H-m-m-m, let's see... the regular price of that stereo I want is $250.00, but it's on sale for 20% off right now. What would the total be with taxes?
NUMERACY FOR ADULT LITERACY LEARNERS
Adult Literacy and Continuing Education 280-800 Portage Avenue Winnipeg, Manitoba R3G 0N4
Manitoba Education and Training
Numeracy for Adult Literacy Learners
Prepared by: Lori Herod, M.Ed., B.A.
October 2000
Adult Literacy and Continuing Education 280-800 Portage Avenue Winnipeg, Manitoba R3G 0N4
Manitoba Education and Training
ACKNOWLEDGEMENTS
The Adult Literacy and Continuing Education, Department of Manitoba Education and Training, would like to thank the following people/organizations for their support of Project Bridges under which this course was produced. ♦ Funding - The National Literacy Secretariat ♦ Administrative Support - St. Agnes Anglican Church, Carberry, Manitoba ♦ Accountant - Ms K. Orchard ♦ Curriculum Development - Ms L. Herod
Manitoba Education and Training Adult Literacy and Continuing Education Level II Certification Course – Numeracy
TABLE OF CONTENTS
Section
Page
Course Information
1
Module 1 - Introduction to Numeracy Teaching and Learning
3
Module 2 - Numeracy for Everyday Living
12
Module 3 - Numeracy for GED/High School Diploma
30
Module 4 - Conclusion
55
References
66
Manitoba Education and Training Adult Literacy and Continuing Education Level II Certification Course - Numeracy
COURSE INFORMATION General Numeracy is an optional course for the Level II Literacy Certificate. It is offered here in correspondence mode and should take approximately ten hours to complete. The material is relevant to all four stages of literacy in the province of Manitoba. Course Objectives The objectives of this course are to: ♦ Outline numeracy levels in adult literacy programs and the equivalent public school grades so that practitioners may prepare students for General Education Diploma follow-on programs. ♦ Introduce and apply the concept of "authentic learning" ♦ Provide an approach to planning instruction and blending numeracy into an overall literacy program ♦ Assist practitioners to identify and plan instruction for students with numeracy learning difficulties Course Outline The topics that will be covered in this course include: ♦
numeracy equivalents of adult literacy levels to public school grades;
♦
mathematical strands and related numeracy skills;
♦
learning outcomes and assessment;
♦
active and authentic learning in numeracy instruction;
♦
numeracy related learning difficulties
♦
text and Internet resources.
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Assignments and Evaluation The course will be evaluated on a “Complete/Incomplete” basis. There are assignments at the end of each module that should be completed at your own pace. Completed assignments should be forwarded as a package to ALCE via one of the following methods: ⇒ Mail:
Correspondence Courses Adult Literacy and Continuing Education (ALCE) 280 – 800 Portage Avenue Winnipeg, Manitoba R3G 0N4
⇒ E-mail:
[email protected]
⇒ Fax:
(204) 948-3104
Questions/Comments/Assistance If you require assistance, clarification or have questions or comments about the materials, please telephone ALCE at: (204) 945-8247 in Winnipeg or 1-800-282-8069 ext. 8247 Toll free
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Manitoba Education and Training Adult Literacy and Continuing Education Level II Certification Course - Numeracy
MODULE 1: INTRODUCTION TO NUMERACY TEACHING AND LEARNING
Module Outline v Introduction v Whole Numeracy Ø Active and Authentic Learning Ø Blending Numeracy into a Larger Literacy Program Ø Problem-solving and Communicating Mathematically v Numeracy "Streams" v Conclusion v Assignment v Appendixes
Introduction Learners will have different needs when it comes to numeracy. Some will want to improve their numeracy skills in everyday living kinds of situations such as shopping, banking, and so on. Others will need very specific instruction in preparation for undertaking a General Education Development (GED) or high school diploma program.
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Whole Numeracy Many adults, not just literacy learners, are anxious about numeracy. Math anxiety stems in large part from the emphasis in the past on the operational or computational side of mathematics. Mathematical ability was seen as something only certain people had. What was missing from this approach, however, was demonstrating to learners that numeracy has "real life" applications that we use daily in our everyday lives. This linking of the operational side of math to its functional side is referred to as a "whole math" or "holistic math" approach and its components are outlined below. Active and Authentic Learning The first thing that must be said about teaching and learning numeracy is that like any learning it should be active and authentic. Research into adult learning has demonstrated that learning is most effective when it is tied to "real life" situations that are meaningful and useful to the student. In addition, the student needs to be actively engaged in his/her learning rather than passively memorizing formulas, multiplication tables, etc. This approach helps adult learners to make abstract concepts concrete; applying knowledge/skills in their own lives/needs is intrinsically motivating. It also encourages critical thinking; that is, the ability to come at problems using a variety of strategies and to know where mathematics fits into a bigger picture. As Archambeault (1993) writes: Whole math activities use real-life and hands-on experiences as the basis for learning mathematical procedures. Adults experience mathematics and number concepts when they shop for groceries, buy gasoline for the car, eat in a restaurant, prepare food, and take medicine. Instructional activities based on these experiences demonstrate an immediate, concrete application of the math concept and also serve to reduce math anxiety. Numeracy: Part of a Larger Literacy Program The second thing that must be said about teaching and learning numeracy is that numeracy which is taught in isolation from other areas such as reading, writing and spelling will not be effective as that which is blended into an overall, larger
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program of developing literacy skills/knowledge. Thus, a mathematics learning session should include components in which the student reads, writes and spells as part of the session.
Reading
Writing
OVERALL LITERACY PROGRAM
Numeracy
Spelling
For example, students are given a task in which they are to read the newspaper and find examples of percents. They write out a list of the areas in which percents are used (e.g., sports, sales flyer, etc), highlighting any words they have difficulty spelling. As a group they discuss areas in their own lives in which they tend to use percents (e.g., shopping, employment income and benefits, etc). Problem-solving and Communicating Mathematically The emphasis in mathematics teaching and learning in the past has been to concentrate on number operations/computation. And while this is indeed crucial, there are other areas now considered to be equally important to developing wellrounded mathematics skills/knowledge that learners can apply in concrete ways in their everyday lives. Two skills in particular are being emphasized: • Problem-solving • Communicating Problem-solving refers to the ability of learners to generate, organize, evaluate and apply mathematics both on paper and in real life situations. Terms that illustrate the reasoning involved in these stages are contained in the following table (Hope & Small, 1993). These are useful for developing questions that will promote students' ability to reason and problem-solve mathematically.
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Generate Data - research - experiment - measure - estimate - calculate - extend - hypothesize - predict - manipulate - explore - invent - discover - associate - brainstorm - survey
Organize Data - tally - arrange - rearrange - order - sequence - classify - match - sort - graph - flowchart - chart - diagram - connect - relate
Evaluate Data - summarize - justify - test - conclude - interpret - explain - reason - infer - question - verify - assess
Apply Data - discuss - investigate - illustrate - describe - display - construct - demonstrate - elaborate - generalize
An example of an activity you might use is as follows: Ø Generate data - Conduct a survey to find out which of the following comedy shows is most popular among your fellow students: § § § §
Third Rock from the Sun Dharma and Greg The 70's Show Frasier
Ø Organize data - Graph your results. Ø Evaluate data - Which show is the most/least popular? Explain how you arrived at your conclusion. Ø Apply data - Discuss factors which might affect a show's popularity rating (e.g., gender, age, culture, time show is on, marketing, etc). In order to problem-solve effectively, we need to teach learner's the basic steps. These include: • • • • • •
Comprehending the problem Developing a plan to solve the problem Implementing the plan Evaluating the results Reflecting on the plan, considering alternatives Communicating the solution(s)
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Communicating mathematically refers to the ability of learners to talk about, discuss, brainstorm, explain, etc., about the numeracy they are using. Educators recognize that math cannot be learned in isolation, "isolation" not only from other areas of learning, but also from other learners: Most of us remember our math classes as being a very quiet time of the school day. Teachers did most of the talking: explaining a concept, asking questions and giving instructions. Students worked independently and silently at their seats. There was little opportunity for "math talk" and student interaction. The new curriculum frameworks recognize that mathematics is a way of communicating .... As children are busy doing their math activities, they need to talk about what they are doing, why they are doing it, how they are thinking and what they are learning (Math Matters, p.14). The importance of being able to "communicate mathematically" can be seen in everyday situations such as trying to point out an error on a dinner or utility bill, mapping out a landscaping diagram or floor plan for our spouse, helping our children do math homework, negotiating a sale of some sort, doing up a budget at home or work, and so on. So, not only do we need to be able to solve math problems, we need to be able to communicate the information to others. Numeracy "Streams" As the above section suggests, one of the first things we want to find out from learners is what their general learning aim is regarding numeracy. What do they want/need to use numeracy skills/knowledge for? Do they want to make everyday living easier by learning about budgeting, banking, etc? Are they planning on obtaining their General Education Diploma or High School Diploma? Do they need a specific mathematical skill for their employment perhaps? A screening form has been included at Appendix A to capture this information. As the screening form will indicate, students will generally fall into one of two basic "streams" regarding their desire to improve their numeracy skills: •
Everyday living numeracy - for use in everyday life
•
Program specific numeracy - bring skills up to level needed for a GED/high school diploma program
These "streams" will be discussed in more detail in Modules 2 and 3 respectively. We also need some background information about their numeracy skills. Did they have problems in mathematics in school. Why? When did they start to experience problems and in what areas? What remedial action, if any, was taken. Did this help? This information is also captured on the form at Appendix A.
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Conclusion Current thinking among educators suggests that numeracy teaching and learning must go beyond a narrow focus on computational/operational skills and knowledge toward integrating mathematics into learners' lives in authentic and active ways. This involves a wider range of skills/knowledge on the part of the learner, in particular the ability to reason and communicate mathematically. It also requires that tutors be aware of how to develop and implement numeracy instruction that is active and authentic. In the following module, we will look at doing so in "everyday living" numeracy.
