for instruction to move students along the trajectory of van Hiele levels.

Learning Trajectories Story problem structures Levels of thinking when solving story problems Ratios and proportional thinking: OGAP trajectory Developmental levels of Geometric thinking… van Hiele Levels

What is a rectangle? What is an appropriate answer to this question… 1. Kindergarten student 2. Grade 3 student 3. Grade 6 student 4. An adult

How do students progress in developing geometric reasoning? Read: pages 310 – 311 of the article Begin with Play by Pierre van Hiele. Stop at the end of the section, “Levels of Geometric Thinking.” As a trio, talk through each of the levels described in the article. Considering the three levels discussed in this article, where would you place the majority of the lessons that you teach?

Clarifying van Hiele Levels Visual Level Recognize figures as total entities, but do not recognize properties. Descriptive Level/Analytic Identify properties of figures and see figures as a class of shapes defined by their properties. Informal Deduction [Analytic] Formulate generalizations about relationships among properties of shapes; Develop informal explanations. The analytic level is in the Progressions (grey box p.3) There has been research done since they have done their initial research

Formal Deduction Understand the significance of deduction as a way of establishing geometric theory within an axiom system. See interrelationship and role of undefined terms, axioms, definitions, theorems and formal proof. See possibility of developing a proof in more than one way. Rigor [Abstract] Compare different axiom systems (e.g., non-Euclidean geometry). Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.

Keeping it Simple Visual – Seeing shapes as a whole: “it’s a rectangle because it looks like a door.” Descriptive – Seeing attributes of shapes: “a rectangle has 4 sides, 4 right angles, 2 equal diagonals…” Informal Deduction – Seeing connections between attributes: “a rectangle is a quadrilateral with 4 right angles.”

Visual Level “See shapes as a whole”

“Describe attributes of a class of shapes” Descriptive Level “Describe attributes of a class of shapes”

Informal Deduction Level “See connections or relationships among attributes of shapes”

Results from your students!! Informal Deduction Results from your students!! Descriptive 902 students took the van Hiele test in May 2016 Grade 6: 200 students Grade 7: 368 students Grade 8: 334 students Visual Precognition Predict: What percent of students placed at each level? Precognition  Visual  Descriptive  Informal Deduction

van Hiele Results: Your Middle School Students (n=902) Grade 6, n=200 Grade 7, n=368 Grade 8, n=334 Criteria: 3 correct per level

van Hiele Results: Your Middle School Students (n=902) Grade 6, n=200 Grade 7, n=368 Grade 8, n=334 Criteria: 3 correct per level

Recall from Day 1 … As numerous studies have shown, U.S. elementary and middle school students are failing to develop an adequate understanding of geometric concepts, geometric reasoning and geometric problem solving. Furthermore, a majority are unprepared to study the more sophisticated geometric concepts and proof in high school. (Clement and Battista, 1992)

Alarming Information! Almost 40% of students finish High School Geometry below Level 2 (Informal Deduction). In fact, because many students have not developed Level 3 (Formal Deduction) thought processes, they may not benefit from additional work in formal Geometry because their knowledge and the information presented will be organized differently. Putting Essential Understanding of Geometry and Measurement into Practice, 3-5, p. 57. NCTM.

Descriptive Level Informal Deduction Level Grade 6, n=200 Grade 7, n=368 Grade 8, n=334 Criteria: 3 correct per level

van Hiele Results: Middle School Students (n=902) Level Grade 6 Grade 7 Grade 8 Precognition 20% (39) 15% (57) 9% (31) Visual 66% (131) 64% (234) 63% (209) Descriptive 7% (14) 14% (50) 14% (47) Informal Deduction 8% (16) 7% (27) Number of Students 200 368 334 Criteria: 3 correct per level

Complete the Tricky Triangle Task Which question do you think will be challenging for students? What language will be expected from student responses? How might this task help you gain further insight into a students’ van Hiele level of geometric thinking? How do you see the van Hiele levels in this task? How might this task help you determine a students’ van Hiele level of geometric thinking?

