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Early algebra

PGoldenberg — Wed, 18 Nov 2009 02:54:44 GMT

When should algebra be taught?:

←Older revision		Revision as of 02:54, 18 November 2009
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	Asking “When should algebra be taught?” is like asking “Is technology harmful or helpful?” There are lots of technologies and lots of uses of them. Some are harmful; some are helpful. Refining the question—asking about a particular use of a particular technology for a particular purpose in particular contexts and at particular stages in one’s learning—makes the question researchable and potentially answerable. Similarly, there are many “algebras”—algebra the course, algebra the discipline, algebraic ideas, algebraic language, early algebra, “patterns, functions, and algebra”—and many different takes on the learning and teaching of each of these. Treating algebra as an indivisible whole obscures the options. It’s more useful to ask what ideas, logic, techniques, and habits of mind algebra entails, and then, about each of these, ask when and to what extent that one item can be learned with intellectual integrity and how a coherent whole can be woven out of these learnings. The answers we get are that some of these ideas do have to wait for eighth or ninth grade, but that others—even including aspects of algebraic language—are already there, early in the primary grades. Moreover, children who get to apply, refine, and strengthen those ideas and skills as they emerge gain the advantage.		Asking “When should algebra be taught?” is like asking “Is technology harmful or helpful?” There are lots of technologies and lots of uses of them. Some are harmful; some are helpful. Refining the question—asking about a particular use of a particular technology for a particular purpose in particular contexts and at particular stages in one’s learning—makes the question researchable and potentially answerable. Similarly, there are many “algebras”—algebra the course, algebra the discipline, algebraic ideas, algebraic language, early algebra, “patterns, functions, and algebra”—and many different takes on the learning and teaching of each of these. Treating algebra as an indivisible whole obscures the options. It’s more useful to ask what ideas, logic, techniques, and habits of mind algebra entails, and then, about each of these, ask when and to what extent that one item can be learned with intellectual integrity and how a coherent whole can be woven out of these learnings. The answers we get are that some of these ideas do have to wait for eighth or ninth grade, but that others—even including aspects of algebraic language—are already there, early in the primary grades. Moreover, children who get to apply, refine, and strengthen those ideas and skills as they emerge gain the advantage.

