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Mathematical Process of the Month: Connections CN Learning in all subjects is predicated on making connections between newUnderstanding Mathematical concepts and existing schema; that is, mental images and ideas, or what the learnerProcess StandardsSuccessful learning in mathematics is already knows. Powerful learning draws on the learners’ experiences and life context.achieved through “relational Students with extensive life experience have more pre-existing schema to createunderstanding”. A conceptual connections with new concepts. This is called “crystalized intelligence”, as opposedunderstanding of math builds on to “fluid intelligence”, which is the natural ability we are born with. Interestingly,humans’ natural desire to make senseof things. It helps learners acquire crystalized intelligence is a greater predictor of student success. This is good news fornew knowledge and apply it in learners, because while fluid intelligence is innate, crystalized intelligence can beunfamiliar situations. Conceptual controlled. Educators capitalize on this by arranging experiences for students throughunderstanding improves students’attitudes toward math because they field trips, interactive learning experiences, projects, and inquiry. Everything acan see math as an accessible subject student has learned in the past becomes the hook on which to hang newthat can be understood. Effective information. As math educators we are very familiar with this analogy, as we knowmathematics instruction incorporatesseven process to ensure that students that math concepts build on each other through the years.develop a conceptual understanding There are several types of connections that are important to learningof math: Communication, mathematical concepts:Connections, Mental Math and Connecting ideas within mathematics. Our curriculum helps us do this byEstimation, Problem Solving,Reasoning, Technology and organizing mathematical themes which are evident in the strands of the curriculum.Visualization. Beside the Outcomes These themes tie mathematical topics together so the student can realize generalin the curriculum document are principles at work and how they are related. Students will encounter these big ideassuggestions for which of theprocesses lend themselves well to the repeatedly and in many different contexts as they develop depth of understandingteaching of that outcome; however, through the grade levels. As educators, we need to ensure that we highlight theseall processes are interrelated and are related concepts to help students build on and expand their prior learning;part of the everyday business in theclassroom. They are the media otherwise, math is perceived as fragmented and compartmentalized. Learning isthrough which we deliver content. through memorization which is low-level and not lasting. Our first job as educators is to become very familiar with the curriculum, especially at our own level but alsoConnections: Because the through the years so that we can understand ideas that are nested within each other,learner is constantly searching and concepts that are threaded and integrated. Ideas must flow naturally fromfor connections on many levels, lesson to lesson and grade to grade.educators need to orchestratethe experiences from which Connections between math and the real world. All learning is achievedlearners extract understanding through anchoring new concepts to existing ideas and experiences that are existing…. Brain research establishes understandings. The things a learner already knows become the “pegs” on which toand confirms that multiple pin new information. A teacher’s task is to illuminate the relations between thecomplex and concrete known and the new. Teachers must always seek opportunities to draw on studentsexperiences are essential for past experiences and understandings to introduce new topics. When students aremeaningful learning and encouraged to contribute their own understandings into the learning, they are moreteaching. (Caine & Caine, 1991,p. 5) (Excerpt from Sask engaged and have a sense of ownership of the learning. Math must be understoodFoundations PreCalc 10 as intrinsic and enmeshed in the fabric of life, physics, and society, not an elite,Curriculum) unobtainable, isolated topic. Teachers and resources must draw out the
(Continued from P. 1) connections between mathematics in the classroom and mathematics in the real world.A simple model for talking about By engaging many senses we create memorable experiences to which conceptsunderstanding is that to are linked. This is the basis of inquiry-based, hands on instruction. Exploringunderstand something is to mathematical topics through experiences, manipulatives, collaborative discussions,connect it with previous presentations, debates, and multimedia create much more memorable learninglearning or other experiences… events than pencil and paper seat work, though the actual content and topics may beA mathematical concept can bethought of as a network of the same.connections between symbols, Connections to other areas of learning. Helping students connect to math inlanguage, concrete experiences, their lives involves highlighting connections to other subject areas. Our curriculumand pictures. (Haylock & documents give suggestions for helping students transfer mathematical knowledgeCockburn, 2003, p. 18, as cited in to other disciplines. Some examples are shapes and tessellations in art, dataSaskatchewan Math 8 interpretation and probability in health and social, data and graphing in scienceCurriculum) education, fractions and music education, timing and statistics in phys ed, and graphsExisting knowledge and as models of behaviours in physics, logarithms as necessary to chemistry and physics,experiences are the anchors to and calculating and measuring in trades classes. In the same way that mathwhich we tie new concepts classrooms draw on literacy and social skills, so should other disciplines require students to apply mathematical reasoning and value mathematical literacy. Connections between symbols and procedures in math. Part of our work in establishing mathematically literate students is helping them gain an understanding of the representations of mathematical ideas. Students must be actively engaged in the work of mathematics to be immersed in the language of math. Word walls and front-loading vocabulary are strategies to assist with connecting to the language of mathematics, as are Frayer