Location: Room H111, Nipissing University North Bay, Ontario, Canada

1. Education in Primary/Junior Division Math & Literacy (EDUC 4605/4615) Workshop 3: Data Management, Probability, Assessment/Evaluation/Reporting, and Planning Instruction Location: Room H111, Nipissing University North Bay, Ontario, Canada Instructor: Daniel H. Jarvis, Ph.D. Email: danj@nipissingu.ca Website: http://www.nipissingu.ca/faculty/danj Office Number: H331 Office Phone: (705) 474 3461 x 4445 Office Hours: by appointment (see embedded schedule)

2. Workshop 3 Agenda • Activity 1: Problem-based Learning (PBL): Measures of Central Tendency • Data Management: Measures of Central Tendency, Graphs, Interpreting • Probability: Key Concepts; Theoretical & Experimental Probability • Activity 2: Crossing the Thames River (Probability) • Assessment, Evaluation & Reporting • Growing Success Document (2010) • RE4MUL8 Highlights (2012)

3. Measures of Central Tendency Definition: A measure of central tendency is a value that can represent a set of data; also called “central measure” Mean: The average; the sum of a set of numbers divided by the number of numbers in the set. [For example, the average of 10 + 20 + 30 is 60/3 = 20] Median: The middle number in a set of numbers, such that half the numbers in the set are less and half are greater when the numbers are arranged in order. [For example, 14 is the median for the set of numbers 7, 9, 14, 21, 39. If there is an even number of numbers, the median is the mean of the two middle numbers.] Mode: The number that occurs most often in a set of data. [For example, in a set of data with the values 3, 5, 6, 5, 6, 5, 4, 5, the mode is 5. A set can have no mode or several.]

4. Central Tendency (Problem-Based) Nine students were given various amounts of M & M candy Based on the statistical measures of central tendency given, determine how many M & M’s each person received Is there more than one possible solution? Lowest Number 0 Highest Number 9 Mean 4 Median 4 Mode 2 Strategy: 9 spaces, 0 first, 9 last, 4 central, two 2s, total 36, etc. Possible Solutions: 022245579, 012245679, 022245669, ???

5. Many Graphing Options Knowing when to use the best graph to represent the data one is analyzing: Tally Charts Pictographs Line Graphs Bar Graphs (Histograms) Circle Graphs (Pie Charts) Scatterplots Box and Whisker Graphs Stem and Leaf Graphs Bar Line Circle

6. Walking and Talking through Graphs Have students select their favourite ice cream flavour and take a token based on a legend written on the board (e.g., brown is chocolate) Have students form long line with four colours grouped together Now have students form a large circle; give each child a long string; teacher/student stands at the center and holds all of the string ends; note the approximate “slices” of colours Record data on the board and/or in an Excel spreadsheet Give students pre-cut strips of oversized grid paper; have them record data with letters or colours for flavours Using the “bulls-eye” handout, have students tape this strip of marked grid paper together (best to leave end square blank) to form a circle Have them place the paper circle on the concentric circle template and mark the divisions for each ice-cream choice then colour and label the circle/pie graph Now have them cut the same strip of paper and line up these pieces vertically on the back of the same handout, which leads to a related histogram (bar graph) Use Dynamic Statistics software like TinkerPlots to import Excel data and view

7. 1. 2. 3. Interpreting Graphs & Creating Narratives Q: If water is poured in at a consistent rate, which time versus water level graph matches each container?

8. 3-D Tic-Tac-Toe/Super Tic-Tac-Toe/Quixo Sudoku/Kakuro/Chocolate Fix Blockus/Tetris/Pentominoes Chess/Checkers/Mastermind/Backgammon Blockers/L-Game/Go/Othello Yahtzee/Boggle/Scrabble/SET Monopoly (Junior & Original) Candyland/Battleship/Sorry/Trouble Euchre/Cribbage/Bridge/Hearts/Golf Uno/Skip-Bo/Wizzard/Phase 10 Ready or Not (Math Scrabble) Other Math/Logic Games? Mathematical Skill Games Q: List as many games as possible that reinforce math concepts?

9. Question 1: What makes a good game? Question 2: Does this game actually focus on what I am trying to teach, or is it merely entertaining (i.e., Where’s the math?! More directly, how does it tie to the curricular expectations)? Question 3: Can this game be extended or modified to teach further developed and/or related skills? Notes: Students love to make their own games up based on new concepts being taught; by encouraging student authoring, we allow for lateral and creative thinking within the math class “Solitaire” versus “FreeCell” philosophy – a continuum between chance and skill – look for math games that encourage active manipulation versus passive play, so that concepts can be reinforced more effectively Where’s the Math?!

