Network Workshop: Writing in Math

1. Network Workshop:Writing in Math Greg Hargreaves

4. This is NOT what is meant by writing in Math!!!

5. Sliced Watermelon A 100 pound watermelon is 99% water. After being sliced and left uncovered it is 98% water. What weight of water has been lost?

6. Watermelon problem solution: 100lb watermelon contains 99% of water which is equivalent to 99lb of water and hence 1lb of flesh After cutting and drying it is 98% water and hence 2% flesh Now, we know that only the water has gone and the amount of flesh at the start and the finish is the same Therefore 1lb is equivalent to 2% of the final weight. This means the final weight is 50lb (1lb is 2%, how much is 100%) So the weight loss is 100lb – 50lb = 50lb!

7. Rope around the earth Imagine wrapping a piece of rope around the earth's circumference, and then having people lined up around the same rope. If everyone along the rope were to pick it up to a height of 1 meter, how long would the gap be between the two ends? What is your guess? Now calculate the actual gap. Show all your work below. (The radius of the earth is about 6400 km.)

8. Solution: Circumference of the earth (C1) is 2πr = 2π.6400 = 40212.38597 km If the rope is lifted 1 m, the new radius is 6400.001km and the new circumference (C2) is 2π.6400.001 = 40212.39225km So the gap in the rope would be C2 – C1 = 40212.39225 – 40212.38597 = 0.006279km or 6.279m Alternatively the gap is C2 –C1 = 2π .6400.001 - 2π.6400 = 2π (6400.001 – 6400) = 2π. 0.001 = 0.00628km or 6.28m

9. The mind is not a vessel to be filled, but a fire to be ignited. (Plutarch)

10. Productive Pedagogies A guide to Productive Pedagogies: Classroom reflection manual lists three degrees of incorporation of Higher-order thinking skills in a “Continuum of practice”: • Students are engaged only in lower-order thinking; i.e. they receive, or recite, or participate in routine practice. In no activities during the lesson do students go beyond simple reproduction of knowledge. • Students are primarily engaged in routine lower-order thinking for a good share of the lesson. There is at least one significant question or activity in which some students perform some higher-order thinking. • Almost all students, almost all of the time are engaged in higher-order thinking. 

11. He who learns but does not think is lost (Chinese Proverb)

12. What is Higher-order thinking? A guide to Productive Pedagogies: Classroom reflection manual states that: “Higher-order thinking by students involves the transformation of information and ideas. This transformation occurs when students combine factsand ideas and synthesize, generalize, explain, hypothesize or arrive at some conclusion or interpretation. Manipulating information and ideas through these processes allows students to solve problems, gain understanding and discover new meaning.”

13. When students engage in the construction of knowledge, an element of uncertainty is introduced into the instructional process and the outcomes are not always predictable; in other words, the teacher is not certain what the students will produce. In helping students become producers of knowledge, the teacher’s main instructional task is to create activities or environments that allow students opportunities to engage in higher-order thinking

14. Original Terms New Terms • Evaluation • Synthesis • Analysis • Application • Comprehension • Knowledge • Creating • Evaluating • Analysing • Applying • Understanding • Remembering

15. BLOOM’S REVISED TAXONOMYCreatingGenerating new ideas, products, or ways of viewing thingsDesigning, constructing, planning, producing, inventing.EvaluatingJustifying a decision or course of actionChecking, hypothesising, critiquing, experimenting, judgingAnalysingBreaking information into parts to explore understandings and relationshipsComparing, organising, deconstructing, interrogating, findingApplyingUsing information in another familiar situationImplementing, carrying out, using, executingUnderstandingExplaining ideas or conceptsInterpreting, summarising, paraphrasing, classifying, explainingRememberingRecalling informationRecognising, listing, describing, retrieving, naming, finding Higher-order thinking

16. A turtle makes progress when it sticks its neck out. (Anon)

17. Remembering The learner is able to recall, restate and remember learned information. • Recognising • Listing • Describing • Identifying • Retrieving • Naming • Locating • Finding   Can you recall information?

19. Remembering cont’ • Listen • Group • Choose • Recite • Review • Quote • Record • Match • Select • Underline • Cite • Sort • List • Memorise • Relate • Show • Locate • Distinguish • Give example • Reproduce • Repeat • Label • Recall • Know • Group • Read • Write • Outline Recall or recognition of specific information • Products include: • Quiz • Definition • Fact • Worksheet • Test • Label • List • Workbook • Reproduction • Vocabulary

20. Teacher roles Directs Tells Shows Examines Questions Evaluates Student roles Responds Absorbs Remembers Recognises Memorises Defines Describes Retells Passive recipient Classroom Roles for Remembering

21. Understanding The learner grasps the meaning of information by interpreting and translating what has been learned. • Interpreting • Exemplifying • Summarising • Inferring • Paraphrasing • Classifying • Comparing • Explaining   Can you explain ideas or concepts?

