Math Relearning/Math Student Simulator/Introduction - Revision history

Andy Culbertson: Added the Essays category.

2016-09-27T12:13:44Z

Added the Essays category.

← Older revision		Revision as of 12:13, 27 September 2016
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	[[Category:Math Relearning]]		[[Category:Math Relearning]]
			[[Category:Essays]]
	[[Category:Complete]]		[[Category:Complete]]

Andy Culbertson: Added a point to the problem solving approach.

2016-09-27T12:11:56Z

Added a point to the problem solving approach.

← Older revision		Revision as of 12:11, 27 September 2016
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	* ''Teaching.'' Explaining things to a student does make them clearer and more memorable to the teacher. But this is essentially what I'm doing when I program. I'm teaching the computer. And in some ways the computer asks better questions than a human student because it needs every logical detail fed to it or else it gives the wrong result or crashes the program.		* ''Teaching.'' Explaining things to a student does make them clearer and more memorable to the teacher. But this is essentially what I'm doing when I program. I'm teaching the computer. And in some ways the computer asks better questions than a human student because it needs every logical detail fed to it or else it gives the wrong result or crashes the program.
	* ''Problem solving.'' Programming is 90% ~~problem solving~~. Much of the time it's problems related to the content, and the problems reveal a lack of clarity in my understanding that gradually resolves as I figure out how the computer should carry out its task. But my program won't give me real-world situations to apply math to, so I'll need to supplement my learning with more typical word problems from regular learning materials.		* ''Problem solving.'' Problem solving is an excellent learning method, but on its own it wouldn't give me a map or even easily reviewable notes. Programming has problem solving covered anyway, because that's 90% of a programmer's work. Much of the time it's problems related to the content, and the problems reveal a lack of clarity in my understanding that gradually resolves as I figure out how the computer should carry out its task. But my program won't give me real-world situations to apply math to, so I'll need to supplement my learning with more typical word problems from regular learning materials.
	* ''Diagrams.'' I would love to be able to represent math knowledge as a series of incremental diagrams that can be understood and reviewed with extreme ease. I hope I can do that someday. But I'd have to find or invent a visual language that could express everything adequately, whereas I'm familiar with programming already. Plus not many diagrams are executable, so they wouldn't carry the advantages of executability. Programs in visual programming languages are executable diagrams, but I don't know those languages, and I suspect they're not the kinds of diagrams I have in mind. Making more diagram types executable is on my future project list.		* ''Diagrams.'' I would love to be able to represent math knowledge as a series of incremental diagrams that can be understood and reviewed with extreme ease. I hope I can do that someday. But I'd have to find or invent a visual language that could express everything adequately, whereas I'm familiar with programming already. Plus not many diagrams are executable, so they wouldn't carry the advantages of executability. Programs in visual programming languages are executable diagrams, but I don't know those languages, and I suspect they're not the kinds of diagrams I have in mind. Making more diagram types executable is on my future project list.
	* ''Pseudocode.'' It's true that if I wrote my program in pseudocode instead of real code, I could approach the explicitness of regular programming without needing so much of the overhead. But it also wouldn't give me the enforced clarity of actual execution. I might leave out important details or make errors in logic that I'd never notice. It's too easy to fudge with pseudocode.		* ''Pseudocode.'' It's true that if I wrote my program in pseudocode instead of real code, I could approach the explicitness of regular programming without needing so much of the overhead. But it also wouldn't give me the enforced clarity of actual execution. I might leave out important details or make errors in logic that I'd never notice. It's too easy to fudge with pseudocode.

Andy Culbertson: Fixed the Number Sense and Memory Improvement links.

2016-09-27T07:28:31Z

Fixed the Number Sense and Memory Improvement links.

