Philipp, R. A., and Vincent, C. (2003). Reflecting on learning fractions without understanding. ON-Math, Vol 2, No. 2, 1-6.
The main idea of this article, “Reflecting on Learning Fractions without Understanding,” is that students learn less if they are taught procedural followed by conceptual, compared to students who learn more from only being taught conceptual instruction. Routine procedures actually inhibit the learning of concepts. This idea was apparent in a study done in a classroom of only conceptual understanding methods. The experiment involved the teacher giving a purely procedure-based lesson about converting between improper fractions and mixed numbers. The experiment was video-taped, students given an assessment, and then interviewed. Students realized that this was a different teaching method as the class went along. A few weeks later, the conceptual side of converting fractions was taught and similar procedures followed. After calculating the results of the assessments, it was apparent that students retained a far greater deal of information after a long period of time when taught conceptually rather than procedurally. When some students were taught procedurally followed by conceptually, results showed that students’ understanding dropped incredibly. One student, Rachel, said she does not really figure out the problems herself when she tries to do problems procedurally; she is just following the way the teacher tells you to figure it out. On the other hand, when she solves problems conceptually, she understands. This is what she had to say: "So when I figure that out, it's easier, and, um, once I figure it out, it's, it stays there 'cause I was the one who brought it there. So, and it is just easier to do when you figure it out yourself, instead of having teachers telling you." When Rachel was asked to solve a problem in the interview, she first attempted it with the procedure taught by her teacher. She arrived at the wrong answer. When told so, she tried the same problem her own way, arriving at the correct answer. When asked why she solved the problem procedurally first, she said it was because that was the way she learned it first. Thus, the conclusion was made that procedure followed by conceptual teaching is not as effective as purely conceptual teaching.
I thought this was an incredible finding. I do not, however, agree that teaching conceptually is the ultimate method of teaching out there. Obviously, as the article outlines, strictly conceptual teaching is better than strictly procedural teaching. I think that the best way for students to learn is by teaching conceptual and procedural hand-in-hand. This is different from what was discussed in the article because the two types of teaching were done weeks apart from each other. We have already discussed why purely conceptual teaching may not work – it takes more time, not all the material gets covered, not all the students get involved, etc. If some procedural is taught along with the conceptual, then students will really understand the material (from the conceptual part of the lesson) and they will know how to quickly arrive at answers (the procedural part of the lesson), and thus they will do well in school and get throughout the required amount of material. For me, I like it when teachers have taught the conceptual side of concepts so that I can truly know what is going on, followed by the procedural side of things so that I might be able to compute answers quickly and efficiently. I think the key part of this idea is for teachers to teach how the conceptual and procedural connect to each other. This will result in optimum understanding, in my opinion at least.
Friday, March 26, 2010
Friday, March 19, 2010
Blog #6 Response: NCTM Article - Proof by Examples
D’Ambrosio, B. S., Kastberg, S. E., and Santos, J. R. (2010). Learning from student approaches to algebraic proofs. Mathematics Teacher, Volume 103, No. 7, 489-495.
In “Learning from Student Approaches to Algebraic Proofs,” a study showed how algebraic concepts are among the most difficult concepts for students to prove. Instead of proving statements algebraically, most students chose to “prove” statements by giving examples. The types of answers to the proof goes as follows: (1) “It’s fact. What is there to prove?” Students said the proof was not necessary, and the statement was self-evident. (2) Some students gave one or more examples that showed the proof held true there, saying this was sufficient for a proof. (3) Others showed the proof held for large numerical examples, therefore implying the proof always held. (4) Some students completely lost sight of the fact that they were actually trying to prove something, not solve something. When they saw the equals (“=”) sign, they assumed they just needed to solve an equation. (5) Very few students proved with algebra, followed by examples, and then the general case. It was apparent that the students who actually looked for the patterns in the examples were on their way to finding the generalized rule for the proof. The overall analysis of the results was this: “The data suggest that students need a much more robust vision of the role of proofs in supporting mathematical activity, making connections between mathematical ideas, and communicating findings. Students seem to have a limited repertoire of meth¬ods to use for proving those statements.” In order to fix this problem, teachers need to analyze their students’ written work carefully. In turn teachers will better be able to draw conclusions on the level of their students’ understanding and foundation in mathematical concepts.
