| Welcome to Livonianeighbors.com. We hope you enjoy your visit. You're currently viewing our forum as a guest. This means you are limited to certain areas of the board and there are some features you can't use. If you join our community, you'll be able to access member-only sections, and use many member-only features such as customizing your profile, sending personal messages, and voting in polls. Registration is simple, fast, and completely free. To ensure your privacy, never use personal information in your screen name or email address ("janedoe@hotmail.com" or "Billysmom" for example). Join our community! If you're already a member please log in to your account to access all of our features: |
| Parents Against Everyday Math; Facebook Group | |
|---|---|
| Tweet Topic Started: Aug 1 2009, 01:28 AM (348 Views) | |
| IlikeLIvonia | Aug 1 2009, 01:28 AM Post #1 |
|
Veteran
|
Please help us get the word out about this group! We'd love to hear your input about how you are making an impact against Everyday Math, TERC, Investigations or other constructivist math programs in your district. http://www.facebook.com/pages/Parents-Against-Everyday-Math/37453309495 ![]() |
![]() |
|
| IlikeLIvonia | Aug 13 2009, 05:28 AM Post #2 |
|
Veteran
|
Discovery learning in math: Exercises versus problems Part 1 Written By: Barry Garelick Columnist EducationNews.org 22-7-09 Discovery learning in math: Exercises versus problems Part I Barry Garelick 7.22.09 Columnist EducationNews.org http://ednews.org/articles/discovery-learning-in-math-exercises-versus-problems-part-1.html What is Discovery Learning? By way of introduction, I am neither mathematician nor mathematics teacher, but I majored in math and have used it throughout my career, especially in the last 17 years as an analyst for the U.S. Environmental Protection Agency. My love of and facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process. I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools. Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retire. I enrolled in education school about two years ago, and have only a 15-week student teaching requirement to go. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses. In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”—to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery. To set this in context, it is important to understand an underlying belief espoused in my school of education: i.e., there is a difference between problem solving and exercises. This view holds that “exercises” are what students do when applying algorithms or routines they know and the term can apply even to word problems. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, we future teachers are told that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms. As someone who learned math largely though mere exercises and who now creatively applies math at work, I have to question this thinking. I believe that students’ difficulty in solving new problems is more likely to be because they do not have the requisite knowledge and/or mastery of skills—not because they were given explicit instruction and homework exercises. Those who make such a differentiation and champion “true” problem solving espouse a belief in having students construct their own knowledge by forcing them to make connections with skills and concepts that they may not have mastered. But, with skills and concepts still gelling students are not likely to be able to apply them to new and unknown situations. Nevertheless, the belief prevails that having students work on such problems fosters a discovery process which the purveyors of this theory view as “authentic work” and the key to “real learning.” One teacher with whom I spoke summed up this philosophy with the following questions: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?” Those are important questions, but I will argue in this article the following points: “Aha” experiences and discoveries can and do occur when students are given explicit instructions as well as when working exercises; and 2) Procedural fluency does not exclude conceptual knowledge—it leads ultimately to conceptual understanding and the two are key for applying mathematics to complex problems. I’m not against asking students to discover solutions to novel and challenging problems—the experience can be quite powerful, but only under the right conditions. A quick analogy may be useful here. Suppose a person who knows how to drive automatic transmission cars travels to a city and is forced to rent a car with a standard transmission—stick shift with clutch. The person in charge of rentals gives our hero a basic 15 minute course, but he has no opportunity to practice before heading out. In addition to this lack of skill in driving a standard transmission, the city is new to him, so he needs to rely on a map to get to where he needs to go. The attention he must pay to street names and road signs is now eclipsed by the more immediate task of learning how to operate the vehicle. But now suppose that prior to his trip he is given instruction in and ample opportunity to practice driving standard transmission cars. With proper training and guidance, he can start off on quiet streets to get the feel of how to coordinate clutch with shifting, working up to more challenging situations like stopping and starting on hills. Over time, as he accumulates the necessary knowledge, and practice, he’ll need less and less support and will be able to drive solo. There will still be problems that he has to figure out, like driving in bumper to bumper traffic that requires starting, slowing, downshifting, and so forth, but eventually, he will be able to handle new situations with ease. Now, given the task of driving in a strange city, he will be able to focus all of his