10. Probability Language and Example Desired Result Possible Result P(event) = Hawaiian Island Group: Transportation Options Plane Boat Plane Boat Boat Helicopter 0 1 Chance of having two plane rides? 1/6 Chance of having three helicopter rides? 0/6 = 0 Chance of having at least one boat ride? 6/6 = 1 PBP BBP PBB BBB PBH BBH

11. Probability Big Ideas (Small) • Probability is a measure of likelihood. It can be expressed qualitatively or quantitatively as a fraction or decimal between 0 and 1 or an equivalent percent. • Unless an event is either impossible or certain, you can never be sure how often it will occur. • To determine an experimental probability, a large representative sample should be used. • To determine a theoretical probability, an analysis of possible equally likely outcomes is required.

12. Car accident in city Chair will not break Probability in Everyday Life Key Emphasis: Probability is part of our/students’ everyday lives; not an abstract concept, but one understood intuitively by even the very youngest children (i.e., babies—what is likely to happen). 1 0

13. Using a Probability Recording Chart Paper / Rock / Scissors Possible Classroom Resources: a die type (4 – 20 sided); spinner; bottle & marbles Crossing the Thames: Moving 6 Boats (2x) Probability Strategy and Analysis Worksheet Attribute Blocks: What are the chances of selecting a red block? a hexagon? a thin block? Experimenting With Probability

14. Crossing the French River Analysis N.B. This is the “bell,” or normal distribution curve from statistics.

15. Five Key Concepts regarding Assessment and Evaluation A Closer Look at the Achievement Chart (2005) & Expectations re. Expectations (OME) “Design-down” Planning & Bloom’s Taxonomy Recording Strategies and Tools Growing Success (2010) New Assessment Document: Assessing and Reporting in ON Growing Up Mathematically: Assessment EQAO Sample Assessments (2010) Workshop: Assessment, Evaluation and Reporting

16. Assessment Meaning The word “assess” comes from the Latin word that means “to sit beside.” In classroom assessment, the teacher is the fundamental assessment instrument as s/he “sits beside” and coaches students toward higher achievement. To be a good assessor, s/he needs a variety of strategies, methods and tools to use with the students over time. The primary purpose of assessment and evaluation is to improve student learning. Program Planning and Assessment (OMET, 1999)

17. KEY # 1: Clear Expectations When a student knows in advance what the expectation of a classroom task is, there is a marked increase in response. RUBRICS/EXEMPLARS CAN HELP TO CLARIFY THE EXPECTATIONS AND THE VARIOUS LEVELS OF ACHIEVEMENT FOR STUDENTS SO THAT THE “MYSTERY”, OR UNKNOWN, ELEMENT IS GREATLY REDUCED

18. KEY # 2: Multiple Forms • Initial Assessment (Diagnostic) • Consider the prior knowledge and experience students must bring to the task and create a list of “look-fors” to determine whether students are ready. This initial assessment can take place while students are brainstorming, playing a game, completing mini-tasks, or solving a problem • Assessment for Learning (Formative) • Use the information gathered during this phase of assessment to plan learning tasks that best address individual needs. Be sure to monitor progress to ensure that the strategies applied are helping students to better understand the concepts • Assessment of Learning (Summative) • During this phase of assessment, consider assessment strategies that address different student needs. Paper-and-pencil tests are not always the best indicator of student understanding and skill level. • Growing Up Mathematically, OAME, 2005, p. 31

19. KEY # 3: Validity & Reliability Validity (On Various Forms of Assessment) • Accurately portrays what it purports to measure • Results are a good reflection of students’ knowledge, attitude or skill development • Ensures a good match between expectations and strategies (Q: Do you “teach to your own tests”? A: I hope so—and that they are quality tests that reflect the spirit/content of new curriculum.) Reliability (Among a Staff) • Amount of agreement that exists among teachers in how they evaluate student work in mathematics • Shared understanding about criteria for assessment about standards to be applied • Avoid idiosyncratic perceptions and biases Earl & Cousins, 1993

20. KEY # 4: Multiple Tools • Paper and Pencil • Performance Assessment • Open-Ended Questions • Investigations • Journals • Observations (Anecdotal) • Conference & Interviews • Self and Peer Assessment

21. KEY # 5: Assessment for Learning • Assessment should support the learning of important mathematics and furnish useful information to both teachers and students. • When assessment is an integral part of mathematics instruction, it contributes significantly to all students’ mathematics learning. • Assessment should become a routine part of the ongoing classroom activity rather than an interruption. • Assessment should be more than merely a test at the end of instruction to see how students perform under special conditions; rather, it should be an integral part of instruction that informs and guides teachers as they make instructional decisions. • Assessment should not merely be done to students; rather, it should also be done for students, to guide and enhance their learning. Principles & Standards (2000), NCTM, pp.22-23

22. Assessment & Evaluation Assessment is often defined as the process of: • Systematically gathering information from a variety of sources • Providing students with descriptive feedback for improvement Evaluation is often defined as the process of: • Judging the quality of student work on the basis of established criteria and against actual examples of student work exemplifying the levels of achievement (e.g., Exemplars) • Assigning a value to represent that quality • letter grades for Grades 1-6 • percentages Grades 7-12

23. Some Key Assessment Questions Do Overall expectations have to be evaluated? If so, how? Do all Specific expectations have to be taught? Do all Specific expectations have to be evaluated? How are Mathematical Process expectations to be treated?

