Forming Problems into Math
ENDEAVOR NOW is a dba. developing math-learning materials. The current project is FORMING PROBLEMS INTO MATH. Part II and part III are classroom ready and available free until 2005 for educational use.
Part II covers analyzing proportion problems. Conventional education approaches this with equation analysis: either as X/A = B/C, requiring algebra procedures or in arithmetic procedures based on the equation: AMOUNT SOUGHT = FRACTION OF AMOUNT KNOWN. Both analyses should be part of math education.
However, the more useful and effective approach is known-step analysis. People often arrive at it on their own and it comes up in schools ad hoc. It is the easiest approach to learn, as no algebra or fraction forming is required. The subject matter is more understandable, as it appears in sentence-line format free of algebra and/or reliance on skill and understanding with fractions. E.g. If eggs cost $.9, how many dozen can be bought with $3? Compare the analyses shown:
It is ignored that most grade 6-12 story-problem analyzing fits one or a combination of two clear analysis forms. Further, utilizing them follows cut and dried procedures, which are clearly formulated and explainable. Learning these and how to apply them is a far more effective approach than the conventional one of picking them up ad hoc through experiences related in other ways, as gifted persons do on their own. To many, analyzing is best left to the subject-area teachers, as it distracts from conventional math study.
The texts are organized around learning these skills singly with how to apply them. Then on parts of problems and finally on whole problems. This is in contrast to the common look at problem, analyze it and do the math for it, then go to the next. The next being one with similar language and/or math to be worked. In this text, the next is related by the analyzing for it. Likely the subject matter is different. Also the math expression the analysis produced can be different. The connection lies in the way the problems are analyzed. Working the math for them is not encouraged, as it creates distraction.
No platitude like “read . . . understand . . . ” Students learn what to read for, structurally unrelated to subject matter, both in text and graphics. They learn the format into which the structural parts fit and how to fit them, the one for proportion already shown and another for forming equations. Problems and parts of them illustrate in contrast to being tasks to be solved or tasks to be done . . .
Part III covers analyzing with equations: relationships leading to them, forming equations from these relationships, and sequencing them and/or the relationships. The first two fit cut and dried procedures. Then the analyzer’s mind is freed up to concentrate on the last, a process of try one, then try another . . . . The format leaves a clear record, eliminating backtracking. The smarter and more experienced the analyzer, the fewer tries needed and the faster they go. The format is flexible and prompts the analyzer what to read for. Problem: Two trains on the same track are 80 miles apart. The first leaves .5 hours earlier than the second at 40 mph. How fast should the second go to reach a side tract in 2/3 hr enabling the first to pass?
Note the unknown turns up, as relationships are identified and formed into math instead of requiring extra seeking and formulation. Gifted students tend to write the x = . . . . after writing the equation or at least after visualizing it. Fussing with this distracts from identifying the relationships. Writing the word equations could be dropped as students progress. But doing so loses part of the analysis record, making explanation and/or identification of mistakes harder and more time consuming.
To reach the analyzing shown: students learn and practice forming equations from isolated statements like these in the example; learn and practice forming such statements; practice identifying the relationships behind them; practice these skills in combinations; finally apply them as needed and practice sequencing the equations.
Part I will cover analyzing problems limited to math leading up to proportion, essentially counts and combining them and relating them. This will be material for 6th grade and below. Parts IV and up will cover the forming of non-equation math expressions as well as equations and utilizing known-step proportion analysis. These are future projects.
An additional project is writing chemistry texts utilizing only known-step analysis for proportion. College and secondary chemistry teachers interested in paid work on this should contract Endeavor Now.
