Vector �

The Official Journal of the British Columbia

Association of Mathematics Teachers

VectorTo see the table of contents, please click on the BCAMT logo.

Fall 2007�

Vector is published by the BC Association of Mathematics Teachers.

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David Tambellini, Vector Editor John Kamimura, Vector EditorBox 445 Kwantlen Park Secondary SchoolChristina Lake, BC 10441 132nd StreetV0H 1E0 Surrey, BC V3T 3V3dtambell@sunshinecable.com kamimura_john@sd36.bc.ca

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Technical Information

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Fall �007 • Volume 48 • Number 3

Vector 3

Inside this issue…

Vector

Official Journal of the BC Association of Mathematics Teachers

Sagar Pooni

Espen Andersen

Brad Epp & LesleeFrancis-Pelton

Marion Smith

Fall 2007 Volume 48 Issue Number 3

Sarah Jimenez

Frank Ho

Jeannie DeBoice

Mike Jacobs

Gary Tupper

11. Fall 2007 Puzzles

12. Fall 2007 Web Sites

15. Closing the Gap: Grade 2 Numeracy Project

18. Equity in Mathematics

24. Street Math

20. A Lesson from Census at School

30. Transformations – An Exploration

40. Musimathics or Mathemusic: Investigating the Effects of Music on Spatial Temporal Reasoning

43. A New Chess Set for Teaching Mathematical Chess

47. The Predicitve Ability of the FSA on Provincial Mathematics 10 Examinations

53. Why You Should Choose Math in Secondary School

David Tambellini56. Metacognition in Mathematics Education

65. Solutions to the Summer 2007 Puzzles

4. The BCAMT Executive 2007–2008

6. Meet Your Executive

Fall 20074

BCAMT Executive

Elementary School Representatives

Selina MillarNumeracy/Science Helping Teacher (Surrey)Work: (604) 592-4322Fax: (604) 590-2588Email: millar_s@sd36.bc.ca

Sandra BallPrimary Numeracy Helping Teacher (Surrey)Work: (604) 590-2255E-mail: ball_s@sd36.bc.ca

Lorill ViningNumeracy Support Teacher (Campbell River)E-mail: lorill.vining@sd72.bc.ca

BCAMT President & Newsletter Editor

Past President

Dave van BergeykSalmon Arm Secondary SchoolSchool: (250) 832-2188Fax: (250) 832-6112E-mail: dlvanb@telus.net

Vice-President

Robert SidleyH. J. Cambie Secondary School (Richmond)School: (604) 668-6430Fax: (604) 668-6132E-mail: rsidley@dccnet.com

Secretary

Rupi Samra-GynanePrincess Margaret Secondary School (Surrey)School: (604) 594-5488Fax: (604) 594-4689E-mail: samra_r@sd36.bc.ca

Membership Chair

Dave EllisE-mail: daveellis@shaw.ca

Secondary School Representatives

Marc GarneauMathematics Helping Teacher (Surrey)Work: (604) 590-2588E-mail: piman@telus.net

Sam MuracaDistrict Coordinator - Numeracy (Langley)Office: (604) 534-9285Fax: (604) 530 2906E-mail: smuraca@sd35.bc.ca

Phil LeeMagee Secondary School (Vancouver)School: (604) 713-8200Fax: (604) 713-8209E-mail: pklee@vsb.bc.ca

Michèle RoblinHowe Sound Secondary School (Squamish)BCAMT Hotline: (877) 888-MATHSchool: (604) 892-5261Fax: (604) 892-5618E-mail: mroblin@sd48.bc.ca

Treasurer

Kathleen WagnerHugh Boyd Secondary School (Richmond)School: (604) 668-6615Fax: (604) 668-6569E-mail: kwagner@richmond.sd38.bc.ca

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BCAMT Executive

Ministry of Education RepresentativeRichard DeMerchantMinistry of Education Office: (250) 387-4416Fax: (250) 356-2316E-mail: richard.demerchant@gov.bc.ca

Peter LiljedahlFaculty of EducationSimon Fraser UniversityWork: (604) 291-5643E-mail: liljedahl@sfu.ca

Post-Secondary Representative

Marc Garneau(see above)

NCTM Representative

Canadian NCTM Regional Representative (Zone 2)

Carol MatsumotoEmail: carol.matsumoto@7oaks.org

Regional Representatives

Chris BeckerPrincess Margaret Secondary School (Penticton)School: (250) 770-7620Fax: (250) 492-7649E-mail: cbecker@summer.com

Brad EppSouth Kamloops Secondary School School: (250) 374-1405Fax: (250) 374-9928E-mail: bradepp@mail.ocis.net

Independent Schools Representative

Chris StroudWest Point Grey AcademySchool: (604) 222-8750E-mail: cstroud@wpga.ca

2007 - 2008

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Name: Dave Van BergeykSchool/District: Salmon Arm Secondary SD #83 (North Okanagan-Shuswap))Role with BCAMT: Vice-presidentDistrict Role: Math Department Head at school; District Numeracy Committee MemberFavourite Mathematician: FermatNon-math Hobbies: Outdoors activities (Running, biking, cross-country skiing, hiking); reading; Scrabble; Wood-working.

