Children doing math illustrations how to#

They were instructed on how to write explanations for their math solutions using a model called вЂњNeed, Know, Do.вЂќ In the problem example given above, the вЂњNeedвЂќ would be вЂњWhat was the original price of the coat?вЂќ The вЂњKnowвЂќ would be the information provided in the problem statement, here the price of the discounted coat and the discount rate. This was not required for all problems given rather, they were expected to do this for two or three problems in class per week, which took up to 10 percent of total weekly class time. In a middle school observed by one of us, the schoolвЂ™s goal was to increase student proficiency in solving math problems by requiring students to explain how they solved them. That is, inspired by what the standards say about understanding, one could ask вЂњDoes the student know why the subtraction operation is done to obtain the 80 percent used in the equation or is he doing it as a mechanical procedure-i.e., without understanding?вЂќ But the above demonstration might, through the prism of the Common Core standards, be considered an inadequate explanation. In fact, for years students were told not to explain their answers, but to show their work, and if presented in a clear and organized manner, the math contained in this work was considered to be its own explanation. X = original cost of coat in dollars 100% вЂ“ 20% = 80% 0.8x = $160 x = $200Д¬learly, the student knows the mathematical procedure necessary to solve the problem. What was the original price of the coat?вЂќДЄ student may show the solution as follows: (And in some instances, as was the case recently in a third-grade classroom, a student would be considered to not understand if he or she drew three groups of five pencils.)Д¬onsider now a problem given in a pre-algebra course that involves percentages: вЂњA coat has been reduced by 20 percent to sell for $160. For example, they might draw a picture illustrating five groups of three pencils. To show they truly understand why 5 x 3 is 15, and why this computation provides the answer to the given word problem, students must do more. In the eyes of some educators, explaining why the answer is 15 by stating, simply, that 5 x 3 = 15 is not satisfactory. Consider a problem that asks how many total pencils there are if five people have three pencils each. While the Common Core leaves this unspecified, current practices are suggestive. But this raises yet another question: What constitutes a satisfactory explanation? The underlying assumption here is that if a student understands something, he or she can explain it-and that deficient explanation signals deficient understanding. One way is to ask the student to justify, in a way that is appropriate to the studentвЂ™s mathematical maturity, why a particular mathematical statement is true, or where a mathematical rule comes from. вЂњBut what does mathematical understanding look like?вЂќ And how can teachers assess it? вЂњStudents who lack understanding of a topic may rely on procedures too heavily,вЂќ states the Common Core website. The Common Core math standards, adopted in 42 states and the District of Columbia and reflected in Common Core-aligned tests like the SBAC and the Partnership for Assessment of Readiness for College and Careers (PARCC), take understanding to a whole new level. Perhaps he or she has simply memorized the method without understanding it and is performing the steps by вЂњrote.вЂќ One distinction is between вЂњknowingвЂќ and вЂњdoing.вЂќ One distinction popular with todayвЂ™s math-reform advocates is between вЂњknowingвЂќ and вЂњdoing.вЂќ A student, reformers argue, might be able to вЂњdoвЂќ a problem (i.e., solve it mathematically) without understanding the concepts behind the problem-solving procedure. What вЂњunderstandingвЂќ in mathematics means, however, has long been a topic of debate. K-12 mathematics instruction involves equal parts procedural skills and understanding. The answer to her question comes down to what the education establishment believes вЂњunderstandingвЂќ to be, and how to measure it. The girl threw her arms up in frustration and said, вЂњWhy canвЂ™t I just do the problem, enter the answer and be done with it?вЂќ A girl pointed to the problem on the computer screen and asked вЂњWhat do I do?вЂќ The proctor read the instructions for the problem and told the student: вЂњYou need to explain how you got your answer.вЂќ At a middle school in California, the state testing in math was underway via the Smarter Balanced Assessment Consortium (SBAC) exam.