Making Connections Using Math Representations

1. High School Mathematics:Making ConnectionsUsing Representations Northgate High School February 10, 2015 Pictures, Diagrams Symbols , Formal Notation Concrete Models Understanding Tables , Charts Graphs

2. Agenda • Making connections using multiple representations • Update from GaDOE • Concerns • Wrap-up Coweta Committed to Student Success

3. Coweta Committed to Student Success

4. Choosing representations The way we represent a mathematical concept or skill will greatly affect … • Students’ understanding of the concept • Students’ attitude towards the concept • The type of connections students make with the concept • The level of access students have to learning the concept • The type of prior knowledge we tap from our students Coweta Committed to Student Success

5. Multiple representations • Thinking tools for mathematics • Used to understand, develop, and communicate different mathematical features of an object or relationship • Engages higher-order thinking • Benefits struggling students and EL students by making math more visual Coweta Committed to Student Success

6. “The depth of conceptual understanding one has about a particular mathematical concept is directly proportional to one’s ability to translate and transform the representations of the concept across and within a wide variety of representational systems.” – Guillermo Mendieta, Pictorial Mathematics Coweta Committed to Student Success

7. Pictures, Diagrams Symbols , Formal Notation Concrete Models Understanding Tables , Charts Graphs Coweta Committed to Student Success

8. Observations • State standardized tests incorporate multiple representations, except for concrete models. • Purposeful inclusion of multiple representations benefit students to develop conceptual understanding and perform better on assessments. Pictures, Diagrams Symbols , Formal Notation Concrete Models Understanding Tables , Charts Graphs Coweta Committed to Student Success

9. Top 10 reasons to use multiple representations in teaching of mathematics Coweta Committed to Student Success

10. 10. The nature of mathematics is about representations. • Mathematics is about representing ideas and relationships through symbols, graphs, charts, etc. • Effective teaching involves the purposeful and effective selection of the representations to engage students. Coweta Committed to Student Success

11. 9. They introduce a change of pace. • Using multiple representations for a given concept introduces a change of pace in our instructional practice. • Students who listen and take notes on a lecture, then work with physical models, and create pictorial representations for their oral presentation get to make a change in the way they use their brains. Coweta Committed to Student Success

12. 8. They help students make connections. • Using multiple representations provides more opportunities for students to make meaningful connections and discover relationships between the concept being studied and their own prior knowledge. • The representations themselves are doors to a whole set of different types of possible connections. Coweta Committed to Student Success

13. 7. The real world is multi-dimensional. • Real world problems do not come neatly packaged in one representation. • Defining the questions and finding alternative solutions often involves reading text, searching on the internet, interpreting graphs, creating tables, solving equations, designing models, and working with others. • Using multiple representations prepares students for the real world of problem solving. Coweta Committed to Student Success

14. 6. Their use increases student engagement and motivation. • Multiple representations increase the level of engagement and the level of motivation of your students. • Some will be more motivated and more engaged when you use models and pictures, while others will connect better to the standard symbolic representations. Coweta Committed to Student Success

15. 5. Their use indicates different approaches are valued. • It conveys the idea that there is not one single way to solve problems; different people, with different perspectives and different strengths may offer a different way approach a problem. • Depending on the context, the audience and other factors, one approach may be more effective than another in any given situation. Coweta Committed to Student Success

16. 4. They facilitate the delivery of differentiated instruction. • Every representation taps a different bank of experiential knowledge and student aptitudes. • By using a wide variety of representations with the key concepts, you are differentiating instruction and building on wider set of students’ strengths. Coweta Committed to Student Success

17. 3. They give students wider access to the same content. • The use of multiple representations give students with different learning styles wider access to the same content. • We all learn differently. Some students who “could not get it or see it” through the traditional symbolic representation will “see it” when you use other representations. Coweta Committed to Student Success

18. 2. Their use increases the depth of students’ understanding. • Research on multiple representations strongly suggests that the depth of students’ understanding of a mathematical concept is directly proportional to their ability to represent, translate, and transform this concept within and across representations. • Different representations of a concept add new layers of understanding for that concept. Coweta Committed to Student Success