Name: Michèle RoblinSchool/District: Howe Sound Secondary (SD #48)Role with BCAMT: PresidentRole - Other: Teacher, Head of Math Department at school, and Member of District Math Leadership TeamMath “Favourites”: Favourite Mathematician: ArchimedesFavourite Books: The Dot and the Line: A Romance in Lower Mathematics by Norton Juster, andThe Math Instinct: Why You’re a Genius (Along with Lob-sters, Birds, Cats and Dogs) by Keith Devlin Non-Math Hobbies: Lake swimming and rock climbing (in summer) and yoga year-round.

Name: Robert SidleySchool/District: Burnaby (SD #41)Role with BCAMT: Past PresidentDistrict Role: Program Consultant (Mathematics & Science)Math “Favourites”: Zero: The Biography of a Dangerous Idea by Charles SeifeNon-Math Hobby: Barbeque baby… its all about the barbeque

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Name: Dave EllisSchool/District: • Retired in June 2007 • Taught 35 years in Vancouver • Math Dept Head at Eric Hamber for 22 yearsBCAMT Role: Membership ChairDistrict Role: Classroom teacherMath “Favourites”: Isaac NewtonNon-Math Hobby: Cycling, Hiking, and watching sports

Name: Rupi Samra-GynaneSchool/District: Princess Margaret Secondary / SD#36 (Surrey)BCAMT Role: SecretaryDistrict Role: Vice-PrincipalMath “Favourites”: Manipulatives!Non-Math Hobby: Indoor Rock Climbing

Name: Kathleen Wagner School/District: Hugh Boyd Secondary, Richmond, BC BCAMT Role: Treasurer District Role: Teacher Math “Favourites”: Graph theory problems

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Name: Sandra BallSchool/District: SD #36 (Surrey)BCAMT Role: Executive memberDistrict Role: Numeracy Helping TeacherMath “Favourites”: • Discovering patterns and relationships of numbers. • Marilyn Burns is my favorite mathematician. • My favourite children’s math book is “One is a

snail, Ten is a Crab” by April Pulley Sayre.Non-Math Hobby: Golf and creating… there’s math there, too!

Name: Selina MillarSchool /District: District Numeracy / Science Helping Teacher, SD #36 (Surrey)BCAMT Role: Elementary Schools Representative Favourite Math Quote:“Students will become confident “doers” of mathematics only if mathematics makes sense to them and if they believe in their ability to make sense of it.” Tranfton and Claus (1994)Favourite Non-Math Activity: Spending time with family and friends.

Name: Lorill ViningSchool/District: Campbell River (SD #72)BCAMT Role: Elementary Schools RepresentativeDistrict Role: Numeracy Resource TeacherMath “Favorites”: • favourite person: Carole Saundry • favourite book: Equal Shmequal • favourite real-world problem - discovering the per-

fect proportions of Baileys in my coffee on Sat-urday morning (still undetermined but a problem worth continuing to investigate!)

Non-Math Hobby: fishing in the waters of Quatsino Sound

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Name: Marc GarneauSchool/District: Curriculum & Instructional Services, SD #36 (Surrey)BCAMT Role: Secondary Schools Representative, NCTM Representative, WebmasterDistrict Role: K-12 Numeracy/Science helping teacherMath “Favourites”: • Favourite mathematicians: Euclid, Descartes,

Newton • Favourite math TV show: Numb3rsNon-Math Hobbies: • family • movies • Old time radio plays

Name: Phil LeeSchool/District: Magee Secondary School / Vancouver School BoardBCAMT Role: Secondary Schools RepresentativeDistrict Role: Physics/Math Teacher“Trigonometry is for me. Probability is for the birds.”

Name: Sam MuracaDistrict: School District #35 (Langley) District Role: CoordinatorBCAMT Role: Secondary School Representative, WebpageFavorite math person: Euler Favorite branch of math: CalculusFavorite math book: Flatland by A. SquareNon-math hobby: Photography, Coaching

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Name: Brad EppSchool/District: South Kamloops Secondary School, Kamloops/Thompson (SD #73)BCAMT Role: Regional Representative and Listserve ModeratorDistrict Role: Teacher & Department HeadMath “Favourites”: The pythagorean relationshipNon-Math Hobbies: Golf, Hockey, & Curling

Name: Peter LiljedahlSchool: Simon Fraser UniversityBCAMT Role: Post-Secondary RepresentativeOther: Assistant ProfessorMath “Favourites”: anything by John Mason and John van de WalleNon-Math Hobby: skiing

Name: Chris StroudSchool/District: West Point Grey Academy, ISEABCAMT Role: Independent School and PIMS liaisonDistrict Role: Grade 7 math and Grade 5 and 6 math enrichmentFavourite math website: www.blokus.com (works on translations, rotations, and reflections)Non-Math Hobby: Mountain biking and being a father.

Name: Chris Becker School/District: Princess Margaret Secondary/Okanagan-Skaha (SD #67)BCAMT Role: Secondary RepresentativeDistrict Role: Teacher/Chairperson Secondary Math CommitteeMath “Favorites”: Favourite math joke: What did one math book say to the other? “I’ve got problems”.Non-math Hobby: Hockey

Vector ��

Fall 2007 Puzzles

Please mail or e-mail your completed answer(s) to:

David Tambellini, Vector Editor Box 445Christina Lake, BC V0H 1E0dtambell@sunshinecable.com

Puzzle 1

Puzzle 2

For each correct solution to these problems that you send in, your name will be entered in a draw to win a BCAMT designer T-shirt.

