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<\/figure>\n\n\n\nWere you\nable to immediately find the answer because you knew that 2 x 80 is 160? Or did\nyou get out your calculator to figure out the answer? If the latter, you might\nbe one of many who could use some help with number sense \u2013 something that\nestimation skills help build. <\/p>\n\n\n\n
Most of our students (and too many of us) have been taught to immediately pick up our pencils and DO something \u2013 anything \u2013 when we have a math problem to solve. We rarely first ask, \u201cShould the answer be larger or smaller than the numbers that I\u2019m working with?\u201d or \u201cIs this a situation where something is growing?\u201d Too often students \u2018solve\u2019 the problem then expect us to tell them whether the answer is correct or not. And, we often wind up asking the student, \u201cDoes this answer make sense?\u201d I think we wouldn\u2019t have to ask them if instead we taught them to reason and then do some estimation to determine what might be a logical answer.<\/p>\n\n\n\n
Let\u2019s look\nat some examples of how reasoning and estimation could really help students\nusing this statement as a starting point:<\/p>\n\n\n\n
Johnetta bought a $34.99 skirt that was half\nprice. The sales tax was 5%. <\/em><\/strong><\/p>\n\n\n\n There\u2019s a variety of questions that could be asked about this statement. Some common ones could be: How much money did she save? What was the amount of tax? What\u2019s the total price she paid including sales tax?<\/em> If a student can reason and use estimation, it doesn\u2019t matter what the question is. She could quickly determine a ballpark answer to a variety of questions, such as the following:<\/p>\n\n\n\n In the first\nsituation, a student who can reason and estimate could determine that half of\n$35 is $17.50. If she knows 10% percent tax on $17.50 is $1.75, she would also\nbe able to figure out that half of that (5% tax) is less than a dollar. So, $17.50\nplus a little less than a dollar in tax will be a little under $18.50. <\/p>\n\n\n\n In the\nsecond situation, a student could use similar reasoning: an estimate of 10% tax\non $35 is $3.50. So, 5% tax would be about $1.75.<\/p>\n\n\n\n In the third\nsituation, it is fairly obvious what the answer should be . . . but ONLY if we\ncan reason. We don\u2019t even have to bother doing any calculations. Clearly the\nanswer has to be more than $35, so there is only one possibility. No\ncalculations needed to this typical test-like question. <\/p>\n\n\n\n And, what\nabout the ubiquitous fraction problems? How does estimation fit? <\/p>\n\n\n\n Our students\ncan never remember to find the common denominator so 9\/11 makes sense if\nthey\u2019ve been taught a procedure. They might not remember exactly what procedure\nto use when, but they know they need to follow a procedure. If instead, they\nare taught how to reason, they can use estimation and eliminate the answer 9\/11\nimmediately. 2\/3 is more than half. 7\/8 is close to 1. So the answer has to be\nat least 1 but less than 2.<\/p>\n\n\n\n But, if they\ndon\u2019t work out the problem and instead use estimation, how do we know they\nunderstand? In estimating, it is clear that the students understand about the\nrelative size of the fractions compared to benchmark fractions. On the other\nhand, what do students really know when they follow a procedure that they have\nbeen told to memorize?<\/p>\n\n\n\n Rather than\nneeding to do a lot of calculating, if students know that \u03c0\nis about 3, they can estimate the answer. Of course, this assumes that they\nknow the difference between area and circumference. Perhaps they would if they\ncould, in their heads, simplify \u03c0 and focus on the different formulae\ninstead. While some of you might be saying that they need to be more accurate,\nlet\u2019s talk about how accurate is accurate enough. After all, 3.14 is only an\nestimate. So is 3.14159. And so is 3.14159265359.<\/p>\n\n\n\n When we\u2019re\ntalking about estimation, we\u2019re not necessarily talking about the traditional rule\n(which we CAN and should break when needed) that says we round up or down,\nbased on whether the number next to the place we want to round to is 5 or more.\nEstimation can be more flexible than that. For example, a person might choose\nto round $4.34 up to $5 (rather than round down to $4) to make sure he has\nenough cash when he gets to the checkout counter. Or perhaps we want to round one\nnumber to make it more compatible with another number; for example, to estimate\nthe size of the fraction 18\/35, one might round the 35 to 36 because mentally\nit\u2019s easier to compare the relationship of 18 to 36.<\/p>\n\n\n\n Are you\nthinking, \u201cWait, my students can\u2019t reason like that!\u201d Well, how about we make\nthat our responsibility \u2013 to teach them to reason and estimate. We understand\nthe pressure to \u201cteach to the test\u201d but think of teaching reasoning and\nestimation skills as a worthy short- and long-term investment. Students with\nthe ability to reason and estimate have an advantage over those who are just\ntaught to memorize, whether on a test or in real life.<\/p>\n\n\n\n ===========================================================================<\/p>\n\n\n\n Donna Curry is the Director of the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center<\/a>, a project managed by the Adult Numeracy Center at AV°ÍÊ¿<\/a>. She has trained teachers nationally, taught and administered ABE classes, and developed adult education curricula and resources for over 30 years.<\/em><\/p>\n<\/div><\/div>\n<\/div><\/div>\n<\/div><\/div>\n![\"\"](\"https:\/\/www.terc.edu\/adultnumeracycenter\/wp-content\/uploads\/sites\/28\/2019\/08\/blog45_img2-1.png?w=572\")
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