19. 1. Using multiple representations increases student achievement. • Constantly using a variety of representations in instruction and requiring students to do so in practice work and on classroom assessments prepares students for standardized tests. • Standardized tests include a large number of questions that focus on interpreting, translating, and transforming mathematical relationships across and within representational systems. Coweta Committed to Student Success

20. Update from the Mathematics Program at GaDOE Coweta Committed to Student Success

21. In progress … • Public comment period for proposed new courses • Foundations of Algebra • Algebra 1 • Geometry • Drafts of course standards and survey links at https://www.georgiastandards.org/Common-Core/Pages/Additional-High-School-Mathematics-Courses.aspx • Surveys will close February 13, 2015, at 5 p.m. Coweta Committed to Student Success

22. Upcoming … • SBOE meets on February 19, 2015 • Public review comments on new courses to be presented • Approval of standards for new courses with tweaks resulting from public comments and recommendations from the Mathematics Advisory Council to be requested • No March SBOE meeting • SBOE meets on April 2, 2015 • Updated graduation rule and the IDA(3) Roster of State-funded Courses presented for approval Coweta Committed to Student Success

23. Traditional pathway • The CCGPS traditional pathway will include support courses and an accelerated pathway. • The course numbers for the ten courses associated with the traditional option will be added to the IDA(3) Roster of State-Funded Courses. • Algebra I Support • Algebra I • Geometry Support • Geometry • Algebra II Support • Algebra II • Pre-Calculus • Accelerated Algebra I/Geometry A • Accelerated Geometry B/Algebra II • Accelerated Pre-Calculus Coweta Committed to Student Success

24. Third and fourth courses • The High School Working Committee will reconvene once the standards are finalized for the ninth and tenth grade courses in both the current and traditional options to assign standards to the third and fourth courses. • Please note that while specific course numbers identify the current third and fourth courses and additional numbers will identify the traditional pathway third and fourth courses, the standards for the third course in either option will be the same and the standards for the fourth pre-calculus courses will also be the same in either option. Coweta Committed to Student Success

25. Foundations of Algebra course • A proposal for a high school core course titled Foundations of Algebra, aimed at students who may have significantgaps in their mathematics achievement, was introduced during the SBOE November 2014 meeting. • It was requested that the course be implemented in the 2015-2016 school year. Coweta Committed to Student Success

26. Foundations of Algebra eligibility • Step I: Middle school administrators/teachers will identify students who Did Not Meet expectations on the 5th, 6th, and 7th grade CRCT. These are students who may have significant gaps in their mathematics achievement. • Step II: Middle schools will administer Individual Knowledge Assessment of Number (IKAN) diagnostic to students identified in Step I (No costs will be incurred by the state or districts.) • Step III: During the scheduling process, high schools will enroll students scoring at IKAN Stage 5 (equivalent to 4th grade mathematics) or below in Foundations of Algebra. Coweta Committed to Student Success

27. Foundations of Algebra tools Assessment tools available for teacher use in 2015-2016: • Course pre- and post- assessments • Module pre- and post- assessments • An instruction manual for administering IKAN’s companion assessment Global Strategy Stage Assessment (GloSS) throughout the course. Coweta Committed to Student Success

28. Foundations of Algebra resources Resources to be posted in TRL by June 1: • Comprehensive Course Overview • Five Instructional Modules infused with spot diagnostics and appropriate intervention mini-lessons Coweta Committed to Student Success

29. Foundations of Algebra PL Professional learning offered during the 2015 Summer Academy Program: • Two 3-hour sessions focused on the administration and usage of the GLoSS assessment tool • Two 3-hour sessions focused on course content • Two 3-hour sessions focused on effective instructional practices for this type of student Coweta Committed to Student Success

30. Foundations of Algebra Q&A • How will this course fit into our high school mathematics sequence? Coweta Committed to Student Success