← Older revision		Revision as of 07:28, 27 September 2016
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	== Current methods ==		== Current methods ==

	Any learning, especially if, like math, it involves developing skills, will require several kinds of learning activities and multiple passes over the material so the information and skills can be processed into long-term memory. See [[~~A_Framework_and_Agenda_for_Memory_Improvement~~]] for more on that. But if a rapid pace of learning is one of your goals, you have to select your learning activities carefully so you learn adequately without wasting a lot of time. You also have to select your material's degree of breadth and depth carefully, since more of either will take a longer time to learn.		Any learning, especially if, like math, it involves developing skills, will require several kinds of learning activities and multiple passes over the material so the information and skills can be processed into long-term memory. See [[A Framework and Agenda for Memory Improvement]] for more on that. But if a rapid pace of learning is one of your goals, you have to select your learning activities carefully so you learn adequately without wasting a lot of time. You also have to select your material's degree of breadth and depth carefully, since more of either will take a longer time to learn.

	What learning activities have I been trying so far? Mainly I've been reading books and other documents that explain math to teachers so they can pass on the understanding to their students. But since a lot of these documents don't explain things in quite the order and manner I'm looking for, I've been analyzing the content to articulate, separate, and trace the conceptual threads as they develop through the grades. Basically I've been expanding and shaping the content from my sources into a structure that I think will give me the kind of knowledge and skill I need. I've been writing my insights as I go. I did a pass over the whole K-12 curriculum to get an overview and then started over with pre-K to learn in detail. As I've come across procedures I'd like to perform with more fluency, I've come up with exercises I think would help me, since I don't need the kinds of exercises for young children the curriculum offers at the early levels.		What learning activities have I been trying so far? Mainly I've been reading books and other documents that explain math to teachers so they can pass on the understanding to their students. But since a lot of these documents don't explain things in quite the order and manner I'm looking for, I've been analyzing the content to articulate, separate, and trace the conceptual threads as they develop through the grades. Basically I've been expanding and shaping the content from my sources into a structure that I think will give me the kind of knowledge and skill I need. I've been writing my insights as I go. I did a pass over the whole K-12 curriculum to get an overview and then started over with pre-K to learn in detail. As I've come across procedures I'd like to perform with more fluency, I've come up with exercises I think would help me, since I don't need the kinds of exercises for young children the curriculum offers at the early levels.
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	Where would I find such conceptual explanations of math? It turns out education researchers have spent decades coming up with them, and their findings have been condensed into resources I can access, such as Chapin and Johnson's <i>[http://www.worldcat.org/title/math-matters-understanding-the-math-you-teach-grades-k-8/oclc/63164894 Math Matters]</i>. However, I had trouble finding a building block approach to conceptual math until I looked into the Common Core Standards and their associated resources, primarily the [http://achievethecore.org/page/254/progressions-documents-for-the-common-core-state-standards-for-mathematics Progressions] and the [https://www.engageny.org/common-core-curriculum EngageNY curriculum]. While building up from basic concepts and experiences is the way children are taught math, adults who are being taught to teach math conceptually don't need things explained in such a stepwise fashion because they already have a broad knowledge of math and only need the concepts better integrated. So, for example, the early books I looked at thought nothing of analyzing place value in terms of exponents, even though children need to know something about place value years before they're ready to know anything about exponents. Common Core is focused on how to teach conceptual math to children rather than adults, so its developers were careful to limit their explanations of math concepts to the level of understanding available at each stage. How well they succeeded is a matter of debate, but I think they did well enough for my purposes.		Where would I find such conceptual explanations of math? It turns out education researchers have spent decades coming up with them, and their findings have been condensed into resources I can access, such as Chapin and Johnson's <i>[http://www.worldcat.org/title/math-matters-understanding-the-math-you-teach-grades-k-8/oclc/63164894 Math Matters]</i>. However, I had trouble finding a building block approach to conceptual math until I looked into the Common Core Standards and their associated resources, primarily the [http://achievethecore.org/page/254/progressions-documents-for-the-common-core-state-standards-for-mathematics Progressions] and the [https://www.engageny.org/common-core-curriculum EngageNY curriculum]. While building up from basic concepts and experiences is the way children are taught math, adults who are being taught to teach math conceptually don't need things explained in such a stepwise fashion because they already have a broad knowledge of math and only need the concepts better integrated. So, for example, the early books I looked at thought nothing of analyzing place value in terms of exponents, even though children need to know something about place value years before they're ready to know anything about exponents. Common Core is focused on how to teach conceptual math to children rather than adults, so its developers were careful to limit their explanations of math concepts to the level of understanding available at each stage. How well they succeeded is a matter of debate, but I think they did well enough for my purposes.

