But guessing blindly and applying the strategy of guess-and-check are two completely different animals. When teachers and students dismiss guess-and-check as problem solving by luck, they are not seeing the sophisticated reasoning and understanding that must be brought to bear to use this approach which has as much right to be called a strategy as any other. Consider the following problem: <\/p>\n\n\n\n
Tori has gotten the following scores on her last four math tests: 79, 86, 92, 88. What does she have to score on her fifth test to have an average score of 85? <\/strong><\/em><\/p>\n\n\n\n Before you start writing an equation to find the answer, also\nconsider why you might pose a problem like this to your students. What is it that\nyou want to know about what they know? Is the important thing that they be able\nto abstract the definition of an average into symbols, or that they understand\nhow averages behave and what they mean? Although it may be marginally faster to\nwrite and solve an equation (assuming one is already skilled at that), consider\nthe reasoning and possibly even learning that can take place when you approach\nthis with guess-and-check.<\/p>\n\n\n\n Let\u2019s put ourselves in Tori\u2019s shoes. She wants to get exactly an 85 average \u2013 no worse and no better. She decides to explore what will happen if she scores an 85 on the next test. This is reasonable because the scores are all fairly close to 85 already. Maybe she can hit the average by aiming right for it. <\/p>\n\n\n Testing out her guess, Tori adds up the five scores to get 430 and then divides by five to get an average of 86. That\u2019s close! And Tori has just demonstrated that she knows how to find an average. Her guess got her close, but the answer was a little higher than what she was aiming for, so a second guess is needed. <\/p>\n\n\n Because 85 was too high, Tori decides to try a lower number. She wants to lower the average by one, so she tries lowering the guess by one (that\u2019s reasoning!). With a guess of 84, adding up the five scores gives a sum of 429 and dividing by five gets Tori to an average of 85.8. <\/p>\n\n\n\n Huh\u2026. that didn\u2019t result in the change she expected, but she also may have just discovered something about the structure of the situation and of averages in general \u2013 making a small change to one number makes an even smaller change to the average. <\/p>\n\n\n Next, Tori decides to try a much lower number. (Maybe if she\ncan get by with a pretty low score, she can hang out with friends instead of\nstudying the night before the test!) This time she tries 75. She arrives at a\nsum of 420 and an average of 84. Oops! That pushed the average too far in the\nother direction. <\/p>\n\n\n\n From here I\u2019ll leave Tori to continue on her own. She knows\nnow that 84 was too high and 75 was too low and I have confidence that she\u2019ll\nhit the solution within a few more guesses. She\u2019ll also have not only practiced\nwith finding averages, but also seen how changing the numbers affects the\naverage. She\u2019ll have both made use of the structure of averages and deepened\nher understanding of it. She may have even noticed that the numbers she was\ntrying contributed different amounts to the sum and if she wanted a sum that\nwas going to give a result of 85 when divided by 5, there was a particular sum\nthat she should be aiming for. <\/p>\n\n\n\n This is actually more thinking and learning than will be done\nby a student who knows how to model the situation by writing and solving an\nequation. There\u2019s nothing wrong with that approach, either, but that student\nhasn\u2019t really engaged in problem solving, only performed an exercise. <\/p>\n\n\n\n In order to use the strategy of guess-and-check, students\nmust at least understand the structure of the problem. Without that\nunderstanding, they cannot check their guesses or make improved guesses. So,\nwhen they successfully navigate a problem with this strategy, neither they nor\ntheir teachers should chalk their success up to luck. Instead, students and\nteachers should appreciate the hard work and reasoning that goes into solving\nwith guess-and-check as well as the learning that can result from it. <\/p>\n\n\n\n ==================================================<\/em><\/p>\n\n\n Aren Lew has been teaching and tutoring math in one form or another since college. They have worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. Aren\u2019s work with the\u00a0SABES Mathematics and Adult\u00a0Numeracy Curriculum & Instruction PD Team<\/a>\u00a0at\u00a0AV°ÍÊ¿<\/a>\u00a0includes developing and facilitating trainings and assisting programs with curriculum development. They are the treasurer for the\u00a0Adult Numeracy Network<\/a>.<\/em><\/p>\n![\"\"](\"https:\/\/www.terc.edu\/adultnumeracycenter\/wp-content\/uploads\/sites\/28\/2019\/02\/guess_check_1.jpeg?w=323\")
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