Using only the numbers 1, 3, 4, and 6, together with the operations +, –, ×, and ÷, and unlimited use of brackets, make the number 24.

Each number must be used precisely once. Each operation may be used zero or more times. Decimal points are not allowed, nor is implicit use of base 10 by concatenating digits, as in 3 × (14 – 6).

As an example, one way to make 25 is: 4 × (6 + 1) – 3.

Find the dimensions of a triangle whose height and 3 sides are 4 consecutive natural numbers.

Fall 2007��

Fall 2007 Web Sites

1) Skillwise

This Web site has some very interesting activities, and not just for number either. Although Skillwise is directed primarily at adult learners, a number of their resources are certainly suitable for younger learners too.

http://www.bbc.co.uk/skillwise/numbers/measuring/

Try it out, and I think you’ll be as surprised as I was.

Also, have a look at this site. It has one of the best math lit lists I have seen for a while.

http://www.sci.tamucc.edu/~eyoung/literature.html

It also as an early childhood link and one for middle and high schools.

Contributed by Pam Hagen

2) Invalid Mathematical Proofs

There is a nice list of invalid mathematical proofs on Wikipedia:

http://en.wikipedia.org/wiki/Invalid_proof

Contributed by Colin McLellan

3) A Plus Math

This Web site was developed to help students improve their math skills interactively.

It features a game room, flash cards, a math word find puzzle, a flashcard creator, worksheets, the “Homework Helper” and more.

http://www.aplusmath.com

4) Free Square and Isometric Dot Paper

Download free square and isometric dot paper and create some dot designs using square translations, square rotations, and hexagonal rotations.

http://www.howe-two.com/nctm/tessellations/dotpaper/index.html

Vector �3

5) Pre-calculus, calculus and linear algebra tutorials

For pre-calculus, calculus and linear algebra tutorials, try

http://www.math.hmc.edu/calculus/tutorials/

The site also has some nice downloadable diagrams for teachers.

6) Elementary Activities

For elementary level brain teasers, math worksheets, flashcards, and more, check out

http://www.pedagonet.com/

7) Elementary level worksheets and drills

For elementary level worksheets and drills, try

http://math.about.com/od/multiplication/ig/Times-Tables-Worksheets/

or

http://www.math-drills.com/

Contributed by Debbie MacCulloch

8) Math Ganes

There are lots of games from other countries that for some reason just haven’t become well known in Canada.

One of my favourites from the UK is a game called “Countdown.” It’s a popular quiz show in the UK and France and the numerical aspect is to use a large number (25, 50, 75 or 100) and 5 small numbers (1 to 9, repeats allowed) and use the four major operations and as many brackets as you want to make a target three-digit number.

http://www.mathsnet.net/puzzles/countdown/index.html

There are lots of other good math diversions on the MathsNet site

http://www.mathsnet.net/puzzles.html

It can easily be used with a class by using the random number generator on a calculator.

Of course, there are some competitors who just completely blow your mind away. Pause it after you see the numbers and have a go yourself.

http://www.youtube.com/watch?v=1XQo6PwS358 Contributed by Richard Izdebski

Fall 2007�4

9) Federal Resources for Educational Excellence

Given the recent discussion on assisting students who need to build their skills, I thought I would pass along this site:

http://www.free.ed.gov/subjects.cfm?subject_id=33

Teacher Workshops (math) provides materials from 2007 summer teacher workshops sponsored by the U.S. Department of Education. Find slides and handouts on teaching everything from algebra to decimals and fractions.

Here are the session materials for math.

http://www.t2tweb.us/Workshops/Sessions.asp?Content=Math

Contributed by Helen Kershaw

10) Interesting Math Web Sites

I thought I would pass on a couple of interesting Web sites that I have found useful.

http://www.homeschoolmath.net/math_resources_1.php

There are lots of great resources here, especially for the elementary level.

http://www.xpmath.com/

l found the math e-books section and the math links interesting.

Contributed by Michael Cornell

11) Math Videos

The ongoing need for balance between a focus on algorithm based methods for solving basic problems and a focus the estimation skills necessary for students to be numerate is cleverly displayed in video take off of the recent Macintosh commercials (“Hi, I’m a Mac, and I’m a PC”).

Check out the following link and watch a couple of familiar faces in their theatrical debuts!

http://www.zapple.ca/page24/page24.html

Contributed by John Pusic

12) Math Glossary

For all teachers K-12, please check out

http://www.ronblond.com/MathGlossary

You will be amazed! It is a wonderful interactive and animated site.

Just for fun, click on Circle and view all grade levels to see the progression of this concept from K-12.

Contributed by Lorraine Baron

Vector ��

Closing the Gap: Grade 2 Numeracy Project

To take part fully in society today, people must be numerate as well as literate. Yet interventions to help struggling learners in math are lacking in many of our schools. Our team – principal Bob Belcher, classroom teacher Liz Jackson, learning assistance teacher Jessie Moore, and myself, district numeracy advisor – focused our inquiry question on how we could raise the lowest achieving primary students in the number strand using formative assessment. At the heart of this inquiry is a social justice issue: we can no longer ignore the gaps that students exhibit and simply hope they’ll ‘get it’ down the road.