-	Any credible claim about habits of mind must surely accord with features of mind: children’s cognitive development. For a charmingly written scientific account of the ways that babies and young children think, read ''The Scientist in the Crib''.<ref name="Scientist">Gopnik, A., Meltzoff, A., and P. Kuhl. The scientist in the crib: what early learning tells us about the mind. New York: HarperCollins. 2000.</ref> The habits of mind approach to curriculum that we first described well over a decade ago<ref name="HoM1"> Cuoco, A., Goldenberg, E. P., & J. Mark. “Habits of mind: an organizing principle for mathematics curriculum” J. Math. Behav. 15(4):375-402. December, 1996.</ref><sup>,</sup><ref name="HoM2">Goldenberg, E. Paul. “‘Habits of mind’ as an organizer for the curriculum” J. of Education 178(1):13-34, Boston U. 1996. (Also “‘Hábitos de pensamento’ …”Educação e Matemática, 47 March/April, & 48 May/June, 1998.)</ref> and have continued to refine <ref name="HoM3">Goldenberg, E. Paul & N. Shteingold. “Mathematical Habits of Mind.” In Lester, F., et al., eds. Teaching Mathematics Through Problem Solving: prekindergarten–Grade 6. Reston, VA: NCTM. 2003. </ref><ref name="HoM4">Goldenberg, E. Paul & N. Shteingold “The case of ''Think Math!''” In Hirsch, Christian, ed., Perspectives on the design and development of school mathematics curricula. Reston, VA: NCTM. 2007. </ref><sup>,</sup><ref name="HoM5"> Cuoco, A., Goldenberg, E. P., and J. Mark. “Organizing a curriculum around mathematical habits of mind.” Mathematics Teacher. (submitted)</ref><sup>,</sup><ref name="HoM6"> Mark, J., Cuoco, A., and Goldenberg, E. P. “Developing mathematical habits of mind in the middle grades.” Mathematics Teaching in the Middle School. (submitted)</ref> does accord well with children’s thinking and became a central design principle behind ''Think Math!'' <ref name="TM">Education Development Center, Inc. (EDC). ''Think Math!'', a comprehensive K-5 curriculum. Boston: Houghton Mifflin Harcourt. 2008.</ref>, the newest NSF-supported elementary curriculum, developed at EDC.	+	Any credible claim about habits of mind must surely accord with features of mind: children’s cognitive development. For a charmingly written scientific account of the ways that babies and young children think, read ''The Scientist in the Crib''.<ref name="Scientist">Gopnik, A., Meltzoff, A., and P. Kuhl. The scientist in the crib: what early learning tells us about the mind. New York: HarperCollins. 2000.</ref> The habits of mind approach to curriculum that we first described well over a decade ago<ref name="HoM1"> Cuoco, A., Goldenberg, E. P., & J. Mark. “Habits of mind: an organizing principle for mathematics curriculum” J. Math. Behav. 15(4):375-402. December, 1996.</ref><sup>,</sup><ref name="HoM2">Goldenberg, E. Paul. “‘Habits of mind’ as an organizer for the curriculum” J. of Education 178(1):13-34, Boston U. 1996. (Also “‘Hábitos de pensamento’ …”Educação e Matemática, 47 March/April, & 48 May/June, 1998.)</ref> and have continued to refine<ref name="HoM3">Goldenberg, E. Paul & N. Shteingold. “Mathematical Habits of Mind.” In Lester, F., et al., eds. Teaching Mathematics Through Problem Solving: prekindergarten–Grade 6. Reston, VA: NCTM. 2003. </ref><sup>,</sup><ref name="HoM4">Goldenberg, E. Paul & N. Shteingold “The case of ''Think Math!''” In Hirsch, Christian, ed., Perspectives on the design and development of school mathematics curricula. Reston, VA: NCTM. 2007. </ref><sup>,</sup><ref name="HoM5"> Cuoco, A., Goldenberg, E. P., and J. Mark. “Organizing a curriculum around mathematical habits of mind.” Mathematics Teacher. (submitted)</ref><sup>,</sup><ref name="HoM6"> Mark, J., Cuoco, A., and Goldenberg, E. P. “Developing mathematical habits of mind in the middle grades.” Mathematics Teaching in the Middle School. (submitted)</ref> does accord well with children’s thinking and became a central design principle behind ''Think Math!''<ref name="TM">Education Development Center, Inc. (EDC). ''Think Math!'', a comprehensive K-5 curriculum. Boston: Houghton Mifflin Harcourt. 2008.</ref>, the newest NSF-supported elementary curriculum, developed at EDC.

	Recognizing, enhancing, and building on developmentally natural habits of mind lets us dissect algebra and sort the resulting bits and pieces in a developmentally natural way, while preserving the content, concepts, and skills that schools (and states, parents, workplaces, and colleges) expect. The fact that it is possible to organize algebraic ideas, logic, and techniques around the development of mind makes clear that we are truly talking about thinking—habits of mind—rather than “features of mathematics” or “idiosyncrasies of mathematicians.” This article describes two of these natural habits of mind.		Recognizing, enhancing, and building on developmentally natural habits of mind lets us dissect algebra and sort the resulting bits and pieces in a developmentally natural way, while preserving the content, concepts, and skills that schools (and states, parents, workplaces, and colleges) expect. The fact that it is possible to organize algebraic ideas, logic, and techniques around the development of mind makes clear that we are truly talking about thinking—habits of mind—rather than “features of mathematics” or “idiosyncrasies of mathematicians.” This article describes two of these natural habits of mind.

