Welcome to the GRED 565 web site. A protected part of the GRED 565 course can be entered by using the user name and password only (please see a hard copy of the syllabus).
Syllabus and schedule of presentations and topics exams. (Potsdam Campus).
Materials for off-campus sections of GRED 565: OTTAWA, WATERTOWN
Excel Files from Helios: Document 1 (in case when Helios is not accessible through helios.potsdam.edu).
Readings for GRED 565.
Common Core Resources.
Printable math worksheets.
Rubrics web site.
Helping your child learn mathematics web site (U.S. Department of Education).
Cool-Math Games web site
NCATE (National Council for Accreditation of Teacher Education).
Principles and Standards for School Mathematics (NCTM).
Show-Me Center (information and resources needed to support selection and implementation of standards-based middle grades mathematics curricula).
The Eisenhower National Clearinghouse
Mathematics education of gifted and talented students.
School Jobs Bulletin Board. See also http://www.teachmath.org/
Digits of PI.
Elementary Mathematics: Content & Methods
Instructor: Dr. Sergei Abramovich
Office: Satterlee 210
Office Hours: M 9:00 a.m. - 10:00 a.m., TU 7:00 p.m.-8:00 p.m., W 9:00 a.m.-10:00 a.m., and by appointment.
Phone: (315) 267-2541 (office)
Background and Rationale
Contemporary mathematics pedagogy measures an educative growth of a student not in terms of the production of correct answers but in terms of the quality and diversity of thinking. Nowadays, a competence in elementary mathematics teaching means much more than the ability to get a right answer to a standard, procedure-bounded problem. The competence includes an in-depth understanding of the concepts behind procedures being taught and awareness of various tools conducive to mediate conceptual development. This course will reflect change and growth in mathematics education set by the National, New York State, and Ontario curriculum standards for school mathematics. It will attempt to increase the confidence level of a future teacher in creating learning situations at the elementary level in which simply stated questions about familiar concepts can generate a considerable amount of inquiry.
To this end, mathematics currently involved in K-6 program as recommended by the New York State Education Department as well as by the Ontario Ministry of Education will be highlighted as a dynamic discipline, built on progressively connected ideas and mutually related concepts. Students will be introduced to current issues and trends in mathematics education such as the use of concrete embodiments (physical manipulatives) and computing technology (including electronic manipulatives), revision of curriculum and professional standards, assessment of authentic performance, and social constructivism.
SUNY Potsdam Education Unit Conceptual Framework
A Tradition of Excellence: Preparing Creative and Reflective Practitioners
GRED 565 course supports the SUNY Potsdam Education Conceptual Framework in several ways. First, through experiences provided in this course students will continue to develop as "well educated citizens" by modeling the skills, attitudes, and values of inquiry relevant for mathematics content and by appropriately using technology such as the Internet, word processing, spreadsheets, and other electronic information technologies. They will continue to develop as 'reflective practitioners" by modeling inquiry, practice, and reflection in their field experiences and journals. They will effectively use research-based models of curriculum, instruction, and assessment as they plan for instruction, design, and teach lessons meeting the diverse learning needs of students, promoting reflective inquiry, critical thinking, and problem solving, incorporating appropriate technology. They will identify national and state learning standards that are related to their lessons. They will develop as "principled educators" by demonstrating
Course Content and Pedagogy
The course content will revolve around the following major topics (recommended by the New York State Education Department): (i) Mathematical Reasoning; (ii) Number and Numeration; (iii) Operations; (iv) Modeling/Multiple representation; (v) Measurement; (vi) Uncertainty; (vii) Patterns/Functions. The study of these topics will be supported by a variety of mathematics education materials. In accord with the National Council of Teachers of Mathematics Professional Standards for Teaching Mathematics, the course pedagogy will focus on developing proficiency in:
To this end, student-centered discussions of selected mathematics education research publications will be a part of the course activities. Students will be expected to read these publications by using the campus (Crumb) library and the Internet resources. Fostering the ability to use such resources is one of educational objectives of this (graduate level) course.
Text book and other required materials
Students are expected to attend, be prepared for, and participate professionally in each class. This includes the ability to support classroom activities by participating in discussions of homework and readings. Professionalism also includes the ability to keep notes of all class discussions and home assignments, using library and the Internet resources to access information, using e-mail and word processing programs as educational tools.
