Erlwanger believes that understanding the reasoning behind the rules is fundamental in learning mathematics. Benny was a sixth-grade student who had been using the IPI method of math since second grade. The IPI method is basically following example problems and then doing similar problems in order to learn a specific rule. Benny has been successful, even one of the top of his class, but when questioned, Benny states incorrect rules. Repeatedly Erlwanger mentions how Benny was only interested with the rules of math, in this case fractions. Benny had been discovering "magical" rules since he began the program in second grade. Benny thought they had been invented by someone and made to be the ultimate standard for doing mathematics. Erlwanger wants us to know that relational understanding is the most beneficial. We need to know the how and the why of the rules in order to completely understand the rules and be able to use them consistently. Benny said he was on a wild goose chase to find the answers in the key because he thought there were multiple right answers, and he was just trying to find the right one. The IPI program had led him to teach himself incorrect rules about fractions and decimals that he blindly followed. He didn't understand that there should be one correct answer. Benny also thinks that all the rules are set in stone, and that there is only one way to do something. If he had a relational understanding, however, he would be able to find more than one way to do a problem because he would understand how the rule worked.
The whole concept of relational understanding and knowing how a rule works is still valid today. When we know how a learned rule works, we can apply it in new situations and not just in the specific case where it was learned. We also might be able to apply a different rule to a given situation because we fully understand how the situation works. There definitely could be more students like Benny who, for one reason or another, end up teaching themselves incorrect rules but get by with high grades. This should also be applied today. Simply because a student has a high grade does not mean they understand the material and are able to get the correct answer all of the time. Benny was one of the top students in his class, but he had gotten by without someone noticing that he didn't actually understand the material. Today we should be more aware of this and catch it at the beginning. By the time Erlwanger talked to Benny, he had been doing this for four years and could not change his thinking even when Erlwanger went back and tried to teach him. We could prevent cases like Benny if we could assure that the students relationally understand the rules, which could be difficult to do.
Monday, January 25, 2010
Thursday, January 14, 2010
Relational Understanding and Instrumental Understanding
A major question in mathematics education today is what qualifies truly understanding the material. There are generally two different approaches to understanding: instrumental understanding and relational understanding. Instrumental understanding is having a mathematical rule and being able to use and manipulate it. Relational understanding is having a mathematical rule, knowing how to use it, and knowing why it works.
From the definitions given, relational understanding includes instrumental understanding and more. Instrumental is simply knowing and applying the rule, while relational is knowing and applying the rule while also being able to know why a rule works and connect one rule with another. Both types of understanding give the correct answers, but relational is much more extensive.
Although relational understanding is often thought of to be a better alternative to instrumental understanding, there are advantages and disadvantages of both. Often the advantages of one type of understanding are the disadvantages of the other. Instrumental has three main advantages. The first is that it is easier to understand, often to a great extent. Some topics are difficult to grasp and can much sooner be learned through just using rules and set computations than through knowing why something works the way it does. The second is that the positive results are instantaneous. Once the rule or algorithm is learned, the student can use it to do many problems in that format and get the correct answers. The third advantage of instrumental learning goes along with the second in that the correct answers can be obtained very quickly and consistently. Instead of relational learning, where the thought process is longer in trying to understand the problem, in instrumental learning, once the student is able to follow a rule or algorithm, they can do problems that apply to that rule rapidly and always get the right answers. All three of these are disadvantages of relational understanding. Having to think about why something works the way it does or how it works is much more difficult than using rules. It is also more time consuming and involves a lot more thought. This applies to both teaching relationally and actually doing the problems. For each problem, the student is relying on his or her understanding of the idea to solve the problem, which may be correct or incorrect. When the students simply learn and use rules that are always there for them (instrumental), they will always get the right answer. There are four general advantages to relational understanding that are disadvantages to instrumental understanding. The first is that relational is easily adjusted when a new task is introduced. The students can take what they have already learned and apply and adjust it to the new idea because they understand why it works. This does not work with instrumental because the students can only do problems that fit within the rule they learned. Although a rule may be very similar to one they already learned, they will not discover it on their own because they don't know why the first rule works. The second advantage to relational is that students can more easily remember what they've been taught. If they know general ideas about why computations work the way they do, they can connect them with other ideas and therefore remember them more easily. On the other hand, in instrumental understanding, a student has to remember many separate rules that seem unconnected from each other. The third is that relational learning turns into its own goal. External benefits are not needed as much as they are with instrumental because just through fully understanding the idea the student is rewarded. The last advantage is relational knowledge naturally grows. Once a student understands relationally what has been taught, he or she will want to expand that knowledge and look for new concepts to apply it to. This does not happen in instrumental where the student learns a new rule and is content until given the next rule because he or she does not know how to make the connections to expand his or her knowledge.
From the definitions given, relational understanding includes instrumental understanding and more. Instrumental is simply knowing and applying the rule, while relational is knowing and applying the rule while also being able to know why a rule works and connect one rule with another. Both types of understanding give the correct answers, but relational is much more extensive.