Our team first met in September 2006 to sketch out our plan. We chose to carefully examine the research on formative assessment, its uses, and its positive impact on student learning, especially in closing the gap between at-risk students and the rest of the class.

Jessie and I began by assessing nine Grade 2 students using First Steps in Mathematics diagnostic tools to pinpoint their misconceptions and gaps in the number strand. Next we reviewed activities from First Steps that are designed to provide the math that’s missing. I plotted these into the unit overview Liz would be using for the next six weeks. It was important to us that these students remained in class so they could participate in group discussions and activities. Jessie and I worked out a schedule in which one of us was in the class every day for six weeks. Where appropriate, we noted any progress we saw in their “gap” areas in a duo tang we could all access. This information was also fed back to the students themselves. Liz created goal cards that she shared with the whole class so everyone could identify the same curricular goals and learning intentions.

After six weeks, I re-assessed the students and found that many had made some gains. Initially, five of the nine children did not “trust the count”: that is, once they had counted eight counters and I repeated the question, ”How many are there?” they had to recount the counters. The other four trusted the count by one’s, but not by two’s. At the end of six weeks, all but one trusted both counts. Three students had made significant gains, as they had also mastered counting by 2’s up to 100.

We then discussed the idea of having the students use each other as learning/teaching resources to improve the weaker students’ skip counting as well as to reinforce the stronger students’ skills. We developed “math coaching kits” that contained counters and laminated 100 charts (with the counts by two, five and ten highlighted). Then we:

• modeled what coaching each other in skip counting looked like; • created a poster about coaching for the classroom; • matched stronger students with slightly weaker students to coach each other; and • coached for five minutes every morning.

The results were exciting: the students identified as having made gains strengthened their skills as they listened carefully to guide their partners. Also, as they switched roles, the weaker students acted as the coaches, and got double practice as they too carefully listened and followed along on the hundred chart.

Jeannie DeBoice is the K - 12 Numeracy Advisor for the Sooke School District. She is currently attending the University of Victoria, working on her masters degree in mathematics education (curriculum and instruction).

Jeannie DeBoice

Fall 2007��

By the next 6-week cycle assessment, all but three students had mastered the skip-counting to 100 and used the skill as a counting strategy. By late February all had mastered skip counting.

The final phase of our project was basic fact mastery. Too often, we see children in Grades 4 and 5 (even Grade 9) who still count on their fingers or make marks in the margin. This is not number sense! Our goal was to build the strategies that would enhance the students’ mental math flexibility and further empower them with number sense.

Together we developed an eight-week project that would also involve the parents at home as other coaching sources. I assessed all the students individually on the strategies they had, clearly describing to them that a basic fact strategy is one in which they could find the answer in their heads, not on their fingers. The students highlighted the facts they had mastered on an addition chart.

Next I created practice cards with the strategy prompts on them and we taught the students how to be coaches for each other ten minutes a day. We made each child a folder that contained the highlighted addition chart and a pack of the practice cards of the strategy they needed to master. Even our most vulnerable student had a much better sense of what he knew, what he still needed to know, and what mastery of basic facts looked like.

John Van de Walle (2000) states that to master addition facts, a child need master only the following four strategies.

• one-more-than/two-more-than • doubles• near-doubles • make-ten.

Then a child can “think addition” to answer most subtraction facts. Liz introduced one strategy at a time to the whole class, using hands-on manipulatives and lots of discussion. We used the first ten minutes of each lesson to practise these strategies. All the children knew the strategy they were working on (by the cards in their folders) and worked with partners using their already established “coaching skills.”

The adults would monitor the work and reintroduce hands-on manipulatives if it appeared that the symbolic prompts on the cards were not enough. This is important for moving struggling learners on, we discovered, as some children need longer with the concrete models than others. The critical piece was to observe closely and recognize when a child needed to return to the concrete and when he/she was ready for the more abstract picture clues. The final stage was to trade the prompt cards with purely symbolic cards – cards with only the addition facts on them.

In addition, we held a parents night to teach these strategies and effective coaching skills for use at home with their children. The emphasis was on using strategies in a playful, fun way (not “drill-and-kill”), and incorporating them into daily activities as opposed to sit-down homework time. The parents used the materials, cards and strategies just as the children did so they could get a feel for the process.

This became an exciting time of growth for some of our weakest students in this phase of the project. It emphasized for us that, to reach that lowest 10 - 20% of the class, you need to “stay the course” and believe that the cycle of assessment, clearly focused feedback and guidance from the teacher to help the learner close the gap will work. I believe it has been the quality of our assessment and descriptive feedback that has helped our most vulnerable students.

One young boy had his breakthrough recently. At the beginning of the project, this child could not count by 2’s, didn’t know what the symbols + and – meant and often looked as though he were a million miles away. Recently, he showed me mastery of his +9’s using a “make ten” strategy. I asked him, (as I often ask all of the students), “How did you answer 8 + 9 so quickly without your fingers? What was happening in your brain?” He replied, “Well, I just took one from the 8, gave it to the 9 to make it a 10, and now 7 + 10

Vector �7

is easy to add in my head!” He giggled with joy and pride as I sat with my mouth hanging open. Then he said, “I’m really becoming a smart math boy, aren’t I?”

