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The brief is grounded in field trials of the RAMPUP summer bridge program, using student examples of reasoning about equivalence, including early understandings of “equals,” proportionality with scaling, and movement toward algebraic notions.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/engaging-english-learners-with-equivalence-as-a-crosscutting-concept-in-mathematics-brief/586047/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/engaging-english-learners-with-equivalence-as-a-crosscutting-concept-in-mathematics-brief/586047.png","ImageObject",300,407,{"name":42,"@type":43},"Jordan Avery","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-15",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",7,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What does the brief say about equivalence as a crosscutting concept in mathematics?","Question",{"text":63,"@type":64},"Equivalence connects ideas across algebra, geometry, and statistics throughout the K–12 span. The brief treats equivalence as something learners can explore while developing language to connect mathematical ideas.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Why is the equals sign challenging for some early learners?",{"text":68,"@type":64},"Some young children interpret “=” as meaning “the answer is coming,” which can lead to incorrect reasoning in arithmetic equations. The brief emphasizes building an understanding of “=” as equality between two expressions.",{"name":70,"@type":61,"acceptedAnswer":71},"How is the brief supported by classroom or program evidence?",{"text":72,"@type":64},"The ideas are informed by field trials from the RAMPUP summer bridge program. 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Students who are bureaucratically designated as “English Learners” can explore equivalence across their K–12 academic careers while developing language to connect ideas . Drawing on what is known generally from theory and research about equivalence, this brief identifies what educators can do to challenge and support English Learners in this area . Equivalence has the potential to be a crosscutting concept that connects ideas from algebra, geometry, and statistics .  \nThe ideas explored in this brief emerged from field trials of a summer bridge program designed to challenge and support rising 9th grade students making the transition from middle to high school. The authors of this brief were part of the team that developed the bridge program, Reimagining and Amplifying Mathematics Participation, Understanding, and Practices (RAMPUP), through the National Research and Development Center to Improve Education for Secondary English Learners at WestEd. RAMPUP is a 3-week program with three modules: patterns, networks, and equivalence (Chu & Hamburger, 2022) . Drawing from the RAMPUP field trials through which we iteratively developed the program, this brief gives examples of students’ reasoning about and explorations of equivalence.  \nEngaging English Learners With Equivalence as a Crosscutting Concept in Mathematics 2  \nThe brief is organized into the following sections:  \n• growing early understandings of “equals”  \n• stretching reasoning about proportionality with scaling  \n• generalizing to algebraic notions of equivalence  \n• developing diverse approaches to reasoning about equivalence  \n• examining examples from RAMPUP  \n• considering what educators can do to challenge and support English Learners  \nGrowing Early Understandings of “Equals”  \nOne of the foundational understandings in early mathematics is whether two arithmetic expressions are equal. The symbol for this relationship is the equals sign (“=”) . Often, young children have learned that the equals sign means “the answer is coming,” as is typical of arithmetic exercises such as  \n2 + 3 =  \nAs a consequence, sometimes students will do “chain arithmetic” in which they record operations serially, as in the following:  \n2 + 3 = 5 + 9 = 14  \nHere, the equals sign is being used to demonstrate that the answer to an operation is coming, but the three expressions are not all equal (Carpenter et al. , 2003) .  \nStudents who conceptualize the equals sign in this way may encounter problems with arithmetic equations such as the following:  \n5 + 9 =   + 8  \nWith an “answer coming” interpretation of the equals sign, students may say the number 14 belongs in the missing space.  \nInstead, it is more generative for students to develop an understanding of “=” as the relationship of equality between two arithmetic expressions . If they understand the equals sign as denoting that relationship, they are able to solve the arithmetic equation above (5 + 9 =   + 8) with multiple forms of reasoning, such as  \n\n| Engaging English Learners With Equivalence as a Crosscutting Concept in Mathematics |  |  |  |  | 3 |\n| --- | --- | --- | --- | --- | --- |\n| • having the same value: “5 + 9 equals 14. And 6 + 8 also equals 14, so 5 + 9 = 6 + 8.”\u003Cbr>• relating the two sides: “I see the 9 and 8 are close. The 9 is bigger by 1, so the missing number on the other side is bigger by 1 as well. The missing number is 6.”\u003Cbr>• transforming both sides: “14 – 6 = 8 because if I add 6 to both sides, 14 = 8 + 6, which is true.”\u003Cbr>Stretching Reasoning About Proportionality\u003Cbr>With Scaling \u003Cbr>Indeed, as students move into late elementary and mi","cbCainsaw1Uq1HUI","https://ap.wps.com/l/cbCainsaw1Uq1HUI","pdf",345887,11,"English","# Growing Early Understandings of \"Equals\"\n## Stretching Reasoning About Proportionality with Scaling\n# Generalizing to Algebraic Notions of Equivalence\n## Developing Diverse Approaches to Reasoning About Equivalence\n## Examining Examples from RAMPUP\n## Considering What Educators Can Do","[{\"question\":\"What does the brief say about equivalence as a crosscutting concept in mathematics?\",\"answer\":\"Equivalence connects ideas across algebra, geometry, and statistics throughout the K–12 span. The brief treats equivalence as something learners can explore while developing language to connect mathematical ideas.\"},{\"question\":\"Why is the equals sign challenging for some early learners?\",\"answer\":\"Some young children interpret “=” as meaning “the answer is coming,” which can lead to incorrect reasoning in arithmetic equations. The brief emphasizes building an understanding of “=” as equality between two expressions.\"},{\"question\":\"How is the brief supported by classroom or program evidence?\",\"answer\":\"The ideas are informed by field trials from the RAMPUP summer bridge program. The program includes modules such as patterns, networks, and equivalence, and the brief uses examples of students’ reasoning from those trials.\"}]","Engaging English Learners With Equivalence as a Crosscutting Concept in Mathematics - Brief | PDF",1792056727,28]