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ASSIGNMENT - MODULE 1 Please answer the following on a separate page. 1) What are two general reasons adult literacy learners tend to want to improve their numeracy skills for? 2) What grade school level in mathematics do learners need to be at for a General Education Diploma (GED) or Adult High School program? 3) What are three examples of how learners might use numeracy in their everyday lives? 4) Why is it important to make learning active and authentic? 5) Construct an activity which includes each stage/component of the reasoning process (i.e., generate data, organize data, evaluate data, and apply data) 6) Describe how you might blend reading, writing and spelling into the above activity.
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Appendix A To Module 1 NUMERACY: BACKGROUND AND LEARNING GOALS
Name: ______________________________ Date: ____________________ LEARNING GOALS v What are your goal(s) as far as improving your numeracy skills? (e.g., everyday living skills, working toward General Education Diploma or High School Diploma, workplace skill required, etc)
BACKGROUND v How many schools did you attend while growing up? (If moved often, may have resulted in gaps in learning)
v What was the highest grade that you completed in school?
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v What areas of mathematics were most/least difficult for you? Why?
v What grade were you in when you started to experience these difficulties in mathematics?
v Did you receive any help for these difficulties? If yes, was the assistance helpful? Why?
v Comments/Other
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Manitoba Education and Training Adult Literacy and Continuing Education Level II Certification Course - Numeracy
MODULE 2: NUMERACY FOR EVERYDAY LIVING v v v v v v v v
Module Outline Introduction Determining Learners' Objectives and Goals Developing a Learning Plan Student-Centered Learning Assessment Videotape Learning Activity Conclusion Appendixes
Introduction Many of our literacy students will have numeracy learning goals that are related to improving their lives -- being better able to help their children with homework, doing up a household budget, figuring out taxes, and so on. The focus of this module will be on techniques and strategies for teaching and learning numeracy for everyday living. Determining Learners' Objectives and Goals We used the screening form in Module 1 to determine whether a learner has a specific (e.g., GED/high school diploma) or a more general (everyday numeracy) learning aim in mind. Once we have determined this, we then need to identify his/her learning objectives and goals. For example, many literacy students want help with their financial skills. Thus, their learning objective is to improve their financial numeracy skills. This is still somewhat general, however, and we next need to determine what specific goal(s) the learner wants to achieve. For example:
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Aim
Improve everyday numeracy skills.
Objective(s)
Improve financial skills.
Goal(s)
Learn how to balance a chequebook.
Numeracy Skills Required Other Literacy Skills Resources
Addition and subtraction, possibly multiplication/division. Printing/cursive writing, capitalization of proper names, short forms (dates), reading. Blank cheques and record pages.
Tutors will need to determine what numeracy skills and resources will be needed for each goal, as well as ways to draw other literacy skills into the learning. A general form has been included at Appendix A for this purpose. It is particularly useful in a one-to-one tutoring situation. A learner's goals may be fairly general or quite specific, and be short, medium, and/or long term. And, of course, the learner may have more than one objective and several goals under each objective. For example, in the above scenario the learner may also wish to develop a household budget, learn about personal banking, and develop a long-term financial plan. Whatever the case may be, the key for learners in this "stream" is to determine their interests/needs and frame numeracy teaching and learning around these. Several examples of active and authentic activities are included at Appendix B and a list of further resources has also been included as Appendix C. Developing a Learning Plan Depending on the nature of each particular program, instruction may involve one-to-one tutoring and/or small group learning. It is recommended that if the program is set up for one-to-one tutoring, that small group activities be included whenever possible. As we have discussed in Module 1 and shall see later in this module, communicating mathematically and problem-solving through input from others are key elements in the development of numeracy skills. In either case, tutors must decide on an appropriate approach to teaching and learning numeracy in this stream. This will depend on many factors including who the learners are, how many there are, what the resources and policies of the program are, and so on. Essentially though, there are three basic approaches: •
Formal - Tutor "maps out" a sequential/formal learning plan that all learners will follow (e.g., GED/high school diploma preparation).
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•
Informal - Tutor is flexible and allows numeracy learning to unfold according to the needs and interests of various students (e.g., one-toone tutoring in everyday living numeracy).
•
Mixed - Tutor develops a "global" plan and fills in the content/detail according to the needs and interest of the learners (e.g., everyday living numeracy, small group learning).
It is likely that in the majority of programs a "mixed" approach will be most useful for everyday living numeracy teaching and learning. The next section will provide an example of "mixing in" details based on the needs/wants of a variety of learners. Student-Centered Learning Juggling the objectives and goals of a variety of students is not a simple matter. The temptation may be to use a formal, sequential plan in which the tutor decides all of the content, activities, assessment procedures, etc. With a little creativity, however, tutors can build a learning plan that addresses a diversity of needs and interests and allows students to have a degree of input into and control over the learning process. One approach is to use an extended activity in which the tutor uses the themes of interest to his/her group and develops a series of active and authentic numeracy "units" such as the one outlined below. v Extended Activity Ø Lead-in lessons covering the following: § Percentages § Checking accounts § Savings accounts § Wages--including deductions § Household expenses § Grocery shopping § Budgets § Reconciling a Bank Statement Ø Objective: Give the students a real life situation that requires the use of all the knowledge they have learned, so they can put it to use in a realistic situation. Note: It is suggested that a fictitious scenario be used rather than students' real life information to protect privacy and confidentiality. Ø Give the students "A Life" § In advance write out different scenarios that a student can have (i.e., Single parent with one child age 4, Married with spouse not working with 3 children ages 4, 5, and 8).
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§ §
Write out different jobs--McDonalds, Sales Clerk, Waitress, Teacher, Nurse, etc., and include what their salaries are and whether they get benefits, how often they get paid and so on. Give each student a chequebook and a certain beginning balance that they each have to start with.
Ø Inform the students that they will have to find a place to live. Have several newspapers ready for them to find a suitable place to fit their "families". They have to write a check to you for rent, deposit, etc. The students will also have to write out checks for all of their bills, extra spending, or anything extra that might pop up. Have the students write 2 checks "per week" for miscellaneous items. Also have them draw as to whether or not they have a car and whether or not they make car payments Ø Tell the students that all cheques go to you. (You are the bank.) Every learning session they receive a different statement about things that can happen to a person in real life (e.g., Child needs braces. Pay orthodontist $100.00 each month for the next four months). At the end of the period (e.g., two weeks) give each student a bank statement and have them reconcile their chequebooks. Ø Items to be turned in: § Budget--per month § Revised budgets--after their first ones didn't quite work § Deposit slips for all deposits going into the chequing account Ø Extend the problem even further to buying a house, developing a long term financial plan, learning about credit and investing, etc. There are a great many opportunities to blend in reading, writing and spelling in this activity. Assessment In that learning in this stream involves improving everyday living, numeracy assessment does not necessarily need to be as formal or specific as in the case in the diploma studies stream. Essentially, the measure of success is the learner's ability to apply the general numeracy they have learned into various aspects of their lives. That said, however, some learners will feel more confident and affirmed if they are given a grade or mark for their efforts. Thus, tutors will need to involve learners in determining how assessment will be conducted. Self-assessment by the learner is an excellent way to promote reflection on learning. For example, Shifter (1996) has her students complete the following exercise at certain points in their learning:
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What I Know About "
" (in this example it is triangles)
1st Week Triangles are shapes
2nd Week Triangles have 3 sides and 3 angles
3rd Week Equilateral triangles have equal sides and angles
They have 3 sides
The sum of the outer angles equals 360 degrees
Isosceles triangles have two sides the same and a different bottom.
Inside angles equal 180 degrees
Triangles are usually symmetrical
4th Week Any triangle can be made by dividing 360 degrees 3 ways with any number in each group. The taller a triangle is, the less degrees there will be in the top angle, the more on the bottom ones.
This method allows students to see ongoing growth in a particular area. The Learning Plan Form at Appendix A allows them to track their general, overall progress. This form captures short, intermediate, and long-term goals and helps learners to see if they are achieving these goals as well as where they have been and where they are going. It "paints the bigger picture" so to speak. There are many ways in which more formalized assessment can be conducted (e.g., pencil and paper quizzes and tests, marked projects, etc). The exact type of assessment tool will, of course, depend on the nature of the numeracy being evaluated and the objectives of the learning. As such, tutors will need to develop their own tools based on the particular situation. Videotape Learning Activity A videotape entitled Adult Numeracy: A New Approach is required to complete this module of the course. Contact Adult Literacy and Continuing Education (see page 2) to borrow a copy of the videotape, as you will be viewing it and completing several activities related to it in this module. The videotape was produced in 1994 by the National Center on Adult Literacy in Philadelphia, Pennsylvania, USA. It was broadcast live in that year by PBS to about 850 sites, and is an excellent example of active and authentic learning. You will note that the activities are based on "real life" applications of mathematics, and the learners are very engaged throughout the tasks.
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In contrast to older approaches to teaching mathematics which stressed learning the operational side of mathematics, the "real life" activities in this video highlight several important differences between the two approaches: §
The focus is on the problem-solving or reasoning process (versus finding the right answer).
§
Instructors serve as guides or facilitators of the problem-solving process (versus delivering/assessing material to be learned).
§
Mathematical communication and active learning are stressed throughout (i.e., learners work in pairs or groups versus individually and there is lots of discussion, versus working on paper and pencil tasks quietly and individually at desks).
§
Teaching and learning are based on concrete/authentic problems or tasks (versus abstract formulae or operations).
§
An integrated approach is used (versus teaching math in a sequential or step-by-step manner).
§
The ability to estimate is emphasized (versus using exact computational or numbers operation skills/knowledge to solve problems).
The videotape is approximately 2 hours in length and covers the following: §
Opening - Students and instructors discuss what learning math means to them
§
Live Classroom Exercise - The Ice Cream Problem
§
Taped Classroom Exercise - Percents
§
Live Classroom Exercise - Discussing the Ice Cream Problem
§
Participant Exercise - The Detective Problem
§
Panel Discussion and Viewer Call-in
§
Closing
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Please turn on the videotape now and watch until the facilitators have explained the "Ice Cream Problem" and the participants are beginning to work. Turn off the tape, complete the Ice Cream Problem (Appendix D to this module), and answer the following questions on a separate page. §
What type of skills/knowledge did you find that you used in solving this problem?
