Spreadsheet
My Family Math Day experience and classroom opportunities to explore the utility of spreadsheets with school age student groups has convinced me that the spreadsheet is an important tool in developing an understanding of mathematics: both the in depth comprehension of quantity as well as the complexities of our number system are reflected quite rapidly in the instruction and the independent activities provided by spreadsheets. Where to begin is always an issue, just as it is in every classroom planning process.
Fifteen years ago, Dr. Robert Zenhausern and I participated in a Saturday program which was attended by a random population of students with at least one parent. We lead students and parents through familiar turf and then introduced new ideas to help them discover the utility of the spreadsheet. The experience was noteworthy because it confirmed that young students, as young as third grade, can develop greater awareness and comprehension of our number system.
From my perspective, start with an introduction that provides young students a feeling of control and success before challenging their knowledge. The challenge is how quickly to move from the known to the unknown: from my perspective, move to the new content as quickly as possible by starting with an obviously simple version based on familiar information, i.e. the lunchroom. This will assure the student that the array displayed is readily knowable: one chair for each student; one table for four students in four classes with twenty-four students each. -
1) I like personalizing word problems. The spreadsheet can do that.
2) I like using real events in math. The spreadsheet can do that, too.
3) I like using materials as a discovery technique. The spreadsheet can do that?
4) I like when kids translate their concrete manipulative objects into equations. The spreadsheet can do that.
Long ago, I trained at the UN School in NYC to use mathematic-manipulative objects to help students see the various patterns in "mathematics". A stack of three green rods is actually quite different from a stack of yellows or reds. Why Children quickly learn that each colored rod represents a different length. The reds equal two white cubes in length but in every other way it is exactly the same. The green is the same as three whites or one red plus two whites is the same as one green andnone white, etc, Using concrete-maniputives students discover multiplication, division and fractions as well as acquire new important skills of logic and reasoning. Take note, the spreadsheet can have the same impact.
From my experience, the spreadsheet is just as effective in specified situations as the utilization of concrete objects in introducing basic concepts. (It works best in developing intuitive understanding of large number patterns.) Once a student can transfer their mathematic skill building work from the manipulative to the digital representation process, he is apparently ready to be introduced to spreadsheets. Perhaps, first at the blackboard, but he can rapidly be moved to the computer where he can see what adding ten or subtracting ten from a column accomplishes.
Numerical - Visualization requires the process of forming an image in ones mind based on the representation of data. To be certain, students need to return frequently to the manipulative stage with ever increasing intervals of contact time as they mature in comprehension. While a few rare students are able to abandon the use of concrete-manipulatives early, teachers should expect that even some of the brightest students may always need to return to the concrete-objects in order to move forward on new concepts. It appears that growth in one does not always predict the growth in th other area. Therefore, concrete-manipulatives can be utilized throughout the instructional environment. (Note, we do not take maps out of the classroom once we have demonstrated comprehension of map skills.)
The need for concrete-manipulatives in tandem with two dimensional representational numbers, in fact, should be viewed as one would view the architect’s tools; he uses both a model and a blueprint. There are times that a model is better than a blueprint for visualizing the planning purposes. Yet, the actual data will eventually be recorded on a blueprint for safe keeping of data. Note that greater understanding of a spreadsheet through discovery and play can be initiated right in the primary classroom. Students participate in the accumulation of data and learn to read legends related to data. The Family Math Day of fifteen years ago was fun because we were able to introduce children and parents simultaneously to the simplicity of the mathematics process and in expanding understandings now that computers are ubiquitous. The kids giggled during my first spreadsheet class; that response reflected the increased awareness and discovery. In fact, they smiled broadly as they proceeded playfully with the basic additive process.
To gain insight into this process take note of their statements: What if we put your age and your father’s age on the left and a "plus" in the box at the top: that tells us your age for next year. (+10) How about your ages in ten years: (+21) how old would each of you be In 21 years?; (+100) in a hundred years? The young little third grade participants giggled as they saw their age ranges change. We also noticed that they teased their parents playfully, too. This is an important noticing for developing thinking skills. Can we capitalize on this new tool for greater insight not accuracy but increased thinking that reflects greater insight and an enhancement of the very joy of learning. Take note of our observations when we played with estimates related to cookies as well as stacks of boxes of raisins:
“Can You Guess My Secret Rule” is a game we have played with early learners. It is fun to play the game on a spreadsheet both with and without a computer. The early childhood psychology expert, Piaget, can provide the educator insight with the range of variables and the issues we need to develop even though the computer spreadsheet was not available for him to explore:
“How does the computer know my rule?” (This Is the question that I would consider after several opportunities to work with the details); “What is my secret?”
Note that learning to be a good observer of quantity is a slow process, but the spreadsheet, can pull data together to provide opportunities to learn important concepts that are hard to illustrate even with concrete-manipulatives:
-How big or tall is each building: We know that it has a column of ten windows or twenty hands going from top to bottom. What information is missing?
-How many cookies do we need if we give each child 2, 3, or 4?
-Use your bank books. Type in the amount that you have saved in the bank.
-If I give each of you ten more dollars and ten cents, just how much money
would your bank book indicate?
Clearly, the observer of students challenged in this way can demonstrate the utility of
employing the spreadsheet for increased comprehension and mathematic awareness.
Seymour Pappert called our attention to the utility of "LOGO" in developing increased comprehension of two dimensional objects; utilizing the spreadsheet that is currently on most computers can have the same impact for our students. Now that we have access to spreadsheets that respond electronically to our challenges, we can, in fact, each demonstrate the complexity of our number system as well as develop an important awareness of how to work with variables. In fact. the variables can be significantly larger than the ones that young students can manipulate using rods or other tactile-manipulatives. I challenge you to experiment. Please share your experiences as well as your challenges in developing increased comprehension of our number system.
