Network Workshop:Writing in Math

Greg Hargreaves

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

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?

Watermelon problem solution:

100lb watermelon contains 99% of water which is equivalent to 99lb of water and hence 1lb of fleshAfter cutting and drying it is 98% water and hence 2% fleshNow, 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!

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.)

Solution:

Circumference of the earth (C1) is 2πr = 2π.6400 = 40212.38597 kmIf the rope is lifted 1 m, the new radius is 6400.001km and the new circumference (C2) is 2π.6400.001 = 40212.39225kmSo 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

The mind is not a vessel to be filled, but a fire to be ignited.

(Plutarch)

Productive PedagogiesA 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. 

 

He who learns but does not think is lost

(Chinese Proverb)

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 facts and 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.”

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

Original Terms New Terms

• Evaluation

• Synthesis

• Analysis

• Application

• Comprehension

• Knowledge

•Creating

•Evaluating

•Analysing

•Applying

•Understanding

•Remembering

BLOOM’S REVISED TAXONOMY

CreatingCreatingGenerating new ideas, products, or ways of viewing thingsDesigning, constructing, planning, producing, inventing.

 EvaluatingEvaluating

Justifying a decision or course of actionChecking, hypothesising, critiquing, experimenting, judging

  AnalysingAnalysing

Breaking information into parts to explore understandings and relationshipsComparing, organising, deconstructing, interrogating, finding

 ApplyingApplying

Using information in another familiar situationImplementing, carrying out, using, executing

 UnderstandingUnderstanding

Explaining ideas or conceptsInterpreting, summarising, paraphrasing, classifying, explaining

 RememberingRemembering

Recalling informationRecognising, listing, describing, retrieving, naming, finding

 

A turtle makes progress when it sticks its neck out.

(Anon)

RememberingThe learner is able to recall, restate and remember

learned information.– Recognising– Listing– Describing– Identifying– Retrieving– Naming– Locating– Finding

  Can you recall information? 

Remembering cont’• List• Memorise• Relate• Show• Locate• Distinguish• Give example• Reproduce• Repeat• Label• Recall• Know• Group• Read• Write• Outline

• Listen• Group• Choose• Recite• Review• Quote• Record• Match• Select• Underline• Cite• Sort

Recall or recognition of

specific information

Products include:

• Quiz

• Definition

• Fact

• Worksheet

• Test

• Label

• List

• Workbook

• Reproduction

•Vocabulary

Classroom Roles for Remembering

Teacher roles

• Directs• Tells• Shows• Examines• Questions• Evaluates

Student roles

• Responds• Absorbs• Remembers• Recognises• Memorises• Defines• Describes• Retells• Passive recipient

UnderstandingThe 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?

Understanding cont’• Restate• Identify• Discuss• Retell• Research• Annotate• Translate• Give examples of• Paraphrase• Reorganise• Associate

• Describe• Report• Recognise• Review• Observe• Outline• Account for• Interpret• Give main idea• Estimate• Define

Understanding of given

information

Products include:

• Recitation

• Summary

• Collection

• Explanation

• Show and tell

• Example

• Quiz

• List

• Label

• Outline

Classroom Roles for Understanding

Teacher roles

• Demonstrates• Listens• Questions• Compares• Contrasts• Examines

Student roles

• Explains• Describes• Outlines• Restates• Translates• Demonstrates• Interprets• Active participant

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?

Applying cont’• Translate• Manipulate• Exhibit• Illustrate• Calculate• Interpret• Make• Practice• Apply• Operate• Interview

• Paint• Change• Compute• Sequence• Show• Solve• Collect• Demonstrate• Dramatise• Construct• Use• Adapt• Draw

Using strategies, concepts, principles and theories in new

situations

Products include:

• Photograph

• Illustration

• Simulation

• Sculpture

• Demonstration

• Presentation

• Interview

• Performance

• Diary

• Journal

Classroom Roles for Applying Teacher roles

• Shows• Facilitates• Observes• Evaluates• Organises• Questions

Student roles

• Solves problems• Demonstrates use of

knowledge• Calculates• Compiles• Completes• Illustrates • Constructs • Active recipient

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

Solution: 21978 x 4 87912

AnalysingThe 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?

Analysing cont’• Distinguish• Question• Appraise• Experiment• Inspect• Examine• Probe• Separate• Inquire• Arrange• Investigate• Sift• Research• Calculate• Criticize

• Compare• Contrast• Survey• Detect• Group• Order• Sequence• Test• Debate• Analyse• Diagram• Relate• Dissect• Categorise• Discriminate

Breaking information down into its component

elements

Products include:

• Graph

• Spreadsheet

• Checklist

• Chart

• Outline

• Survey

• Database

• Mobile

• Abstract

• Report

Classroom Roles for AnalysingTeacher 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

EvaluatingThe 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?

Evaluating cont’• Judge• Rate• Validate• Predict• Assess• Score• Revise• Infer• Determine• Prioritise• Tell why• Compare• Evaluate• Defend• Select• Measure

• Choose• Conclude• Deduce• Debate• Justify• Recommend• Discriminate• Appraise• Value• Probe• Argue• Decide• Criticise• Rank• Reject

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

Classroom Roles for EvaluatingTeacher roles

• Clarifies• Accepts• Guides

Student roles

• Judges• Disputes• Compares• Critiques• Questions• Argues• Assesses• Decides• Selects • Justifies• Active participant

CreatingThe 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?