Although relational understanding is often thought of to be a better alternative to instrumental understanding, there are advantages and disadvantages of both. Often the advantages of one type of understanding are the disadvantages of the other. Instrumental has three main advantages. The first is that it is easier to understand, often to a great extent. Some topics are difficult to grasp and can much sooner be learned through just using rules and set computations than through knowing why something works the way it does. The second is that the positive results are instantaneous. Once the rule or algorithm is learned, the student can use it to do many problems in that format and get the correct answers. The third advantage of instrumental learning goes along with the second in that the correct answers can be obtained very quickly and consistently. Instead of relational learning, where the thought process is longer in trying to understand the problem, in instrumental learning, once the student is able to follow a rule or algorithm, they can do problems that apply to that rule rapidly and always get the right answers. All three of these are disadvantages of relational understanding. Having to think about why something works the way it does or how it works is much more difficult than using rules. It is also more time consuming and involves a lot more thought. This applies to both teaching relationally and actually doing the problems. For each problem, the student is relying on his or her understanding of the idea to solve the problem, which may be correct or incorrect. When the students simply learn and use rules that are always there for them (instrumental), they will always get the right answer. There are four general advantages to relational understanding that are disadvantages to instrumental understanding. The first is that relational is easily adjusted when a new task is introduced. The students can take what they have already learned and apply and adjust it to the new idea because they understand why it works. This does not work with instrumental because the students can only do problems that fit within the rule they learned. Although a rule may be very similar to one they already learned, they will not discover it on their own because they don't know why the first rule works. The second advantage to relational is that students can more easily remember what they've been taught. If they know general ideas about why computations work the way they do, they can connect them with other ideas and therefore remember them more easily. On the other hand, in instrumental understanding, a student has to remember many separate rules that seem unconnected from each other. The third is that relational learning turns into its own goal. External benefits are not needed as much as they are with instrumental because just through fully understanding the idea the student is rewarded. The last advantage is relational knowledge naturally grows. Once a student understands relationally what has been taught, he or she will want to expand that knowledge and look for new concepts to apply it to. This does not happen in instrumental where the student learns a new rule and is content until given the next rule because he or she does not know how to make the connections to expand his or her knowledge.
Tuesday, January 5, 2010
Blog Entry #1
Mathematics is everywhere around us. It is a very precise science that relies on formulas and concepts. I'm having a hard time coming up with an exact definition, but it includes measurements, numbers, and symbols. Like other sciences, it is used to explain natural laws.
I learn math the best when a teacher explains the concept and why it is that way. I like to know why we use certain formulas because it helps me to better understand what we're doing. Even though proofs can be long, they are also very helpful when one topic is building on another. I also learn by example. I like to be shown how to do something before I really know how to do it because otherwise I might end up practicing doing it incorrectly. I like to be shown the complete process while someone can explain why they are doing each step. It is not beneficial when someone just does the problem without any explanation.
My students will learn mathematics through a step-by-step explanation given of what the concept is followed by several examples of how to apply the concept. I've found that practicing the concepts is beneficial to remembering and fully understanding the ideas taught. My students will also learn through the opportunity to explain the concepts to each other. That way they will know what part they understand and what part they are still having trouble with.
One of the most important things is having ample time to ask questions. Like we do here, my high school had a math lab where students could go to get help. This provided a way for students to get individualized assistance with questions they had and ideas they were confused with. This can happen if homework time is made available during the class, as well. Students will also learn better if they are interested in the topic. Finding ways, such as news articles or example problems, to connect mathematical concepts to real life promotes students' learning in mathematics.
Some current practices that are detrimental to students' learning of mathematics is when only the teacher is speaking and explaining the concepts with a wall between them and the students. The classroom needs to be an interactive environment and open to questions from the students. Often teachers will just try to get the class through some material until they reach their favorite part of the curriculum without explaining the importance of what had been taught so far. Teachers need to try to keep everything relevant, as much as possible.
I learn math the best when a teacher explains the concept and why it is that way. I like to know why we use certain formulas because it helps me to better understand what we're doing. Even though proofs can be long, they are also very helpful when one topic is building on another. I also learn by example. I like to be shown how to do something before I really know how to do it because otherwise I might end up practicing doing it incorrectly. I like to be shown the complete process while someone can explain why they are doing each step. It is not beneficial when someone just does the problem without any explanation.
My students will learn mathematics through a step-by-step explanation given of what the concept is followed by several examples of how to apply the concept. I've found that practicing the concepts is beneficial to remembering and fully understanding the ideas taught. My students will also learn through the opportunity to explain the concepts to each other. That way they will know what part they understand and what part they are still having trouble with.
One of the most important things is having ample time to ask questions. Like we do here, my high school had a math lab where students could go to get help. This provided a way for students to get individualized assistance with questions they had and ideas they were confused with. This can happen if homework time is made available during the class, as well. Students will also learn better if they are interested in the topic. Finding ways, such as news articles or example problems, to connect mathematical concepts to real life promotes students' learning in mathematics.
Some current practices that are detrimental to students' learning of mathematics is when only the teacher is speaking and explaining the concepts with a wall between them and the students. The classroom needs to be an interactive environment and open to questions from the students. Often teachers will just try to get the class through some material until they reach their favorite part of the curriculum without explaining the importance of what had been taught so far. Teachers need to try to keep everything relevant, as much as possible.
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