Author Biographies Sinan Kanbir is an Assistant Professor of Mathematics Education, University of Wisconsin—Stevens Point. He received his Ph.D. in mathematics education from Illinois State University, a masters degree in computer and educational technology from Marmara University, and a bachelors degree in mathematics education from Gazi University. Prior to his appointment at UW-Stevens Point, he had taught mathematics at the middle- and high-school levels, and university-level courses for prospective middle- and elementary-school teachers. His main research interests include the development of students’ and teachers’ algebraic reasoning, particularly in the elementary and middle grades, and the development of pre-service and in-service teachers’ mathematical knowledge for teaching. He currently teaches elementary and middle-school mathematics pre- service teacher-education courses. His main research has been in the areas of algebra education and the history of school mathematics. M. A. (Ken) Clements’s masters and doctoral degrees were from the University of Melbourne, and at various times in his career he taught, full-time, for a total of 15 years, in primary and secondary schools. He has taught in six universities, located in five nations, and is currently Professor in the Department of Mathematics at Illinois State University. He has served as a consultant and as a researcher in Australia, Brunei Darussalam, India, Malaysia, Papua New Guinea, South Africa, Thailand, the United Kingdom, the United States of America, and Vietnam. He served as co-editor of the three International Handbooks of Mathematics Education—published by Springer in 1996, 2003 and 2013—and, with Nerida Ellerton, co-authored a UNESCO book on mathematics education research. He has authored or edited 33 books and over 200 articles on mathematics education, and is honorary life member of both the Mathematical Association of Victoria and the Mathematics Education Research Group of Australasia. © Springer International Publishing AG 2018 223 S. Kanbir et al., Using Design Research and History to Tackle a Fundamental Problem with School Algebra, History of Mathematics Education, DOI 10.1007/978-3-319-59204-6 224 Author Biographies Author Biographies (Continued) Nerida F. Ellerton has been Professor in the Department of Mathematics at Illinois State University since 2002. She holds two doctoral degrees—one in Physical Chemistry and the other in Mathematics Education. Between 1997 and 2002 she was Dean of Education at the University of Southern Queensland, Australia. She has taught in schools and at four universities, and has also served as consultant in numerous countries, including Australia, Bangladesh, Brunei Darussalam, China, Malaysia, the Philippines, Thailand, the United States of America, and Vietnam. She has written or edited 18 books and has had more than 150 articles published in refereed journals or edited collections. Between 1993 and 1997 she was editor of the Mathematics Education Research Journal, and between 2011 and 2014 she was Associate Educator of the Journal for Research in Mathematics Education. In 2016 she received an award for top researcher at Illinois State University. List of Appendices Appendix Page A Protocol for Algebra Interviews with Seventh-Graders 227 B Algebra Test (Three Parallel Versions are Reproduced) 229 C “Questionnaire” Completed by Seventh-Grade Students at School W at the Beginning of the Algebra Workshops on “Structure” 237 D Statement of Instructional Aims for the Structure Workshops with the Seventh-Grade Students at School W 239 E Detailed Lesson Plans for Four Workshops on “Structure” for Seventh- Grade Students at School W, Including Homework Challenges for Each Workshop 241 F Detailed Plans for Group Tasks in the Modeling Workshops: Finding Recursive and Explicit Rules for Patterns 257 G Classroom Observation Schedule 287 H Pre-Teaching to Post-Teaching “Growth” with Respect to the Five Basic Cognitive Structure Components 289 I Generalization Categories (After Radford, 2006) 295 © Springer International Publishing AG 2018 225 S. Kanbir et al., Using Design Research and History to Tackle a Fundamental Problem with School Algebra, History of Mathematics Education, DOI 10.1007/978-3-319-59204-6 Appendix A Protocol for Algebra Interviews with Seventh-Graders 1. The interviews should be tape-recorded. 