Making Connections in Mathematics
For Eighth grade math exhibitions, students consider a topic in depth for an eight week period. During this time they select a topic, do extensive research, prepare an outline, write an essay, prepare a 15 to 20 minute talk, and create supporting slides and artifacts. Their classmates do peer edits of their essays. They do dry runs. Constructive, refining comments come from classmates and teachers. Everything gets improved, so that final essays can be written and final talks readied for the big day.
The big day was last Thursday. Eighth grade students presented their exhibitions to parents, teachers, administrators and middle school peers. Excellence was on display as students shared their work on Pascal's triangle, graphing techniques in 1-, 2-, 3- and more dimensions, order of operations, the Pythagorean Theorem, solving linear systems, and other topics.
An additional goal was to make connections. Making connections from topic to topic is a huge part of thinking mathematically. It is a habit we are working on at all points in the mathematics curriculum. And it is a powerful goal as students think deeply about their chosen topic.
For example, one student prepared an excellent exhibition on the order of operations. His first draft was substantive, interesting, and quite complete. But rather than stop, he chose to take on an additional challenge presented by his teachers: explain complications that arise when variables are introduced. He used this idea as a springboard to explain the distributive law and the commutative law. This was a powerful connection, rarely made by first year algebra students:
Another student solved a problem involving a system of linear equations. She showed a complete solution using linear combination:
Then she extended her work by showing an alternate solution. She graphed the two equations on the Cartesian plane, and showed how the two lines intersect at exactly the point she found using linear combination. This is a wonderful connection for a middle school student. The abstract, procedural method (linear combination) connects clearly and precisely to the concrete, visual graph of the two equations. This is the power of René Descartes' graphing inventions from the 1600's:
Another exhibition involved a seemingly impossible equation, requiring the student to think deeply about fractions, addition of fractions, multiplication of fractions, and infinite series:
This student showed how the equation connected to a beautiful, creative visual image. The image creates a mental picture that makes all the complicated aspects of the equation simple and clear:
Having shown her audience how it worked, she asked them to try it themselves using a square instead of a triangle:
Connections, connections! Mathematics abounds!
The big day was last Thursday. Eighth grade students presented their exhibitions to parents, teachers, administrators and middle school peers. Excellence was on display as students shared their work on Pascal's triangle, graphing techniques in 1-, 2-, 3- and more dimensions, order of operations, the Pythagorean Theorem, solving linear systems, and other topics.
An additional goal was to make connections. Making connections from topic to topic is a huge part of thinking mathematically. It is a habit we are working on at all points in the mathematics curriculum. And it is a powerful goal as students think deeply about their chosen topic.
For example, one student prepared an excellent exhibition on the order of operations. His first draft was substantive, interesting, and quite complete. But rather than stop, he chose to take on an additional challenge presented by his teachers: explain complications that arise when variables are introduced. He used this idea as a springboard to explain the distributive law and the commutative law. This was a powerful connection, rarely made by first year algebra students:
Another student solved a problem involving a system of linear equations. She showed a complete solution using linear combination:
Then she extended her work by showing an alternate solution. She graphed the two equations on the Cartesian plane, and showed how the two lines intersect at exactly the point she found using linear combination. This is a wonderful connection for a middle school student. The abstract, procedural method (linear combination) connects clearly and precisely to the concrete, visual graph of the two equations. This is the power of René Descartes' graphing inventions from the 1600's:
Another exhibition involved a seemingly impossible equation, requiring the student to think deeply about fractions, addition of fractions, multiplication of fractions, and infinite series:
This student showed how the equation connected to a beautiful, creative visual image. The image creates a mental picture that makes all the complicated aspects of the equation simple and clear:
Having shown her audience how it worked, she asked them to try it themselves using a square instead of a triangle:
Connections, connections! Mathematics abounds!