attention on navigating through new streets (having already achieved driving mastery of the vehicle that will take him where he needs to go). Whether in driving, math, or any other undertaking that requires knowledge and skill, the more expertise one accumulates, the more one can depart from the script and successfully take on novel problems. It’s essential that at each step, students have the tools, guidance, and opportunities to practice what they learn. It is also essential that problems be well posed. Open-ended, vague, and/or ill-posed problems do not lend themselves to any particular mathematical approach or solution, nor do they generalize to other, future problems. As a result, the challenge is in figuring out what they mean—not in figuring out the math. Well-posed problems that push students to apply their knowledge to novel situations would do much more to develop their mathematical thinking. Some Examples To make this discussion more concrete, let’s take a look at three math problems. The first is a discovery-type problem that the target students do not have the necessary knowledge to solve. The second is an ill-posed problem. The third, in contrast, is a well-posed problem that relies upon prior knowledge and is mathematically meaningful. The first problem comes from the first-year textbook of the Interactive Math Program series (IMP, 1997) and is given to students who have just started algebra. These students have had limited exposure to systems of linear equations—for example, two equations with two unknowns. “You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in all possible combinations of two: bales 1 and 2, bales 1 and 3, bales 1 and 4, bales 1 and 5, bales 2 and 3, bales 2 and 4, and so on. The weights of each of these combinations were written down and arranged in numerical order, without keeping track of which weight matched which pair of bales. The weights in kilograms were 80, 82, 83, 84, 85, 86, 87, 88, 90, and 91. Your initial task is to find out how much each bale weights. There may be more than one possible solution; if this is so, find out what all of the solutions are and explain how you know.” (IMP, 1997; p. 27) Students who are just beginning algebra do not have the prerequisites to solve this problem efficiently. They would probably have to use “guess and check,” a method that might result in the right answer, but that is not likely to deepen students’ understanding. A key observation necessary to solve the problem efficiently is that no two bales are equal in weight because no two of the sums provided in the problem are equal. Such a relationship is not easy for beginning algebra students to see, and is only one of several different kinds of reasoning required to solve such a problem. How would beginning algebra students with little foundation in systems of linear equations and mathematical reasoning feel when confronted with such a problem? David Klein, a mathematics professor from California State University at Northridge, (personal communication with author, October 29, 2008) commenting on this problem, said, “It is an annoying problem and has little educational value. If I had been given such problems at that age, I think that I would have hated math.” Why such strong words? Because it is unlikely that guess-and-check will provide any insight that can be transferred to other problems or result in a deeper understanding of mathematics. Meanwhile, the foundational knowledge that comes with mastery of different types of algebraic problems over time has not been learned. 2. Our second example is not a discovery-type; rather, it is an ill-posed problem that can be interpreted many ways and, as a result, is not educational. This problem comes from the “Ten-Minute Math” section of a teacher’s guide for TERC’s “Investigations in Number, Data, and Space” for fourth grade. (Russell et al., 2008) In this particular activity, students decompose numbers in an exercise that is ultimately designed to get students to think beyond place value. The guide explains that decomposing numbers “is more than just naming the number in each place. It includes understanding, for example that while 335 is 3 hundreds, 3 tens, and 5 ones, it is also 2 hundreds, 13 tens, and 5 ones” (Russell et al., 2008; p. 27). It then proceeds with the following instructions (for the teacher) and questions: Step 1 Write or say a number. Write a number on the board (or say it and have students write it.) For example: 1,835 Step 2 Ask: "How many groups of _______ (10, 100, 1,000, etc.) are in the number? For example, ask students how many groups of hundreds are in 1,835. If students think that eight is the only answer, ask them to consider a context such as money. If this were money, how many hundred dollar bills would we have if we had $1,835? Establish with students that there are 18 hundreds in 1,835. The problem is poorly worded and constructed given the answer the authors seek, and I feel sorry for the dedicated student who tries to make sense of it. Students who have an understanding of place value and see the 8 in 1,835 as representing 8 hundreds are now confronted with conflicting information: there is more than one answer, and 8 is not the answer being sought. They are then guided to make a “discovery” that there are 18 hundreds by making a connection that if there are 18 hundred dollar bills contained in $1,835, then there are 18 hundreds in 1,835. Those are two different statements. Worse, the latter is mathematically incorrect in the context of the question asked. Since the previous “Ten-Minute Math” focused on decimals, students may reason—correctly—that although there are 18 hundred dollar bills in $1,835, there are actually 18.35 hundreds contained in 1,835. Asking how many hundreds are in 1,835 is a division problem (1,835/100), but the activity calls it a place value problem, and the result is an incorrect answer. If students are not already profoundly confused by all this, they will be soon: the activity then asks them to “make up five different combinations of place values [sic] that equal 1,835: 15 hundreds + 33 tens + 5 ones; 16 hundreds + 23 tens + 5 ones; and so on.” While the problem may result in students thinking of different answers, it does not encourage mathematical thinking, does not push students to further their knowledge of mathematics, it incorrectly characterizes place value and in so doing, it confuses more than enlightens. 