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Putting the Pieces Together

What does it mean to think mathematically?  

It means using math vocabulary, language and symbols to describe or interpret mathematical concepts, procedures and to discover relationships among ideas.  Therefore when a student problem solves, they use previous knowledge, skills, and understanding of concepts to solve a problem.  This process might include formulating problems, applying a variety of strategies, or interpreting results.

What can we do to help our students become better mathematical thinkers?  

We can teach and model problem solving strategies.  We can remember and plan our lessons to involve the three stages of conceptual development: concrete, pictorial, and abstract. We can have the students talk or write about how they got an answer either with the class or with a partner.  We can use writing in the mathematics classroom (such as math journals) to allow the students to practice expository writing and show their understanding.  We can exhibit math word walls and have the students use the glossary in their book to write and define terms. 

We can also create a positive and safe classroom atmosphere for problem solving... 

By being enthusiastic and allowing the students to take risks without consequences.  By emphasizing the process as well as the answer, the students may be willing to try unconventional or different ways to solve the problem.  I always tell my students that there isn't just one right way to get an answer which surprises many of them.  In fact, this is one of the posters that hangs in my classroom.

As math teachers, let's continue to emphasize problem solving so that all students will acquire confidence in using mathematics meaningfully. But most of all, let's have fun while we are doing it!

$7.75 on TPT
If you are interested in having a math dictionary for your students, check out A Simple Math Dictionary. It is a 30 page dictionary which uses easy and clear definitions as well as formulas and examples so that students can learn and understand new math words without difficulty or cumbersome language. Most definitions include diagrams and/or illustrations for the visual learner. Over 300 common math terms are organized alphabetically for quick reference.

Problem Solving Top Ten List #3

The first step in the problem solving process is to correctly identify the problem. The next is to explore, identify, and choose a problem solving strategy. The third step in the process is to correctly implement the strategy chosen. But what happens when a student swears his/her strategy isn’t working? Usually, they have a problem solving habit that I might categorize as “malfunctioning” (not effective). Let’s look at the worst problem solving habits that some of your students just might have.

  1. Trying to do it all in your head; not writing anything down.
  2. Arbitrarily choosing a strategy.
  3. Staying with a strategy when it is not working.
  4. Giving up on a strategy too early.
  5. Getting fixated on a single strategy and trying to use it for everything.
  6. Not asking yourself: “Does this make sense?”
  7. Being afraid to ask for help.
  8. Not checking your answer.
  9. Not noticing patterns.
  10. Going through the motions instead of thinking.
The student should be asking...
  1. Have I shown an adequate amount of work to demonstrate what strategy I have used?
  2. Is there more than one strategy which I could use to solve this problem?
  3. Does choosing one strategy over another make the implementation easier?
  4. Does the strategy I have chosen use any tables, charts, formulas or properties I need to review
  5. What technology or manipulatives could I use to help me solve the problem?
As mathematics teachers, what can we do in the
classroom to guide this kind of thinking?

A Go Figure Debut for a Clip Artist Who Is New!

Chermaine is the first Teachers Pay Teachers seller I have featured that lives outside of the United States. She is Malaysian; however, she resides in Singapore. For five years, she has taught science in secondary school, specializing in chemistry and physics. She currently has left school to start her own science and math learning center. She is also a local band vocalist, performing for charity shows regularly; so, singing is her forte!

In her store, The Cher Room, are 116 resources, five of which are free. Most of her items are clip art, with only about 5-6 items being learning aid materials. Presently, she is fully focusing on clip art (I love her work!) She already has quite a full assortment for biology and physics while chemistry, earth science and math sets are currently in the process of expansion and development.

To show her sincerity and determination in making Math Clip Art, her featured free item is a pizza fraction set that contains 12 graphics - six that are colored and six that are line art. Included in this set:
Free Item
  • Full Pizza
  • Half 
  • One-quarter
  • Three-quarters
  • One-eighth
  • Seven-eighths

Mega Bundle
One of her most popular sets is a non-science home set that Chermaine has chosen to be featured as her paid product. It is a mega bundle in which you get the ENTIRE HOME SERIES clip art for a total of 12 rooms/areas. Bundles are always great money savers on TPT!  She also sells this clip art in smaller bundles of four.

As a math instructor, I particularly love her perimeter and area clip art because she includes nine real life applications which are always hard to find. She also has 2-D and 3-D shape clip art, volume clip art and even clip art for the ever popular Pythagorean Theorem...something that is very difficult to obtain!

So if you are creating items for your science or math classroom or you just want quality clip art at a reasonable price, head on over to Chermaine's store where you will find a wealth of materials!

Effective 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 ten 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 research 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.

ConcreteBring 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 multi-link cubes.

PictorialGive 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.

AbstractHave 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.

If you want more examples and suggestions about using this model to write lesson plans, click on the link below the illustrationIt will take you to Graphing without Paper or Pencil in which is appropriate for grades K-5 and is based on the Conceptual Model of Development: concrete to pictorial to abstract.