Length

PGoldenberg — Tue, 17 Nov 2009 15:10:31 GMT

←Older revision		Revision as of 15:10, 17 November 2009
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		+	{{bluebox\|<big>'''See [[length, width, height, depth]] for the use of these terms with two- or three-dimensional objects.'''</big>}}
		+
	__TOC__		__TOC__

-	''Length'' is used two ways: it is the name we use for the [[measure]] of a [[one-dimensional]] object, the name of the one measurement that can be made; when used in connection with two- or three-dimensional objects, its use is more casual. ~~(See [[length, width, height, depth]].)~~	+	''Length'' is used two ways: it is the name we use for the [[measure]] of a [[one-dimensional]] object, the name of the one measurement that can be made; when used in connection with two- or three-dimensional objects, its use is more casual.

Early algebra

PGoldenberg — Sat, 14 Nov 2009 02:44:44 GMT

When should algebra be taught?:

←Older revision		Revision as of 02:44, 14 November 2009
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	<center>'''Language, mathematics, and habits of mind'''</center>		<center>'''Language, mathematics, and habits of mind'''</center>
	<center>E. Paul Goldenberg, June Mark, and Al Cuoco</center>		<center>E. Paul Goldenberg, June Mark, and Al Cuoco</center>
-	<center>Education Development Center, Inc. (EDC)</center>	+	<center>Education Development Center, Inc. (EDC)<ref name = "NSF">This work was supported in part by the National Science Foundation, grant numbers ESI-0099093, DRL-0733015, and DRL-0917958. The opinions expressed are those of the authors and not necessarily those of the Foundation.</ref></center>
	<center>'''[http://thinkmath.edc.org/downloads/AlgebraOfLittleKids_final.pdf Download PDF version of this paper]'''</center>		<center>'''[http://thinkmath.edc.org/downloads/AlgebraOfLittleKids_final.pdf Download PDF version of this paper]'''</center>
	==When should algebra be taught?==		==When should algebra be taught?==
	Asking “When should algebra be taught?” is like asking “Is technology harmful or helpful?” There are lots of technologies and lots of uses of them. Some are harmful; some are helpful. Refining the question—asking about a particular use of a particular technology for a particular purpose in particular contexts and at particular stages in one’s learning—makes the question researchable and potentially answerable. Similarly, there are many “algebras”—algebra the course, algebra the discipline, algebraic ideas, algebraic language, early algebra, “patterns, functions, and algebra”—and many different takes on the learning and teaching of each of these. Treating algebra as an indivisible whole obscures the options. It’s more useful to ask what ideas, logic, techniques, and habits of mind algebra entails, and then, about each of these, ask when and to what extent that one item can be learned with intellectual integrity and how a coherent whole can be woven out of these learnings. The answers we get are that some of these ideas do have to wait for eighth or ninth grade, but that others—even including aspects of algebraic language—are already there, early in the primary grades. Moreover, children who get to apply, refine, and strengthen those ideas and skills as they emerge gain the advantage.		Asking “When should algebra be taught?” is like asking “Is technology harmful or helpful?” There are lots of technologies and lots of uses of them. Some are harmful; some are helpful. Refining the question—asking about a particular use of a particular technology for a particular purpose in particular contexts and at particular stages in one’s learning—makes the question researchable and potentially answerable. Similarly, there are many “algebras”—algebra the course, algebra the discipline, algebraic ideas, algebraic language, early algebra, “patterns, functions, and algebra”—and many different takes on the learning and teaching of each of these. Treating algebra as an indivisible whole obscures the options. It’s more useful to ask what ideas, logic, techniques, and habits of mind algebra entails, and then, about each of these, ask when and to what extent that one item can be learned with intellectual integrity and how a coherent whole can be woven out of these learnings. The answers we get are that some of these ideas do have to wait for eighth or ninth grade, but that others—even including aspects of algebraic language—are already there, early in the primary grades. Moreover, children who get to apply, refine, and strengthen those ideas and skills as they emerge gain the advantage.

