The Difference Between Solving a Problem and Learning Some Math From It

“Problem solving must be the focus of school mathematics.” This call opened the National Council of Teachers of Math (NCTM)’s Agenda for Action: Recommendations for School Mathematics of the 1980s. The Agenda helped launch math educators into a decade of intense interest in problem solving. Researchers could also claim credit for this growing excitement. In the years leading up to NCTM’s Agenda, problem solving had emerged as a vibrant area of research in cognitive science and experimental psychology.1.

In the early 1970s, John Sweller found himself needing a change. After finishing graduate school he had accepted a position as a psychology lecturer for a teacher training program. The first problem was the location — a small town, far away from Sweller’s family. Second, Sweller was unused to teaching, and the time it took away from his research activities. Finally, his research was on learning in rats, and he was finding this work unproductive. After just one year, Sweller left for Sydney, where he reinvented himself as a researcher in the emerging field of human problem solving.2

In one of his early problem-solving studies, Sweller tasked his undergraduates with a number puzzle.3 “I am going to give you one or more problems to solve,” he told participants. “You will be given an initial number and asked to transform it into a final number by multiplying 3 and/or subtracting 69 as many times as is required.” The game, however, was rigged. The numbers were carefully chosen so that each initial number could easily be transformed into the final number by alternating multiplication with subtraction. For example, the first problem asked participants to get from 60 to 111 – simply multiply by 3 and subtract 69. The second problem went from 31 to 3 – multiply, subtract, multiply, and finally subtract once more.  The third problem could again be solved by alternating between multiplication and subtraction. Would the players of this game discover this winning strategy all on their own?

Sweller found that most participants never discovered this rule. Instead, they used a different technique to attack the puzzle – at each turn they performed whichever move would make their number closer to the goal. Suppose a participant was tasked with turning 54 into 210. 54 is less than 210, so they would multiply to get closer to 210. That gave 162 — still too small. OK, multiply again. That gives 486, which is too large! Subtract, then subtract and subtract again until you are below 210. Continue this process – “means-ends search” in Sweller’s parlance – until the puzzle is solved. (In contrast, alternating between multiplication and subtraction would solve the puzzle in four moves.)

Sweller hypothesized that this wasn’t just a bad strategy for solving the puzzle, but that it would be awful for ever discovering a better approach. After all, if you’re always comparing the number you have to the number you want, you’re completely ignoring all of your prior moves. This ignorance of past moves eliminated any chance that a participant might notice patterns that would lead to the successful strategy. The means-ends search is not only slow, but it directs all of one’s attention away from what matters for learning.

To Sweller, these results underscored the huge difference between solving a problem and learning something useful from that experience: “After an enormous amount of problem-solving practice, subjects could remain oblivious of a simple solution rule.” 4

If problem solving was ineffective for learning to win a simple game, then it would likewise be trouble for learning something more complex, such as an algebraic procedure. Sweller designed experiments that allowed him to observe novices attempting to solve mathematics problems. He saw the same thing: beginners chose “search” strategies that drew attention away from the sorts of observations that might lead to obtaining a more powerful strategy. If teachers wanted to foster expertise, they would need techniques to circumvent these learning-killing search strategies.  5

The first alternative to problem solving Sweller championed was “goal-free problems.” Despite their name, Sweller’s goal-free problems do have goals, but those goals are nonspecific (“find as many angles as you can”) rather than specific (“find angle x”). Sweller pitted goal-free and conventional problems against each other and compared the learning that resulted. The winner: goal-free problems. 6

The advantage of problems with nonspecific goals is that they allowed novices to avoid fixating on those goals. When Sweller asked participants to find the value of a particular angle in a diagram, novices were more likely to work backwards from the “goal” angle, constantly checking their progress towards the goal and how they might get closer to it. (This is the same means-end search that Sweller observed with his number puzzle.) Too much of a novice’s attention was consequently devoted to the goal angle and how close they were to deriving its value. As in his number puzzle experiments, even when participants successfully solved these goal-specific problems, little learning resulted.