3. Our third example offers a sharp contrast to the other two. This problem comes from the fourth grade textbook in the series called Primary Mathematics from Singapore. (Primary Mathematics, Standards Edition, 2008). It is well posed and requires students to apply their prior knowledge. “What is the value of the digit 8 in each of the following? a) 72,845 b) 80,375 c) 901,982 d) 810,034 e) 9,648,000 f) 8,162,000” Students cannot escape the lesson about place value since they cannot simply note where the 8s are, they must know what the various positions of the 8s mean. Preceding this problem in the Singapore text are other problems that introduce the concept of a number being a representation of the sum of smaller components of that number by virtue of place value; i.e. 1,269 can be expressed as 1,000 + 200 + 60 + 9. Similarly, students are asked to express written out numbers, such as ninety thousand ninety, using numerals in the standard form (i.e., 90,090). They are also asked to write numbers in numeral form, such as 805,620, in words. In short, students are asked no ambiguous questions, and the underlying concept of place value is indicated clearly via examples that can be applied directly to problems. By the time students reach the problem asking for the value of “8” in the various numbers, they have a working knowledge of what the numbers in various positions represent. This problem pushes them to apply that knowledge, thereby revealing any confusion they may have and also providing enough guidance for them to see that the position of the number dictates its value. Advocates of complex problems that get students “off the script” may think this problem is not challenging enough. After all, any discovery students make is inherent in the presentation of the problem and the solution clearly comes from work that the students have just completed. But as anyone recalls from the early days of having to learn something new, it feels a whole lot different answering questions on your own, even after having received the explanation. In fact, such experience constitutes discovery. So I have to ask, what is wrong with acquiring incremental amounts of knowledge through well-posed problems? It is, after all, much more efficient than discovery-type problems that require Herculean sense-making efforts and leave most floundering for a solution, without a clear sense of whether they are right or wrong. Contrast the approach of the first two examples with the third problem. This problem is well posed and it requires students to connect what they have just learned about place value to this new application. Instruction that uses such problems isn’t “handing it to the student.” To the contrary, it’s providing the support and guidance that students need to grow. I see it as a staging; a way to get students to apply easier problems to solve harder ones, and a way for procedural fluency to lead to understanding. I am not alone in such thinking. According to a study by Liping Ma, Chinese teachers interweave conceptual and procedural knowledge of mathematics. They believe that “a conceptual understanding is never separate from the corresponding procedures where understanding ‘lives.’ ” (Ma, 1999) The key is for problems to be carefully sequenced such that they incrementally increase in difficulty and require students to use their knowledge in new ways—and that’s the key to making a meaningful discovery. |
![]() |
|
| IlikeLIvonia | Aug 13 2009, 05:36 AM Post #3 |
|
Veteran
|
Discovery learning in math: Exercises versus problems Part II Written By: Barry Garelick Columnist EducationNews.org 26-7-09 Discovery learning in math: Exercises versus problems Part II http://ednews.org/articles/discovery-learning-in-math-exercises-versus-problems-part-ii.html Barry Garelick - July 26, 2009 Columnist EducationNews.org Making Meaningful Discoveries To better explain the effectiveness (and efficiency) of carefully sequenced, well-posed problems, let me use my favorite example: my daughter. When she was 10, I tutored her on addition and subtraction word problems, and I found that her understanding deepened as we worked on her procedural fluency with problems that demanded new, harder applications. Using the Saxon math series, Math 65, (Hake, et. al., 2001) we were on a chapter that explained how to do certain types of word problems. Specifically, these were called “Some and Some More” and “Some Went Away.” Some and Some More was defined by way of an example: “Before he went to work, Tom had $24.50. He earned $12.50 more putting up a fence. Then Tom had $37.00.” Some Went Away was defined similarly: “Tom took $37.00 to the music store. He bought a pair of headphones for $26.17. Then Tom had $10.83.” I asked my daughter if she understood what was going on in each type. “Yeah, you add the numbers together for the Some and Some More and you subtract for the Some Went Away.” She seemed very pleased with this and added, “Seems pretty easy.” The next few practice problems, however, weren’t quite as easy as she thought. They weren’t exactly like the sample problems; she needed to combine the new procedures with prior knowledge in