I think this will allow for teachers to be able to lesson plan much more accurately. When teachers analyze their students’ work and find out where their weak points are, e.g. proving statements algebraically, then they will know which concepts need to be taught more fully in the classroom. Once students get a handle on proofs, they will have a much easier time with connecting and communicating their mathematical ideas. More specifically, once students gain a better understanding on how to prove mathematical concepts, they will have a greater “development of stu¬dent understanding of one another’s work and move toward the use of proof as a tool in exploring connec¬tions, relationships, and the veracity of mathematical statements and in communicating their findings.” I do not agree that giving examples as a solution to a proof is in reality a proof, but I can see why students would assume it to be true. They have not been educated on the matter. Teachers need to teach why giving only examples as answers to a proof is not sufficient for all cases. When teachers teach concepts only through examples, the processes may stick easier in the minds of the students. However, if teachers do give proofs behind the mathematical thinking, then students will have an easier time convincing others and themselves of the connections they make between their next mathematical ideas and the previous ones. It all comes down to the basic ideas discussed earlier this semester: the importance of relational teaching/understanding.
In “Learning from Student Approaches to Algebraic Proofs,” a study showed how algebraic concepts are among the most difficult concepts for students to prove. Instead of proving statements algebraically, most students chose to “prove” statements by giving examples. The types of answers to the proof goes as follows: (1) “It’s fact. What is there to prove?” Students said the proof was not necessary, and the statement was self-evident. (2) Some students gave one or more examples that showed the proof held true there, saying this was sufficient for a proof. (3) Others showed the proof held for large numerical examples, therefore implying the proof always held. (4) Some students completely lost sight of the fact that they were actually trying to prove something, not solve something. When they saw the equals (“=”) sign, they assumed they just needed to solve an equation. (5) Very few students proved with algebra, followed by examples, and then the general case. It was apparent that the students who actually looked for the patterns in the examples were on their way to finding the generalized rule for the proof. The overall analysis of the results was this: “The data suggest that students need a much more robust vision of the role of proofs in supporting mathematical activity, making connections between mathematical ideas, and communicating findings. Students seem to have a limited repertoire of meth¬ods to use for proving those statements.” In order to fix this problem, teachers need to analyze their students’ written work carefully. In turn teachers will better be able to draw conclusions on the level of their students’ understanding and foundation in mathematical concepts.
I think this will allow for teachers to be able to lesson plan much more accurately. When teachers analyze their students’ work and find out where their weak points are, e.g. proving statements algebraically, then they will know which concepts need to be taught more fully in the classroom. Once students get a handle on proofs, they will have a much easier time with connecting and communicating their mathematical ideas. More specifically, once students gain a better understanding on how to prove mathematical concepts, they will have a greater “development of stu¬dent understanding of one another’s work and move toward the use of proof as a tool in exploring connec¬tions, relationships, and the veracity of mathematical statements and in communicating their findings.” I do not agree that giving examples as a solution to a proof is in reality a proof, but I can see why students would assume it to be true. They have not been educated on the matter. Teachers need to teach why giving only examples as answers to a proof is not sufficient for all cases. When teachers teach concepts only through examples, the processes may stick easier in the minds of the students. However, if teachers do give proofs behind the mathematical thinking, then students will have an easier time convincing others and themselves of the connections they make between their next mathematical ideas and the previous ones. It all comes down to the basic ideas discussed earlier this semester: the importance of relational teaching/understanding.
Tuesday, February 16, 2010
Blog #5 Response: Teaching Without Algorithms
There are a number of advantages that teaching math without teaching algorithms can bring. One that is outlined in Warrington's essay is that students learn to construct knowledge and find meaning in sophisticated areas that are difficult to truly understand. This is brought about through the careful creation of an environment where students are not afraid to discuss with each other and think for themselves. Students are allowed to be creative and are not "shackled by rules and conventions" that limit creativity. This was illustrated in the experience of Warrington giving the class the word problem about chocolate-covered peanuts.
