Higher Level Thinking Questions and van Hiele Levels

1.  What level(s) of Bloom’s Taxonomy most closely align with the level(s) of the van Hiele Model? Justify your thinking.

Level 0:  Concrete is most similar to Bloom’s knowledge.  The student can identify and compare geometric shapes.  Comparing shapes at this level is not necessarily at Bloom’s understanding level.  I think it is more of a visual comparison.

Level 1:  Analysis is similar to Bloom’s understanding level.  This is where the student could examine the shape and classify it by its properties.  Analysis could also be compared to Bloom’s apply level.  It’s at this point that students can apply formulas and compute areas.  They could also look at different shapes and compare their perimeters and area.

Level 2:  Informal deduction corresponds to Bloom’s analysis level.  Students begin to analyze and differentiate properties of shapes and apply them to one another.

Level 3:  Deduction must be at Bloom’s synthesis level.  Using deduction requires a thorough understanding of geometric principles and combining them in such a way to formulate theorems.

Level 4:  Rigor is at the highest level of thinking so I would compare it to Bloom’s evaluation level which is also is the top of the scale. 

2.   Answer the question asked in the article: “How can you use the van Hiele levels to help students learn mathematics?”

I think van Hiele’s levels relate closely to Vygotsky’s zone of proximal learning.  Students can not learn something unless they are ready for that stage of learning.  van Hiele’s levels demonstrate this as well.  We can not ask students to perform at a level higher than they are currently on.  As teachers, we must evaluate our students and where they fall within this framework.  Activities must be presented to them at their levels.  As the previous article showed, one lesson may cover more than one level.  Students will simply think at different levels and be able to apply concepts in a variety of ways. 

3.  Review the “Guiding Questions for Group Discussion.” Using the Questioning Cue Words from Module 4 < link to table from previous module>, develop additional questions that you could ask students if you were to use this lesson in your classroom. Use the Bloom’s Question worksheet you used in Module 4.

Use the tiles to configure two different shapes with the same perimeter.  Do you think that their areas will be equivalent as well?  Test your prediction.  Compare the two areas? 

Construct a 3, 4, and 5 triangle.  Using each of the sides construct squares.  Evaluate a² + b² = c².  Now using the same 3, 4, and 5 triangle construct triangles from each of the sides.  Predict whether the same holds true?  Does a(triangle) + b(triangle) = c(triangle).  Discuss why or why not?

Pythagorean Puzzles

Puzzle #1

 I spent a lot of time working on the first figure.  After a many attempts, I found out that it would work if I rotated the square.  Once, I did this, then the other pieces fell into place.  The second one was fairly easy.  I placed the square in the upper right corner, and I could then see two rectangles that I needed to make using the 4 triangle pieces.

 Puzzle #2

The first one came to me easily.  I aligned the triangles around the four sides and rotated the square to fit in the middle.  The second figure was a little more difficult, but it didn’t take long to figure out that I needed to create two rectangle with the 4 triangle pieces leaving two square areas.  A small one and a larger one.

      This type of exercise would be useful to students because they could compare similar shapes and congruent shapes.  They will see how different shapes fit together to form new shapes.  Students can discover that the area of one particular shape might be equivalent to the area of a different shape.  This also allows the students to think outside of the box.  They will learn to look at the tans from different aspects by rotating and flipping them.  This exercise is also a great problem solving exercise. 

     I’m not sure that the hands-on manipulatives or the virtual manipulatives are better than the other one.  However, the virtual manipulatives might be easier to work with since they will stay in place and not move on you.  When demonstrating to the class, the virtual manipulatives would work better since you can link your computer screen to an overhead projector.  On the other hand, I don’t know how many math classrooms have more than a couple of computers in them.  It might be easier just to use the hands-on manipulatives.

Tangrams

     Notice that the perfect squares formed along leg A and leg B of the small red triangle required 2 of the small triangular units to form each one of them.  Notice that the perfect square formed along the hypotenuse required 4 triangular units.  This is twice as many as it took to form the squares on legs A & B.

     On the legs of the yellow, mid-size triangle, it took 4 triangular units to form each of the squares.  On the hypotenuse of the mid-size triangle it took 8 of the triangular pieces to form a perfect square.  Although using a different size right triangle, it still took twice as many pieces to form the square on the hypotenuse as it did on either leg.

     On the legs of the large, blue triangle, it takes 8 triangular pieces to form a square.  On the hypotenuse of the large triangle, I used all 7 pieces of the tangram to form a square.  Based upon data from the small and medium triangle, we can assume that it would take 16 small triangular pieces to form a square along the hypotenuse.

 

 

     In all right triangles, we notice that it takes the square of A + the square of B to = the square of C(hypotenuse).  Hence, A² + B² = C ².  Students have just discovered the Pythagorean Theorem!
     This is a great introduction to square roots.  This hands on activity walks the students step-by-step through the process of really understanding square roots. 
     I would present this activity as it was presented to us.  I’d have them follow the 6 steps.  Students would be placed in pairs and each pair could work together to unravel the Pythagorean Theorem.  We would use the Tangram Blackline Master, make copies, color and cut out the shapes.  I would use heavyweight paper so the students wouldn’t get as frustrated as I did when placing the pieces together.  They kept moving on me!