To discover a pattern or a rule, one needs to look away from the goals and their present progress, and instead turn to work in the past. What moves have you already tried? Which combinations of moves work particularly well together? Which angles in a diagram, when derived, help you calculate other angles? By eliminating a single, clear goal for participants to fixate on, participants were free to notice patterns in their past moves. (And if there was a gap between their current status and a goal? They could discard the goal and choose another, instead of working backwards to derive it.) This freedom to think about the past is precisely what is needed for discovering useful, expert-like shortcuts. Sweller’s results showed that these discoveries did, in fact, take place more frequently when problems were given with nonspecific goals. Therefore, nonspecific goals were better for learning than conventional problems.

These results were important for Sweller, but he was interested in a more fundamental result. Goal-free problems were still problems, if unconventional ones.  7 What if problems were totally unnecessary for learning?

Worked examples are not problems – they are explanations of how a problem is correctly solved. Goal-free problems function by eliminating means-end search, instead drawing participants’ attention to their past successes. Was there any reason why the participants had to generate these past successes themselves? Sweller hypothesized that this was unnecessary. If people learned from studying and generalizing from their own examples of problem-solving success, it would be equally effective if these problems were presented by an instructor instead of generated by the learner.

In another series of experiments, Sweller carefully tested this idea. His results confirmed the hypothesis: the quality of learning was the same whether students learned via worked examples or self-discovered solutions. The major difference was time – problem solving took a lot of it! Worked examples took far less time. In this sense, explanations were more efficient than discovery. 8

With these results in hand, Sweller began to take his results to the math education world. A 1989 piece in the Journal for Research in Mathematics Education asked, “Should Problem Solving Be Used as a Learning Device in Mathematics?” Their answer was unambiguously, “no”:

Students may learn more by solving goal-free problems or by studying their problem solutions than by solving the problems in the first place. This of course begs the question: Why solve the problem in the first instance? 

This wholesale skepticism of the value of problem solving put Sweller at odds with many in the educational establishment.  9

Given how slippery the term “problem” has proven, it’s worth checking-in to see what Sweller means in his usage. After all, he does advocate for goal-free problems. If a student solves a goal-free problem, what should we call that? Apparently, not “problem solving.” What sort of teaching is he opposed to, then?

Sweller never defines “problem” in his 1989 paper, but he does give an example of the sort of mathematics instruction he is railing against at this stage of his thinking:

The conventional mode of mathematics teaching is stereotyped. New material is presented and one or two worked examples using the new materials are demonstrated, followed by a reasonably large number of problems or exercises…Solving many conventional problems may not be the best way of acquiring this knowledge.

It’s easy to imagine advocates of problem solving nodding along with Sweller. They too were opposed to unnecessarily long sets of conventional problems students are often tasked with in math classes. Alan Schoenfeld — researcher, educator and chamption of problem solving — advocated for work with genuinely difficult, perplexing problems, not conventional work. Further, Schoenfeld later declared the movement of the 1980s “superficial,” adding that it had failed to incorporate the “deeper findings about the nature of thinking or problem solving.” 10 Sweller and his opponents could find common ground in their dissatisfaction with the way math was conventionally being taught.

Sources: Problem Solving in the 1980s, Sweller’s Research Journey

  1.  NCTM, 1980; Schoenfeld, 1992. 
  2. Sweller, 2016. 
  3.  Sweller, et al., 1982. 
  4. Sweller & Cooper, 1985. 
  5.  Sweller, et al., 1983, Sweller & Cooper, 1985. 
  6.  Sweller & Levine, 1982, Sweller, et al., 1983, Owen & Sweller, 1985. 
  7.  Or were they? “Problem” and “problem solving” have historically been fantastically tricky terms to pin down. 
  8.  Sweller & Cooper, 1985, Cooper & Sweller, 1987. 
  9.  Owen & Sweller, 1989. 
  10.  Schoenfeld, 2004. 

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