1. Equity and Excellence: Understanding Ratios and Proportional Reasoning Presented at 48 th NCSM Annual Conference April 12, 2016 Susie W. Håkansson, Ph.D. President, TODOS: Mathematics for ALL

3. Description We want ALL students, particularly English learners to develop mathematical proficiency in proportional reasoning. In order for this to occur, we as teachers need to Address the language needs of students Address our own conceptual understanding of proportional reasoning Adopt best practices in working with ELs

4. Question How do we as teachers acquire the following: (1) understanding of the language needs of English learners and (2) our own conceptual understanding of proportional reasoning?

5. Frontloading Language Self-efficacy Conceptual understanding DiscourseMetacognition Zone of proximal development Equity and excellence

6. Challenges Facing Students Access to the language of mathematics Access to the mathematics content ExpectationsSelf-efficacy Equity and excellence

7. Why Is English So Hard? The soldier decided to desert his dessert in the desert. Upon seeing the tear in the painting, I shed a tear. After a number of injections, my jaw got number. A minute is a minute part of a day.

8. Why Is English So Hard? They were too close to the door to close it. I did not object to the object. We must polish the Polish furniture. The farm was used to produce produce. The bandage was wound around the wound.

9. Why Is English So Hard? There is no egg in eggplant and no ham in hamburger. How can a slim chance and a fat chance be the same, while a wise man and a wise guy are opposites? Did you say thirty or thirteen? Did you say two hundred or two hundredths? Did you say fifty or sixty?

10. “Even” Social register The floor is even (smooth/liso) The picture is even with the window (leveled/nivelado) Sleep provides even rhythm in our breathing (regular/uniforme) The dog has an even temperament (calm/calmado) If we share equally, we will be even (balance/igual) I looked sick and felt even worse (comparative/aún) So simple, even a child could do it (comparative/incluso)

11. Social register Got even To be even Even out Break even Not even Even-steven “Even”

12. Mathematics register Number: even numbers (e.g., 2, 4, 36, 58) Number: even amounts (e.g., even amounts of flour and sugar) Measurement: exact amount (an even pound) Function: even function (e.g., y = 5x 2 – 3; y = cos x) “Even”

13. Math WordMeaning 1 (Math)Meaning 2 SolutionThe answerTwo or more substances mixed together ExerciseMath problems to solvePhysical movement to stay fit ProductThe answer when you multiply two or more numbers Something made by humans or machines ExpressionMath statement with numbers and/or variables To show emotion Dual Meaning Words

14. Mathematics Language vs. English Multiplying... English: repeated addition—larger Mathematics: larger, smaller, or neither Dividing... English: cut into pieces Mathematics: same as multiplication (dividing by a non-zero number is multiplying by its reciprocal) “Ameobas multiply by dividing”

15. Teachers learn to amplify and enrich—rather than simplify—the language of the classroom, giving students more opportunities to learn the concepts involved. Aída Walqui, Teacher Quality Initiative

16. MP1: Make Sense of Problems and Persevere in Solving Them DO STUDENTS [DO TEACHERS] Explain a problem to themselves, determine what it means, and seek possible entry points? Analyze what’s given, constraints, relationships, and goals? Make conjectures about what the solution might look like? Plan a solution pathway instead of jumping into a solution?

17. MP1: Make Sense of Problems and Persevere in Solving Them Use multiple representations (verbal descriptions, symbolic, tables, graphs, diagrams, etc.)? Check their answers using different methods? Continually ask “Does this make sense?” Understand the approaches of others and identify correspondences between different approaches?

18. Frontloading Language Proportional Reasoning Direct proportion Inverse proportion InvariantCovariation Multiplicative thinking

19. Mathematics Content Let’s go to mathematics content. We must always begin any lesson with the mathematics content we want our students to understand. We then ask the following questions: What do we want our students to know and be able to do? What mathematics language do we expect students to know before the lesson? What mathematics language do we plan to use? Are there words that have multiple meanings or interpretations? How do we address them? What instructional materials would support the students in learning the material? How do we plan to assess what students learned?

20. Reasoning Reasoning suggests that we use common sense, good judgment, and a thoughtful approach to problem solving, rather than plucking numbers from word problems or any problem and blindly applying rules and operations.

21. Proportional Reasoning In colloquial terms, proportional reasoning is reasoning up and/or down in situations in which there exists an invariant (constant) relationship between two quantities that are linked and varying together.

22. Proportional Reasoning Many students who have not developed their proportional reasoning ability have been able to compensate by using rules in algebra, geometry, and trigonometry courses, but, in the end, the rules are a poor substitute for understanding. We need to get away from having students start with a/b = c/d. We need students to understand proportional reasoning, and we need to develop proportional reasoning in students by understanding proportional reasoning ourselves.

23. Proportional Relationships Proportional relationships involve some of the simplest forms of covariation, where two quantities are linked to each other in such a way that when one changes, the other one also changes in a precise way with the first quantity. Let’s look at a couple of problems.

24. Laundry Detergent If a box of detergent contains 80 cups of powder and your washing machine recommends 1¼ cups per load, approximately how many loads can you do with one box?

25. Laundry Detergent 1¼ cups 1 load 1¼ cups 1 load 5 cups 4 loads 5 cups 4 loads 40 cups32 loads 40 cups32 loads 80 cups64 loads 80 cups64 loads The ratio of cups to loads is 1¼:1 or 5:4 In a direct proportion, the direction of change in the related quantities is the same. We say that “y is directly proportional to x” or that “y varies as x.