Disadvantages of teaching math without teaching the algorithms are abundant. As Warrington presented each new question to the class, she would allow for as much discussion as was necessary for the students to get the right answers. If this was done in all classrooms, much material would have to be cut due to lack of time. I do not think cutting necessary topics from the curriculum is an ideal choice of action. A second disadvantage of teaching math without teaching the algorithms is that only the truly interested student will learn and participate. I do not feel every single student in this sort of classroom setting would be fully engaged or comfortable in thinking completely independently, and this could become frustrating and trivial for a number of students. Algorithms and easier to hold on to, and the less-creative mind may have an easier time getting through the course materials with them.
Disadvantages of teaching math without teaching the algorithms are abundant. As Warrington presented each new question to the class, she would allow for as much discussion as was necessary for the students to get the right answers. If this was done in all classrooms, much material would have to be cut due to lack of time. I do not think cutting necessary topics from the curriculum is an ideal choice of action. A second disadvantage of teaching math without teaching the algorithms is that only the truly interested student will learn and participate. I do not feel every single student in this sort of classroom setting would be fully engaged or comfortable in thinking completely independently, and this could become frustrating and trivial for a number of students. Algorithms and easier to hold on to, and the less-creative mind may have an easier time getting through the course materials with them.
Wednesday, February 10, 2010
Blog #4 Response: Constructivism
I think what Glasersfeld meant by constructing knowledge is to take in what you hear and see through your own lens of experience and calling that knowledge. A teacher may try to teach a certain subject, but the student may not interpret what is being taught the way the teacher meant it to be. The student constructs their own knowledge based on their own experiences. It is not acquiring knowledge since it is not the information being presented is never quite the same as what is actually being stored in the brain.
If I believed in constructivism, I would teach in a way to rigorously interpret the understanding of my students. If they are all taking in information in different ways, I want to know how much of that is correct or incorrect. I will do this though frequent quizzes, homework assignments, and class activities to see what is really being understood correctly by my students. Possibly it would be wise to have a couple of student-teacher interviews each year so that I might talk with my students one-on-one to see how they are interpreting what is being taught.
If I believed in constructivism, I would teach in a way to rigorously interpret the understanding of my students. If they are all taking in information in different ways, I want to know how much of that is correct or incorrect. I will do this though frequent quizzes, homework assignments, and class activities to see what is really being understood correctly by my students. Possibly it would be wise to have a couple of student-teacher interviews each year so that I might talk with my students one-on-one to see how they are interpreting what is being taught.
Saturday, January 23, 2010
Blog #3 Response: Erlwanger's Paper on Benny
Why would someone wish to publish an article on the deficiency of a child's understanding? I believe Erlwanger did this not only to exploit the deficiencies in the Individually Prescribed Instruction (IPI) program itself, but also to stress to the reader the importance of a teacher-student relationship. These two ideas are manifest (1) in the way Erlwanger portrays the unfortunate situation of Benny's tragic misconceptions of decimals and fractions, with the problem partly being due to the fact that the IPI program provided example problems without rules as to why the answers to the examples were such. Benny was forced to concoct rules to solve the problems; and since answers could be represented in different ways, Benny came to believe with confidence that all his answers and rules were correct. (2) Much of Benny's misconceptions could have been avoided if Benny but had one to supervise his progress at a more intimate level. Erlwanger said that "in IPI, teachers are prevented by their role perception from understanding the pupil's conception of what he is doing." The teachers just "go by the key ... what the key says" and don't actually critique the work themselves. If teachers were more involved throughout the course of Benny's past four years of education, they could have had a true perception of Benny's progress.
One main point that is still valid today is the fact that teachers need to be involved with their students' education. For an illustration, I had an algebra class in high school that ran solely on a computer program called Aleks. It allowed students to move at their own pace, be independent, and, similar to Benny's situation, it basically eliminated the role of the teacher. I sympathized with Benny when he said that finding an answer was a chase, a search for patterns. I do not feel I learned as much as I could have if I had had an actual teacher. I appreciated how I could move at my own pace, but when comparing that year to the next (same teacher but in an ordinary class setting), I realize I remember more things from the second year rather than the first. Now, I realize Aleks gave me a more short-term instrumental understanding, while having an actual teacher gave me a more permanent relational understanding.