§
How might you integrate the ice cream problem into other areas of your learners' literacy program (i.e., reading, writing and spelling)?
§
What might you add to extend the ice cream problem?
Now, please turn the tape back on and watch it to the beginning of the "Detective Problem." Please turn the tape off and answer the following questions. §
The learners worked in pairs for the "Ice Cream Problem" and in groups during the pre-taped session on percents. What is the value of having learners work together?
§
How might you encourage "mathematical communication" when you are tutoring a single learner versus a group?
Please turn the tape back on and listen to the "Detective Problem. Turn the tape off and write out you how you would approach the problem. (Please include your notes when you send in the assignment.) Turn the tape back on once you have completed the task and watch it until the end. Then answer the following questions. §
In the "Detective Problem," what different responses did the viewers come up with that you hadn't thought of?
§
What might be some of the concepts/operations you could discuss with learners that were involved in this exercise?
§
Give one example of a survey that you might have your learners do that would help them learn about statistics and probability (i.e., collecting, organizing, evaluating and communicating data).
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Conclusion This module covered numeracy for everyday living and looked at identifying the student's learning objectives and goals. It was suggested that programs offering one-to-one tutoring, include small group activities as much as possible in order to promote communication and problem-solving strategies. A videotaped activity allowed participants to see the importance of these two elements, as well as experience active and authentic learning themselves. In the next module we will look at teaching and learning preparatory numeracy for GED/Adult High School diploma studies.
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Appendix A To Module 2 NUMERACY LEARNING PLAN Name: ____________________________________ Date: ________________ Aim: ____________________________________________________________ Objective: _______________________________________________________ Short Term
Intermediate
Long Term
Goal(s)
Numeracy Skills Required
Resources
Activities
Literacy Skills
Progress
Other
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Appendix B To Module 2 EXAMPLES OF ACTIVITES FOR EVERYDAY LIVING NUMERACY v Newspaper Activities (taken from Revised Newspaper Math by M. Terry) Ø Lottery Dreamer Gambles $50,000 & Loses - Lynn Borisoff wanted to be rich. First she sold her house in Hamilton for $50,000. Then she spent this money on lottery tickets. The tickets cost $10.00 each. Lynn spent her summer vacation driving to different towns to buy the tickets. "With so many tickets, I figured I had to win big" she said. All she won was $1,300.00. "I think the lottery hooks people with lots of small cash prizes. How many people get close to the really big money?" She grabbed a handful of tickets and said, "You know, these tickets look like money, but they're just garbage!" Lynn sees some hope. "I spent all my money and only won peanuts. I'm pretty broke right now, but I can make more money by working hard." §
It cost Lynn $50,000 to make $1,300. What was her net loss?
§
Ticket sellers make $0.15 on each ticket they sell. How much money did they make from the tickets Lynn bought?
§
Suppose it took Lynn 12 seconds to check each lottery ticket to see if she had won any money. How many hours and minutes will it take Lynn to check all of her tickets?
§
The big winners were a group of thirteen people. They have to split a million dollars. How much will each winner receive?
Ø Complete the Headlines §
Melbourne, Australia - An Australian physical fitness expert is really not walking! He walked for 71 hours at an average of 5km/h. After completing his "stroll," he said he felt fine. He went to the hospital for a check-up and was released after treatment for blistered feet. Complete this headline: Australian Completes Walk of ______ Km
§
Montrose, USA - Jim and his fiancée planned to be married. Unfortunately, the plans fell apart. Jim received his ring back and 12 cookies for every day they had been together. His former fiancee gave him a total of 4,380 cookies. Complete this headline: Couple Breaks Up After _____ Days Together
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v Blended Activities Ø Student Developed Problems- Hicks and Waddlington (1994) have their numeracy students write math problems based on their personal lives. For example, one student came up with the following: §
My name is ___________. I ride the bus Mody, Wensday, Friday to GED class at Trinity Episcopal church on Jackson Avenue. I pay at 90 cents coming and going. For week would be? For a month? For three mons? For a year?
There is lots of opportunity here for the tutor to work on various literacy skills in the context of a meaningful problem that the learner has developed him/herself. Ø Employment - Your learners have indicated that they would like to work on literacy and numeracy related to employment. You give them the following two newspaper advertisements: §
Person Friday wanted - $250/week. A good junior job. Must type 50wpm. Also involves filing, switchboard, mail, copying and supplies. Hours are 8:30 AM to 5:00 PM with 60 minutes lunch and two15 minute breaks. Phone Barb at 586-9834.
§
Receptionist/Typist - Needed immediately in Fallingbrook area. This person will type 60 wpm, be well-groomed and have a pleasant demeanor. Good company benefits. Salary $24,000 annually. Hours are 8:45 AM to 5:15 PM daily, 60 minute lunch and 2 30 minute breaks. Send resume to Box 609, Stn "C," Winnipeg, MB R7B 2K2 before March 15th. •
Which position has the longest working day and by how much?
•
How much will the Person Friday position pay yearly? (52 weeks)
•
How much will the Person Friday position pay weekly? (5 days)
•
Which position pays more per hour and by how much?
•
Writing, reading and spelling activities - develop a resume, write a cover letter, write a "Job Wanted" advertisement, read about job interviews, etc.
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Ø Taking Medicine (taken from Archambeault, 1993) - Ask learners to write a short paragraph describing a recent illness (their own, a family member or friend). Use a medical reference book to read about the illness. Discuss ways to measure liquid medicines. Use water to demonstrate the difference in quantities measured by teaspoons, soupspoons, and tablespoons. Compare the differences in quantities measured by tableware and measuring spoons. Read the labels on non-prescription medicine bottles. Prepare a chart showing the time of day when the medicine should be taken and the dose. Calculate the total number of doses over a period of days. Ø Working in Pairs §
You and a friend want to share an apartment. Each roommate must find a job in the "help wanted" section of the newspaper's classifieds. List the name of a job and monthly salary. •
Job 1: ___________________________ $___________monthly
•
Job 2: ___________________________ $ ___________monthly
•
Total income for two roommates
•
Subtract 35 % for deductions (taxes, etc) $ ___________
•
Final income:
$ ___________
$ ___________
§
Using the total monthly income above as a guide, find an apartment you can afford. Look in the "real estate" or "apartments for rent" section. List the monthly rent. $ ___________
§
Water, electricity and TV cable cost money. Estimate their cost as 10 percent of the rent money. $ ________ monthly
§
You will probably also want a telephone.
§
How much do you have left over for food and entertainment? $________ monthly
§
Write a list of the things you will need for the apartment (e.g., furniture, dishes, etc)
§
Write a list of the "rules" for living together.
$________ monthly
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v Other Themes (adapted from Archambeault, 1993) - Only one example is shown for each of the following "themes," but of course, there are many, many activities that can be built around each. Ø Grocery Shopping - Use newspaper flyers from various grocery stores and have students prepare a list of five to ten items that are available in each flyer, noting the price at each store. Calculate the total cost for each store and prepare a chart that compares the savings. Ø Restaurants - Use menus from local restaurants to order meals for several people. Total the bill and calculate the taxes and tip. Divide the total among the number of people who ate. Ø Catalogue Shopping - Use mail order catalogues and select several items to order. Fill out the order form, calculate the total cost of the items, and the taxes and shipping costs. Compare the costs of the items to buying in a local store. Ø Cooking - Write out recipes for favourite dishes (the class can trade, put together as a cookbook project, or have a potluck). Double or triple the ingredients to increase the number of servings. Reduce quantities to decrease the number of servings. Use hands-on activities to measure and compare quantities (e.g., 1/2, 1/3, 1/4, etc). Convert Imperial to metric quantities and vice-versa. Ø Buying Gasoline - Determine the distance from your city to other cities using a road map. List this information on a chart. Estimate the kilometers/litre of gas for your car. Calculate the cost of gas to drive between various cities. Prepare a chart to record speedometer readings and litres of gas purchased to confirm the estimated km/gal for your car over a period of time.
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Appendix C To Module 2 RESOURCES: EVERYDAY LIVING NUMERACY There are lots of resources to build authentic learning activities. A few to get you started, however, are listed below. Text Resources Author
Date
Title
Publisher
Bolt, B.
1987
Even More Mathematical Activities
Cambridge: Cambridge University Press.
Devlin, K.
1998
Life by the Numbers
John Wiley & Sons
Dunlap, N.
1991
San Antonio: Education Service Center
Goldman, S.
1989
The Adult Literacy and Mathematics Curriculum Development and Teacher Training Special Project Strategy Instruction in Mathematics
Lee, M. & Miller, M.
1997
Scholastic Trade
Muschla, J.
1996
Real-Life Math Investigations: 30 Activities That Apply Mathematical Thinking to Real-Life Situations Hands-On Math Projects With RealLife Applications: Ready-To-Use Lessons and Materials for Grades 612
Muschla, J. & Muschla, G.
1996
Hands-On Math Projects With Real-Life Applications: Ready-ToUse Lessons and Materials for Grades 6-12
Prentice Hall
Pappas, T
1991
Wide World Pub
Stewart , I.
1998
More Joy of Mathematics : Exploring Mathematics All Around You. The Magical Maze : Seeing the World Through Mathematical Eyes
Learning Disability Quarterly,12. 4355.
