Further my post dt. Jun 22, 2013 8:20 AM: > <snip> > > I agree with you (Joe Niederberger, I believe) that > children should NOT skip mastering arithmetic (and other skills) to give them more time for deep thought. > I believe that most of the issues involved (in the ongoing discussion here) would more or less resolve themselves quite satisfactorily if we'd learn to work with simple models involved using the transitive relationship "CONTRIBUTES TO", as follows:
"Mastering basic arithmetic skills MAY CONTRIBUTE TO enhancing 'deep thought' on issues"
"Mastering basic logical skills MAY CONTRIBUTE TO mastering basic skills of arithmetic" ... ... ...
"Mastering the alphabet (in, say English/any other language) MAY CONTRIBUTE TO ability to read and understand simple words (in English/other language), which in turn MAY CONTRIBUTE TO ability to read and understand simple declarative statements, which in turn MAY CONDTRIBUTE TO ability to read and understand simple logical statements, which in turn MAY CONTRIBUTE TO read and understand more complex logical statements, which in turn MAY CONTRIBUTE TO read and understand complex logical arguments, ..., ..."
"To read and understand simple logacal statements MAY CONTRIBUTE TO analyse simple logical statements, ...."
"To analyse simple logical statements MAY CONTRIBUTE TO synthesise simple logical statements for simpler 'elements', ..., ..."
All of the above is much easier to DO (and understand) if one uses what I call 'prose + structural graphics' (p+sg), about which I've often written here. MUCH more complex arguments can be handled pretty effectively (and very easily indeed), via p+sg. I am here trying to explain via 'pure prose' something that is properly communicated using p+sg, which is not a procedure to be highly recommended. I'm forced to do this because Math-teach does not readily enable arguments using p+sg.
To provide a real-life instance, a young college freshman student used p+sg very successfully to tackle his quite complex 'Mission':
M: "To understand thoroughly all topics of my math syllabus and THEREBY to improve, very significantly, my results in my math exams, tests, and quizzes".
It turned out that I did not at all have to PUSH him to learn his math. He did all the needed PUSHING for himself to make himself get over the many difficulties he faced. All I did was to ENCOURAGE him (to the effect that he could quite easily understand and successfully do all the math involved in his college math courses; that he could easily overcome all the difficulties he might face).
NO PUSHING AT ALL! ONLY ENCOURAGEMENT!!! (at various stages of his model-making).
In fact, I taught that student NO MATH AT ALL! I only showed him how to:
- -- 'write appropriate elements' at various stages of development of his models for the above-noted Mission; - -- how to develop his models from 'elements' as they arose; - -- how to interpret his models; - -- what to do next at various stages of model development). (Some of his elements included stuff like: "To get help from my peers who are very good at math to tackle specific tough problems"; "To get needed knowledge from my math professors/lecturers"; etc).
Everything became very much easier for me AND for that student because ENCOURAGEMENT is in fact a MUCH stronger positive behavioural reinforcement than is PUSHING [at least so I believe (assuming I'm using these technical terms correctly)].
In conclusion: One may need to PUSH donkeys - I believe human beings respond much better to ENCOURAGEMENT.
I'd also suggest that, for donkeys, carrots may be more effective to ENCOURAGE them than sticks to PUSH them.
In the specific instance: I believe that Joe Niederberger's statement is 100% correct. (He had argued to the effect that "the notion that children must skip over mastering arithmetic, so they have more time for 'deep thought' is silly").