-	Any credible claim about habits of mind must surely accord with features of mind: children’s cognitive development. For a charmingly written scientific account of the ways that babies and young children think, read ''The Scientist in the Crib''.<ref name="Scientist">Gopnik, A., Meltzoff, A., and P. Kuhl. The scientist in the crib: what early learning tells us about the mind. New York: HarperCollins. 2000.</ref> The habits of mind approach to curriculum that we first described well over a decade ago<ref name="HoM1"> Cuoco, A., Goldenberg, E. P., & J. Mark. “Habits of mind: an organizing principle for mathematics curriculum” J. Math. Behav. 15(4):375-402. December, 1996.</ref><sup>,</sup><ref name="HoM2">Goldenberg, E. Paul. “‘Habits of mind’ as an organizer for the curriculum” J. of Education 178(1):13-34, Boston U. 1996. (Also “‘Hábitos de pensamento’ …”Educação e Matemática, 47 March/April, & 48 May/June, 1998.)</ref> and have continued to refine <ref name="HoM3">Goldenberg, E. Paul & N. Shteingold. “Mathematical Habits of Mind.” In Lester, F., et al., eds. Teaching Mathematics Through Problem Solving: prekindergarten–Grade 6. Reston, VA: NCTM. 2003. </ref><ref name="HoM4">Goldenberg, E. Paul & N. Shteingold “The case of Think Math!” In Hirsch, Christian, ed., Perspectives on the design and development of school mathematics curricula. Reston, VA: NCTM. 2007. </ref><sup>,</sup><ref name="HoM5"> Cuoco, A., Goldenberg, E. P., and J. Mark. “Organizing a curriculum around mathematical habits of mind.” Mathematics Teacher. (submitted)</ref><sup>,</sup><ref name="HoM6"> Mark, J., Cuoco, A., and Goldenberg, E. P. “Developing mathematical habits of mind in the middle grades.” Mathematics Teaching in the Middle School. (submitted)</ref> does accord well with children’s thinking and became a central design principle behind Think Math! <ref name="TM">Education Development Center, Inc. (EDC). Think Math! comprehensive K-5 curriculum. Boston: Houghton Mifflin Harcourt. 2008.	+	Any credible claim about habits of mind must surely accord with features of mind: children’s cognitive development. For a charmingly written scientific account of the ways that babies and young children think, read ''The Scientist in the Crib''.<ref name="Scientist">Gopnik, A., Meltzoff, A., and P. Kuhl. The scientist in the crib: what early learning tells us about the mind. New York: HarperCollins. 2000.</ref> The habits of mind approach to curriculum that we first described well over a decade ago<ref name="HoM1"> Cuoco, A., Goldenberg, E. P., & J. Mark. “Habits of mind: an organizing principle for mathematics curriculum” J. Math. Behav. 15(4):375-402. December, 1996.</ref><sup>,</sup><ref name="HoM2">Goldenberg, E. Paul. “‘Habits of mind’ as an organizer for the curriculum” J. of Education 178(1):13-34, Boston U. 1996. (Also “‘Hábitos de pensamento’ …”Educação e Matemática, 47 March/April, & 48 May/June, 1998.)</ref> and have continued to refine <ref name="HoM3">Goldenberg, E. Paul & N. Shteingold. “Mathematical Habits of Mind.” In Lester, F., et al., eds. Teaching Mathematics Through Problem Solving: prekindergarten–Grade 6. Reston, VA: NCTM. 2003. </ref><ref name="HoM4">Goldenberg, E. Paul & N. Shteingold “The case of ''Think Math!''” In Hirsch, Christian, ed., Perspectives on the design and development of school mathematics curricula. Reston, VA: NCTM. 2007. </ref><sup>,</sup><ref name="HoM5"> Cuoco, A., Goldenberg, E. P., and J. Mark. “Organizing a curriculum around mathematical habits of mind.” Mathematics Teacher. (submitted)</ref><sup>,</sup><ref name="HoM6"> Mark, J., Cuoco, A., and Goldenberg, E. P. “Developing mathematical habits of mind in the middle grades.” Mathematics Teaching in the Middle School. (submitted)</ref> does accord well with children’s thinking and became a central design principle behind ''Think Math!'' <ref name="TM">Education Development Center, Inc. (EDC). ''Think Math!'', a comprehensive K-5 curriculum. Boston: Houghton Mifflin Harcourt. 2008.</ref>, the newest NSF-supported elementary curriculum, developed at EDC.
-	</ref>, the newest NSF-supported elementary curriculum, developed at EDC.	+