Prentice Hall
John Wiley & Sons
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Internet Resources You Work Hard for your Money By the Numbers: Consumer Mathematics Hands on Math Activities Math Links Consumer Math Adult Education Teachers Place: Math Education Lifelong Learning: Family, Community, Work and Other Workplace Education Center The Mathematical Skills and Abilities Adults Need To Be Equipped for the Future
Http://cls.coe.utk.edu/lpm/workhard.html#1.2 Http://www.kindermagic.com/BTN/BTNhome.html Http://www.net1plus.com/users/devenslc/problems. html Http://www.bhs-ms.org/mlprealg.htm Http://www.cis.yale.edu/ynhti/curriculum/units/1982/ 6/82.06.11.x.html Http://forum.swarthmore.edu/teachers/adult.ed/ Http://www.otan.dni.us/cdlp/lllo/home.html Http://www.abc-canada.org/wec/wecindex.html or Http://www.std.com/anpn/framewk.html
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Appendix D To Module 2
THE ICE CREAM PROBLEM The Situation You have been asked to manage a local ice cream store. It is a very small operation that has been losing money. The owner of the store thinks that the store may have been offering the wrong flavours of ice cream, and that sales could be improved by offering more popular flavours. For you first order, the ice cream distributor will deliver 40 containers of ice cream to your store. You must decide what flavours to order and how many containers of each kind. Your display case holds ten containers of ice cream, so you can offer up to ten flavours at a time, and you probably want to offer enough variety to satisfy your customers. The other 30 containers will be stored in a back room storage area (see enclosed drawing). Your Task Decide what flavours to order and how many containers of each flavour. (You may use the enclosed newspaper article with the pie chart and data to help you decide.) Be prepared to explain your decision and your reasoning process later on. Ice Cream Flavours Chocolate Pistachio Raspberry Chocolate Chip Mint Chocolate Chip Vanilla Fudge Swirl Rocky Road
Swiss Chocolate Strawberry Oreo Cookie Peach Coffee Butter Pecan
Vanilla Skor Bar Crunch Black Cherry Lemon Bubble Gum Pralines and Cream
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What's the Scoop? By George Lyons America leads the world in per capita production of ice cream. According to the International Ice Cream Association, in 1993 U.S. ice cream production exceeded 1.5 billion gallons, which translates to an average of 23.6 quarts per person. A survey conducted by the IICA found that while chocolate was once the second most popular flavour of ice cream, it now accounts for only 8 percent of all retail ice cream sales. Vanilla remains the most popular flavour at 28 percent of ice cream retail sales, followed by fruit flavours at 15 percent and candy mix-in flavours at 13 percent. The leading toppings are hot fudge and chocolate fudge. At the local Yummy Ice Cream Shop there is quite a range of exotic flavours to choose from, everything, it seems but plain chocolate. "Flavours that have chocolate sell more than plain chocolate," said the owner, Jean Baker. "I think people are looking for the unusual and that's what we try to provide here." Her best sellers are Raspberry, Chocolate Chip, Oreo Cookie, and, of course, plain vanilla.
Top Ice Cream Flavours in the U.S.
Other 25%
Vanilla 27%
Chocolate 8% Fruit 14%
Nutty 13%
Candy 13%
Based on retail sales in 1993
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Ice Cream Storage Area - 30 Containers
Display Counter - 10 Containers
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Manitoba Education and Training Adult Literacy and Continuing Education Level II Certification Course - Numeracy
MODULE 3: NUMERACY FOR GED/HIGH SCHOOL DIPLOMA
Module Outline v v v v v v v v v v v
Introduction Numeracy Levels Mathematics Strands Problem-Solving and Communicating Mathematically Learning Outcomes Resources Professional Development Assessment Conclusion Assignment Appendixes
Introduction As discussed in Module 1 there are two "streams" which literacy students typically fall into regarding numeracy learning. The first involves numeracy for everyday living and was the subject of discussion in Module 2. The second deals with numeracy leading to GED or high school diploma programs and will be the focus of this module. Numeracy Levels Numeracy teaching and learning in this "stream" will be far more comprehensive and specific than that of the "everyday living" stream. GED/high school diploma mathematics focuses on the operational or computational side of high school level mathematics. In order to prepare our students to go on to GED/high school diploma studies, they will need to be at Stage 4 or Grade 9.
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Mathematics "Strands" In terms of Grade 9 mathematics/Level 4 numeracy, learners must be competent in certain areas of mathematics. There are four specific mathematics "strands" currently being taught in the western Canada public school education system that students in this stream need to focus on. These include: • Patterns and Relations Strand - This refers to the ability of the learner to investigate, establish and communicate rules for both numerical and nonnumerical patterns, and use those rules to make predictions. (patterns, variables, equations, relations, and functions). • Shape and Space Strand - This refers to the ability to describe, classify, construct and relate 2-D shapes and 3-D objects. • Number Strand - This strand deals with both number concepts (estimation, informal computation), and number operations or the ability to formally compute numbers. • Statistics and Probability Strand - This refers to the ability of learners to collect, display and analyse data and make predictions based upon that data. Problem-Solving and Communicating Mathematically As discussed in Module 1, both problem-solving and communicating mathematically must be interwoven into each strand. The ability to problem-solve or reason in numeracy is crucial to learning since it involves the ability to use or apply mathematics. Communicating mathematically is important to the learning process because it provides the opportunity to examine a problem from a variety of angles. In addition, the act of oral discussion about numeracy aids learners to process material and helps to make abstract concepts more concrete. An example (taken from Wells, 1997) of a Grade 9 activity in which students must problem-solve and communicate is shown below: Activity "Debit Card Use Explodes, So Does Financial Worry" This headline and the following facts appeared in the Winnipeg Free Press on June 11, 1997, and could be used in a mathematical communication group activity: G G G
Banks have issued 28 million debit cards. There are about 21 million adults in Canada. There were 677 million debit card transactions in Canada last year.
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G G G G G G G
Interact expects there will be as many as one billion transactions this year. 11 million people use their debit card at least once a month. Canadians perform 80 million debit card transactions a month. Debit card use soared 360 per cent between 1994 and 1996. About 190,000 merchants in Canada accept debit cards. There are a total of 260,000 Interact terminals in service. Charges for debit card use range from 30 cents to 45 cents per transaction, except for consumers who have arranged flat-fee service with their banks.
Task 1) Use these facts to create mathematical problems. 2) Find and explain multiple solutions to your problems.
Learning Outcomes The general learning outcomes for Grades 1 to 9 in each of the above strands have been included as Appendix A to this module. While the specific learning outcomes for each strand by grade are too numerous to include in this course, one example is included at Appendix B. These are intended to provide tutors with a general guide around which to frame a learning plan and develop specific content/activities. An example of multi-level measurement activities in the "shape and space" strand has been included as Appendix C. The following is an example of an activity for single-level learners: Grade: 9 (Level 4) Strand: Statistics and Probability General Learning outcome: Explains the use of probability and statistics in the solution of complex problems. Specific Learning Outcomes • Recognize that decisions based on probability may be a combination of theoretical calculations, experimental results, and subjective judgements. • Demonstrate an understanding of the role of probability and statistics in society.
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• Solve problems involving the probability of independent events Problem The City of Winnipeg has undergone a change in population over the last few decades. According to the Statistics Canada census, the population was: 1966 1976 1986 1996
-
320,000 482,000 573,000 661,000
(1) Graph the results of the census. Identify the axes. (2) Estimate what the population would have been in 1983. Explain. (3) Predict what the population of Winnipeg would be in 2006. Explain your reasoning. (4) A researcher claims that by the year 2026 the population of Winnipeg will be 1,322,000. Is this claim valid? Explain your reasoning. What factors could affect the actual population in 2026? Resources In that this is a basic, introductory course on numeracy teaching and learning, the mathematics operations are not covered. As such, programs will need to accumulate resources that deal with the specific operational or computational aspects of mathematics. Toward this end, an Annotated Bibliography of Numeracy Materials has been included with the course material for tutors to keep. Note that there are also some good resources listed for the everyday living numeracy stream. It is also recommended that you also check with your ALCE provincial coordinator regarding available resources. A word of caution is offered here about the use of textbooks. Although the focus of numeracy in this stream virtually necessitates the use of texts, and involves far more sequential instruction, as much as possible learning needs to be active, authentic and concrete.
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Professional Development Teaching and learning numeracy at the lower levels of the GED/high school stream and/or for everyday living is unlikely to pose a problem for most programs in terms of developing content and instructing. However, the middle and upper end of this stream does require that tutors have a certain level of numeracy ability themselves. For example, Grade 9 mathematics in the public school system has 11 units: • • • • • • • • • • •
Mathematical Reasoning Statistics Polynomials Spatial Geometry Linear Relations Similarity and Congruence Probability Powers and Exponents Trigonometry Measurement Transformational Geometry
As such, program directors need to recruit individuals who already possess these numeracy skills, and/or tutors will need to undertake professional development in the areas of mathematics. Assessment A word of caution is offered here about assessment. As was mentioned in Module 1, literacy learners often have had negative experiences with learning and are particularly anxious about being assessed. In addition, many people in general suffer from math anxiety, a fear that they do not have the ability to do math. Thus, it is worthwhile discussing this with the learner first, and if such is the case, spending some time and effort building trust and a sense of ease before undertaking any assessment. Once the learner is comfortable, we need to do an initial assessment of his/her numeracy skills. A good starting point is the screening form we used in Module 1 to collect information about the student's background and learning goals. One question was about the grade he/she began to experience difficulties and in what areas. If, for example, a student began to have problems in Grade 5, we will need to go back a year or two and look at what numeracy skills are strong/weak in each strand. Two general forms have been included at the end of this module (See Appendixes D and E) to assist tutors in this regard. The first is a general form that can be used to assess the overall grade level a learner is at. The other can be used to assess specific learning outcomes in a particular strand for a particular learning outcome. Much like content, specific evaluation material will
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need to be developed by individual tutors/programs. Both forms may be used for ongoing assessment purposes as well. Methods of assessing numeracy ability may include: • Observation, discussion, structured interviews - used primarily to determine the learner's understanding and ability to communicate a concept mathematically. • Paper and pencil quizzes/tests - used primarily to determine the learner's ability in operational/computation numeracy. • Individual or Group Projects - used primarily to determine learner's ability to problem-solve (develop, implement, evaluate a plan and communicate the results). Conclusion This module dealt with teaching and learning numeracy in preparation for GED/high school diploma studies for which students will need to be at a Grade 9 level of mathematics. Due to the emphasis on the operational side of mathematics, tutors must have appropriate mathematical skills, as well as sufficient resources to be able to develop instructional activities and evaluation materials at the various levels. In the final module, we will look at several factors for tutor/programs to consider when planning for numeracy teaching and learning.
