Middle School Mathematics: Making Connections Using Representations

1. Middle School Mathematics:Making Connections Using Representations Smokey Road Middle School February 24, 2015 Pictures, Diagrams Symbols , Formal Notation Concrete Models Understanding Tables , Charts Graphs

2. Agenda • Multiple representations • Recommendation guidelines • Front Row differentiated math practice • Wrap-up Coweta Committed to Student Success

3. Multiple representations for making connections Coweta Committed to Student Success

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

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8. 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

9. 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

10. “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

11. 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

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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. Guidelines for math course recommendations for upcoming 9th graders Coweta Committed to Student Success

24. Math course recommendations For all students, it is important to err on the side of caution by recommending the year-long course. It is more feasible to make a change in students’ schedules from year-long to term than vice versa. If a student is registered for the one-term class, the student may not be scheduled to take the course until second semester. Second semester one-term students do not have the capability of being changed to a year-long class when the course becomes too difficult because the year-long class are half-way through the course. The inability to make a change can be extremely problematic for a student. If there is a student you decide to recommend taking the term course but you have any reservations, please make a note on the form that the student should take the course first semester. Coweta Committed to Student Success

25. Front Row Coweta Committed to Student Success

26. FrontRow

27. Why use Front Row? Once the diagnostic test is completed for each domain section, Front Row will adapt it's levels and practice questions based on individual student results. Students can easily work at their own pace on what they need.

30. After the diagnostic… Teachers can assign certain standards to practice or let Front Row select based on student needs.

31. Adaptive vs. Assigned… • Adaptive Practice: For each domain, students start with a pre-test. This pre-test determines their level, and students work from that point onward. There is no set number of questions it takes to reach mastery - it's a more involved process that takes timing, order, difficulty, and accuracy into account • When students struggle with a topic, Front Row automatically remediates them. This means that we use the Front Row adaptive algorithm and the standard breakdowns to work through material that supports understanding of with whatever the student is having trouble • Front Row automatically reviews concepts the student had trouble with in the past. When a student has trouble with something, the program first works towards remediating the student and helping him/her master it. Then, it regularly reviews that topic with the student in the future. Every time a student sees a "bonus question" with extra coins - that's what's happening.

32. Adaptive vs. Assigned… • Assigned Practice: Following teacher direction, students select a particular domain and level that they will practice. • Students will be given 10 questions at the level selected. If students answer a question incorrectly, the program will still give them several attempts to correctly answer the question, but it will be scored as incorrect in the Assigned Feed View. • Once students have answered 10 questions at that level, the program will take them out of Assigned Practice and have them continue working on Adaptive Practice. • Questions are selected randomly and pulled from the standard that correlates with the level the student selected.

33. Assigned Practice…

34. Using Front Row… Some questions are multiple choice and some are free response. There is also a play button allowing them to play a tutorial if they get stuck. The problems are on the left, and students can use the right side as scratch paper or drag in a manipulative to help them solve the problem.

35. Individual Student Report Card Details which specific standards a student is both strong and weak in. Can be used to help differentiate instruction. This can be sent to parents also.

36. Analysis by Standard This report is individual by student. Allows teachers to quickly tell level of mastery for each standard and which one the student is currently working on. This can also be used to help group students.

37. Grade Level at a Glance… At a quick glance teachers can tell how students are performing overall. This can also help teachers target lower-level students and accelerate higher-achieving students.

38. More detailed breakdown… Teachers can see at a quick glance where their students are. This can help with grouping and differentiation.

39. Let Front Row help with Groups Groups are listed by domain and broken down by specific standards. Front Row will give you groups based on similar levels or it can mix levels using stronger students mixed with weaker ones.

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

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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