Mathematics for Diverse Learners Gatekeeping Revealed

1. Mathematics for Diverse LearnersGatekeeping Revealed Marcia M. Burrell, Ph.D.State University of New York College at Oswego Curriculum and Instruction Department marcia.burrell@Oswego.edu • International Conference of The Mathematics Education for the Future Project • Challenges in Mathematics Education for the Next Decade • September 10–15, 2017 • Hotel AnnabellaBalatonfüred, Lake Balaton, Hungary Gatekeeping Revealed

2. Overview/Abstract • How does one ensure the rigorous teaching of mathematics for all, while preparing a diverse population for success in mathematics? • This presentation will review student writings about their perceptions, expectations and behaviors around their understandings of their roles as open, flexible teachers with an array of tools, needed to engage their students in learning. Gatekeeping Revealed

3. Theoretical Framework Multicultural perspective required for success • Children are competent • Provide students with scaffolding with what they know and what they don’t know • Instruction during class • Extend student thinking • Exhibit in depth knowledge as well as subject matter knowledge • Ladson-Billings Gatekeeping Revealed

4. Mathematics: inclusive Instrument • Performance Data: • Attrition Rate from Mathematics • Decline in perceived achievement • How do People Learn Mathematics • Brain Based Learning • Standards Based Curriculum • Problem Solving Approach Gatekeeping Revealed

5. Mathematics as an inclusive instrument • “How might mathematics educators ensure that gate keeping mathematics becomes an inclusive instrument for empowerment rather than an exclusive instrument for stratification?” • Stinson, 2004, p.9 • How can we as mathematics educators transform the status quo in the mathematics classroom? • Why does U.S. education not provide all students access to a quality , advanced (mathematics) education that would empower them with economic access and full citizenship? Gatekeeping Revealed

6. Mathematics Involves Process Mathematics is a language. It is concise , beautiful, orderly and the most efficient method of communicating complex patterns, ideas and relationships. Numbers are at most a small corner of mathematics…When we teach children arithmetic facts, it is to facilitate communication, not eliminate it. Kerhonkson, L., W. (1999). Gatekeeping Revealed

7. Math Proficiency is tied to… • Conceptual understanding • Procedural fluency • Strategic Competence • Adaptive reasoning • Productive disposition National Research Council, Adding it up: Helping children learn mathematics. Washington, DC: National Academy press, 2001 Gatekeeping Revealed

8. Characteristics of Constructivism • Use cognitive terminology such as classify, analyze, predict and create • Encourage student inquiry • asking meaningful questions • Using meaningful contexts • Encourage students ability to ask questions • Seek elaboration of student’s initial response • Engage students in experiences that enhance understanding • Allow wait time after posing questions • Provide time for students to construct relationships Gatekeeping Revealed

9. Characteristics of Constructivism (continued) • Encourage and support student autonomy and initiative • Use raw data and primary scores • Use manipulatives, interactive and physical materials • Allow student responses to drive lesson • shift instructional strategies • alter content accordingly • Inquire about student’s understanding of concepts before sharing yours Gatekeeping Revealed

10. Mathematics Myths & Anxiety • Myths prevent us from learning to our capacity(and teaching to our capacity). • Instructional behaviors cause students to have anxiety in class(some covert and some overt). • Only small percentages(7%) of students had only positive experiences with mathematics from Kindergarten through College. (Jackson & Leffingwell, 1999) Gatekeeping Revealed

11. Mathematical Practices—Ways of Knowing Gatekeeping Revealed

12. Provide Students with CIA centered on their experiences… • Failing to provide African American students with curriculum, instruction, and assessment that are centered on their experiences, culture, and traditions is a major obstacle to providing them with an empowering mathematical experience Tate, 1994 CIA= Curriculum Instruction and Assessment Gatekeeping Revealed

13. Different Approaches to Teaching • Pose open ended questions • Make explicit connections • Provide multiple pathways for learning Gatekeeping Revealed

14. Instruction Supports Challenges • Allow students to use their own informal problem-solving strategies. • Encouraging math talk. • Design instructional activities that effectively bridgeconceptions and math understandings. Gatekeeping Revealed

15. Tennison, 2007 • Heterogeneous grouping • Less tracking • Curriculum that is connected • Use of practical real world problems • Challenging mathematics for all • More higher order thinking problems • Problem solving approach with open ended problems • Central problems organized into units— • Multiple branches of mathematics Gatekeeping Revealed