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ASSIGNMENT - MODULE 3 On a separate page, please answer the following. 1. Although numeracy in this stream necessitates the use of mathematics textbooks, why is it important to avoid teaching solely from these and ensuring that active and authentic activities are included as much as possible? 2. Pick one general outcome for Grade 4 in the "Numbers" strand and develop a math question that would help you to assess the learner's ability in these areas. 3. Develop an activity for one general learning outcome in each of the four strands for the Grade 4 level. 4. You are teaching a multi-level class. Develop activities for one of the general learning outcomes in one of the strands (see Appendix C for an example).
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Appendix A to Module 3 GENERAL LEARNING OUTCOMES General learning outcomes for each of the four numeracy strands by grade level /literacy stage are outlined on the following pages. These are adapted from Hill (1995). As this course focuses on a basic approach to teaching and learning mathematics, specific outcomes have not been included for all grade levels. However, an example has been provided of specific outcomes for Grade 9 at Appendix C. PATTERNS AND RELATIONS STRAND LITERACY STAGE GRADE LEVEL Identifies, creates and compares patterns arising from daily experiences in the classroom Identifies, creates, describes and translates numerical and nonnumerical patterns arising from daily experiences in school and on the playground Investigates, establishes and communicates rules for and predictions from numerical and non-numerical patterns including those found in the community Constructs, extends and summarizes patterns, including those found in nature, using rules, charts, mental math, calculators Uses relationships to summarize, generalize and extend patterns, including those found in music/art Uses informal and concrete representations of equality to solve problems
Gr 1 X
LS1 Gr Gr 2 3
Gr 4
LS2 Gr Gr 5 6
LS3 Gr Gr 7 8
LS4 Gr 9
X
X
X
X
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X
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Expresses patterns, including those used in business and industry, in terms of variables, and uses expressions containing variables to make predictions Uses variables and equations as problem-solving tools to express, summarize, and apply relationships in a restricted range of contexts Uses patterns, variables and expressions together with graphs to solve problems Solves and verifies 1 and 2 step linear equations with rational number solutions Generalizes, designs and justifies mathematical procedures using appropriate patterns, models and technology Solves and verifies linear equations and inequalities in one variable SHAPE AND SPACE STRAND LITERACY STAGE GRADE LEVEL Estimates, measures and compares using whole numbers and non-standard units of measure Explores and classifies 3-D objects and 2-D shapes according to their properties Orally describes the relative position of 3-D objects and 2-D shapes Uses measurement concepts, appropriate tools, and the results of measurements to solve problems in everyday contexts
X
X
X X
X
X
Gr 1
LS1 Gr Gr 2 3
Gr 4
LS2 Gr Gr 5 6
LS3 Gr Gr 7 8
LS4 Gr 9
X
X
X
X
X
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Uses co-ordinates to describe the position of objects in 2 dimensions; and describes motion in terms of a slide, flip or turn Solves problems involving perimeter, area, surface, area, volume and angle measurement Uses visualization and symmetry to solve problems involving classification and sketching Creates patterns and designs that incorporate symmetry, tessellations, translations and reflections Solves problems using the properties of circles and their connections with angles and time zones, and problems involving area and perimeter Links angle measures to properties of parallel lines Creates and analyses patterns and designs, using congruence, symmetry, translation, reflection, and rotation Applies indirect measurement procedures to solve problems; generalizes measurement patterns and procedures; solves problems involving area, perimeter, surface area and volume Links angle measures and properties of parallel lines to the classification and properties of quadrilaterals Creates and analyzes design problems and architectural patterns using the properties of scaling, proportion and networks Uses spatial problem-solving in building, describing and analyzing geometric shapes
X
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X
X
X
X
X
X
X
X
X
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Specifies conditions under which triangles may be similar or congruent and uses these conditions to solve problems Uses trigonometric ratios to solve problems involving a right angel Describes the effects of dimension changes in related 2-D shapes and 3-D objects in solving problems involving area, perimeter, surface area and volume Applies coordinate geometry and pattern recognition to predict the effects of translations, rotations, reflections and dilations on 1-D lines and 2-D shapes
X
X
X
X
NUMBERS STRAND LITERACY STAGE GRADE LEVEL Recognizes and applies whole numbers 0 to 100, and explores halves Applies informal methods of addition and subtraction on whole numbers (maximum sum of 18) Recognizes and applies whole numbers up to 1,000, and explores fractions (halves, thirds, fourths) Develops a number sense for whole numbers up to 1,000, and explores fractions (fifths and tenths) Applies an arithmetic operation (addition, subtraction, multiplication and division) on whole numbers, and illustrates its use in solving problems Uses and justifies an appropriate calculation strategy or technology to solve problems
Gr 1 X
LS1 Gr Gr 2 3
Gr 4
LS2 Gr Gr 5 6
LS3 Gr Gr 7 8
LS4 Gr 9
X
X
X
X
X
X
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Demonstrates a number sense for whole numbers from 0 to 10,000 and explores proper fractions Applies arithmetic operations on whole numbers, illustrates their use in creating and solving problems, and demonstrates understanding of addition and subtraction of decimals (tenths and hundredths) Develops a number sense for whole numbers, decimals and common fractions, and explores integers Demonstrates a number sense for decimals, common fractions, integers and whole numbers Demonstrates a number sense for rational numbers, including common fractions, integers and whole numbers Applies arithmetic operations on whole numbers and decimals, and illustrates their use in creating and solving problems Applies arithmetic operations on whole numbers and decimals in solving problems Applies arithmetic operations on decimals and integers, illustrates their use in solving problems, and illustrates the use of rates, ratios, percentages, and decimals to solve problems in meaningful contexts Applies arithmetic operations on rational numbers to solve problems; and applies the concepts of rate, ratio, percentage and proportion to solve problems in meaningful contexts
X
X
X
X
X
X
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X
X
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Develops a number sense for powers with integral exponents and rational bases Generalizes arithmetic operations from the set of rational numbers to the set of polynomials Explains how exponents can be used to bring meaning to large and small numbers, and uses calculators or computers to perform calculations involving these numbers Explains and illustrates the structure and interrelationship of the sets of numbers within the rational number system Generalizes, designs and justifies mathematical procedures using appropriate patterns, models and technology Uses a scientific calculator or a computer to solve problems involving rational numbers Explains and illustrates the structure and interrelationship of the sets of numbers within the rational number system Generalizes arithmetic operations from the set of rational numbers to the set of polynomial numbers STATISTICS AND PROBABILITY STRAND LITERACY STAGE LS1 GRADE LEVEL Gr Gr Gr 1 2 3 Collects, organizes and X describes, with guidance, data based on first-hand information Collects, displays and describes data, independently, based on X first-hand information Collects first and second hand information, displays the results in more than one way, and X interprets the data to make
X
X
X
X
X
X
X
X
Gr 4
LS2 Gr Gr 5 6
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LS4 Gr 9
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predictions Collects first and second hand data, assesses and validates the collection process, and graphs the data Develops and implements a plan for the collection, display and interpretation of data Develops and implements a plan for the collection, display and analysis of data gathered from appropriate samples Develops and implements a plan for the collection, display and analysis of data using measures of variability and central tendency Collects and analyzes experimental results expressed in two variables using technology as required Describes the concept of chance using ordinary language Uses simple experiments to illustrate chance Uses simple probability experiments designed by others to explain outcomes Designs/uses simple probability experiments to explain outcomes Predicts outcomes, conducts experiments and communicates the probability of single events Uses numbers to communicate the probability of single events from experiments and models Creates and solves problems using probability Compare theoretical and experimental probability of independent events Explains the use of probability and statistics in the solution of complex problems
X
X
X
X
X
X
X
X
X X
X
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X
X
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Appendix B To Module 3 SPECIFIC MATHEMATICAL LEARNING OUTCOMES: GRADE 9 The following learning outcomes for Grade 9 numeracy are taken from Well (1997). GENERAL OUTCOME Statistics and Probability Strand Explains the use of probability and statistics in the solution of complex problems
Collects and analyzes experimental results expressed in two variables using technology as required
Number Strand Develop a number sense for powers with integral exponents and rational bases
Generalize arithmetic operations from the set of rational numbers to the set of polynomials
SPECIFIC OUTCOME - recognize that decisions based on probability may be a combination of theoretical calculations, experimental results, and subjective judgements - demonstrate an understanding of the role of probability and statistics in society - solve problems involving the probability of independent events - assess the strengths, weaknesses and biases of samples and data collection methods - critique ways in which statistical information and conclusions are presented by the media and other sources - create scatterplots for discreet and continuous variables - illustrate power, base, coefficient and exponent using rational numbers or variables as bases or coefficients - determine the value of powers with non-negative integral exponents using the exponent laws - etc. - determine equivalent forms of algebraic expressions by identifying common factors - find the quotient when a polynomial is divided by a monomial - etc.