2. The interviewer should have the following: (a) A sheet of paper with 482 + (18 + 300) on it. [See (2) below.] (b) A sheet of paper with value of 4 × (¼ × 128) on it. [See (3) below.] (c) A sheet of paper with Tn = 2n + 3 on it. [See (4) below.] (d) A sheet of paper with Tn = 5n – 2 on it. [See (5) below.] (e) A sheet of paper with 15 – (5 – x) = (15 – 5) – x on it. [See (6) below.] (f) A sheet of paper showing the Table in Question 7 below. (g) A sheet of paper showing the illustration in Question 8 below. 3. Ask each student the questions listed below, and make notes as you go. 1. I am going to say two words and, as soon as I say them, I want you to say something, or draw something, or do something—do the first thing that comes into your head after I say the words. The words are … “distributive property.” Here are the words again: “distributive property.” 2. Without using a calculator, find the value of 482 + (18 + 300). [Once an answer is given ask for an explanation of where that answer came from.] 3. Without using a calculator, find the value of 4 × (¼ × 128). [Once an answer is given ask for an explanation of where that answer came from.] 4. If we write Tn = 2n + 3, then we can say T5 equals 13, because 2 times 5 plus 3 equals 13. What would T11 equal? [When the pupil gives an answer, ask her or him to write down how she or he obtained that answer. Also, ask the student to explain what she or he thought, in words.] 227 228 Appendix A: Protocol for Algebra Interviews 5. Give the pupil a piece of paper with Tn = 5n – 2 on it, then ask her or him to say which values of n would make Tn greater than 20. [When the pupil gives an answer, ask her or him to write down how she or he obtained that answer. Also, ask the student to explain what she or he thought, in words.] 6. Give the pupil a piece of paper with the equation 15 – (5 – x) = (15 – 5) – x on it, and then, pointing to the x, say: “Which numbers could x equal so that you would get a true statement?” [When the pupil gives an answer, ask her or him to write down how she or he obtained that answer. Also, ask the student to explain what she or he thought, in words.] 7. Give the pupil a piece of paper with the following table on it: First 1 2 3 4 5 … n Value Second 3 5 7 9 ? … ? Value Then ask (pointing): What number should we place under the 5 in the table? Then ask (pointing): What do you think we should we put under the n? 8. The diagram below shows how tables and chairs are arranged in a school cafeteria. One table can seat 4 people, and tables can be pushed together (but always in a straight line). When two tables are pushed together, 6 people can sit around the table (as shown), etc. A. If 10 tables were pushed together (in a straight line), how many people could sit around them (assuming the pattern shown above)? B. If Pn represents the number of people who can sit when n tables are pushed together (in a straight line), what is the rule giving Pn in terms of n? Appendix B: Algebra Test (Pre-Teaching and Retention Version) 229 Appendix B Algebra Test (Three Parallel Versions are Reproduced) Algebra Test (Pre-Teaching Version, and also the Retention Version) 1. If Tn = 13 – 3n, where n can be any whole number, which values of n would make the values of Tn positive? 2. A really important property for numbers and for algebra is called the associative property for multiplication. Describe this property in your own words. 3. Suppose you were asked to calculate the value of 940 + (60 + 403) in your head (without writing anything down, or using a calculator). How would you do it, and which property would you be using? 4. If Sn = 101 + 50n, where n can be any whole number, what is the value of S4? 5. A student is creating towers out of unit cubes. Each unit cube, by itself, has 6 square faces, but when two unit cubes are stuck together, one exactly on top of the other, there are only 10 faces in the tower (including the top and the bottom). The first tower has 1 unit cube and 6 faces. The second tower has 2 unit cubes, one on top of the other, and the third tower has 3 unit cubes, etc. We say that the surface area of the first tower is 6 units, of the second tower is 10 units, etc. n = 1 n = 2 n = 3 n = 4 What is the surface area of a tower with 50 cubes? 6. If Tn = 2n + 5, what is value of Tn + 1 – Tn? 230 Appendix B: Algebra Test (Pre-Teaching and Retention Version) 7. Which values of x would make this statement true? 5(x – 3) = 5x – 10. Explain how you got your answer. 8. Without using a calculator find the value of (72 × 5) × 2, and explain how you got your answer. 9. What must x equal if 12 – (8 – 4) = (12 – x) – 4? Explain how you got your answer. 10. If 96 = 16y + 32y, what must y equal? Explain how you got your answer.