	Recognizing, enhancing, and building on developmentally natural habits of mind lets us dissect algebra and sort the resulting bits and pieces in a developmentally natural way, while preserving the content, concepts, and skills that schools (and states, parents, workplaces, and colleges) expect. The fact that it is possible to organize algebraic ideas, logic, and techniques around the development of mind makes clear that we are truly talking about thinking—habits of mind—rather than “features of mathematics” or “idiosyncrasies of mathematicians.” This article describes two of these natural habits of mind.		Recognizing, enhancing, and building on developmentally natural habits of mind lets us dissect algebra and sort the resulting bits and pieces in a developmentally natural way, while preserving the content, concepts, and skills that schools (and states, parents, workplaces, and colleges) expect. The fact that it is possible to organize algebraic ideas, logic, and techniques around the development of mind makes clear that we are truly talking about thinking—habits of mind—rather than “features of mathematics” or “idiosyncrasies of mathematicians.” This article describes two of these natural habits of mind.

EarlyAlgebra

PGoldenberg — Sat, 14 Nov 2009 02:36:20 GMT

Redirecting to Early algebra

New page

#REDIRECT [[Early algebra]]

Early algebra

PGoldenberg — Sat, 14 Nov 2009 02:35:35 GMT

←Older revision		Revision as of 02:35, 14 November 2009
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-	~~'''[http://thinkmath.edc.org/downloads/AlgebraOfLittleKids_final.pdf Download PDF version of this paper]'''~~	+	<center><big>'''The algebra of little kids'''</big></center>
-		+	<center>'''Language, mathematics, and habits of mind'''</center>
-	<center><big>The algebra of little kids</big></center>	+
-	<center>~~A mathematical-~~habits-of-mind ~~perspective on elementary school~~</center>	+
	<center>E. Paul Goldenberg, June Mark, and Al Cuoco</center>		<center>E. Paul Goldenberg, June Mark, and Al Cuoco</center>
	<center>Education Development Center, Inc. (EDC)</center>		<center>Education Development Center, Inc. (EDC)</center>
		+	<center>'''[http://thinkmath.edc.org/downloads/AlgebraOfLittleKids_final.pdf Download PDF version of this paper]'''</center>
	==When should algebra be taught?==		==When should algebra be taught?==
	Asking “When should algebra be taught?” is like asking “Is technology harmful or helpful?” There are lots of technologies and lots of uses of them. Some are harmful; some are helpful. Refining the question—asking about a particular use of a particular technology for a particular purpose in particular contexts and at particular stages in one’s learning—makes the question researchable and potentially answerable. Similarly, there are many “algebras”—algebra the course, algebra the discipline, algebraic ideas, algebraic language, early algebra, “patterns, functions, and algebra”—and many different takes on the learning and teaching of each of these. Treating algebra as an indivisible whole obscures the options. It’s more useful to ask what ideas, logic, techniques, and habits of mind algebra entails, and then, about each of these, ask when and to what extent that one item can be learned with intellectual integrity and how a coherent whole can be woven out of these learnings. The answers we get are that some of these ideas do have to wait for eighth or ninth grade, but that others—even including aspects of algebraic language—are already there, early in the primary grades. Moreover, children who get to apply, refine, and strengthen those ideas and skills as they emerge gain the advantage.		