models, carrol diagrams, and other concept attainment activities and graphical representations. Teachers promote mathematical literacy by introducing many representations, modelling different approaches, and arranging opportunities for students to compare, explore, reason with and talk about mathematical approaches and representations. Saskatchewan Renewed Mathematics Curriculum Glanfeild, F. (2007). Reflections on research in school mathematics. Toronto, Pearson.Communication works together NCTM Web Site, http://www.nctm.org/with reflection to produce new Ontario Association for Mathemamtics Education, http://www.oame.on.ca/main/index1.php?lang=en&code=home New Jersey Mathematics Curriculum Framework (1996) Standard 3-Mathematical Connectionsrelationships and connections. Manitoba Mathematics Curriculum Framework, Grade 8 Curriculum Support Document,Students who reflect on what http://www.edu.gov.mb.ca/k12/cur/math/support_gr8/full_doc.pdfthey do and communicate withothers about it are in the bestposition to build usefulconnections in mathematics.(Hiebert et al., 1997, p. 6) Every new idea is connected to pre-existing knowledge and experiences SUM conference: May 3-4, Sciematics: The Changing Face GSSD Divison-wide PD day, Saskatoon. Featuring Dan Meyer of Education. Saskatoon, May Feb 1 2013. Math topics: and Marian Small. 9-11, 2012, College of Benchmarking with Susan http://www.smts.ca/sum- Agriculture and Biosciences, U Muir, Exploring Math conference of S. Instruction with Cindy Smith http://www.sciematics.com/
Formative Assessment Feature Frayer Model: Like many strategies that may be introduced as “formative assessments”, The Frayer model (one example is shown at left) is not only useful for formative assessment but also as a routine instructional practice. It has been described as a concept map, graphic organizer, and vocabulary development tool. It can be used to introduce topics or concepts, front load vocabulary, or check for understanding. It can also be used for clarifying mathematical symbols. The Frayer model helps students make sense of words or concepts, and connect them to pre-existing understandings. It requires critical thinking to establish deeper understanding, and it creates a visual reference for concepts and vocabulary. Students can work on a Frayer model individually, in pairs, or collaboratively in groups. You could have students include pictures to help make connections with the concept. After making Frayer models have students examine each other’s work, compare and discuss. Frayer models have more to them than meets the eye! Customizable, Downloadable Frayer Model Templates are available here: http://www.worksheetworks.com/miscellanea/ graphic-organizers/frayer.html Shown at left is a table of instructional practices (Marzano) that are shown to have significant impacts on student learning. The Frayer model incorporates several of these! “Contextualizing and making connections to the experiences of learners are powerful processes in developing mathematical understanding. When mathematical ideas are connected to each other or to real-world phenomenon, students can begin to view mathematics as useful, relevant, and integrated.” –WNCP 2006 K-W-L Chart: This is often used as a pre-assessment or entrance slip. A chart has three columns, where students record what they already know (K), what they want to know (W) and what they have learned (L). This can be done on a small piece of paper, for entrance slips, or on three separate poster papers for a more collaborative activity. This activity requires students to activate prior knowledge, and apply higher-order thinking strategies in order to construct meaning. Having students do this at the start of a lesson or unit can be a focussing activity, and having them repeat it at the end gives a visualization of learning and progress, which encourages motivation by creating an awareness ofA downloadable K-W-L Chart is available at achievement.http://www.educationworld.com/tools_templates/kwl_nov2002.doc
Relational Understanding: Instrumental Understanding: Conceptually based Rule based Knowing both “how” and “why” Knowing “how” but not “why” Acquired by sense-making Acquired by rote Interconnected knowledge Isolated knowledge Easier to remember Harder to remember Involves fewer principles of more general Involves a multiplicity of rules application Inflexible, not readily adaptable to new tasks Flexible, more adaptable to new tasks Cool Stuff to try: Have you heard of three-ring? It allows Prototype Departmentals for WAM 30, Foundations 30 and PreCalc 30 you to quickly create digital are on line at folders for all your students, http://www.education.gov.sk.ca/pro and upload documents, screen totypes snips, photos and videos to each file. It’s a way of creating a digital portfolio. What a coolDid you know? You have access to way to give teachers a snapshot“Destination Math”, a web-based learning of their child’s work in yourtool for k-9 student practice. You can tailorassignments to fit curriculum concepts and to class, or to track formativedifferentiate. Destination Math is a assessment data.responsive program, so it tracks studentanswers, gives like problems to reinforceconcepts it finds a student is having troublewith, or introduces a tutorial to get thestudent back on track. It can also track time Learn how to set up destination math foron task and correct answers. your class from the Destination Math Webinar and other Webinars by Michelle Morley, which can be viewed at http://central.gssd.ca/math/?page_id=1 520 The link to Destination Math Destination Math is popular is http://success.gssd.ca/lms as a Pod activity. Math Webinars. SMART Math Tools – Gary, Jan. 23 ~ Screen Casting – Michelle, Math Coach March 6 ~ Photo Story – John, April 17 ~ Building a Personal Learning Community - Michelle. These Please visit my blog at webinars are free. See Michelle Morley’s blog for log in info www.blogs.gssd.ca/csmith/ This site has useful resources, but it is a work in progress. Please email me if you haveWeb Resources: ideas or requests for thisPreCalc, Foundations resources and screen captures newsletter.http://arthurmathwarman.blogspot.ca/p/pre-calculus-30.htmlQuick Draw: Class starter activity that generates good discussion and math vocabulary:http://pages.cpsc.ucalgary.ca/QuickDraw/Effective use of math Word Walls Rubric http://blogs.gssd.ca/smuir/?tag=word-wallsIpad math Apps for middle years: http://www.teachthought.com/apps-2/12-of-the-best-math-ipad-apps-of-2012/