Tuesday, December 13, 2011

Lesson Plans and Research

We often hear of research based strategies and how to use them in our classrooms.  Having worked at two colleges in the past eight years, I have discovered that some who are doing this research have never been in a classroom or taught anyone under the age of 18!  (Sad but True)  Then there are others who truly understand teaching, have done it, and want to make it more effective for everyone. That's the kind of research I am anxious to use.  I came across the Conceptual Development Model while teaching a math methods class to future teachers.  It was one of the first researched models that I knew would work. 

The Conceptual Development Model involves three stages of learning – 1) concrete or manipulative, 2) pictorial, and 3) the abstract.  The concrete stage involves using hands-on teaching which might involve the use of math manipulatives or real items.  Next, the pictorial stage utilizes pictures to represent the real objects or manipulatives. A visual such as a graphic organizer would also fit in this stage.  Last, the abstract stage of development entails reading the textbook, using numbers to compute, solving formulas, etc.  Let's look at two classroom examples.


Example #1:  You are a first grade teacher who is doing an apple unit.  You decide to have the children graph the apples, sorting them by color.

Concrete:  Using a floor graph, the children use real apples to make the graph.

Pictorial:  The children have pictures of apples that they color and then put on the floor graph.

Abstract:  The children have colored circles which represent the apples.


Example #2:  You are a fifth grade teacher who wants to teach how to find the volume of a cube or rectangular solid.

Concrete: Bring a large box into the classroom, a box large enough for the children to climb inside, OR have the students build 3-D objects using multilink cubes.

Pictorial: Give the students pictures of 3D objects which are drawn but shows the cubes used to make the solid. Have the students count the cubes to determine the volume.

Abstract: Have students use the formula l x w x h to find volume.

Requiring my perspective teachers to think about this model and to use it when planning a math unit dramatically changed the quality of instruction which I observed in the classroom. 

Writing Lesson Plans
Now that I teach mathphobics on the college level, I am finding this model to be a crucial part of my planning.  Most of my students started math at the abstract level, "Open your books to page...." without any regard to the other two stages of development.  Using manipulatives and graphic organizers have changed my students' ability to learn math, and some have even ended the semester by saying, "I like math".  Maybe this is a model we should all consider implementing before we start back to school in January.

If you want more examples and suggetions about using this model to write lesson plans, click under the Conceptual Development model on your left. 

Wednesday, December 7, 2011

I'm Pro-Tractor!

Using a protractor is supposed to make measuring angles easy, but somehow some students still get the wrong answer when they measure. Here are a few teacher tips that might help.

1)  Make sure that each student has the SAME protractor.  (To avoid having many sizes and types, I purchase a classroom set in the fall when they are on sale.)  If each student's protractor is the same, you can teach using the overhead or an Elmo, and everyone can follow along without someone raising their hand to declare that their protractor doesn't look like that!  (Since the protractor is clear it works perfectly on the overhead. No special overhead protractor is necessary.)

2) Show how the protractor represents 1/2 of a circle.  When two are placed together with the holes aligned, they actually form a circle.

3) Talk about the two scales on the protractor, how they are different, and where they are located.  It's important that the students realize that when measuring to start at zero degrees and not at the bottom of the tool.  They need to understand that the bottom is actually a ruler. 


I use a couple of word abbreviations to help my students remember which scale to use.

4)  When the base ray of an angle is pointing to the right, I tell the students to remember RB which stands for Right Below.  This means they will use the bottom scale to measure. 

5) When the base ray of an angle is pointing to the left, I tell the students to remember LT which are the beginning and ending letters of LefT. This means they will use the top scale to measure the angle.

6) Of course the protractor has to be on the correct side.  It's amazing how many students try to measure when the protractor is backwards.  All the information is in reverse!

7)  Make sure the students line up the hole with the vertex point of the angle, aligning the line on the protractor that extends from the hole, with the base ray.  Even if they choose the correct scale, if the protractor is misaligned, the answer will be wrong.

8)  Realize that the tools the students use are massed produced, and to expect students to measure to the nearest degree is impossible.  To purchase accurate tools such as engineer uses would cost more than any of us are willing to spend!

Have fun as your class measures and constructs angles.  And don't forget what protractors really do.....They always vote "yes" to donate tractors to farmers!

If you would like supplementary materials for angles, check out these two products: Angles: Hands On Activities  or  Geometry Vocabulary Crossword Puzzle.