This child was now engaged in his learning and proud of his success. As Black and Wiliam state: “The learner first has to understand the evidence about this gap and then take action on the basis of that evidence. Although the teacher can stimulate and guide this process, the learning has to be done by the student” (Black et al, 2003, p. 14). Ethically, all children should be guided to take their learning as far as is possible, yet I fear that sometimes in our profession we give up on some children far too soon. This boy could easily have been one of those children.

Liz, Bob and Jessie have been wonderful teachers with whom to work and I know we’ve all gained tremendously from the partnership. I’ve really learned first hand the power of collaboration as we’ve shared with each other ideas for strategies, ways to help specific children and our joy at their success.

References

Black, P., Harrison, C., Lee, C., Marshall, B., & Wiliam, D. (2003). Assessment for learning: putting it into practice. New York: Open University Press.

Pearson Professional Learning. (2006). First steps in mathematics. Toronto: Pearson Education.

Van de Walle, J. A. (2000). Elementary and middle school mathematics: Teaching developmentally. Toronto: Pearson Education.

Fall 2007�8

Equity in Mathematics

Sagar Pooni is a student in the Technology and Critical Thinking (TACT) elementary education program at the University of British Columbia.

Equity is essential when it comes to assuring that all students are given the opportunity to reach their mathematical potential in the classroom. In order to achieve even-handedness in math classrooms, we must negate any cultural bias, achieve gender impartiality and allow children to work to their abilities.

To combate a disparate level of achievement among multicultural students, Ladson–Billings (1995) recommends using culturally appropriate material. In her article, she provides an example of a math problem where students are asked to calculate the better option between two monthly fares. The quandary emerges in the cultural context of the question that asks the children to calculate the number of trips a person must make to their place of occupation in a single month. The children were unable to determine what was being asked since they were not familiar with the customary 9 to 5, five days a week, work routine.

In mathematics, the word problem is of utmost importance; children must have an equal opportunity to comprehend, assess and evaluate such inquiries. It is the math problem that allows students to apply their technical skills, conceptual understanding and critical thinking to real-life situations. If the word problem is unnecessarily and carelessly clouded in irrelevant context in regards to its audience, children will be missing out on a great opportunity to learn practical mathematics.

Another line of reasoning put forward by Ladson-Billings is the powerful notion of cultural relevance. She cites the case of a California teacher who used the history of mathematics in relation to the ethnicity of his students. For example, he referred to the awesome mathematics that the Mayans used to create their advanced civilization and as a result his Latino students were exposed to a personal connection to mathematics.

Teachers in our B.C. educational environment can use this same strategy. Since both India and China have profound histories that have impacted the evolution of mathematics, it would be advantageous to rely on these merits to encourage Asian children to appreciate and pursue math in a more effectual manner. One can never underestimate the value of having a vested interest in a subject resulting from a sense of one’s own identity. Once children see that their own culture has a proud intellectual history of mathematics, they are more likely to want to understand these contributions and perhaps make some of their own.

In terms of gender, there is much speculation and study on discrepancies between boys and girls in mathematical ability. Some experts claim that males have an innate advantage over females in mathematical aptitude. The Science Daily (2006) published an article where UBC researchers investigated the power of perception among women in regards to math.

Heine and Dar-Nimrod found the worse math performances belonged to women who received a genetic explanation for female underachievement in math or those who were reminded of the stereotype about female math underachievement. Women who received the experiential explanation performed better – on par with those who were led to believe there are no sex differences in math.

Sagar Pooni

Vector ��

This study emphasizes that stereotypes can truly affect students in the classroom. Girls who believe that they are inferior in any way will live out that belief in an academic setting. This random experiment proves that once girls are empowered to shed the skin of misconception and prejudice they are able to reach their own potential; a potential that has no scientific basis for being less than that of a male.

As teachers we must ensure that we do not perpetuate any stereotypes such as girls being less gifted in math. We must push our girls just like we push our boys and we must maintain the same expectations for both genders. It is also critical that parents demand excellence in math from their daughters as well; stereotypes are perpetuated by all aspects of society in this case.

When we think of diversity in the classroom for mathematics it is too often effortless to look over exceptional students. Davis and Rimm (1998) state the following.

The boredom and frustration of even the average-scoring SMPY [Study of Mathematically Precocious Youth] contestants when incarcerated in a year-long algebra class is difficult to appreciate. Often, highly able youths themselves are not aware of the extent of their slowdown, because it has been their lot from Kindergarten onward… Often, they take off like rockets when allowed to do so. (p. 118)

Here the authors speak of high-end exceptional students who are being constrained to their age appropriate grade levels despite the fact they are years ahead of their peers; the detriment of being stifled for years is prevalent in many schools from grade to grade.

SMPY is a program that tests 7th and 8th graders to seek students who have exceptional mathematical reasoning abilities; these students go on to pursue advanced mathematics in special programs. According to Davis and Rimm (1998), some of the advantages of such a program include a higher motivation for learning math, an enhanced sense of accomplishment, reduced arrogance as they are humbled by their intellectual peers, and better preparation for future education.