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Explain how exponents can be used to bring meaning to large and small numbers, and use calculators or computers to perform calculations involving these numbers
Explain and illustrate the structure and interrelationship of the sets of numbers within the rational number system Generalize, design and justify mathematical procedures using appropriate patterns, models and technology Use a scientific calculator or a computer to solve problems involving rational numbers Explain and illustrate the structure and interrelationship of the sets of numbers within the rational number system
Generalize arithmetic operations from the set of rational numbers to the set of polynomial numbers
Patterns and Relations Strand Generalize, design and justify mathematical procedures using appropriate patterns, models and technology Solve and verify linear equations and inequalities in one variable
- understand and use the exponent laws to simplify expressions with variable bases and evaluate expressions with numerical bases - use a calculator to perform calculations involving scientific notation and exponent laws - give examples of situations where answers would involve the positive square root, or both positive and negative square roots of numbers - use logic and divergent thinking to present mathematical arguments in solving problems - document and explain calculator keying sequences used to perform calculations involving rational numbers - give examples of numbers that satisfy the conditions of natural, whole, integral and rational numbers, and show that these numbers comprise the rational number system - describe orally and in writing whether or not a number is rational - identify constant terms, coefficients, and variables in polynomial expressions - represent and justify the addition and subtraction of polynomial expressions using concrete materials and diagrams - etc. - model situations that can be represented by first-degree expressions - write equivalent forms of algebraic expressions or equations with rational coefficients - illustrate the solution process for a first-degree, single-variable equation using concrete materials or diagrams - solve algebraically first-degree inequalities with one variable, display the solutions on a number line, and test the solutions
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Shape and Space Strand Use spatial problem-solving in building, describing and analyzing geometric shapes
Specify conditions under which triangles may be similar or congruent and use these conditions to solve problems
Uses trigonometric ratios to solve problems involving a right angel
Describes the effects of dimension changes in related 2-D shapes and 3D objects in solving problems involving area, perimeter, surface area and volume
Apply coordinate geometry and pattern recognition to predict the effects of translations, rotations, reflections and dilations on 1-D lines and 2-D shapes
- recognize and draw the set of points in solving practical problems - draw the plan and elevations of a 3-D object from sketches and models - sketch or build a 3-D object given its plan and elevation views - recognize when and explain why two triangles are congruent, and use the properties of congruent triangles to solve problems - relate congruence to similarity in the context of triangles - explain the meaning of sine, cosine and tangent ratios in right triangles - demonstrate the use of trigonometric ratios in solving right triangles - calculate an unknown side or angle in a right triangle using appropriate technology - etc - calculate area and perimeter to solve design problems in two dimensions - relate expressions for volumes of pyramids to volumes of prisms, and volumes of cones to volumes of cylinders - calculate volume and surface area to solve design problems in three dimensions - etc - identify the single transformation that connects a shape with its image demonstrate that a triangle and its dilation image are similar - etc.
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Appendix C To Module 3 ACTIVITIES FOR A MULTI-LEVEL CLASS Strand: Shape and Space General Learning Outcome: Measurement Grades 1-2 Objectives: Students will learn to measure objects with and without a ruler. Skills and Knowledge: Students will be able to • Recognize the need for knowing how big or long something is. • Identify the uses of measurement in mathematics. • Identify what a ruler is and how to use it. Materials and Resources: Rulers, students desks, small objects. Activity: Students will be asked to first measure the top of their desks without a ruler. Have students pair off, measure the small objects and record their findings. Conduct a discussion about the need for measurement in our lives. Grade 3 Objective: To introduce students to estimated and accurate measurements. Skills and Knowledge: • Make estimations of length • Measure length Materials and Resources: Rulers, measuring tapes, paper, pencil Activities: Review the lines and numbers on a measuring tape. Have students pair off and estimate each other's height. Record this information. Then have each student measure the other for that person's actual height. Record this information. Have several other objects (e.g., table, floor tile, piece of string, etc) for students to estimate the length of and then measure.
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Grade 4 Skills and Knowledge: The students will be able to • Accurately use a metric ruler to measure length. • Accurately read and record measurements taken in centimeters and millimeters. • Find a sum of multiple metric measurements. • Compare and order individual measurements. • Use a histogram to graph their results. Materials and Resources: butcher paper, graph paper, metric ruler, crayons or markers, pencils, paper. Activities: Divide the class into groups of four. Each student will measure and record the length of each person's smile in their group. The students need to check their results against the results of the rest of the group. If there are any discrepancies the students should verify the results as a group. When an accurate measurement has been obtained the results are recorded and then ordered from least to greatest. Each student graphs the results. Grade 5 Objective: To use measurement to demonstrate division. Materials and Resources: A meter stick, a ball of cord, a pair of scissors Activities: Have the students pair off. Have one student measure and the other cut. Measure and cut a piece of cord 110 cm long. Start at one end of the cord and cut pieces that are 8 cm long. Cut as many pieces as possible. Measure carefully. Count the number of 8 cm pieces you have cut. Measure the piece that is left over. It should measure 6 cm. Repeat the process for other lengths of string. Make and complete a table showing: • • • • •
Length of single piece Length of single pieces to be cut from single piece Number of exact-size pieces that can be cut from single piece Size of left-over piece (the remainder) Division sentence (Give the division sentence this represents)
Repeat the above with other examples.
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Grade 6 Objective: This activity is used to help understand vertical and horizontal measurement of large objects Skills and Knowledge: • Demonstrate measurement of the trunk, crown, and height using vertical and horizontal measurement. • Compare results with other groups. • Create a graph of their findings for the trunk, crown, and height of the tree. • Define horizontal, vertical, and circumference. Materials and Resources: String, ruler, paper pencil, meter stick, trees. Activities: • Trunk: Measure from the ground to 4 1/2 feet high on the trunk. At that height, measure the trunk's circumference. Use a string around the trunk and measure the length of the string. Round to the nearest inch. Record the number and label as circumference. • Crown: Find the tree's five longest branches. Put markers on the ground beneath the tip of the longest branch. Find a branch that is opposite it and mark its tip on the ground. Measure along the ground from first marker to the second marker. Record the number and label as crown. • Height: Have one partner stand at the base of the tree. Back away from the tree, holding your ruler in front of you in a vertical position. Keep your arm straight. Stop when the tree and the ruler appear to be the same size. (Close one eye to help you line it up.) Turn your wrist so that the ruler looks level to the ground and is in a horizontal position. Keep your arm straight. Have your partner walk to the spot that you see as the top of the ruler. Be sure the base of the ruler is kept at the base of the tree. Measure how many feet he or she walked. That is the tree's height. Round to the nearest foot and record your answer as the height. • Tying it all together: Allow time for groups to compare answers and then measure the tree again as needed. Usually it takes several measurements. Be sure and allow time for each person to take several measurements since they will be working with partners.
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• In the classroom: Have students make bar graphs using information gathered outside. Have students locate the biggest tree, smallest tree of the same species. Grade 7 Objective: Students will be able to describe the difference between area and volume and also be able to understand how various units of measure relate to one another. Materials and Resources: Newspaper, scissors, masking tape, rulers and meter sticks, cardboard, markers to identify finished models. Activities: Following an introduction to area and volume students will work in groups to build models of square centimeters, square inches, square feet, square meters, and then cubic centimeters, cubic inches, cubic feet, and cubic meters. This becomes a good cooperative team effort at problem solving. Students are provided with materials, but no initial instruction is given on how to build their models. When the groups have completed their projects they will send a spokesperson to the front of the room to share with the class what they have built, what it is called, and how it compares to some of the other models built by other groups. This activity leaves students with a lasting memory of these ideas which are otherwise hard to grasp. Grade 8 Objectives: Many students tend to memorize, without understanding, formulas that we use in geometry or other mathematics areas. This particular activity allows students to discover why pi works in solving problems dealing with finding circumference. Skills and Knowledge: • • • •
Measure the circumference of an object to the nearest millimeter. Measure the diameter of an object to the nearest millimeter. Explain how the number 3.14 for pi was determined. Demonstrate that by dividing the circumference of an object by its diameter you end up with pi. • Discover the formula for finding circumference using pi, and demonstrate it Materials and Resources: Round objects such as jars, lids, etc., measuring tapes, or string and rulers, paper, pencil, calculator
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Activity: Have the students pair off and hand out materials. Have student teams make a table or chart with a column for name or number of the object, circumference, diameter and ?. Measure and record each object's circumference and diameter, then divide the circumference by the diameter and record result in the ? column. Find the average for the ? column and compare to other groups in the class to determine a pattern. Students can then find the average number for the class. Explain to the students that they have just discovered pi, which is very important in finding the circumference of an object. (You may wish to give some historical information about pi at this time or have students research the information.) Have students come up with a formula to find the circumference of an object knowing only the diameter of that object, and the number that represents pi. Students must prove their formula works by demonstration and measuring to check their results. Have students write their conclusions for the activities they have just done. Students may also share what they have learned with other members of the class. Grade 9 Objective: Students will understand how to apply trigonometry in real-world measurement situations. Skills and Knowledge: • Understands and applies basic and advanced properties of the concepts of measurement • Understands and applies basic and advanced properties of the concepts of geometry • Understands the basic concepts of right triangle trigonometry (e.g., basic trigonometric ratios such as sine, cosine, and tangent) • Selects and uses an appropriate direct or indirect method of measurement in a given situation (e.g., uses properties of similar triangles to measure indirectly the height of an inaccessible object) Materials and Resources: Protractor, tape measure, scientific calculator with trigonometric functions, writing materials Activity: Students will work in groups of two or three members. Each group will be assigned an object to measure the height of (e.g., flag pole, building, tree). The group will select a method to determine the angle from their position to the top of the object using the protractor (e.g., using a ruler, piece of paper, or arm to sight along and then determining the angle from the horizontal). They will then use the tape measure to determine the distance from their position to the base of the object. After the measurements have been completed, the students should
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draw a diagram showing the angle of elevation, distance, and the height at which the angle was measured (i.e., the eye level height of the student measuring the angle). The students will then decide which trigonometric function (i.e., sine, cosine, tangent) to use and solve for the height of the object. Note: This activity is designed to follow an introduction to the basic concepts of right triangle trigonometry. If tape measures are not available, students may measure distance in 'paces' and use a yard/meter stick to determine the length of each 'pace'.
