Standard 4, instructional strategies. A teacher must understand and use a variety of instructional strategies to encourage student development of critical thinking, problem solving, and performance skills.

The math lesson below provides details about a public school Montessori math classroom, and follows a fifth grade boy who struggles with math.  The lesson concludes with an in-depth reflection about the teaching of the lesson.

Math Reflection Paper

Part 1

The 4^th/5^th Grade Classroom Environment and Students

Exploring the Context for Learning

The School

This public Montessori magnet school serves an population of city children.  The school serves half white and half children of color.  Half of the students receive free or reduced lunch.  A quarter of the students in the school are African American students whose home language is not English.  About a quarter of the student body is composed of East African immigrant children.  One quarter of the children in the school receive ELL instruction.  Eight percent of children in the school receive special education services. 

Observations of Teaching

The Classroom

This is a packed grade 4-5 classroom made up of 34 students.  Montessori materials crowd bookshelves that line every wall.  The teacher conducts small group lessons at a low Montessori platform while the rest of the students work independently on Montessori work.  The use of Montessori math materials for the many English language learners helps them acquire math ideas more easily because much of the work is done with physical objects and Montessori materials that help to convey concepts in a hands-on fashion.  The small group milieu allows for ample discussion time of children’s math thought processes and questions.  At this public Montessori school some classrooms are almost pure Montessori, and some are not.  This class is a traditional Montessori room.  The teacher is devoted to the Montessori philosophy.  But to meet the 4^th and 5^th grade Minnesota math standards and help children perform on the state MCA test she sometimes supplements the curriculum with Math Investigations and Everyday Math material.

There is a two-hour block devoted to Montessori math daily.  As a non-certified Montessori teacher, I cannot teach Montessori lessons, but I can learn the lesson and conduct follow-up assistance and guidance with the students.  I can also augment Montessori lessons with some material from Everyday Math and Math Investigations as deemed appropriate by the teacher.

Problem Solving

The Montessori math curriculum for 4^th and 5^th graders contains many aspects of the problem-solving philosophy.  Students focus on ideas and sense making as they explore Montessori metal and wooden geometric shapes and extend past knowledge when they apply it to new lessons they are learning.  Because students work independently at their own pace on geometry lessons, and because many geometry assignments have an open-ended discovery format, students develop confidence that they can do math and that it can make sense on their terms.  Student-teacher small group lessons provide time for the teacher to double-check student thinking and make corrections.  Students’ context for much of their math learning—the Montessori geometric inset materials—is familiar and accessible to the students, and provides points of entry for different types of students.  The Montessori method requires students to reason, make connections and physical representations, communicate their findings to other students, and come up with solutions to problems.

One geometry lesson that I observed for 4^th grade students was entitled, “The Equivalency of a Rhombus to a Rectangle.”  (See Appendix A notes.)  In this work a red rectangle and a red rhombus are inset next to each other in a flat green metal frame.  The red rhombus is cut up and contains an equilateral triangle and two scalene right triangles.  In the lesson the teacher guided the students to ask questions and make observations about the relationship between the two figures.  Students were invited to take turns moving the shapes around and comparing how they fit into the rectangle and rhombus.  Later, students were to assemble a booklet with cut-out paper rectangles and rhombuses, and their written observations about the similarities of the two shapes and the relationship between the two shapes. 

Student observations and findings included:

“The small equilateral triangle fits into the rectangle in the middle.  The two scalene right triangles can be fit in on either side of the equilateral triangle.”

“Hey!  The space inside both shapes is the same!”

(Teacher) “What do we call that inside space?”

“The AREA!”   

(Teacher)  “The area of the rhombus is equivalent to the area of the rectangle.”

“The base of the triangle is the same as the base of the rectangle.”

“The height of the triangle is the same as the height of the rhombus is the same as the height of the rectangle.”

“The diagonal of the triangle bisects the rhombus in half.”

(Teacher)  “Talk to me more about the bisector.”

“It has to be a straight line!  It has to divide the rhombus in half, into two equal halves!”