order to solve them. Along the way, her understanding deepened substantially. For example, one problem was “After losing 234 pounds, Jumbo weighed 4,368 pounds. How much did Jumbo weigh before he lost the weight?” Another was “The price went up from $26 to $32. By how much did the price increase?” We read the first problem. I asked her what type of problem it was. Silence. “Where are you stuck?” I asked. “It doesn’t tell me how much the elephant weighs,” she said. This was correct; it was different than the initial presentation. “Does the elephant lose some weight?” I asked. She nodded. “So is it a Some and Some More?” I asked. She replied, “No, it’s a Some Went Away because the weight goes away.” Because this was an example problem, the solution was given in the text. It was represented as follows: Before: Jumbo weighed… W pounds Then: Jumbo lost… –234 pounds Now: Jumbo weighs… 4,368 pounds I showed her the set-up, and she knew immediately what to do. “Oh, you add 4,368 and 234 to get what he weighed before.” She knew this because in previous lessons she had solved problems with missing numbers in subtraction (e.g. F – 15 = 24). With these carefully sequenced lessons, she had been well prepared; she knew how to solve a missing number in a subtraction problem. Now, she discovered how the Some Went Away scheme worked in conjunction with her prior knowledge and was able to make the connection. Similarly with the second problem, now knowing how the schemes connected with prior knowledge, she readily identified the problem as falling in the Some and Some More category. From previous experience she knew how to solve such problems. “You just subtract: $32 minus $26 is $6.” I asked if she could write it out as a “missing number in addition” problem using a letter for the missing number. She reluctantly complied: X + $26 = $32. I noticed that she resisted putting the problems in this form. When I asked her why, she said “Why bother? I can just add or subtract and get the answer. Besides, that’s algebra and I’m not ready for it.” I decided not to argue that point given that I was tired, things were going well, and the steps for solving simple equations in algebra were clearly falling into place. By contrast, if all the subtraction word problems had been in the form of “John had $30 and spent $15; how much did he have left?” there would have been no discovery. (This is originally what my daughter was expecting—and hoping—would happen). If an assignment is constructed properly, as it was in the case of these problems, students should have an unscript-like understanding of the material by the time they are done—whether conducted in class or as homework They should also have some level of “aha” experience brought about by procedural fluency. With my daughter, I’ve seen that the procedural fluency resulting from the exercises helps clarify the concept, even if it wasn’t fully understood before starting the problem set. Scaffolding This process of extending previously learned material to slightly more difficult problems, in the jargon of education, is called “scaffolding”. When done properly—whether in a set of homework problems or in a classroom—it is an extremely effective method to bring about meaningful discovery. Algorithmic Procedures and the Development of Critical Thinking Mathematics demands mastery of foundational steps in order to build upon them. As such, it is relentlessly linear. Without such mastery or foundation students will not be prepared to solve new and complex problems. Yet some practitioners believe that the above examples are “inauthentic work”—mere exercises—and as such, they do not lead students to learn how to do the “authentic work” of solving problems. In fact, the application of learned and mastered material in new, off-the-script context does not happen immediately—nor is it brought about by giving students problems which they‘re not equipped to solve. Daniel Willingham, a cognitive scientist who teaches at the University of Virginia, maintains that it takes time and effort for knowledge to accumulate to the point that connections between learned material and new and difficult problems can be made. (Willingham, 2002). Willingham refers to the difficulty that novices have with thinking critically as “inflexible thinking,” which he characterizes as perfectly normal and to be expected among students. Specifically, the critical thinking that allows for the application of prior knowledge to new, unfamiliar type problems requires recognition of an underlying principle that can be used to solve the problem. The underlying principle can be found in any number of simpler problems to which a person has been exposed—the challenge is in looking beyond what Willingham calls the surface structure to see the deep structure. For example, the following two problems are based on the same underlying principle—rate—and in fact the same equation is used to solve both of them: 1) John can mow the lawn in 20 minutes while his brother Bob can mow the same lawn in 30 minutes. If both mow at a constant rate, how long does it take for both of them to mow the lawn together? 