The late Grant Wiggins was adamant about “authentic problems” and “authentic problem solving”. He felt that scaffolding problems was a cheat and that it short-circuited understanding. That is not the case.
In solving word problems, worked examples provide students a direct access to solving problems that are similar, and in the same category. By scaffolding such problems–that is, varying the problems slightly beyond the initial worked example–students are forced to stretch and to make connections.
Students do best with explicit instruction, starting with simple problems. They then begin to develop the knowledge and skills to solve increasingly more difficult problems with novel twists. Without explicit instruction in problem solving, many just give up and don’t try the problems. Students benefit by seeing how to think about the problem before actually working it. Imitation of procedure therefore becomes one of imitation of thinking.
While people may criticize this as mere imitation and rote learning it is not. As anyone knows who has learned a skill through initial imitation of specific techniques such as drawing, bowling, swimming, dancing and the like, watching someone doing something and doing it yourself are two extremely different things. What appears easy often is difficult–at first. So too with math. Imitation of thinking is a level of understanding as one goes up the scale from novice to expert.
For example, students may be shown how to solve this type of problem: Two trains, 360 miles apart, head toward each other, one going at 100 mph and the other at 80 mph. How long will it take them to meet? The student can be shown that the sum of the two distances represented by 100t and 80t (where t is the time traveled by each train) makes up the initial 360 miles. A variation of this problem is: After the trains pass each other, how long will it take for them to be 90 miles apart? In this case, the same concept is at work: the sum of the two distances represented by 100t and 80t makes up the future distance of 90 miles.
In the words of Dylan Wiliam (Emeritus Professor of Educational Assessment at the University College of London Institute of Education): “For novices, worked examples are more helpful than problem-solving even if your goal is problem-solving”
traditional math 
“Successful School Restructuring* was published in the mid-90s by the Association for Supervision and Curriculum Development out of Washington, DC, but was prepared by the Wisconsin Center for Education Research for millions of dollars. It was an argument to replace traditional school methods by programs of “authentic” instruction. This word has been usurped by the education industry to misinform the public (e.g., “authentic” assessment that is anything but) along with “standards”, to a point where their use has become a red flag that there will be nothing authentically documentable nor any real standard involved. They have become code words that essentially say “all constructivist rhetoric is valid” and data to the contrary will not be collected and will be ignored if collected. I have often said, exaggerating only slightly, that the NCTM will not publish anything that does not genuflect to its Standards five times on every page as if using the word “standards” could compensate for the fact there is not a single standard in the entire document.
‘With that history, I started to count the number of times the word “authentic” appeared on each page in Restructuring. The numbers were impressive, 2, 5, even 10 times (pages 7 and 11) until I reached page 17 where a shaded box highlighted the “standards” of the “authentic instruction” that were being endorsed. Page 19 is classic self-parody. On the small (5.5 by 8.5) page, “authentic” was used only twice but the word “standards” was used 17 times along with the following statement: “The Center’s standards for authentic intellectual work are silent about the specific content students should be expected to learn in any subject or grade level.”
I could not have said it better myself.
* U.S. Department of Education, Office of Educational
Research and Improvement (Grant No. R117Q00005-95) [means 1995] and by the Wisconsin Center for Education Research, School of Education, University of Wisconsin, Madison]
Click to access ED387925.pdf
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