Special note regarding the weekend offering of the course. Because of intensive nature of studies in such a format, students who miss any weekend (or the time-equivalent combination of classes) will be asked to drop the course and take it at a different time.Those students who miss one full day (either Friday or Saturday) due to a family emergency (which must be documented) may be allowed to make up work in some form (TBA) at the discretion of instructor.
Students are expected to read the text book, course materials, and selected mathematics education publications as assigned by the instructor. Throughout the semester, please plan to watch carefully for assignments given. Some assignments may involve the use of a computer and students must plan for time to work in a Mac computer lab on such assignments. Students are expected to have (or acquire) a minimum knowledge of Microsoft Word, Microsoft Excel, and Dynamic Geometry programs. As stated in the New York State Education Department Core Curriculum for Mathematics, "software that allows students to explore, conjecture and investigate mathematics, provides unique opportunities for students learning mathematics. Teachers should take full advantage of this resource, if available" (see the web site of NY State Education Department at http://www.nysed.gov/rscs/resguide/Mathcc1.pdf, Resource Guides, Mathematics, Elementary Grades, p. 12). Note that these high expectations are set for students (and their teachers alike) as early as in elementary grades.
The course activities will include (depending on a final enrollment) up to eight student-centered discussions of research publications relevant to elementary mathematics curriculum. (These publications have been put on GRED 565 reserve at the Crumb library). To this end, teams of at most 4 students in each team will be created. Each team will be responsible for doing one such discussion. More specifically, this will include the following collaborative activities:
Each such summary will be put on the course web site on a week preceding a discussion with understanding that the whole class can be prepared for discussion by reading this summary via the Internet. (In order to count the number of words in a typed document, one can use "Word Count" feature from Tools menu of MS Word program). One copy of each document is required for a team. In evaluating a presentation, the following rubric will be applied:
Group e-summary of not less than 600 words submitted on time - 20%
Individual reflection of not less than 300 words submitted on time - 20%
The use of a computer during the presentation - 20%
The use of manipultives during the presentation - 20%
The use of transparencies and/or handouts, auxiliary literature, conducting whole class discussion during the presentation - 20%
Submitted individual reflections will be available for a pick up (with my evaluation of the whole presentation) in the plastic box attached to the door of 210 Satterlee Hall on the day of presentation (unless an emergency occurs). Please pick up your graded reflections promptly.
Two topic exams will be given during the semester. These topic exams (to be arranged) will be based on readings, homework, and activities presented in the class. A reading list will be given a week before a topic exam. There will be no make-up topic exams given unless illness or family emergency occur (these must be documented).
A final exam for the course will be replaced by work on a final project. A final project may take different directions. One such direction is to develop a lesson based on one of the key ideas from mathematics core curriculum as recommended by New York State Department of Education (see mentioned above GRED 565 Course Materials). The second direction is to structure a project as a journal that reflects on one's experience in observing an elementary mathematics classroom in the field. Reflections should describe one's observations in terms of their connection to ideas studied in the course. Finally, the third direction is to reflect on a possible involvement in teaching mathematics during a field experience. Regardless of a direction chosen, an underlying philosophy of a final project should be structured by the following basic assumptions of contemporary classroom discourse:
The length of a final project is expected to be three to five pages (not fewer than a 1000-word document typed on a computer). Team projects (not more than three students in a team) are welcome, but collaboration on a project is not required. On the cover sheet of the project please type your e-mail address (it should include e-mail addresses of all team members if it's a collaborative project).
If your final project is an observation journal, please follow guidelines provided in the "pink" document (Field Experience Guidelines). Of particular interest is any information related to students' asking questions during a lesson, the discussion of more than one way to solve a problem, the availability of manipulatives and computers in the elementary classroom and their use by a host teacher. If the use of these tools was never observed, please write about that including grade level(s) observed. Information submitted in your final project based on classroom observations will be considered strictly confidential.
If you are involved in student teaching during your practicum, please write in your final project about topic taught and instructional materials used; describe most interesting episodes from your teaching experience.