It is crucial that students be challenged and be given the opportunity to push their talents to new levels.We cannot sacrifice a child’s untapped potential of achieving new heights in math for the sake of a flawed ideology of forced equity. When students in a classroom are distinctly operating at far greater levels of mathematical ability than their peers, the principles of equity do not necessitate their hindrance. Rather, equity calls for these students to receive those resources and challenges that their intellectual capacities exact in order to achieve their potential. An egalitarian mode of teaching does not require an artificial levelling of group capabilities: instead it demands that each child be given an equal challenge to his/her specific ability.

Mathematics has propelled our civilization to amazing heights. It is imperative that equality reign in our classrooms so that these students can use mathematics to help make this world more equitable as well.

Bibliography

Davis, G. & Rimm, S. (1998). Education of the gifted and talented (4th Ed.). Boston: Allyn & Bacon.

Ladson-Billings, G. (1995). Making mathematics meaningful in multicultural contexts. In W. Secada, E. Fennema, & L. Byrd Adajian (Eds.), New directions for equity in mathematics education. Cambridge: Cambridge University Press.

Science Daily. (October 20, 2006). Women’s Math Performance Affected by Theories on Sex Differences. Retrieved from http://www.sciencedaily.com/releases/2006/10/ 061019161245.htm .

Fall 2007�0

A Lesson from Census at School

Marion Smith is an Education Consultant with Statistics Canada.

Preamble

You can now teach data analysis with the Census at School online project from Statistics Canada. From the Census at School site, you can collect survey responses from your students via an online questionnaire, and access a downloadable spreadsheet of answers ready for analysis. By manipulating their own raw data, students learn how to describe a population with statistics.

Students can use their own data to investigate a question or hypothesis and draw conclusions. They can also learn about various types of graphs; frequency tables; mean, median and mode; fractions; ratios; percentages; estimating and forecasting; and bias. The online questionnaire includes a measurement component (e.g., height, arm span, foot length) relevant to the Vitruvian theory of body proportions. There are two survey questionnaires: Grades 4-8, and 9-12.

Your class’ spreadsheet of responses is password-protected, downloadable only by the classroom teacher. All responses are added to a national database of Canadian data that will be available for large-sample analysis next school year. Previous years’ databases are currently available for comparing class data to national and provincial results. Individual responses cannot be identified by class, school, or city.

Find the bilingual Web site at: http://www.censusatschool.ca or http://www.recensementecole.ca. Teaching activities are located at http://www19.statcan.ca/02/02_004_e.htm, where you will also find the new Teacher’s Guide to Data Discovery, with information on Canadian datasets, choosing appropriate graphs, and calculating basic statistical measures.

New environment questions on the 2007-2008 questionnaire, include:

• the importance of environmental issues, • whether households participate in recycling and composting, and • whether households use fluorescent lights.

The Royal Statistical Society in the U.K. developed Census at School to support the teaching of data analysis through technology and the Internet. This program runs in the U.K., Australia, New Zealand, South Africa and Canada. Other countries are also interested in participating.

Contact Marion Smith, marion.smith@statca.ca, phone (604) 666-1148, to order a free Vitruvian Man poster in English or French.

We have reorganized StatsCan math pages. Resources are sorted by grade level (Kindergarten to Grade 5, Grades 6 to 8, and Grades 9 to 12). Within each grade level, there are three sections: Lessons, Key Resources, and Data.

Marion Smith

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Lessons:

These are divided into curriculum-relevant mathematics topics (e.g., data management; graphing; functions, relations and modelling), and by sub-topics (for example, under data management: data analysis, data collection and data manipulation). Under each topic, lessons are further divided by the type of Statistics Canada resource they use (Census at School, Census of Canada, E-STAT, and Other).

Key Resources:

This page lists exercises, tools, and other resources, such as articles, function modelling resources, reference material, and external links.

Data:

This section links to aggregate-level data tables and individual-level microdata files.

See http://www.statcan.ca/english/kits/courses/math.htm, or go to http://www.statcan.ca/menu-en.htm then click on the link “Learning resources” and then inside the box labelled “Quick links” click on the link “Resources by school subject” and finally on the link “Mathematics.”

A Sample Lesson from Census at School

Statistics allow us to describe quite complex populations simply, with numbers. Census at School provides introductory lessons for this branch of mathematics, allowing students to participate in the collection, classification and interpretation of their own data, then reporting the results in either numerical or graphical format.

One lesson, Circle and bar graphs, provides a circle graph of pet ownership, based on survey results for 58 people. Using their observations of the graph, students are asked to estimate how many people own a dog, cat, gerbil, etc. By dividing the graph into fractions, students can hone in on the actual numbers. One clue is that at least one person owns a reptile. Students must also keep in mind that their estimates cannot total more than 58.

The teacher’s notes suggest having the students make estimations on their own before sharing with a partner. Then, in a class discussion, students can share their initial observations, comparisons and justifications before refining their estimates.

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In the next part of this lesson, students analyze modes of travel to school (by school bus, car, walking, cycle, public transit, other). They are given a table that shows the various travel modes and numbers of students per mode. At this point they examine the data, asking themselves if this data could represent their school, and what modes of travel would conceivably be included in the “other” category. They can also try to draw some conclusions about whether the school in question is located in a rural, urban or suburban area.