	Why do I need to do all this work analyzing and expanding these sources' content? To form a good map, all the necessary information needs to be present, explicit, and direct. My sources don't really share my goal of building a complete mathematical edifice, so their information doesn't always follow this pattern. This was especially true of the pre-Common Core sources. Some of the information I wanted to know was missing, so as I read and tried to fill in my picture of math, I had to answer certain questions myself, such as what is a number? This was fun but took a lot of time, though in the case of defining numbers I was pleased with the results (see [[/~~Math_Relearning/Number_Sense~~]]). Fortunately, the Common Core sources answered a lot more questions for me.		Why do I need to do all this work analyzing and expanding these sources' content? To form a good map, all the necessary information needs to be present, explicit, and direct. My sources don't really share my goal of building a complete mathematical edifice, so their information doesn't always follow this pattern. This was especially true of the pre-Common Core sources. Some of the information I wanted to know was missing, so as I read and tried to fill in my picture of math, I had to answer certain questions myself, such as what is a number? This was fun but took a lot of time, though in the case of defining numbers I was pleased with the results (see [[Math Relearning/Number Sense]]). Fortunately, the Common Core sources answered a lot more questions for me.

	But then as I began reading EngageNY, I thought it'd be helpful to spell out all the propositional concepts in each module so I could trace the progression of mathematical ideas in a clear and logical order. Many of these propositions were implicit in the source material, since the logical progression I had in mind took very small steps, similar to a proof. And statements in any work are rarely written as directly as I'd like. Spelling these out or rewriting them took only a moderate amount of time, but it still felt like too much considering the amount of uncertainty I felt about the result. More on that in the Notes section below.		But then as I began reading EngageNY, I thought it'd be helpful to spell out all the propositional concepts in each module so I could trace the progression of mathematical ideas in a clear and logical order. Many of these propositions were implicit in the source material, since the logical progression I had in mind took very small steps, similar to a proof. And statements in any work are rarely written as directly as I'd like. Spelling these out or rewriting them took only a moderate amount of time, but it still felt like too much considering the amount of uncertainty I felt about the result. More on that in the Notes section below.

Andy Culbertson: Raised the Reservations section a level.

2016-09-27T07:21:20Z

Raised the Reservations section a level.

← Older revision		Revision as of 07:21, 27 September 2016
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	Third, this project can act as a framework for learning projects in other subject areas. I'm picturing a multi-level framework containing a software framework, an information design framework, a learning process framework, and maybe others. A general notetaking app based on these models is a tempting future project.		Third, this project can act as a framework for learning projects in other subject areas. I'm picturing a multi-level framework containing a software framework, an information design framework, a learning process framework, and maybe others. A general notetaking app based on these models is a tempting future project.

	=== Reservations ===		== Reservations ==

	With a description of programming's characteristics and benefits under our belts, it's worthwhile to look at some objections someone might make to the notion of backing up in my math relearning project to take this new approach. Given my goal of learning a lot of math deeply and quickly, is it really a good idea?		With a description of programming's characteristics and benefits under our belts, it's worthwhile to look at some objections someone might make to the notion of backing up in my math relearning project to take this new approach. Given my goal of learning a lot of math deeply and quickly, is it really a good idea?

Andy Culbertson: Added the article.

2016-09-27T07:19:30Z

Added the article.

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