If your final project is a lesson plan (supported by New York State or Ontario standards and core curriculum for mathematics – please see the course materials), it should be relevant to the elementary classroom. Your lesson plan may be based on one of the key ideas/strands from the core curriculum. In your lesson plan please address such issues as the use of manipulatives and information technology, the promotion of reflective inquiry and diversity of thinking among students.
Home assignments 20%
The use of technology 10%
Topic exams 30%
Final Project 20%
An interactive chart titled Calculation of Grade is attached to the password-required domain of the course web site. Note: Blue numbers related to exams are subject to change.
According to the chart: range 100%-94% - 4.0; range 87%-93% - 3.7; range 80%-86% - 3.3; range 73%-79% - 3.0; range 66%-72% -2.7; range 59%-65% - 2.3; range 52%-58% - 2.0; below 52% - 0.0.
It is expected that all work will be the students own otherwise documented. Failure to credit others for direct quotations and ideas will be considered plagiarism and will result in the student receiving a grade of 0.0 for that assignment.
List of publications put on reserve at Crumb Library
National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics, Chapter 4: Standards for Grades Pre-K-2. Reston, VA: NCTM.
Conference Board of the Mathematical Sciences. (2001). The Mathematical Education of Teachers, Chapters 1-3. Washington, D.C.: Mathematical Association of America.
Becker, J.P., & Selter C. (1996). Elementary School Practices. In A.J. Bishop et al. (eds)., International Handbook of Mathematics Education, 511-564. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Schifter, D. (1998). Learning Mathematics for Teaching: From a Teacher's Seminar to the Classroom. Journal of Mathematics Teacher Education, 1, pp. 55-87.
Common Core State Standards Initiative (2011). Common Core Standards for Mathematics. http://www.p12.nysed.gov/ciai/common_core_standards/.
Anderson, A., Anderson, J., and Shapiro, J. (2004). Mathematical Discourse in Shared Storybook Reading. Journal for Research in Mathematics Education, 35(1), pp. 5-33.
Nunes, T. (1992). Ethnomathematics and everyday cognition. In D. A. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning . New York: MacMillan.
Fuys, D.J., and Liebov, A.K. (1993). Geometry and Spatial Sense. In R. Jensen (ed.), Research ideas for the classroom: Early childhood mathematics. NCTM: Reston, VA.
Schedule of presentations and topic exams (Spring 2013)
Monday 10:00 a.m. section
Monday 7:10 p.m. section
Tuesday 4:30 p.m. section
1) Textbook: Read section 1, chapters 1, 2, 3. Be ready for discussion.
2) Textbook, p.50: problem with blue square tiles (see figure). Solve this problem. Find as many solutions as you can. Explain your reasoning. Formulate a similar problem and solve it.
3) Textbook, page 15: "Start and jump numbers: searching for patterns" problem. Explore this problem. Be ready to share your ideas with the class.
1) Be prepared for the discussion of the NCTM Standards for preK-2 (summary is on the course web site).
2) Course materials, pp. 33-36, Classroom ideas for grades 1-2: 1B, 1C, 1D, 3C. Read these activities and be ready to present/discuss them in class.
3) Read the textbook pp. 272-275 (about patterns).
4) Create 5 examples of repeating patterns using pictures, describe your patterns using A, B, C, ... terminology.
1) Be prepared for the discussion of Mathematical Education of Teachers (summary is on the course web site).
2) Textbook to read: Chapter 9 (p. 149-164): Developing meanings for the operations.
3) Performance Task 1: "I used two identical shapes to make a rectangle. What might they have been?" Find several solutions to this task and describe the presentation of the task, i.e., educational objective, materials used, difference in presentations at different grade levels.
4) Performance Task 2: Find ways to add consecutive counting numbers in order to reach sums in the range 1 through 15. (For example, 3, 4, and 5 are consecutive numbers and 3+4+5=12). Formulate 5 questions that may stem from this performance task and answer your own questions.
5) Performance Task 3: If the 5 key on your calculator were broken, how could you find the sum 458 + 548 + 354? Is there more than one way of doing that?
1) Read summary for the third discussion topic "Elementary school practices" on the web.
2) Present the following addition problems in a pictorial form (drawing base-ten blocks on small place value charts - see Figure 11.2 on page 195 or Figure 11.10 on page 201 of the textbook) and decide whether or not a problem involves regrouping:
Support each of these addition problems with two types of real-life situations introduced in class.