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Appendix D To Module 3 GED/HIGH SCHOOL PREPARATION: GENERAL ASSESSMENT
Name: _________________________________ Date:____________________ Level/ Grade _________ Performs below the standard
Patterns and Relations Strand
Statistics and Probability Strand
Shape and Space Strand
Number Strand
Performs At the standard
Performs above the standard
(Form adapted from Well, 1997)
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Appendix E to Module 3 GED/HIGH SCHOOL PREPARATION: SPECIFIC ASSESSMENT
Name: ________________________________ Date:____________________ Level/Grade: ________________
Strand: ____________________________
Learning Outcome: ________________________________________________
Low
Medium
High
Mathematics The response shows limited understanding of the concept and/or contains errors in computation
The response shows a good understanding of the concept(s), contains no major errors in reasoning, and/or contains a correct answer with a minor calculation error The response shows a thorough understanding of the concept(s), shows mathematical insights above and beyond the norm, and/or contains no calculation or conceptual errors
Communication The response includes minimal, sparse and/or unclear explanation, contains graphs, charts and/or diagrams that are not labeled well The response includes a reasonable clear explanation, and/or contains a graph, chart or diagram with most elements explained The response contains an exemplary explanation that is clear and concise, includes a clearly labeled and fully explained diagram, chart or graph, and/or includes an explanation of what was done and why
Problem-solving The response shows inappropriate or incorrect strategies and/or contain irrelevant information The response shows a good understanding of the problem-solving process, and/or shows appropriate use of strategies with minor errors All important elements of the problem are identified and understood, uses appropriate and systematic strategies to solve the problem, and/or has a thorough under-standing of the problem-solving process (beyond the requirements of the problem)
(Form adapted from Hill 1995)
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Manitoba Education and Training Adult Literacy and Continuing Education Level II Certification Course - Numeracy
MODULE 4: CONCLUSION
v v v v v v
Module Outline Introduction Other Considerations Putting It all Together Conclusion Assignment Appendixes
Introduction This last module will look at several other factors programs/tutors must consider in regard to numeracy. A final section with look at putting together the various key elements covered in this course into a bigger picture of teaching and learning numeracy in an adult literacy program. Other Considerations Dyscalculia (Math Learning Disability) - One important consideration is the possibility that the learner has a disability called "dyscalculia." While limited ability in numeracy may by the result of many things (e.g., gaps in learning caused by moving around, illness, leaving school as a child, second language difficulties, etc., it may also be the result of a learning disability called "dyscalculia" which Sharma & Brazil (1997) define as follows: Dyscalculia is an inability to conceptualise numbers, number relationships (arithmetic facts) and outcomes of numerical operations (estimating the answers to numerical problems before actually calculating.) While literacy practitioners are not learning disability specialists and must take care not to diagnose problems, it is important that the possibility of dyscalculia be checked prior to beginning any numeracy instruction. Otherwise, both tutor and learner are likely to experience a great deal of frustration when difficulties are encountered. Dyscalculia, like other learning disabilities, requires the development and use of different teaching and learning strategies.
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The form in Appendix A is intended as an informal screening tool versus a diagnostic tool. If the screening does indicate the possibility of dyscalculia, the student may use the completed form (perhaps with a letter from the program), as a means of arranging for a formal assessment by a trained learning disabilities specialist. If it does appear that a learner has dyscalculia, this will require some extra effort, patience and knowledge on the part of the tutor. It is highly recommended, therefore, that practitioners/volunteers who will be tutoring students who are suspected as having dyscalculia, complete the following Level ll courses: •
Learning Styles and Strategies; and,
•
Learning Differently: An Introduction to Learning Disabilities and Adult Literacy
Newman (1998) suggests that in general, numeracy instruction for students with dyscalculia needs to focus more on the development of conceptual mathematical abilities than for learners who do not suffer from dyscalculia. Some examples include: •
The ability to follow sequential directions.
•
A sense of directionality or spatial orientation and organization (e.g., ability to tell left from right, north/south/east/west, up/down, forward/backwards, horizontal/vertical/diagonal).
•
Pattern recognition and its extension.
•
The ability to visualize: the ability to conjure up pictures in one's mind and manipulate them.
•
The ability to estimate: the ability to form a reasonable educated guess about size, amount, number, and magnitude.
•
Deductive reasoning: the ability to reason from the general principal to a particular instance.
•
Inductive reasoning: the ability to see patterns in different situations and the interrelationships between procedures and concepts
Translating these conceptual areas into actual numeracy instruction will take a capable and willing tutor. Not all practitioners/volunteers will be willing and/or able to deal with a learner who has dyscalculia, thus program directors will need to take extra care in matching tutor and learner.
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A fictional letter from a student with dyscalculia (Newman, 1997) has been included as Appendix B to help put tutors in the "shoes of the learner" so to speak. Particularly noticeable in the letter is the emotion attached to learning math. Many adult learners with dyscalculia, having failed repeatedly to master this subject, experience very high levels of math anxiety. As such, much of the practitioner's efforts will need to be directed toward helping reduce this anxiety in order for learning to take place. Dyscalculia-related resources have been included as Appendix C. Individual versus Group Learning - Although not all programs can or do offer group instruction, it is recommended in the case of numeracy that group instruction occur as much as possible since communicating and problem-solving are considered essential to developing well-rounded numeracy skills. In addition, from a practical standpoint this offers the best use of resources in that it is likely that many programs will have only a limited number of tutors with the necessary numeracy skills. The problem with group learning, however, is the fact that learners will have different abilities and needs (e.g., everyday living versus GED numeracy), and juggling these will require some flexibility and creativity on the part of tutors. In a one-to-one situation where no group work is possible or feasible, tutors will also need to be creative about stimulating communication with and problemsolving through others. For example, learning sessions could include a lot of discussion and brainstorming by the tutor and learner, activities could be designed that involve family, friends or coworkers, if a computer is available a cyber learning buddy could be located, and so on. Computers - Programs who do not have access to computers/Internet should be reassured that their teaching and learning will not suffer because of this. There are a wide variety of excellent print resources available to conduct an interesting and effective numeracy program. That said, the advent of cheaper computer technology offers the opportunity to explore and use a wealth of material that is varied, readily available, generally free of charge, and often interactive. One particularly valuable aspect of using computers is that learners are not as dependent on a tutor and can be somewhat more self-directed in their learning. Although this has long been recognized in adult education as increasing motivation, self-confidence, etc., as with most things, there is a flip side to consider. The nature of computers can lead to learning in isolation, and since communicating mathematically and problem-solving with others are key to effective numeracy learning, over-reliance on this technology needs to be avoided. Additionally, while interactive, computers cannot offer the same degree of responsiveness that a tutor can.
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Putting It All Together As we have seen throughout this course there are several elements that are important to effective numeracy teaching and learning. These are summarized below. Identifying Learner Needs - Learners are likely to fall into two general "streams" including: a) numeracy for everyday living; and, b) preparatory numeracy for GED/high school diploma studies. The problem for programs/tutors becomes how to best accommodate the differing needs of learners within the resources of the program. Authentic and Active Learning - In Module 2 you watched a videotape of learners involved in several tasks (i.e., the "Ice Cream" and "Detective" problems). These tasks involved computational, reasoning and communication skills that would be encountered in a real life situations; that is, they were authentic and active learning tasks. The learners could relate to the problems and extend or apply them in their own lives and this needs to be the thrust of numeracy for adult learners in literacy programs. While this is somewhat easier to do in the everyday living numeracy stream, tutors preparing students for GED/high school diploma studies should also strive to use a similar approach as much as possible. Blended Activities - In the assignment for Module 2, you were asked to extend the "Ice Cream Problem" into other literacy areas (e.g., reading, writing and spelling tasks); that is, blend numeracy into the larger, overall literacy program. The reason for this is that learning is much more effective if subjects are not taught in isolation from one another. This holds true for the various areas within a subject. For example, in the GED/high school diploma preparation stream, there are approximately 11 mathematical areas that need to be covered for Grade 9 or Level 4 numeracy. While the temptation may be to teach these as separate, discrete units, tutors must strive constantly to connect or blend one area to another, as well as to other literacy skills in order for learners to develop "big picture" thinking. Assessment - It is crucial to understand that in addition to having negative emotions associated with learning in general, numeracy is a particular source of anxiety for many literacy learners, not to mention many adults in general. Thus, assessment needs to be timely and as "gentle" as learners require. For example, rather than assessing numeracy skills immediately, allow some time to pass in which the focus is on making any learning enjoyable, successful and building a sense of trust and ease between tutor and learner(s). Completing a fun activity that includes an element of numeracy may be a way of indirectly observing the learner(s) ability to start and then moving toward more formal assessment. Depending on the student's learning objectives and goals, formal assessment may not even be necessary.
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Resources - One point that tutors should keep in mind is that it pays to be resourceful about resources. There are many, many resources available and from a wide variety of sources. Some suggestions are discussed below. •
Print - Some print resources will, of course, have a cost and the program will need to budget appropriately. Other resources, however, are available free of charge. For example, Human Resources Development Canada has a series called "Working Solutions" available free of charge from any employment office. There are six booklets in the series, one of which is called "Preparing a Realistic Budget." Programs are encouraged to look at alternate avenues for material. For example, in Manitoba, the Association of Home Economists has a videotape and workbook series available called "Basic Skills for Living," one topic of which is entitled "Learning About Money: A 3-Part Series." It's an excellent resource with a workbook and videotape that covers banking, drawing up a spending plan and credit.