23. Understanding cont’ • Describe • Report • Recognise • Review • Observe • Outline • Account for • Interpret • Give main • idea • Estimate • Define • Restate • Identify • Discuss • Retell • Research • Annotate • Translate • Give examples of • Paraphrase • Reorganise • Associate Understanding of given information • Products include: • Recitation • Summary • Collection • Explanation • Show and tell • Example • Quiz • List • Label • Outline

24. Teacher roles Demonstrates Listens Questions Compares Contrasts Examines Student roles Explains Describes Outlines Restates Translates Demonstrates Interprets Active participant Classroom Roles for Understanding

25. Applying The learner makes use of information in a context different from the one in which it was learned. • Implementing • Carrying out • Using • Executing  Can you use the information in another familiar situation?

27. Applying cont’ • Paint • Change • Compute • Sequence • Show • Solve • Collect • Demonstrate • Dramatise • Construct • Use • Adapt • Draw • Translate • Manipulate • Exhibit • Illustrate • Calculate • Interpret • Make • Practice • Apply • Operate • Interview Using strategies, concepts, principles and theories in new situations • Products include: • Photograph • Illustration • Simulation • Sculpture • Demonstration • Presentation • Interview • Performance • Diary • Journal

28. Teacher roles Shows Facilitates Observes Evaluates Organises Questions Student roles Solves problems Demonstrates use of knowledge Calculates Compiles Completes Illustrates Constructs Active recipient Classroom Roles for Applying

29. Find the numerical values of the letters A, B, C, D & E if the following is true: A B C D E x4 E D C B A

30. Solution: 21978x 4 87912

31. Analysing The learner breaks learned information into its parts to best understand that information. • Comparing • Organising • Deconstructing • Attributing • Outlining • Finding • Structuring • Integrating Can you break information into parts to explore understandings and relationships?

33. Analysing cont’ • Compare • Contrast • Survey • Detect • Group • Order • Sequence • Test • Debate • Analyse • Diagram • Relate • Dissect • Categorise • Discriminate • Distinguish • Question • Appraise • Experiment • Inspect • Examine • Probe • Separate • Inquire • Arrange • Investigate • Sift • Research • Calculate • Criticize Breaking information down into its component elements • Products include: • Graph • Spreadsheet • Checklist • Chart • Outline • Survey • Database • Mobile • Abstract • Report

34. Teacher roles Probes Guides Observes Evaluates Acts as a resource Questions Organises Dissects Student roles Discusses Uncovers Argues Debates Thinks deeply Tests Examines Questions Calculates Investigates Inquires Active participant Classroom Roles for Analysing

35. Evaluating The learner makes decisions based on in-depth reflection, criticism and assessment. • Checking • Hypothesising • Critiquing • Experimenting • Judging • Testing • Detecting • Monitoring Can you justify a decision or course of action?

37. Evaluating cont’ • Choose • Conclude • Deduce • Debate • Justify • Recommend • Discriminate • Appraise • Value • Probe • Argue • Decide • Criticise • Rank • Reject • Judge • Rate • Validate • Predict • Assess • Score • Revise • Infer • Determine • Prioritise • Tell why • Compare • Evaluate • Defend • Select • Measure Judging the value of ideas, materials and methods by developing and applying standards and criteria. • Products include: • Debate • Panel • Report • Evaluation • Investigation • Verdict • Conclusion • Persuasive speech

38. Teacher roles Clarifies Accepts Guides Student roles Judges Disputes Compares Critiques Questions Argues Assesses Decides Selects Justifies Active participant Classroom Roles for Evaluating

39. Creating The learner creates new ideas and information using what has been previously learned. • Designing • Constructing • Planning • Producing • Inventing • Devising • Making  Can you generate new products, ideas, or ways of viewing things?

41. Creating cont’ • Formulate • Improve • Act • Predict • Produce • Blend • Set up • Devise • Concoct • Compile • Compose • Assemble • Organise • Invent • Compile • Forecast • Devise • Propose • Construct • Plan • Prepare • Develop • Originate • Imagine • Generate Putting together ideas or elements to develop a original idea or engage in creative thinking. • Products include: • Film • Story • Project • Plan • New game • Song • Newspaper • Media product • Advertisement • Painting

42. 100 Problem • Separate 100 into FOUR numbers, so that by • Adding 4 to the first • Subtracting 4 from the second • Multiplying the third by 4 and • Dividing the fourth by 4 • ALL THE RESULTS WILL BE THE SAME

43. Solution: 12 20 4 64 12 + 4 = 16; 20 – 4 = 16; 4 x 4 = 16; 64 ÷4 = 16 12 + 20 + 4 + 64 = 100

44. Teacher roles Facilitates Extends Reflects Analyses Evaluates Student roles Designs Formulates Plans Takes risks Modifies Creates Proposes Active participant Classroom Roles for Creating

45. A good teacher makes you think even when you don’t want to.

46. The responses we get from students – both written and verbal, often depend on how we ask the question

47. Lower and Higher Order Questions • Lower level questions are those at the remembering, understanding and lower level application levels of the taxonomy. • Usually questions at the lower levels are appropriate for: • Evaluating students’ preparation and comprehension • Diagnosing students’ strengths and weaknesses • Reviewing and/or summarising content

49. Lower and Higher Order Questions • Higher level questions are those requiring complex application, analysis, evaluation or creation skills. • Questions at higher levels of the taxonomy are usually most appropriate for: • Encouraging students to think more deeply and critically • Problem solving • Encouraging discussions • Stimulating students to seek information on their own

50. This world is but a canvas for our imaginations. (Henry David Thoreau)