What reasoning strategy is used in each case? (see textbook, chapter 10, pp. 174-178)
3) Write a subtraction question for each of the following situations.
a. Mr. Wilson had 16 boys in a class of 25 children.
b. Joan weighed 68 pounds and John weighed 95 pounds.
c. Mrs Bennett bought 24 oranges and served 16 of them to girls after soccer practice.
d. Bob had 34 cents. The whistle costs 63 cents.
Identify the type of subtraction question in each case.
Present each subtraction fact as coloring activity on a place-value chart (for reference see worksheets "Subtraction as coloring" (1 & 2), file "GRED 565 spreadsheet files.xls", on the abramovsclass volume of Helios server).
4) Read and understand activities in the Course Materials (be ready for discussion):
p.34, #2C; p. 39, ## 6A and 6B.
1) Prepare for the next discussion topic "Learning mathematics for teaching" (read summary on the course web site). Note: for Spring 2013 the topic is "Common Core State Standards Initiative (2011). Common Core Standards for Mathematics."
Write a story that illustrates the fact 3X5=15 through repeated addition. Write another story to show that 5X3=15 ( multiplication is commutative).
Represent the fact 4X7=28 in the form of a diagram with blocks (or dots); that is, by using blocks (or dots) show multiplication as repeated addition.
Then draw a new picture which would represent 28 as a base-ten number; that is, through tens and ones.
Draw a picture that would represent 28 as base-six number, that is through sixes and ones.
Explore the multiplication table and find patterns in the table. Describe your patterns.
3) Long division at the concept level with base-ten blocks: p. 244 of the textbook. Using the structure of Figure 13.10a (p. 244), represent the fact 426÷3=142 through long division at the concept level and write a story that leads to this division fact (see stories/problems (in bold) on p. 243).
1) Prepare for the next discussion topic: "Common Core State Standards Initiative (2011). Common Core Standards for Mathematics." (summary is on the web).
2) Download "Hair ribbons and Wanda's cake" excel file (see a web link on the protected part of the course website, before summaries) and save it on your disk.
3) Fractions. Answer questions on Figures 15.10 and 15.11, pp. 301-302 of the textbook.
4) Compare two fractions by drawing diagrams (area model for fractions; also paper folding): Which is more, a) 3/8 or 2/5? b) 5/6 or 4/5?
5) Solve the following problems:
a) Ron has several one, two, and five-dollar coins in his pocket (at least three coins of each denomination). If he takes three coins (one by one) out of his pocket, how much money could have he taken? In how many different ways could the total of $12 have he taken?
b) 4/15 of a distance from the downtown Post Office to a Shopping Mall equals 12/5 of a mile. What is this distance? Justify your answer by drawing a picture/diagram and write a number sentence (an arithmetical fact) that corresponds to your picture.
c) I eat 2/3 of a cup of cottage cheese each day for lunch. I have 4 and 2/3 cups of cottage cheese in my refrigerator. How long will that last? Justify your solution by drawing a detailed labeled picture and write a number sentence (an arithmetical fact) that corresponds to your picture.
Day 8. (Spring 2013: Homework is given in hard copy).
GRED 565 Topic Exam 1 Material to be reviewed.
1. This topic exam will consist of nine tasks based on:
a) Classroom ideas for pre-K-K and grades 1-2 that we discussed (more specifically, see tasks 1C and 1D mentioned on p. 33 of the Course Materials)
b) Tasks related to pictorial representations of numbers and operations with numbers using blocks or dots (see item 2 of homework for Day 6 ); problems listed in item 5 of homework for Day 7 (not for spring 2013).
c) Discussions dealing with comparison of and operations on fractions (multiplication, division) using pictorial representation (diagrams of the area/region model).
d) Textbook, pp. 301-302, material of Figures 15.10, and 15.11
e) Recognizing patterns (see homework for Day 3).
2. Tips for taking the test. Some tasks will consist of short response mathematics questions (multiple choice), other tasks will consist of extended response mathematics questions. Read each question carefully and think about answer before writing a response. Make sure to show your work and/or justify your answer when required. Please be prepared to draw simple pictures.