24. Some Key Assessment Answers Yes: The Overall expectations are evaluated by the students’ achievement of the related Specific expectations. The Overall expectations are not evaluated in and of themselves. Yes: All specific expectations must be accounted for in either instruction or assessment. However, the teacher will determine the amount of time and the degree of depth for each. A teacher will often be covering a group, or cluster, of Specific expectations at any given time. No: Teachers select the Specific expectations that, in their professional judgement, will allow the students to demonstrate achievement of Overall expectations. Mathematical Process expectations are to be treated as Specific expectations which must each be taught (but not all individually assessed). Note: In mathematics, if a problem-based learning approach is taken, then students are consistently afforded opportunities in class to demonstrate the Mathematical Process expectations (see Jarvis, 2008 article).

25. Achievement Chart Categories (OC Math: Revised, 2005) think . . . “KUTCA" Knowledge/Understanding: Subject-specific content acquired in each grade (knowledge), and the comprehension of its meaning and significance (understanding) Thinking: The use of critical and creative thinking skills and/or processes Communication: The conveying of meaning through various forms Application: The use of knowledge and skills to make connections within and between various contexts

26. Achievement Levels Defined • Level 3: The “provincial” standard which identifies a high level of achievement of the provincial expectations. Parents of students achieving at Level 3 in a particular grade can be confident that their children will be prepared for work at the next grade. • Level 1: Identifies achievement that falls much below the provincial standard. • Level 2: Identifies achievement that approaches the standard. • Level 4: Identifies achievement that surpasses the standard. It should be noted that achievement at Level 4 does not mean that the student has achieved expectations beyond those specified for a particular grade. It indicates that the student has achieved all or almost all of the expectations for that grade, and that he or she demonstrates the ability to use the knowledge and skills specified for that grade in more sophisticated ways than a student achieving at Level 3. Q: Should we refer to “Level 3 Questions”? “Level 3 student”? A: Better to have questions that allow for four levels of achievement, and to refer to students who generally achieve at a Level 3 in area(s).

27. “Design Down” Planning Expectations Assessment Tasks Teaching/Learning Strategies Accommodations/Modifications

28. Using Bloom’s Taxonomy Knowledge Comprehension Application Analysis Synthesis Evaluation Rich Learning and Assessment Tasks (see Flewelling) are planned and implemented to reach higher-order Bloom levels. This usually represents a “trade-off,” as assessment is more involved.

29. Student presentation to parents and teacher Student takes ownership Goal setting becomes a key stage Preparation time is essential Student-Led Interviews Student Teacher Parent

30. GROWING SUCCESS Assessment, Evaluation, and Reporting in Ontario Schools First Edition (2010) Covering Grades 1-12

31. Reporting Student Achievement • Elementary Progress Report Card • Issued between October 20 and November 20 • Emphasis on development of learning skills and work habits as: • Excellent • Good • Satisfactory • Needs Improvement • Indicates progress students are making towards achievement of the curriculum expectations for each subject/strand as: • Progressing Very Well • Progressing Well • Progressing With Difficulty

32. Reporting Student Achievement • Elementary Provincial Report Card • Indicates a student’s development of the learning skills and work habits as: • Excellent • Good • Satisfactory • Needs Improvement • Indicates a student’s achievement of the curriculum expectations in all subjects and strands as: • Letter grades for Grades 1-6 • Percentage marks for Grades 7-8

33. Reporting Student Achievement • Elementary Provincial Report Card • Issued twice in the school year: • January 20 to February 20 – reflects student’s achievement of curriculum expectations introduced and developed from September to January/February • Towards the end of June – reflects student’s achievement of curriculum expectations introduced or further developed from January/February to June

34. Reporting Student Achievement • Elementary Provincial Report Card • Language, all four strands are reported for both reports • French, one mark, no strands are reported • Native Language, a space is provided to indicate the native language, one mark, no strands are reported • Mathematics, at least four of the five strands are reported for each report, each strand is reported at least once per year • History and Geography, history and/or geography are reported for both reports, each is reported at least once per year • Health Education and Physical Education, both are reported for both reports • The Arts, at least three of the four strands are reported for each report, each strand is reported at least once per year

35. DVD Reflections Think: Based on the OAME video clip on Rich Assessment: (i) What role does questioning by the teacher play in each phase of assessment? (ii) What might the teacher do if the information gathered using different assessment strategies is inconsistent (e.g., if the student communicates understanding during an interview but not on a test)? Pair: Discuss these impressions with a partner or with a small group. Share: Whole-group discussion, sharing “Assessment involves the teacher’s gathering of information to make instructional decisions and to inform students about the progress of their mathematics learning and their achievement of curriculum expectations. Rich assessment involves the use of various assessment strategies (e.g., observation, interview, performance assessment, work-sample analysis) and tools (e.g., anecdotal notes, checklists, rating scales, rubrics) that fit the task, the learning context, and the students’ readiness for assessment.”