16. Goals for Rigor • Create a curriculum organized around habits of mind… • Close the gap between what the users and makers of mathematics do and what they say. • Creating • Inventing • Conjecturing • Experimenting • Give students an authentic research/problem solving experience • Certain habits of mind should cross curriculum Gatekeeping Revealed

17. TIMMS Performance Data • U.S. Fourth Graders Score Above Average in Math! • U.S. Eighth Graders Score Below Average in Math! • U.S Twelfth Graders Among the Lowest in the World! Math wars. Retrieved October 11, 2008, from http://www.educationworld.com/a_curr/curr071.shtml Gatekeeping Revealed

18. Steady Decline in Achievement TIMSS revealed that students in the U.S. experience a steady decline in math achievement levels between the fourth and twelfth grades Math wars. Retrieved October 11, 2008, from http://www.education-world.com/a_curr/curr071.shtml Gatekeeping Revealed

19. Attrition Rate from Mathematics • From 9th grade on… • 50% per year • Black and Latino students significantly larger • 12% grade 8 • 11% grade 12 • 5% bachelor’s degree • 2% for Masters and Ph.D. Scheonfeld, 2004 Gatekeeping Revealed

20. Richard Riley, U.S. Secretary of Education “You need to provide traditional basics, along with more challenging concepts, as well as the ability to problem-solve, and to apply concepts in real world settings." Math wars. Retrieved October 11, 2008, from http://www.education-world.com/a_curr/curr071.shtml Gatekeeping Revealed

21. Status Quo PRINCIPLE #1: Teachers must engage students’ preconceptions PRINCIPLE #2: Understanding requires factual knowledge and conceptual understandings PRINCIPLE #3: A metcognitive approach enables student self-monitoring Donavan, 2005 Mathematics is a subject that is rarely taught in a way that makes use of the three principles that connect to how people learn mathematics and how people learn in general. (Bransford, 2003) Gatekeeping Revealed

22. Engage Preconceptions—Build on Existing Knowledge Challenges: • It is not always about computation and following rules, but about relevant quantitative problems. • We need to link formal mathematics training with students’ informal knowledge and problem-solving capacities? Gatekeeping Revealed

23. Engage Preconceptions &Build on Existing Knowledge Challenges: • It is not always about computation and following rules, but about relevant quantitative problems. • We need to link formal mathematics training with students’ informal knowledge and problem-solving capacities? Gatekeeping Revealed

24. -1. Engage Preconceptions2. Factual and Conceptual frameworks3. Metacognitive approach to self-monitoring - How People Learn Gatekeeping Revealed

25. Problem Solving with George Polya George Polya described his experiences of problem solving in his book, How to Solve It. A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive facilities, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Gatekeeping Revealed

26. Habits of Mind/Mathematical Ways of knowing • Cuoco, 1996 • An Organizing Principle of Mathematics Curricula • Pattern Sniffers • Experimenters • Describers • Tinkerers • Inventors • Visualizes • Conjecturers • Guessers Gatekeeping Revealed

27. Difference Related to theStruggle • Teachers in the U.S. simplify things so that students need not struggle. • Students need to learn to memorize procedures with a full understanding of the of conventions adopted over time for efficiency. Gatekeeping Revealed

28. Door to No Return Hard Work—Time on Task—Effort—Struggle VS Ability tied to standardized aptitude tests Gatekeeping Revealed

29. Technology as an Inclusive Instrument • As teachers use technology, and begin to think differently about their teaching practices, they transform or begin to think differently about their role as teachers and therefore provide their students with different opportunities to demonstrate what they know as well • Student questions around the technology may create a space for additional interactions to explain the content. • The technology may also present the teacher with the opportunity to reflect on his or her practice or to change his or her teaching plans to accommodate the students in a way that may not have occurred previously (Carney, 1998). Gatekeeping Revealed

30. Technology as an Inclusive Instrument • The teacher may begin to form different ideas about effective practice, effective use of technology, better student-to-teacher interactions, and possibly greater student achievement (Grayson & Martin, 1998). • The technology may be a catalyst for additional student-to-teacher interactions. • Convincing the teacher that these interactions make a difference is part of how constructivism and technology may be interconnected. • The pressure to use technology is part of what teachers have come to expect, but the training and professional development for how to appropriately use the educational software takes time (Dugdale et al., 1998; Jones, 2001) and may not be a part of the ongoing system of teacher training and professional development (Baron, 2001; Paul, 2002). Gatekeeping Revealed

