No doubt once it was real progress

No doubt once it was real progress when developers and teachers offered learners tangible material in order to teach them arithmetic of whole number. (In the case of fractions it never did work properly.) Even now I would not like do without it. Whether unstructured, at the start of arithmetic, or structured, as on the abacus, it is indispensable as part of the primordial reality. It is a small part, to be sure, one source among many others of whole number arithmetic. Relying solely or too strongly on it is dangerous, and since this is the habit in some kinds of instruction I feel obliged to warn against it. Low achievers have great difficulty detaching arithmetic from the palpable material -- according to remedial teachers and de- velopers of remedial teaching. This may unfortunately be true, that is, for a kind of inflexible instruction where from the start onwards learners have been tied too tightly to palpable material. Flexibility should be allowed, and if need be, taught rather than fought. Moreover, one may ask, what is the use of low achievers learning an arithmetic they are judged in advance unable to detach from tangible material.