31. Foundations of Algebra Q&A • How will adding Foundations of Algebra affect graduation requirements? • Foundations of Algebra will be a core course. • The mathematics graduation requirements will remain virtually unchanged: • Four units of core credit in mathematics shall be required of all students, including Mathematics I or GPS Algebra, or its equivalent and Mathematics II or GPS Geometry, or its equivalent and Mathematics III or GPS Advanced Algebra or its equivalent. Additional core courses needed to complete four credits in mathematics must be chosen from the list of GPS/CCGPS/AP/IB/dual enrollment designated courses. Coweta Committed to Student Success

32. Foundations of Algebra Q&A • Will students who complete Foundations of Algebra, Coordinate Algebra, and Analytic Geometry (or Foundations of Algebra, Algebra 1, and Geometry) be eligible for admission to USG/TCSG institutions? • USG and TCSG have provided assurance that students completing either one of these sequences will be eligible for admission to some of their institutions. • USG suggests that the completion of four years of mathematics core courses is strongly advised. Coweta Committed to Student Success

33. Foundations of Algebra Q&A • Will Foundations of Algebra be assessed via an End of Course assessment? • Will Foundations of Algebra be limited to entering ninth grade students? • Will there be a support course connected to Foundations of Algebra? • Could you teach Foundations of Algebra in eighth grade for high school credit? • No • Yes, for 2015-2016 • No • No Coweta Committed to Student Success

34. Foundations of Algebra Q&A • What do we do with students who fail Foundations of Algebra? • Is there a limit on the class size for this course? • Not sure; looking for suggestions … • If it is coded as an REP class, the class size is somewhat reduced. If it is also offered as a collaborative class, you could have two teachers in the class. Coweta Committed to Student Success

35. Milestones assessments • Georgia Milestones will include four mathematics assessments in 2015-2016 • Coordinate Algebra • Analytic Geometry • Algebra I • Geometry • Each assessment will be aligned to the standards of the tested course Coweta Committed to Student Success

36. Depth of trig instruction in Advanced Algebra Placement recommendations for upcoming 9th graders Concerns Coweta Committed to Student Success

37. Advanced Algebra trig standards MCC9‐12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. MCC9‐12.F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.   MCC9‐12.F.TF.2Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.  MCC9‐12.F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. MCC9‐12.F.TF.8 Prove the Pythagorean identity (sin A)2 + (cos A)2 = 1 and use it to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle.  Coweta Committed to Student Success

38. Trig in Pre-Calculus • Extend the domain of trigonometric function using the unit circle. • Coordinates of special points • Symmetry and periodicity of trig functions • Model periodic phenomena with trigonometric functions. • Inverse trig functions • Trig equations • Prove and apply trigonometric identities. Coweta Committed to Student Success

39. Placement recommendations • Some current 9th graders are inappropriately placed • Guidelines for making recommendations are provided to middle school math teachers • Suggestions for changes to reflect the lack of state test scores this year? • Other suggestions for improvement? Coweta Committed to Student Success

40. This is the last regularly scheduled workshop for this school year. Spring/summer workshops are being planned. Your input for professional learning needs is solicited. Wrap-up Coweta Committed to Student Success

41. Coweta Committed to Student Success

42. References • Georgia Department of Education. (2015). Additional high school mathematics courses. Retrieved from https://www.georgiastandards.org/Common-Core/Pages/Additional-High-School-Mathematics-Courses.aspx • Georgia Department of Education. (2015, January 27). Mathematics in Georgia: District mathematics supervisors. Retrieved from https://attendee.gotowebinar.com/recording/1816347393726818818 • Gudder, S. (1994). A mathematical journey. New York, NY: McGraw-Hill. • Mendieta, G. (2006). Pictorial mathematics: An engaging visual approach to the teaching and learning of mathematics. Santa Cruz, CA: Meaningful Learning Research Group. • Virginia Department of Education. (2014). Making mathematical connections and using representations. Retrieved from http://www.doe.virginia.gov/instruction/mathematics/professional_development/index.shtml Coweta Committed to Student Success