Student Interview

What is a Triangle? In your notebook… Write a definition of a triangle. Draw a 4-square on your paper. 1. 2. 3. 4.

What is different about the triangles? What is similar? 1. Draw a triangle. 2. Draw another triangle that is different from the triangle you just drew. 3. Draw a third triangle that, again, is different than the previous triangles. 4. Finally, draw a fourth triangle that is different than any of the previous triangles. What is different about the triangles? What is similar?

“I believe that development is more dependent on instruction than on age or biological maturation and that types of instructional experiences can foster, or impede, development.” Pierre M. van Hiele

Is the figure a triangle or not a triangle? Explain. The Triangle Task Is the figure a triangle or not a triangle? Explain. B A C D F E G H I M J K L

Triangle Task Results, Grades 1-8, Percent Correct Core Math Partnership, May 2016 (n = 535 students; 28 classes) H C I J

Figure C Figure I Grade n 1 15 2 78 3 102 4 93 5 95 6 53 7 99 Triangle Task Results, Grades 1-8, Percent Correct Core Math Partnership, May 2016

Figure J Figure H Grade n 1 15 2 78 3 102 4 93 5 95 6 53 7 99 Triangle Task Results, Grades 1-8, Percent Correct Core Math Partnership, May 2016

Review your Tricky Triangle Student Work Reflect on your classroom expectations – What constitutes a quality geometric explanation? Step 1: What trends do you see in the student work? student understandings student misconceptions Taking turns, present your observations. Then discuss: What common themes emerged?

Trends in Student Language/Descriptions Step 2: Review student descriptions. As you review your student work consider… What pleases you? What concerns you? In what ways does the work reflect or not reflect your expectations for a quality geometric explanation? Share out what you noticed with your table group. Step 3: In what way do students’ descriptions become more sophisticated as students grow older?

Reflection Reflecting on the triangle task results, what are some ideas and understandings, in “student friendly terms,” that you hope your students will be articulating and owning by the end of next school year?

Figure H It has one little slant. (2) Because it is a long top. (2) Because it doesn’t have straight sides (3). It is not because it does not have even sides (4). Lines are different (4). Because it’s too long (7). Long narrow shape (7). It is not a triangle because it does not have 3 equal sides (7). Not even sides (7). Sides need to be the same sizes (7). Not a triangle because it does not have 3 even sides (7).

Figure J Too too skinny. (2) It’s too long. (2) It points to the side not straight up. (2) Because it is not fat (3). It does not look like a triangle (3). Does not have even sides (4). It is too long (4). Not equal sides (4). Too skinny (4). It’s too long (5). It is not a triangle because it does the sides are not equal (7). The shape is wrong (7). Sides are too big and the bottom too short (7). Not a triangle because of uneven sides (7).

Triangle Ideas Students are Expected to Own and Communicate A triangle is still a triangle no matter how it is sitting. It doesn’t have to be flat on the bottom; it is okay if it points downward. A triangle has 3 sides; the sides do not have to be equal; sides can be different lengths. Triangles must have straight lines that touch each other and the lines don’t stick out. Corners can’t be curved. Triangles can be many sizes and point in different directions. Triangles can be skinny or fat. Triangles can be tall or short. Triangles can be straight or slanted.

Learning Intention We are learning... one research-based model (van Hielel levels) for the development of students’ understanding of shapes and their attributes. We will be successful when... We can describe the van Hiele levels of students’ geometric thinking about 2-dimensional shapes and their attributes. We can explain some implications for instruction to move students along the trajectory of van Hiele levels.

PRR: van Hiele Levels Considering the work we’ve done today and the information from the van Hiele article. How do we support students with a diverse set of experiences in doing meaningful geometric work that moves them from their current van Hiele level of reasoning toward Informal Deduction and beyond?

Core Mathematics Partnership Project Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.