In “The Equivalency of a Rhombus to a Rectangle” student-teacher discussions I witnessed the beginnings of Level 3 Deductive reasoning according to Van Hiele’s geometric thought construct.  The 4^th graders were going beyond merely the properties of shapes.  However, they were at a level of intuition primarily, but beginning to develop some logical arguments to support and prove their observations, e.g., (in the rhombus and rectangle) “The height is the same, and the base is the same.  Hey!  They have the same area even though they look different!”  The 4^th grade students in this classroom are engaged in discovering geometric relationships that will lead to later theorems and proofs.

 

Part 2, Math Instruction

Polygon Capture Lesson Planning Outline

Student description	Simon is a 4th grader who struggles in school.  He does not have an IEP or special education designation.  Simon benefits from a scaffolded lesson that provides plenty of time to review prior knowledge and warm-up with practice.
Purpose	The purpose of the game is to encourage the student to look at the relationships between geometric properties.
Objectives Student will be able to:	·      Describe, classify, and understand relationships among two-dimensional objects using their defining properties; ·      Make beginning “If-then” statements about polygons.  For example, “If a quadrilateral has these _______ properties, then it must be a __________.”  (Level 2, Informal Deduction) (Van de Walle et al., 2010).
NCTM Standards	·      Classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes; ·      Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions;
MN Standards	·      4.3.1.1: Describe, classify and sketch triangles, including equilateral, right, obtuse and acute angles. ·      4.3.1.2: Describe, classify and draw quadrilaterals, including squares, rectangles, trapezoids, rhombuses, parallelograms and kites.  Recognize quadrilaterals in various contexts.
Materials	·      Everyday Mathematics Student Reference Book Grade 5 Polygon Capture directions, p. 328, Appendix B ·      Polygon Capture pieces ·      Polygon Capture property cards ·      “Geometry and Constructions” reference section from Everyday Mathematics ·      Scissors
Before Phase	With Simon: ·      Look over the definitions of “polygons” in the Geometry and Constructions reference section of Everyday Mathematics ·      Discuss convex, nonconvex, and regular polygons. ·      Go over prefixes that are added to –gon that tell the number of sides the polygon has, e.g., hexagon.  ·      Refer to the tree diagram in Everyday Mathematics that shows how different Quadrangles are related. ·      Talk about quadrangles that are and are not parallelograms. ·      Assess Simon’s familiarity with the terms parallel, perpendicular, acute, obtuse, right angles.  Ask him to illustrate these terms. ·      Cut out Polygon Capture pieces and Polygon Capture Property cards. ·      Match property cards with polygon capture pieces as a “warm-up” to tomorrow’s polygon capture game.  ·      As Simon matches cards to shapes, ask him why he thinks his solution is correct and reasonable.  Ask him why he did it that way (Van De Walle et al., 2010, p. 60).
During Phase	·      One of the main goals of this phase is to again probe Simon’s thinking and asking him to explain what he is doing as he plays polygon capture. ·      Look for Level I Analysis—an ability to talk about all rectangles, for example. ·      Look for and encourage Level II Informal Deduction—if a shape has sides that are not parallel then it is not in the parallelogram family, for example.
After Phase	·      Show Simon his observations that you have written down. ·      Ask him what he notices about his thinking. ·      Refer back to the tree diagram of Quadrilaterals in order to discuss relationships between shapes.
Assessment	·      Note Simon’s level of thinking.  Is he at Level I or Level II?

Reflection About Teaching Math Lesson

Before Phase

I explained to Simon that we would eventually be playing the polygon capture game and showed him the game.  I said that we would first do some geometry review.  It was very helpful to review with Simon key 4^th grade geometry math terms: polygon; vertex; parallel; perpendicular; obtuse, acute and right angles; convex, concave and regular polygons; and the prefixes that are often attached to polygon shapes (e.g., hexagon, octagon).  This was done using clear and valuable Everyday Math reference material (See Appendix C) and drawings that Simon and I made.

