Our role is to continually be "helping students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics: creating, conjecturing, exploring, testing, and verifying" (Lester et al., 1994, p.154).
Problem solving strategies vary, but the focus is always on the "thinking" that is occurring. When we "coach" our students through different problem solving puzzles, we are allowing them to develop their mathematical thinking.
The students take charge of their learning and construct personal ideas about mathematics.
We can't expect students to solve every problem with numbers. In fact, that can be counter-productive.
Children need to experience how to deal with math problems on their own terms and in concrete ways before moving to the abstract.
Kids also need a chance to talk about their math discoveries and share insights with each other. That's when they refine and extend their thinking.
At the core of solving math problems is the necessity to have a deep foundation in number sense. From there, students begin to apply that understanding to basic word problems and often begin using a Model Drawing approach.
This is a pictorial representation that helps students internalize and visualize math.
These are other heuristics, or methods for solving a problem. Students
must be taught how to use a variety of methods to solve the same
problem. Only then can we see the results of their conceptual
understanding of applying mathematics.
1. Make A Pattern
When we tell elementary students to make a pattern, they often do not understand. Essentially, a pattern is a figure that repeats itself, such as a number or word. It is best to use manipulative to study pattern making to provide a concrete understanding.
two basic types of patterns: repeating and growing. A repeating pattern
has an identifiable core that repeats over and over, such as
ABACABACABAC. A growing pattern repeats a mathematical process that
makes the figure or number grow. Fibonacci numbers are a great example of growing patterns and are found throughout nature.
2. Make A Table
Often a table is used to help organize information found in Making a Pattern. A table helps to quantify a pattern so students can visualize the growing numbers.
Students note changes in stages and explain how the change in the table is occurring. This is done by identifying the repeating element.
You can use picture books to support teaching how to make a table. A fun one that children love is Two of Everything. Here is an elementary problem solving worksheet to go with it: Magic Pot table (courtesy of MathWire.com).
3. Make an Organized List
When we talk about making a list as part of problem solving strategies, we generally refer to a Tree Diagrom. This is important because part of efficient problem solving techniques is to not make random guesses.
tree diagram is useful to determine the number of given outcomes
(possibilities) from a set situation. You can tell the children that
the trunk of the tree is the problem. Each variable is a branch.
Little branches grow from larger branches, and to calculate the final
answer you only count the final number of little branches.
4. Guess and Check
This is one of the more difficult maths problem solving strategies for children to understand, but is often one of the first taught and used. They get the "Guess" part but have difficulty with the "Checking." Often they aren't sure what they are checking for.
always point out that a good way to check an answer is to re-read it and
see if it actually makes sense. Is the given answer actually possible?
The guess and check problem solving techniques helps
students to think logically, make predictions and use mathematical
equations. It all leads to deeper understanding.
5. Draw a Picture
Drawing a picture is a problem solving technique that has students make a visual representation of what the problem is. It really helps solidify concrete thinking.
Drawing a picture is the step between the visual and symbolic language of math. Pictures and diagrams are problem solving strategies that many students learn at the earliest stages of math development.
This is a good strategy as it is a way to
communicate mathematical thinking. With just a bit of encouragement,
most students will draw pictures.