•
Computer Software Programs/Internet - The resources available in terms of software and numeracy information and activities on the Internet are virtually endless, but a few are listed below to get you started. No particular program or site is recommended. Rather, tutors/learners are encouraged to explore and use "whatever tool works" in their particular situation. Adult Learners Math Skills Toolbox By the Numbers The Math Forum Numeracy E-List Adult Numeracy Network Boston Adult Numeracy Homepage Numeracy "Down Under" AWE: Adult Numeracy FREE Math Math Lessons Database Math and Technology Instructional Units
http://www.cdlr.tamu.edu/tcall/toolkit/ch07.htm http://www.kindermagic.com/BTN/BTNhome.ht ml http://www.forum.swarthmore.edu/ http://www.std.com/anpn/numeracy.html http://www.std.com/anpn/ http://www2.wgbh.org/MBCWEIS/LTC/CLC/nu mintro.html http://sunsite.anu.edu.au/languageaustralia/numeracy/ http://www.resnet.wm.edu/~sscroa/ http://forum.swarthmore.edu/teachers/adult.ed/ http://www.ed.gov/free/s-math.html http://www.mste.uiuc.edu/mathed/queryform.ht ml http://www.indiana.edu/~techprep/
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K- 12 Resources Awesome Library of Math Resources The Math Room Boone's Math Lessons St. Francis Xavier University Math Links Canadian Mathematical Society The Math Forum Math Internet Projects and Activities Cornell Math Gateway Math Archives Math Gems Ask Dr. Math Mega Mathematics Math Advantage Interactive Math Canada's Schoolnet Daisy's Math Word Problems For Kids PBS Mathline
http://www.neatschoolhouse.org/Library/Materials_Search/Le sson_Plans/Math.html http://www.sol.com.sg/cgi-bin/mathquote.cgi http://www.crpc.rice.edu/CRPC/GT/sboone/Le ssons/ http://juliet.stfx.ca/~x94emj/math.htm http://camel.math.ca/Education/ http://forum.swarthmore.edu/ http://www.glenbrook.k12.il.us/gbsmat/glazer/ ho.html http://www.tc.cornell.edu/Edu/MathSciGatewa y/math.html http://archives.math.utk.edu/ http://www-sci.lib.uci.edu/SEP/math.html#7 http://forum.swarthmore.edu/dr.math/drmath.e lem.html http://www.c3.lanl.gov:80/mega-math/ http://www.harcourtschool.com/menus/math_ advantage.html http://www.cut-the-knot.com/ http://www.schoolnet.ca/home/e/resources/m athematics/index.html http://www.openuniversity.com/ http://www.stfx.ca/special/mathproblems/welc ome.html http://www.pbs.org/teachersource/math/index. html
• Human - As noted in Modules 2 and 3, programs will need tutors who are able to teach the various levels of mathematics. While Stages 1 and 2 are most likely within the ability of most tutors, Stages 3 and 4 do require greater ability in computational or "hard" math skills. Additionally, learners with dyscalculia will require tutors with are both willing and capable of dealing with this learning disability. While this may be accomplished through professional development and/or recruiting tutors who possess the necessary skills, programs can also seek out members of the community who may be willing to conduct a course or teach specific sessions. Willing tutors may be found in the business community, in local educational institutions, etc.
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Conclusion The focus of mathematics instruction has gradually changed in the past decade or so. While emphasis is still placed on computational or operational skills and knowledge, there has been a growing awareness of the importance of also emphasizing problem-solving and communication skills, as well as tying numeracy more to the "real world." This newer emphasis on making numeracy teaching and learning active and authentic serves to emphasize the fact that numeracy is a life skill rather than an 'academic' skill only some people are capable of.
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Assignment - Module 4 Please answer the following on a separate page. 1) What is "dyscalculia" and what are three areas tutors can help learners with to improve their numeracy skills? 2) Why is it important to avoid diagnosing "dyscalculia"? 3) Identify the numeracy skills used in grocery shopping and briefly outline a learning plan for teaching these skills. 4) For each of the following statements, please indicate if you agree or disagree and explain why. a) People learn numeracy best by focusing on operational/computational skills. b) Being able to apply numeracy in everyday living contexts will naturally "fall out of" learning operational/computational math skills. Therefore, that is what we should concentrate on in our programs. c) Estimating really means "guessing" and is not really a useful numeracy skill to develop. d) For many literacy learners, assessing numeracy skills needs to be "timely" and gently" done. e) Talking about numeracy helps people to learn mathematics. f) "Problem-solving" skills are important to solving both operational and everyday living numeracy problems. g) It is too difficult to try and plan learning and develop activities that meet diverse learners' needs and wants.
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Appendix A To Module 4 DYSCALCULIA SCREENING FORM This is intended for use as an informal screening checklist versus a diagnostic tool. Three or more "Yes" answers may indicate that learning problems are the result of dyscalculia versus gaps in learning, second language difficulties, etc. DOES THE LEARNER .... Commonly make substitutions, transpositions, omissions, and reversals with numbers?
YES
NO
COMMENTS
Obtains inconsistent results in addition, subtraction, multiplication and division? (e.g., can sometimes do the math while other times cannot or has difficulty) Have ongoing problems grasping and remembering math concepts, rules, formulas, and order of operations?
Have limited or poor ability to do math in his/her head? (e.g., difficulty mentally figuring change due back, amounts to pay for tips, tax) Have difficulty grasping and using everyday living math skills? (e.g., financial planning or budgeting, balancing chequebook, long term financial planning). Have difficulty keeping score during games like bowling, often looses track of whose turn it is during games like cards and board games, and/or has limited strategic for games like chess? (Adapted from Newman, 1997)
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Appendix B to Module 4 LETTER FROM A LEARNER WITH DYSCALCULIA (Excerpted from Newman, 1997) Dear Math Professor: On tests, please allow me scrap paper with lines and ample room for uncluttered figuring. I need instant answers and a chance to do the problem over once if I get it wrong the first time. Often my mistakes are the result of "seeing" the problem wrong. To AVOID this, you would have to watch as I went through each problem and correct any mistakes in recording as they happened. Problems written too closely together on the page cause me mental confusion and distress. Please make the test problems pure, testing only the required skills. They must be free of large numbers and unnecessary distracting calculations. These side-track me into a frenzy! Please allow me more than the standard time to complete problems and please check to see that I am free of panic (tears in my eyes, mind frozen). If possible, please allow me to take the exam on a one-to-one basis in your presence. Most importantly, never forget that I WANT to learn this and retain it! But realize that math is very DIFFERENT than other subjects for me. It is traumatic! The slightest misunderstanding or break in logic overwhelms me with tears and panic. Please understand that I have attempted and failed many times and math is a highly emotional subject for me. Pity will not help me at all, but your patience and individual attention will. I do not know why this is so hard for me. It is like my math memory bank keeps getting accidentally erased. And I cannot figure out how to correct the system errors! I ask that we work together after class on the material just presented. Or, if that is impossible, sometime that day for at least an hour. I ask that extra problems be given to me for practice and maybe a special TA (teaching assistant) be assigned to me. I know that working with me may be just as frustrating for you. There are no logical patterns to my mistakes. A lot of them are in recording or in "seeing" one part of a problem in another. Sometimes I read 6x(x+3) as 6(x+3). After you work with me a couple of times, I'm sure you'll realize how important it is to keep problems as simple as possible because my brain creates enough of its own diversions. It is typical for me to work with my teacher until I know the material well- and then get every problem wrong on the test! Then 5 minutes later, I can perform the test with just the teacher, on the chalkboard, and get all problems correct. So, please, do be patient with me, and please do not give up on me! When presenting new material, I must be able to WRITE each step down and TALK it through until I understand it well enough to teach it back to you. Maybe you could go over the upcoming lesson with me. Then the lecture would be more of a review and I would not be sitting there in tears. Lastly, I am sure you know by now that I am not trying to "get out of" doing what is required of the rest of the class. I am not making excuses for not "pulling my load." I am willing to put WAY more into this class than is required of the average or better student. I am not lazy, and I feel really smart in everything but math. That is what frustrates me the most! Everything is easy for me to learn, but Math makes me feel stupid! Why is this one subject so hard? It doesn't make sense. Even trying harder and studying more is futile. I probably will forget everything I learned once this class is over. (That has been my experience with numbers in general- they just slip my mind.) But I wish to apply myself as fervently as necessary to achieve an above average grade in this class. Thanks in advance for all your help along the way :-)
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Appendix C To Module 4 DYSCALCULIA RESOURCES Text Abrohms, A. (1992). Literature-Based Math Activities. New York: Scholastic Books. Bley, N. & Thornton, C. (1989).Teaching Mathematics to the Learning Disabled. Austin, Texas: PRO-ED. Dino-Durkin, C. (1997). Brain Boosting Math Activities. New York: Scholastic Books. Kaye, P. (1987). Games for Math. New York: Pantheon Books. Kenschaft, P. (1997). Math Power. Reading, MS: Addison-Wesley. Streding, L. (1997). Ten Minute Math Mind Stretchers. New York: Scholastic Books Internet Dyslexia & Dyscalculia Support http://www.shianet.org/~reneenew/dss.html Services of Shiawassee County Dyscalculia International http://www.shianet.org/~reneenew/DIC.html Consortium LD Online (Math) http://www.ldonline.org/ld_indepth/math_skills/ math-skills.html Dyscalculia: What it is and what http://www.hopkins.k12.mn.us/Pages/North/L it isn't D_Research/dyscalculia.htm Myths and Realities about Math http://www.naspweb.org/services/spr/jfmm263 Interventions for Students with .html Learning Disabilities
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REFERENCES Archambeault, E. (Spet/Oct 1993). Holistic Mathematics Instruction, Adult Learning. 5(1). Evans, D., Klippenstein, S. & Skogan, S. (1992). Annotated Bibliography of Numeracy Materials. Manitoba: National Literacy Secretariat CommunityBased Literacy Program Partnership Project. Hicks, K.F. & Wadlington, E. (May/June 1994). Making Life Balance: Writing Original Math Problems with Adults, Adult Learning 5 (5). Hill, P. (1995). Manitoba Curriculum: Framework of Outcomes and Grade 3 Standards. Winnipeg: Manitoba Education and Training. Hope, J. and Small, M. (1993). Interactions. Canada: Ginn Publishing Newman, R. (1998), Diagnosing Math Learning Disabilities/Recommended Practices. Retrieved May 1999 from the World Wide Web. http://www.shianet.org/~reneenew/Edu502.html Newman, R. (1997), Dyscalculia Symptoms. Retrieved May 1999 from the World Wide Web. http://www.shianet.org/~reneenew/calc.html Schifter, D. (1996). What's Happening in Math Class? Newark, DE: International Reading Association. Sharma, M. and Brazil, P. (1997). Dyscalculia. Retrieved May 1999 from the World Wide Web. http://www.shianet.org/~reneenew/BerkshireMath.html Terry, Marion (date unknown). Revised Newspaper Math. Wells, A. (1997). Senior 1 Mathematics: Manitoba Curriculum Framework of Outcomes and Senior 1 Standards. Winnipeg; Manitoba Education and Training.
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