During our discussion Simon re-discovered that a true polygon cannot have curved sides or open sides, and that the points have to meet (vertices).  He noted that, “Parallel lines never meet.  They keep going.”  He drew parallel lines.  He drew and identified acute, obtuse and right angles.  He remembered that a quadrilateral has four sides.  He enjoyed pointing out that a hexagon was related to a hexagonal prism, and this led to a deeper discussion about the square area of flat shapes (2-D), and the cubic area of three-dimensional shapes.  As we held this discussion Simon helped me cut out the polygon capture pieces.     

The next day we continued our warm-up discussion.  We examined an Everyday Mathematics tree diagram of Quadrangles (Quadrilaterals) and discussed these terms as referring to shapes with four angles and these same shapes have four sides.  “How are polygons and quadrilaterals similar and different?”  I asked Simon.  He struggled but eventually remembered that, “A polygon can have many sides, but a quadrilateral has just four sides.”  I urged him with prompts to make a similar if-then statement, and he eventually came up with, “If a polygon has four sides, then it is a quadrilateral.”  He noted that, “A square and a rhombus are similar because they have two sets of parallel lines, and they are quadrilaterals.  They are different because their angles are different.”  At first it was hard for him to see that they also have in common four sides that are all the same length.     

During Phase and After Phase

I explained the Polygon Capture game to Simon and asked him if he would like to play it.  He declined, saying that he would rather match the different cards with the shapes.  I asked him to choose to begin with either side cards or angle cards, and he chose angle cards.  He chose to work with a limited number of shape cards (about nine).  He chose the card “At least one angle greater than 90 degrees.”  He was proud to immediately choose a hexagon and declare, “It has obtuse angles that are bigger than 90 degrees.”  He chose the card, “All angles are right angles,” and matched this card with two rectangles and two squares.  I drew a scalene right triangle and asked if this could be matched with the card, and he said, “No.  It has only one right angle.  It has only three sides.  If it had another 90 degree angle it would start turning into a square.”  He was proud of his statement and his deeper understanding.  Our discussion of right angles and rectangles and squares led to a further discussion and examination of the tree diagram of Quadrangles and the parallelograms, rectangles, rhombuses and squares.  We attempted an if-then statement about the square and its relationship to rhombuses, and its relationship with rectangles, but this was difficult.  He noted again, “The square and rhombus are related because they have four sides that are all the same length!”  Then Simon chose the card, “Only one pair of sides parallel” and matched it with the trapezoid.  With prompts and references to the Quadrangle tree diagram he was able to say, “If a shape has only two pair of sides parallel like a trapezoid, it is not in the parallelogram group.”                    

Levels of Geometric Thought

Simon is definitely at Level 1 Analysis as he is usually able to list all the properties of different shapes, but he has a harder time consistently seeing that shapes can be subclasses of one another (Level 2, Informal Deduction).

Simon made several “if-then” statements with prompts.  He is moving toward Level 2 thinking.

 Lesson Plan Changes

It seemed constructive to drop the game-playing plan and opt for the card matching idea that Simon suggested.  The more Simon is able to choose math activities, the more his math anxiety goes down.  The more he is able to discuss his thinking, the more he appears to learn.  If I had time I would extend this lesson to playing the Polygon Capture game eventually, preferably including another student, too. 

What Have I Learned?

Students benefit from having lessons broken down into smaller and smaller pieces.  I should have had an assortment of pattern blocks so that Simon could physically play with and handle these pleasing shapes in addition to the paper shapes.  For whole class instruction after an all-class lesson I would break the class into smaller groups and have differentiated small group activities prepared that would allow for discussion, comparison, give-and-take.  I would spend most of my time with the group that needed the most help.  I would rate the groups for their on-task behavior and quiet voices and reward them using our class reward system (e.g., Teddy bear tickets that kids can redeem).  I am feeling more and more comfortable with geometry after taking an excellent geometry and measurement course, in addition to this course.  I can see that it is vital that the teacher possess a deep understanding of math content so that she can see what her students are trying to get at, and is able to lead them forward based on their glimmers of understanding.     

Bibliography

Van De Walle, J., Karp, K., & Bay-Williams, J. (2010).  Elementary and middle school mathematics, 7^th ed.  Boston: Allyn and Bacon.