Asking “When should algebra be taught?” is like asking “Is technology harmful or helpful?” There are lots of technologies and lots of uses of them. Some are harmful; some are helpful. Refining the question—asking about a particular use of a particular technology for a particular purpose in particular contexts and at particular stages in one’s learning—makes the question researchable and potentially answerable. Similarly, there are many “algebras”—algebra the course, algebra the discipline, algebraic ideas, algebraic language, early algebra, “patterns, functions, and algebra”—and many different takes on the learning and teaching of each of these. Treating algebra as an indivisible whole obscures the options. It’s more useful to ask what ideas, logic, techniques, and habits of mind algebra entails, and then, about each of these, ask when and to what extent that one item can be learned with intellectual integrity and how a coherent whole can be woven out of these learnings. The answers we get are that some of these ideas do have to wait for eighth or ninth grade, but that others—even including aspects of algebraic language—are already there, early in the primary grades. Moreover, children who get to apply, refine, and strengthen those ideas and skills as they emerge gain the advantage.
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	Having confidence that [[Image:Hands0.png\|50px]] and [[Image:Hands2.png\|50px]] represent the same quantity is not the same as knowing the commutative property of addition. The commutative property is not about the arrangement of physical objects in space, but about the behavior of a particular element (here, the + sign) in a formal syntactic system of written symbols. In some contexts, children can make perfect sense out of written symbols—even significant parts of algebraic notation—but most young children cannot make sense of formal operations on a string of symbols. So, at this stage, commutativity remains largely an intuitively obvious idea about the “physics of mathematics”: the nature of aggregation, not the nature of symbols. Even so, we, as educators, can support the young child’s logic better if we recognize that it is already relying on the underlying ideas that formal mathematics will later codify. The fact that children see that the principle applies regardless of the numbers means that it captures the essential algebraic aspect of the structure of addition that commutativity is about.		Having confidence that [[Image:Hands0.png\|50px]] and [[Image:Hands2.png\|50px]] represent the same quantity is not the same as knowing the commutative property of addition. The commutative property is not about the arrangement of physical objects in space, but about the behavior of a particular element (here, the + sign) in a formal syntactic system of written symbols. In some contexts, children can make perfect sense out of written symbols—even significant parts of algebraic notation—but most young children cannot make sense of formal operations on a string of symbols. So, at this stage, commutativity remains largely an intuitively obvious idea about the “physics of mathematics”: the nature of aggregation, not the nature of symbols. Even so, we, as educators, can support the young child’s logic better if we recognize that it is already relying on the underlying ideas that formal mathematics will later codify. The fact that children see that the principle applies regardless of the numbers means that it captures the essential algebraic aspect of the structure of addition that commutativity is about.

-	===Logical precursors of the distributive property of multiplication over addition: ===	+	===Logical precursors of the distributive property of multiplication over addition ===

	Pick a number. Multiply it by 5; also multiply it (your original number) by 2; now add those results. You get the same answer you’d get if you multiplied your original number by 7. The [[distributive property]], a general statement of that fact, is possibly the most central idea in elementary arithmetic, key to understanding the algorithms, at the core of fluent mental calculations (e.g., 102 × 27 can be computed in two parts, as 100 × 27 + 2 × 27), and the logical basis for many “rules” of algebra that might otherwise seem arbitrary.		Pick a number. Multiply it by 5; also multiply it (your original number) by 2; now add those results. You get the same answer you’d get if you multiplied your original number by 7. The [[distributive property]], a general statement of that fact, is possibly the most central idea in elementary arithmetic, key to understanding the algorithms, at the core of fluent mental calculations (e.g., 102 × 27 can be computed in two parts, as 100 × 27 + 2 × 27), and the logical basis for many “rules” of algebra that might otherwise seem arbitrary.