One main point that is still valid today is the fact that teachers need to be involved with their students' education. For an illustration, I had an algebra class in high school that ran solely on a computer program called Aleks. It allowed students to move at their own pace, be independent, and, similar to Benny's situation, it basically eliminated the role of the teacher. I sympathized with Benny when he said that finding an answer was a chase, a search for patterns. I do not feel I learned as much as I could have if I had had an actual teacher. I appreciated how I could move at my own pace, but when comparing that year to the next (same teacher but in an ordinary class setting), I realize I remember more things from the second year rather than the first. Now, I realize Aleks gave me a more short-term instrumental understanding, while having an actual teacher gave me a more permanent relational understanding.
Friday, January 15, 2010
Blog #2 Response: Instrumental v. Relational Understanding
I found this reading assignment to be rather interesting. It brought up ideas I never have thought about before, yet ideas that actually have been quite applicable in my experience of schooling. While reading through the article, I was able to draw some distinctions between the the topics of discussion: instrumental and relational understanding.
Instrumental understanding is the type where students have been taught the rules and shortcuts to doing mathematics. Although they have these basic tools necessary to solve problems, they do not always know where they are applicable -- that is where relational understanding comes into play.
Relational understanding is more or less the concepts behind mathematical operations. It is the understanding of why, when, and how. It is possible one may get through a mathematical homework assignment with just the instrumental understanding, but they most likely will not know why they are doing the particular steps, or how to apply it to related problems. If one understands the instrumental AND relational, however, they more likely will be able to apply what they know to related problems. Also, they will be able to retain the knowledge for a longer period of time.
In my own experience, teachers who have taught instrumental understanding are the ones who, for example, would teach for you to just memorize the formula a^2+b^2=c^2 for whenever you see a right triangle. Those that have taught relational understanding are the ones who teach us the formula, but then also teach how the formula came about, what a, b, and c represent, why the formula works, and when to use it. When this is done, I have found that even when I have forgotten what the formula was, the relational understanding enables me to contrive it once again.
Instrumental understanding is the type where students have been taught the rules and shortcuts to doing mathematics. Although they have these basic tools necessary to solve problems, they do not always know where they are applicable -- that is where relational understanding comes into play.
Relational understanding is more or less the concepts behind mathematical operations. It is the understanding of why, when, and how. It is possible one may get through a mathematical homework assignment with just the instrumental understanding, but they most likely will not know why they are doing the particular steps, or how to apply it to related problems. If one understands the instrumental AND relational, however, they more likely will be able to apply what they know to related problems. Also, they will be able to retain the knowledge for a longer period of time.
In my own experience, teachers who have taught instrumental understanding are the ones who, for example, would teach for you to just memorize the formula a^2+b^2=c^2 for whenever you see a right triangle. Those that have taught relational understanding are the ones who teach us the formula, but then also teach how the formula came about, what a, b, and c represent, why the formula works, and when to use it. When this is done, I have found that even when I have forgotten what the formula was, the relational understanding enables me to contrive it once again.
Wednesday, January 6, 2010
Blog #1 Response
- What is mathematics? Mathematics is a language in which to calculate patterns and change, broken down into different categories such as algebra, geometry, and calculus, each represented by their unique symbols. Symbols on a page is no more mathematics than sheet music is music - it takes a learned mathematician to take those symbols off the page and apply them to create real mathematics, as it takes a learned musician to take notes off a page to create music.
- How do I learn mathematics best? I learn mathematics best by a) having a great teacher that can put across the ideas adequately, and b) studying notes and homework to solidify those ideas. If I do not review, much is lost from my learning.
- How will my students learn mathematics best? My students will learn mathematics best by me going through the material carefully and clearly, asking questions and giving quizzes frequently to make sure they understand the material. That way I know what they still need to work on, and when we can move on.
- What are some of the current practices in school mathematics classrooms that promote students' learning of mathematics? I think one of the main methods today is testing. The curriculum today is so focused on standardized testing and "getting the grade" that much of the curriculum is aimed towards passing those big tests. In order to pass the big tests, teachers give regular tests in class, which promote students to study and achieve.
- What are some of the current practices in school mathematics classrooms that are detrimental to students' learning of mathematics? A practice that is detrimental to students' learning today is when teachers ONLY focus on testing. What I mean by this is that some teachers give homework with the aim of giving students practice to prepare for the test, then they do not collect it with the aim to grade it. This generally leads to students slacking off up until right before the test, where there it no time left to adequately learn all the material. Small quizzes and graded homework is important for the learning of the students.
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