Creating cont’• Compose• Assemble• Organise• Invent• Compile• Forecast• Devise• Propose• Construct• Plan• Prepare• Develop• Originate• Imagine• Generate

• Formulate• Improve• Act• Predict• Produce• Blend• Set up• Devise• Concoct• Compile

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

100 ProblemSeparate 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

Solution:

12 20 4 64

12 + 4 = 16; 20 – 4 = 16; 4 x 4 = 16; 64 ÷4 = 16

12 + 20 + 4 + 64 = 100

Classroom Roles for CreatingTeacher roles

• Facilitates• Extends • Reflects• Analyses• Evaluates

Student roles

• Designs• Formulates• Plans• Takes risks• Modifies• Creates• Proposes• Active participant

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

The responses we get from students – both

written and verbal, often depend on how we ask the question

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

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

This world is but a canvas for our imaginations.

(Henry David Thoreau)

The diagram below shows the results on a Science test for two groups, labeled as Group A and Group B. The mean score for Group A is 62.0 and the mean for Group B is 64.5. Students pass this test when their score is 50 or above.

Looking at the diagram, the teacher claims that Group B did better than Group A in this test. The students in Group A don’t agree with their teacher. They try to convince the teacher that Group B may not necessarily have done better. Give one mathematical argument, using the graph, that the students in Group A could use.

How can we change this question so that it differentiates our students?

Consider the following:

Question 1

How many students were in each of the classes?Class A __________ Class B _____________

Question 2How many more students got 60 - 69 in Class B compared to Class A? _________

Question 3a) What is the Modal range for each class?Class A __________ Class B ____________b) Explain how you got this answer.

Question 4a) What is the median Score range for each class? Explain your answer.Class A __________ Class B _____________b) Explain how you got this answer.

Question 5Represent the information in the graph on one or more pie charts.

Question 6Compare your pie chart to the graph given in the question – which do you think is the most effective way to represent the data? Justify your answer.

Question 7

Looking at the diagram, the teacher claims that Group B did better than Group A in this test.

The students in Group A don’t agree with their teacher. They try to convince the teacher that Group B may not necessarily have done better.

Give at least two mathematical arguments, (using either of the graphs) that the students in Group A could use.

We can use graphs like this to determine slopes, equations, intercepts etc – how can we use it to stimulate students to write?

For this problem you need to imagine that you hovering above a straight road in a helicopter. The road runs north-south, and looking down you see different coloured cars travelling along the road. A telephone box is visible and serves as a reference point. You can see from the telephone to a point 240 metres north along the road. You cannot see to the south of the phone box. Your observations were made for a period of 15 seconds and the results are recorded graphically.

Describe in English the movements of each car giving as much detail as possible (eg. speed, direction, does it pass other cars and if so when & where does this happen, etc.)

Write a story about one or more of the interacting cars.

Use the following stimulus material to write scaffolded questions.

The classroom should….Allow for flexible groups

Foster the students responsibility for their own learning

Provide a mechanism for students to get help when teacher is unavailable

Provide ongoing learning activities for students as they complete assigned work

Display assignments/project schedules/scoring rubrics/general procedures

Instruction should….Be less teacher lecture and more investigation and group sharing

Be less whole group and more small group or individual

Be aimed at different learning styles

Allow for a variety of responses

The Learning Environment Is the physical arrangement supportive of both teacher directed activities and student-student interaction and does it allow access by the teacher to all students?

Is there evidence of daily routines?

Are there problem solving strategies posted and visible to students?

Is there a word wall or vocabulary list?

Are the relevant standards clearly displayed and referred to during the lesson?

Assignments should…Vary in content and difficulty

Vary in time allocated

Contain clear directions for students and parents

Problem PresentationIs the learning objective displayed for students and does it reflect the main focus of the lesson?

Is there evidence of academic rigor and high expectations placed on the students?

Are there word problems with a variety of structures and solution paths?

Is there a mix of individual & collaborative work?

Is there evidence of connections between the task and the real world?

Is there assessment of understanding before students are sent for group work?

Group ActivityAre there structures to ensure that each student is required to participate?

Is the group focusing on the task as they manage their own work?

Is careful though given to the selection of group members?

Are students interacting with each other as well as working independently?

ShareAre students provided with opportunities to explain, clarify & justify their thinking?

Are students given the opportunity to share the same problem so different thinking can be examined?

CommunicationDoes the teacher select multiple-solution strategies and orchestrate student presentations in order to build connections?

Does the teacher routinely incorporate writing practices into the lesson?

Are students held accountable for their verbal explanations?

AssessmentAre students clear on the expectations for success?

Do teachers and students use rubrics as guidelines for improvement, assessment and grading?

Is the teacher using assessment results to guide instruction?

Has the teacher implemented a method to record student performance data on a regular basis?

Are errors in answers used as learning sites?

Are assessments used to reflect the way the math is being taught, stressing understanding not just rote learning?

Can students read and comprehend problems?

Is there evidence of the use of assessment tools like item analysis and diagnostic testing?

Is there a summary at the end of the lesson to assess students learning?