The students complete the table, converting the raw data to fractions, decimals and percents. Using a blank pie chart divided into 100 increments, provided with the lesson, students then graph their results. Even if your students haven’t yet encountered percentages, if they can convert the fractions to decimals, they can work out the proportions on the pie chart.

After completing the pie chart, students are asked to compare it to a vertical bar chart based on the same data, and critique the two graphs with regard to which graph better displays the data, which one is easier to read, and which one would be easier to create.

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This part of the lesson can be used by itself: that is, you can use it without having your students complete the online Census at School survey. However, with your own class’ survey responses, it is possible to build on these activities by having students graph their own information.

Students may begin by posing a question for research, such as: “What is the most common eye colour in the class?”, and predict the results. They will need to create a frequency table for the four eye colour choices (blue, brown, green, other). They could further break down the data, graphing eye colour by gender.

Students should consider whether their chosen data is best graphed with a bar chart or a circle graph.

Once the graphs are completed, students can discuss and write about their findings. Did their predictions hold true? Ask them to describe their class using their newly discovered statistics.

Census at School activities relate to various learning outcomes such as the following.

• Name and record fractions for the parts of a whole or a set. (Grade 4)

• Describe and represent decimals (tenths and hundredths) concretely, pictorially, and symbolically. (Grade 4)

• Relate decimals to fractions (to hundredths). (Grade 4)

• Construct and interpret pictographs and bar graphs. (Grade 4)

• Differentiate between first and second hand data. (Grade 5)

• Construct and interpret double bar graphs to draw conclusions. (Grade 5)

• Demonstrate an understanding of percent (limited to whole numbers) concretely, pictorially and symbolically. (Grade 6)

• Create, label and interpret line graphs to draw conclusions. (Grade 6)

• Graph collected data and analyze the graph to solve problems. (Grade 6)

• Construct, label and interpret circle graphs to solve problems. (Grade 7)

A number of the questions (travel to school, breakfast food choices, participation in sports) are directly relevant to current issues around children’s health, nutrition and fitness. They provide an opportunity for students to discuss these issues: for example, how many students walk to school or eat a nutritious breakfast, without zeroing in on any particular individual’s habits. Similarly, the new environmental questions will provide content for discussions on household practices such as recycling, composting and the use of fluorescent light bulbs.

For teachers looking for technology applications (the use of spreadsheets, choosing and sorting columns of data, graphing with a computer) the Census at School spreadsheet provides a wealth of variables for students to investigate.

Fall 2007�4

Street Math

Mike Jacobs is a mathematics consultant for the Durham Catholic District School Board.

(Editors’ Note: This article first appeared in the September 2007 issue of the Ontario Mathematics Gazette, and is reprinted here with permission.)

Have you ever stopped to think about how closely the math you learned in school (especially elementary) correlates to the math that you use in real life? Try these two questions.

1) You have two cheques to deposit, one for $298 and one for $309. How much do you deposit in total?2) What is the cost of 5 CDs at $8.99 each?

Now ask yourself, how did you get the answers? Even better, give these questions to your students and ask them how they get their answers. For the first question, did you align the numbers in columns and add the digits starting with the units? Or did you mentally add the hundreds, then the tens and then the units? Perhaps you used $2 from the $309 and in effect did $300 + $307. I am convinced that there are more ways of doing these problems than I have actually taught. I also know that the way I solve them does not involve the traditional algorithms that I was taught when I was in elementary school in England. This strikes me as curious.

I am not alone in abandoning my learned algorithms when doing mental math. Similar problems were presented to elementary teachers of the Durham Catholic District School Board and the vast majority used a non-standard approach. Keith Devlin, in his excellent book The Math Instinct, refers to the work of Nunes, Schliemann and Carraher (1993) in highlighting the math used by young Brazilian street vendors and how this differed markedly from the math that they where taught in school. This street math reminded me of a former student of mine who refused to do the simplest of money addition problems in class. However, I saw the same student out of school working in the local street markets and being able to add five or more prices mentally and work out the change after the items had been paid for.

Why is this? Why would a student, whom I have seen do multi-step problems in his head, believe that he cannot do easier problems in class? Perhaps the answer lies in the notion that our approach to teaching number sense has been algorithm-driven at the expense of getting our students to develop a good repertoire of mental math strategies. This certainly was my experience as a student when I learned algorithms long before I learned about mental math strategies or even before I learned about how to make a good estimate. To be honest, this is probably how I first taught as well. Now I realize that when it comes to teaching number sense, maybe algorithms are the last things that should be taught. It is interesting to note that most adults do not use algorithms in everyday life.

Now don’t get me wrong. I am not a mathematical anarchist who is against rules and algorithms of any type. What I do believe, however, is that for these algorithms to be understood and used effectively with meaning, there are many other skills that first need to be in place. Students who are being asked to do addition and subtraction must fully understand place value and how to partition numbers. Students who are being asked to do multiplication and division must fully understand addition and subtraction.

Mike Jacobs

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Research by Kamii and Dominick (1998) provides strong evidence that algorithms actually work against the development of place value and number sense in students. In fact, students who had not been taught any algorithms (but instead had been encouraged to invent and refine their own strategies) were significantly more successful than students who had been taught an algorithm. As Pascal said, “We are usually convinced more easily by reasons we have found ourselves than by those that have occurred to others.”