26. Men Working It takes 6 men 4 days to complete a job. How long will it take 8 men to do the same job, assuming that they all work at the same pace? Think: 6 men work 4 days 1 man doing the work of 6 men all by himself takes 24 days 8 men, dividing up the work that 1 man did, take 3 days

27. Men Working # men# days# man days 6424 1 83 46 21224

28. Men Working In inverse proportion, the direction of change in the related quantities is not the same. We say that “y is inversely proportional to x” or that “y varies inversely as x.” # men# days# man days 124 21224 46 64 83

29. Multiplicative Thinking Differentiate between additive and multiplicative situations and apply whichever transformation is appropriate. Process of addition is associated with situations that entail adding, joining, subtracting, separating, and removing. Process of multiplication is associated with situations that involve such processes as shrinking, enlarging, scaling, duplicating, exponentiating and fair sharing.

30. Your Turn: Getting Started Six men can build a house in 3 days. Assuming that all of the workmen work at the same rate, how many men would it take to build the house in 1 day? If 6 chocolates cost $.93, how much do 22 cost? Between them, Jose and Marisol have 32 marbles. Jose has 3 times as many as Marisol. How many marbles does each of them have? Guillermo can mow Ms. Rivas’ lawn in 45 minutes. Guillermo’s little brother takes twice as long to do the same lawn. How long will it take them if they each have a mower and they work together?

31. Your Turn: Getting Started Yolanda wants to buy a CD player costing $210. Her mother agreed to pay $2 for every $5 Yolanda saved. How much will each contribute? A company usually sends 9 people to install a security system in an office building, and they do it in about 96 minutes. Today, they have only three people to do the same size job. How much time should be scheduled to complete the job? A motor bike can run for 10 minutes on $1.30 worth of fuel. How long could it run on $.91 worth of fuel? Washington Academy boasts a ratio of 150 students to 18 teachers. How can the number of teachers be adjusted so that the academy’s student- to-teacher ratio is 15 to 1?

32. Proportional Reasoning Proportional reasoning refers to the ability to scale up and/or down in appropriate situations and to supply justifications for assertions made about relationships in situations involving simple direct proportions and inverse proportions.

33. Delivery of Instruction What did I do to support the language, the content, and the belief that you can succeed? Engaged you in a high cognitive demand task. Engaged you in discourse. Assessed by walking around (ABWA). Provided access to language and content. Allowed for discourse among yourselves. Allowed you to explore the why—the metacognition. Provided feedback knowing you can solve the problem.

34. Key Practices for English Learners Provide high cognitive demand tasks Provide high cognitive demand tasks Scaffold the language by enhancing and enriching the language to access the content Scaffold the language by enhancing and enriching the language to access the content Expect students to “do” the mathematics Expect students to “do” the mathematics Access prior knowledge and build on prior knowledge Access prior knowledge and build on prior knowledge Understand flexibility in ways students respond Understand flexibility in ways students respond

35. Encourage and expect mathematical talk; increase discourse Encourage and expect mathematical talk; increase discourse Ask students probing questions to clarify and draw out their thinking Ask students probing questions to clarify and draw out their thinking Have students share and justify their reasoning and process they used to solve the problem Have students share and justify their reasoning and process they used to solve the problem Provide opportunities for students to work individually, pair share, and in small and whole groups Provide opportunities for students to work individually, pair share, and in small and whole groups Key Practices for English Learners

36. Frontload language (don’t over frontload) Frontload language (don’t over frontload) Use language as a resource for learning not only as a tool for communicating but also as a tool for thinking and reasoning mathematically Use language as a resource for learning not only as a tool for communicating but also as a tool for thinking and reasoning mathematically Provide diverse avenues of action and expression Provide diverse avenues of action and expression Be aware of multiple meanings of words Be aware of multiple meanings of words Have students Think, Ink, Pair, Share Have students Think, Ink, Pair, Share Key Practices for English Learners

37. Equity and Excellence We want to provide access to ALL students. We also want ALL students to make sense of rigorous, high quality, and cognitively demanding mathematics. We want them to approach the zone of proximal development, not the zone of minimal effort. We want equity AND excellence. Equity without excellence is meaningless. Excellence without equity is unjust. We must always ask ourselves, what can we do to incorporate both?

38. Summary Our Next Steps: Provide greater access to high cognitive level mathematics to ALL students by Ensuring that we understand proportional reasoning (or any mathematics content), Understanding the language needs of ALL students, and Acquiring the knowledge that understanding the content and language needs of students, equity and excellence, become a part of our DNA.

39. TODOS 2016 Conference Ensuring Equity and Excellence in Mathematics For ALL June 23-25, 2016 Scottsdale Plaza Resort, Phoenix, AZ Classroom Teachers Attending TODOS 2016 Conference is co-sponsored by NSF-funded Arizona Master Teachers of Mathematics (AZ-MTM), award #1035330, administered by the Department of Mathematics at The University of Arizona. Participants will leave with important tools, strategies, ideas, and models for their own settings so they can advocate for and enact mathematics teaching that increases Equity, Access, and Achievement for ALL students through rigorous and coherent mathematics. http://www.todos-math.org Registration Limited

40. TODOS Information JOIN TODOS for only $25 for a one-year membership, $70 for three years! Visit TODOS Booth in Hallway Read Social Justice Position Statement TODOS Website: http://www.todos-math.org

41. Contact Information Susie W. Håkansson, Ph.D. shakans@g.ucla.edu See me for a business card! Join TODOS! “Teaching Fractions and Ratios for Understanding” by Susan J. Lamon

42. Questions?