MATH 5375
Constructing Whole Number and Operation

For the course syllabus, click here and here.
For problems worked or assigned in class, click here.
For a math ed web resources page, click here.
For links to the online readings, see below.
For student interviewing tips, click here.
For the sample interview write-up, click here.
For tips on writing papers on math problems, click here.
For tips on writing the 2-Problem Paper in this course, click here.

NOTE: Technical information such as prerequisites, text materials, course format and assignments, and other details can be found in the syllabus, a copy of which is provided in a link at the top of this page.

"College-level" problems

• Session 5: Numeration Systems -- use the summative question 7, supported by the other questions on these pages, to develop a cohesive, comparative explanation of the Roman and Babylonian systems in particular, and symbol value and place value systems in general.
• Session 5: Zeroes and Ones -- use the summative question 7, supported by the other questions on this page as well as the class discussion and responses, to explain options for designing numeration systems using only two meaningful characters (including true binary), and compare their properties.
• Session 6: Sticks and Stones, together with Mayan addition and subtraction from Session 8 -- use summative question at the end of the latter set, supported by the problems on both these pages, to develop a cohesive explanation of the Mayan system and how regrouping works in it, including a comparative analysis (relative to the Hindu-Arabic system).
• Session 7: Introduction to Bases -- use the summative questions at the end of this activity to give clear sets of instructions for conversion to base n using concrete [proportional] and symbolic representations, and compare the two. Of course, you may use some of the problems in this activity as examples, but be sure your instructions are stated in purely general terms.
• Sessions 8-14: Mayan addition and subtraction, Driving in Septobasiland, Regrouping in Other Bases, Mayan multiplication, Making Groups in Other Bases, Money in Septobasiland, Mayan division, Division in Other Bases, Arithmetic in Other Bases -- Use any one of these activities as a basis for developing a general explanation of the traditional algorithm for performing any one of the four arithmetic operations (choose one!) in terms of a general base (i.e., not ten). Be careful to couch your general set of instructions in terms of a general base, and not any specific one (although of course all your examples or illustrations will use particular bases).
• Session 15: Russian Peasant algorithm -- Although the two-problem paper is due before Session 15, some participants may be interested in analyzing this intriguing alternative algorithm for multiplication. Work through the questions in the handout, and write a paper which (a) briefly explains the steps in the algorithm, (b) explains mathematically why the algorithm is correct, (c) describes the mathematical skills needed and not needed, and (d) uses these to speculate why a person might prefer this algorithm over the traditional (partial products) multidigit multiplication algorithm.

Links to related sites

• learner.org's online courses "Learning Math" include Number and Operation

Links for specific class meetings

• Session 1
  • DMI sample interview write-up (from BST)
• Session 2
  • Video on the need to decompose numbers (from learner.org)
• Session 4
  • the Marshmallows video (from Annenberg)
• Session 5
  • Numeration Systems
  • pp.104-108 of Bassarear's text, from Google books
  • A history of the Sumerian/Babylonian system
  • Zero in the Babylonian system
  • Video on the history of binary (base two) numeration in computers (from learner.org)
  • the Pumpkin Seeds video (from Annenberg)
• Session 6
  • Video of children using DigiBlocks (from learner.org)
  • Case 13 from the old edition of BST
• Session 7
  • Video of a middle school classroom discussion of converting from base ten to base five (from learner.org)
  • Video of the same middle school classroom discussing how to convert symbolically from base five to base ten (from learner.org)
  • Mini-cases (a) and (b)
  • Mini-cases (c) and (d)
• Session 9
  • Base four addition applet (from learner.org)
  • optional reading, Chapter 2 from the prior edition of BST
• Session 11
  • reading: "Children's invention of multidigit multiplication and division algorithms" by Ambrose, Baek, and Carpenter
  • the multiplication half of this article made simple: "Children's mathematical understanding and invented strategies for multidigit multiplication" by Baek
• Session 14
  • reading: alternative division algorithms (read Ed's before Susan's)