31. Using Concept Mapping-- • Teaching teachers to incorporate Concept Maps into their lesson planning • Write a concept map for the Unit of interest. Gatekeeping Revealed

32. Using History— • Teaching teachers to incorporate History into their lesson planning • Write a 2 to 4 page reaction paper. Think about how to analyze and synthesize the material using the following headings (individual) Gatekeeping Revealed

33. Using Questioning-- • Teaching teachers to incorporate Questioning into their lesson planning • Write questions connected to objectives • What does it mean?… Gatekeeping Revealed

34. System of Gatekeeping • Who is the gatekeeper? • One who controls the gate? • One who keeps people in or out? • What are the conditions by which the gatekeeper keeps people in or out? • Who tells the gatekeeper what to do? • Who tells the gatekeeper who to let in or out? • What are the conditions by which the gatekeeper keeps people in or lets people out? • What is on the other side of the gate and why might someone want to keep someone in or out. • How are you a gatekeeper? Gatekeeping Revealed

35. Mathematics for Diverse Learners course • 3 week course/Hybrid • Some mathematics from proficiency • History of mathematics • Teaching methods • Research • Lesson Development • Collaboration with peers • 3years in MAT program • Process Standards Reminders (ARRCC) • Accuracy • Reasoning • Representation • Communications • Connections Gatekeeping Revealed

36. Teacher Training has to respond to the GATEKeeping Concerns • The teacher is a part of the larger system responsible for gate keeping in mathematics. • Teachers are primarily the last connection to the pipeline in math. • Teacher training and teacher behaviors lead to successful student behaviors in mathematics. • Instruction that assures that all students are capable of mastering the subject. • That is culturally relevant • That is teaching from a multicultural perspective • Teachers must teach to reach their students. It is the educators responsibility. • Ladson-Billings, 1994 Gatekeeping Revealed

37. Plan for Gatekeeping • Have more students study algebra and geometry by 8th and 9th grade. • Raise state and local standards of academic performance. • Measure student performance against rigorous standards. • Offer a challenging curriculum and encourage students to take demanding courses. • Improve the teaching of mathematics by improving teacher training and reducing the number of teachers teaching out-of-field. • Destroy the myth that advanced mathematics is only for a few students. Math wars. Retrieved October 11, 2008, from http://www.education-world.com/a_curr/curr071.shtml Gatekeeping Revealed

38. History of Stratification • Plato, 2300 years ago • 1890’s • 1920’s • 1890- 1940’s • 1926 • 1950’s • Plato believed that all students needed to learn arithmetic, but he reserved advanced mathematics for those who would serve as philosopher guardians. • Mathematics education as a separate field • NCTM argued for math to be a part of the curriculum • Growth in school population—perception that intellectual capabilities of students decreased • No need for mathematics in the curriculum • Education attacked for not educating for business & industry and then Sputnik • All events lead to one question: • “The questions were not only what mathematics should be taught and how, but more importantly, who should be taught mathematics. (p. 10, in Stinson, 2004) Gatekeeping Revealed

39. ACCESS or Barrier? Gatekeeping Revealed

40. Gatekeeper Recommendations • Use Visualizations to solve Realistic applications • explorations and experimentation • Balance between conceptual and procedural knowledge in algebra instruction • Change in Curriculum materials • fewer problems • greater depth • emphasis on small group work • analysis of graphical representations • Intelligent use of Technology • Teachers cast in non-traditional roles • Teachers attend workshops to help move beyond transmission of knowledge • Teachers conduct research in their own classrooms Gatekeeping Revealed

41. Targeted Competencies for Novice Teachers PLANNING • Planning for content understandings • Using knowledge of students to inform teaching • Planning assessments to monitor and support student learning INSTRUCTION • Engaging students in learning • Deepening student learning during instruction ASSESSMENT • Analyzing student work • Using feedback to guide further learning • Using assessment to inform instruction REFLECTION • Analyzing Teaching Effectiveness ACADEMIC LANGUAGE • Identifying Language Demands • Supporting students’ academic language development • Evidence of language use 18 Gatekeeping Revealed

42. Can your private thoughts and expectations influence how well a rat runs a maze? Gatekeeping Revealed

43. Expectations in your head act to influence behaviors Gatekeeping Revealed

44. Perceptions found in student writings: • Reluctance to engage in diversity and worry about CRT’s relevance, • Worry about implementing culturally relevant teaching, • Acknowledgment of access to mathematics is a civil rights issue, • Development of social-cultural approaches to teaching, • Investigations of studies on race and Math, • Socio-economic status (SES) playing a role in learning. Gatekeeping Revealed