3. Assessment. 2-and 3-point holistic rubrics (a scoring guide suggested by the New York State Department of Education) will be applied to assess your performance on the test. The description of the rubrics can be found on pp. 130-132 of the Course Materials.
1) Read the summary of the next presentation (#6) "Mathematical discourse in shared storybook reading " on the course website.
2) Textbook: read pp. 315-324 (Chapter 16).
3) Using models shown on Figures 16.3 (p. 320) and 16.5 (p. 323) add and subtract fractions:
a) 3/8+1/4 (using fraction circles)
b) 5/6-1/2 (using fraction circles)
c) 2/3+1/6 (using rectangles)
d) 1and 2/3 -1/6 (using set model).
4) Using area model for fraction, multiply fractions on pictures:
Write a story (similar to those presented in class) to support each of the above multiplication situations in a)-d).
5) Compare fractions using 10x10-grids (for reference see such grids on p.344 (Figure 17.8) of the textbook and the abramovsclass file "GRED 565 spreadsheet files.xls", sheet Compare fractions as decimals ): 2/5 and 1/2; 7/10 and 4/5; represent these fractions as decimals and as percent.
6) Textbook: Chapter 18: Read pp. 357-360 about the concepts of ratio and proportion.)
1) Read summary on the course web site related to the next presentation (Ethnomathematics and Everyday Cognition).
2) Textbook: p. 354: solve percentage problems ##1, 2, 3, 4 (in bold) by drawing a picture (similar to Figure 17.8, p. 354).
3) Solve the following problems by drawing a picture to enhance their instructional presentation:
a) In the set of 60 calculators, 16 calculators are TI-81s, 29 calculators are TI-84s and the rest of calculators are TI-92s. What percent of the calculators is TI-92s? What is the ratio of TI-92s to the rest of the calculators?
b) Kelly's class of 30 students has 18 boys. What is the boys to girls ratio in the class? After spring recess three girls were transferred to another school. What is the girls to boys ratio in Kelly's class now?
c) Ron was given 1/3 of the collection of 15 counters. 3/5 of what he got turned out to be the only red counters in this collection. How many counters were red? What is the ratio of red counters to non red counters in what Ron has gotten?
1) Read summary for the presentation "Geometry and spatial sense."
2) Computer assignment "Fraction circles." The assignment will be collected and graded.
3) Work on the final project.
MEMO ON THE FINAL PROJECT
1. Your final project (see p. 5 of the syllabus for details) is due on Day 13.
2. It should be a three to five-page document (not fewer than 1000 words) typed on a computer and submitted in hard copy (NOT through the TaskStream).
3. Team projects (not more than three students in a team) are welcome, but collaboration on a project is not required.
4. If your project is an observation journal, please follow the guidelines provided in the "pink" document. Of particular interest is information related to students asking questions during a lesson, classroom discussion about more than one way to solve a problem, the availability of manipulatives and computers in the elementary classroom and their use by your host teacher. If you have never observed the use of these tools, please write about that. It should include grade level observed; if you observed multiple grade levels, please list all grades you were involved with.
5. If you were involved in student teaching during your 100-hour field experience (practicum), please write about what you have taught, what materials you’ve used, describe most interesting episodes from your teaching experience.
6. You may also present a lesson plan (supported by NY State or Ontario standards and core curriculum for mathematics – please see the course materials booklet) relevant to the elementary classroom. Your lesson may be based on one of the key ideas/strands from the core curriculum. In your lesson plan please describe how you plan to address such issues as the use of manipulatives and information technology, developing mathematical connections, promoting reflective inquiry and diversity of thinking among students.
7. As mentioned in the syllabus, regardless of the type of your final project, its underlying philosophy should be structured by the following basic assumptions of contemporary pedagogy: conceptual development (emphasis on conceptual understanding vs. operational understanding – teaching through problem solving in context rather than through worksheets), reflective inquiry (creating a learning environment in which students feel comfortable to ask "what if" and "why" questions and reflect on their work and that of their peers), making connections among different concepts (e.g., between counting and addition, between addition and subtraction, between addition and multiplication, between subtraction and division, between multiplication and division), and the use of technology (e.g., physical manipulatives, computers, the Internet, computer-generated worksheets, overheads, and so on). It is strongly recommended that you make four separate sections in your project to write about each of the four assumptions.