2) It takes John 20 minutes to walk to school from home, while it takes his sister 30 minutes, both walking at constant rates of speed. John starts walking from school to home at the same time that his sister starts walking from home to school. How long will it take for them to meet? (1/20 + 1/30)X = 1 For the first problem the rate is the portion of the job completed per minute. Specifically, John's rate is 1/20 of the job per minute and Bob’s rate is 1/30 of the job per minute. The amount accomplished in X minutes is (1/20)X for John and (1/30)X for Bob. Therefore, the portion of the lawn mowed (or job completed) in X minutes by both John and Bob working together is (1/20 + 1/30)X. In setting it up this way, we are adding up their accomplishments in terms of how much of the job is completed in X minutes. Since the problem asks how long it takes to mow the lawn, we are interested in exactly one job, and want to find what value of X is needed to get this done. To do this, we set the above expression equal to 1 and solve for X: (1/20 + 1/30)X = 1 The second problem is solved using the same reasoning as the first. The key to solving it is to see the connection between distance and time, and that this is still a rate problem. In this problem, however, the rate is the portion of the total distance walked per minute. In one minute John walks 1/20 of the way between school and home, while his sister walks 1/30 of the way. The solution follows as described above for the lawn mowing problem, and results in the same equation. Beginning algebra students may understand how to solve the first problem but may not make the connection that the same concept of rate underlies the second problem as well. In fact, as Willingham explains, it is unlikely that students will make such connections readily until they have developed true expertise. Only experts see beyond the surface level of a problem to its deeper structure. So how do you teach students to make such connections; i.e., to think critically? Is it a failure of the math program—or the teacher—if they do not? Willingham (2002) argues that understanding the deep structures of a discipline such as mathematics is an important goal of education, “but if students fall short of this, it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots.” Rather, they are making the small steps necessary to develop better mathematical thinking. Simply put, no one leaps directly from novice to expert. While there is no direct path to learning the thinking skills necessary to apply one’s knowledge and skills to unfamiliar territory, Willingham (2002) argues that one way to build a path from inflexible to flexible thinking is to use examples. This approach could be used for rate problems such as the two problems just described. In fact, this is what was done in the two well crafted lessons I observed. Students extended their knowledge along scaffolding built from examples—examples that fit on the underlying structure. Although it does not necessarily happen automatically, thinking becomes more flexible as more knowledge and experience are acquired. Think of the girl in the high school geometry class who solved the triangle problem. What had become a mechanical or algorithmic habit for her—drawing in the altitudes of triangles—ultimately led to the solution? Many problems in mathematics involve evaluating their form through algebraic manipulation, or in the case of geometric figures, adding supplementary lines. Such analysis leads to insights about a problem’s underlying structure. The girl’s habit, which some might consider algorithmic thinking (and therefore “inauthentic”) was part and parcel of her flexibility in thinking and applying a previously learned principle in a novel way to a new problem. Students should certainly be given increasingly challenging problems—but problems that draw upon what they have learned. As discussed above, applying what has been learned to new problems one hasn't seen before does not simply happen. It comes about from practice and acquiring tools and experience using them. If you look at an expert "problem solver", someone who is a mathematician, engineer, or scientist, say, you'll find someone who has command over many procedures and skills and has memorized many things. Such experts have so much extensive background knowledge that many problems—for them—are like exercises. People who know math can look at a problem and based on their knowledge and experience will break the problem down and apply a straightforward, mathematical solution. For people who haven’t mastered what they need to know in math, just about everything is a problem that is not easily broken down. Students given well-defined problems that draw upon prior knowledge, as described in this article, are doing much more than simply memorizing algorithmic procedures. They are developing the procedural fluency and understanding that are so essential to mathematics; and they are developing the habits of mind that will continue to serve them well in more advanced, college level mathematics courses. Poorly-posed problems with multiple “right” answers turn mathematics into a frustrating guessing game. Similarly, problems for which students are expected to discover what they need to know in the process of solving it do little more than confuse. But well-posed problems that lead students in manageable steps not only provide them the confidence and ability to succeed in math; they also reveal the logical, hierarchical nature of this powerful and rewarding discipline. |
![]() |
|
| IlikeLIvonia | Aug 13 2009, 06:21 AM Post #4 |
|
Veteran
|
President of American Mathematical Society's letter to the school board (Parents For Quality Math Education): Attached please find the letter from George Andrews, the President of American Mathematical Society, Evan Pugh Professor of Mathematics at Penn State, to the board of our school district, urging a change in the math program. Thanks to President Andrews for taking time for this important matter. I thank him for giving me permission to post his letter. Wen Shen, PhD. LetterAndrew.pdf http://groups.google.com/group/parents-for-quality-math-education/attach/10ad22adb3e418fd/LetterAndrew.pdf?part=2 Edited by IlikeLIvonia, Aug 13 2009, 06:23 AM.
|
![]() |
|
![]() Our users say it best: "Zetaboards is the best forum service I have ever used." Learn More · Register Now |
|
| « Previous Topic · Livonia Neighbors Forum · Next Topic » |