45. Expectationsfound in student writings: • Understanding CRT is important, • Social Justice issues have a place of teaching, • Questioning approaches to teaching are part of teaching, • Using real data in teaching aids understanding, • Get to know students, as it improves student learning. Gatekeeping Revealed

46. Behaviors Revealed in student writings: • Understanding the vocabulary in mathematics affects student learning, • More support around open-ended questions could improve student learning • Social justice approaches to learning can improve learning environments in mathematics classrooms, • Teaching methods matter in the classroom. Gatekeeping Revealed

47. Discussion Culturally Relevant Teaching is important for pre-service teachers. The task of creating better (more open) learning environments requires teacher introspection about their future behaviors as teachers. Giving students time to struggle with math concepts, using technology to mediate learning, explaining that math is more than memorization and arithmetic, and using Polya problem-solving approaches are revealed through pre-service teacher writings. Gatekeeping Revealed

48. Interpretations • a CRT course help teacher candidates move towards opening the gate to providing their students with access to mathematics • The student writings provide an insight to the type of homework that could open the gate to learning. • The writings help teachers operationalize their role as gatekeepers, then even the type of homework they assign could be more useful. Schmitz & Perels (2011) discuss the sense of independence students experience when they have the appropriate guidance to be in charge of their learning. • Documenting the requisite behaviors and intentions necessary for success in mathematics is important. • We should develop resources for classroom teachers to support learning when students are in and away from the classroom environment. Gatekeeping Revealed

49. When it comes to MATHEMATICS Learning You have 9 lives Gatekeeping Revealed

50. Selected References • Brandsford, J. D., Brown, A. L., & Cocking, R. R. (1999). How people learn: Brain, mind, experience, and school. National Academy Press. • Culturally relevant teaching. (n.d.). Retrieved January 14, 2017, from https://en.wikipedia.org/wiki/Pedagogy • Dee, T. S. (2007). Teachers and the gender gaps in student achievement. Journal of Human Resources, 42(3), 528-554. • Gutstein, E., & Peterson, B. (Eds.). (2005). Rethinking Mathematics: Teaching Social Justice with numbers. Milwaukee: Rethinking Schools, Ltd. • Huitt, W., & Cain, S. (2005). An overview of the conative domain. Educational psychology interactive, 1-20. • Ladson-Billings, G. (1992). Culturally relevant teaching: The key to making multicultural education work. Research and multicultural education: From the margins to the mainstream, 106-121. • Lampert, M. (1990). When the problem is not the question, and the solution is not the answer: Mathematical knowing and teaching. American educational research journal, 27(1), 29-63. • Larson, M. (2016). The Need to Make Homework Comprehensible. National Council of Teachers of Mathematics. • Midgley, C., Feldlaufer, H., & Eccles, J. S. (1989). Student/teacher relations and attitudes toward mathematics before and after the transition to junior high school. Child development, 981-992. • Miller, L. & Spiegel, A. (Hosts). (2015, January 23).Invisibilia: How to become Batman pt. 1[Radio broadcast episode]. http://www.npr.org/programs/invisibilia/378577902/how-to-become-batman • Nasir, N. S., & Cobb, P. (Eds.). (2007). Improving access to mathematics: Diversity and equity in the classroom. New York Teachers College Press. • Polya, G. (1973). How to solve it: A new aspect of mathematical method. Princeton: Princeton University Press. • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. Handbook of research on mathematics teaching and learning, 334-370. • Schmitz, B., & Perels, F. (2011). Self-monitoring of self-regulation during math homework behavior using standardized diaries. Metacognition and Learning, 6(3), 255-273. • Shapka, J. D. (2009). Trajectories of math achievement and perceived math competence over high school and postsecondary education: Effects of an all-girl curriculum in high school. Educational Research and Evaluation, 15(6), 527-541. • Stigler, J. W., & Hiebert, J. (1999). Understanding and improving, comparing standards internationally: Comparing standards internationally: Research and practice in mathematics and beyond, (pp.119 -134). • Stinson, D.W (2004). Mathematics as gate-keeper: Three theoretical perspectives that aim toward empowering all children with a Key to the Gate, The Mathematics Educator14 (1), 8–18. • Tait-McCutcheon, S. L. (2008, June). Self-efficacy in mathematics: Affective, cognitive, and conative domains of functioning. In Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia (pp. 507-513). Gatekeeping Revealed