8. Information submitted in your projects based on classroom observations will be considered strictly confidential.
9. Please visit the protected part of the course web site at http://www2.potsdam.edu/abramovs/gred565site (see password in the syllabus) to read a number of exemplary final projects on all three topics.
1) Assignment: Exploring geometry on grid paper (a hard copy of this assignment was given in class). Note: square is considered to be a special case of rectangle with equal adjacent sides. The assignment will be collected and graded.
2) Solve the following two problems without using formulas for area and perimeter.
a) Among all rectangles with whole number sides and perimeter 24 cm, one rectangle has the largest area and one rectangle has the smallest area. What is the largest area and what is the smallest area? Explain your answer.
b) The area of a rectangle is 84 square centimeters. One side of the rectangle is 14 cm long. Find the length of the other side of this rectangle. Find perimeter of this rectangle. Show your work.
3) Activity on constructing symmetrical shapes (use worksheet given in class).
4) Textbook: read Chapter 20, pp. 402-414.
5) Bring scissors and a glue stick for a hands-on activity.
1) Final projects are due.
2) Exploring arrangements:
a) In how many ways can a doctor schedule appointments for eye exams for Alan, Betsy, and Christina?
b) In how many ways can a doctor schedule appointments for eye exams for Alan, Betsy, Christina, and Derek?
c) How many 2-digit numbers can be made using numerals 3 and 4?
d) How many 3-digit numbers can be made using numerals 3 and 4?
e) How many 3-digit numbers can be made using numerals 0, 3, 4?
(You can use tree diagrams to answer these questions.)
Review problems for topic exam 2.
There will be 9 tasks (extended response mathematics questions) with multiple-choice answers.
1) What percent of 40 is 12? Answer this question using a picture.
2) What percent of 40 is 28? Answer this question using a picture.
3) In the set of 60 calculators 16 are TI-81s, 29 are TI-84s and the rest are TI-92s. What percent of the calculators are TI-92s? What is the ratio of TI-92s to the rest of the calculators? Answer these questions using a picture.
4) Kelly's class of 30 students has 18 boys. What is the boys to girls ratio in the class? After spring recess three girls were transferred to another school. What is the new girls to boys ratio in Kelly's class? Answer these questions using a picture.
5) Among 20 buttons, 6 are red. What percent of the buttons are red? What percent of the buttons are not red? Answer these questions using a picture.
6) A school has students to computers ratio equal 5/2. If the number of students is 100, how many computers are in the school? 10 new computers were purchased. Find the new students to computers ratio? What is the new computers to students ratio? Use a chart to solve this problem.
7) Among 40 students, 16 are boys. What percent of the students are boys? What percent of the students are girls? Use a diagram to answer these questions.
8) There are 12 girls and 8 boys in Jerry's class. What would be the probability of drawing a boy's name from a hat containing the names of all class members? Two girls from the class were transferred to another school after a break. Find the probability of drawing a boy's name from a hat containing the names of all class members except the two girls.
9) Consider numbers (numerals) 9, 8, and 7. How many 3-digit numbers can be made using these numerals? How many 4-digit numbers can be made using numbers 0, 9, 8, and 7? Explain your answer.
10) The area of a rectangle is 56 square centimeters. One side of the rectangle is 8 cm long. Find the length of the other side of this rectangle. Find the perimeter of this rectangle. You may not use geometric formula for area.
11) Among all rectangles with whole number sides and perimeter 64 cm, one rectangle has the largest area and one rectangle has the smallest area. What is the largest area and what is the smallest area? Explain your answer.
12) Draw three geometric shapes with at least one line of symmetry. Draw three geometric shapes with no lines of symmetry.
13) A spinner pictured below is spun. What is the probability of landing on
14) Course Materials: Problems 7F and 7H on p. 64.
Monday/Tuesday, May 9/10.
1) Problems from NYS Testing Program in Mathematics, Grade 4:Course Materials, p.96, #9; p.97, #11; p.98, #13; p.100, #17; p. 101, # 19; p.102, #20; p.106; #27; p.108, #30; p.117, # 38; p.129, #48.2) Bring scissors and glue for a hands-on activity.3) Read textbook, p. 357-358 (section TESSELLATIONS).
4) Final projects are due.