The First Steps in Mathematics framework, in which we have trained all teachers at eight of our Board’s elementary schools, is an extremely useful tool for placing students on a mathematical developmental continuum. It allows teachers to see, irrespective of the curriculum demands, if students are ready to understand certain concepts and algorithms. In the example below, a student has answered the question simply by counting “2+3” and then “3+4” on his fingers.

32 +43 75

The student has no sense of the quantity 7 nor whether or not his answer is reasonable. The question has to be asked, “Should any student need to use the standard algorithm for adding two 2-digit numbers?” I would suggest not, and that introducing a standard algorithm too early and for “easy” numbers is counter-productive. As Fosnot and Dolk (2001) have noted, “Using algorithms, the same series of steps with all problems, is antithetical to calculating with good number sense.”

Algorithms become problematic for students because of the myriad of often-subtle rules that must be followed for success. These rules make sense only when the student has developed a thorough working knowledge of such big ideas as place value and operations. Think how confusing the “rules” for a division algorithm can be if a student has no understanding of place value.

OK, so 8s into 2 doesn’t go which is kinda weird ‘cos my teacher said that 2 means 2 thousands and I would’ve thought that there must be at least a few 8s in 2000. Anyway, I know I don’t put the 0 down for the “no 8s” as that’s what you do if you get a 0 to start with and now you say 8s into 24. That’s 3 which for some reason we have to put above the 4. Now we say 8s into 5. This is 0 but we do put the 0 down this time as it’s in the middle of the number and carry the remainder 5 and now say 8s into 56. This is something that finally makes sense to me as I know that that is 7… or is it 8? Hmmm…

Other problems will arise if students are exposed only to algorithms and not to mental math strategies. The standard algorithm for finding the cost of the CDs in the initial example is inefficient compared to simply thinking “5 items at $9 gives $45 now subtract 5¢ to get $44.95.”

Van de Walle (2001) noted that invented strategies are more concerned with quantities than most algorithms, which are digit-orientated. He also noted that invented strategies are flexible and change with the numbers involved; algorithms are rigid, however. By encouraging, modelling and practising mental math and self-invented strategies, students will be able develop a repertoire of skills that will allow them the flexibility to deal with real-life situations.

How can this be done? I would suggest that we start believing that math is a social activity and a fantastic opportunity for students to feed off each others’ ideas. Give your students a verbal question and ask them to display their answers using “number fans.” (See figure 1.) At a glance you will see who has the correct answer and you can then ask these students to explain what they did. By listening to how others approach a

)8 2456307

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problem, students will realize that there is more than one way to crack an egg. Given sufficient opportunity to practise (the National Numeracy Strategy in the U.K. recommends 10 minutes daily) students will begin to develop the flexible repertoire of mental math skills that they so desperately need.

For improving mental math skills, The 24 Game is another fantastic resource that your students will enjoy using. (See figure 2.) Students have to use all four of the numbers on a given card exactly once, but with any operations, to get the answer 24. If the numbers are 2, 3, 4 and 4 they could say (4 – 2) x (3 x 4). (Is there another way?) A more difficult example has the numbers 1, 6, 4 and 3. I’ll leave this one for you to answer!

Figure 1. A Number Fan

As students become proficient in their mental math skills, we can increase the complexity of the questions so that they might need to use informal pencil-and-paper methods to jot down their workings. This is useful because sometimes there are too many numbers to hold in our head. For example, to do 45 x 14 it might be easier to think of 90 x 7 and to jot these numbers down on paper before figuring out that 630 is the answer. I have often seen carpenters and joiners use informal jottings on the wood they are using to figure out a problem. Eventually the questions we give will be too complex and perhaps it is only at this point that we should introduce formal written algorithms.

Figure 2. A Card from The 24 Game

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What makes a good algorithm though? Is there any one algorithm that is the Holy Grail of long multiplication so that it will be the most efficient, and yet understood by all? I doubt it, but would love to know otherwise. To me, a good algorithm has to:

1) be efficient and simple to use;

2) be fully understood by the user;

3) connect with the user’s previous and future knowledge; and

4) provide the right answer!

This does mean that one algorithm might be good for one student yet bad for another. We should bear in mind, also, that algorithms can be culturally biased. If we just consider multiplication, compare our ‘traditional’ algorithm with these others below.

a) Multiplication by use of the Distributive Property

(See how this connects to the “array method.”)

28 × 27 = 20(20 + 8) + 7(20 + 8)= 20 × 20 + 20 × 8 + 7 × 20 + 7 × 8

Hence 28 × 27 = 400 + 160 + 140 + 56= 756

28×27

56140160400756

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b) Elizabethan, Lattice or Gelosia Multiplication (Napier’s Bones)

1× 27 = 272 × 27 = 544 × 27 = 1088 × 27 = 216

16 × 27 = 432

Hence 28 × 27 = (4 × 27) + (8 × 27) + (16 × 27)= 108 + 216 + 432= 756

c) Egyptian Multiplication

d) Russian Peasant Multiplication

My father, an E.A. in a school in the south of England, was taught the Russian peasant method by a student with special needs.

Hence 28 × 27 = 108 + 216 + 432= 756

Can you see why this works? Is it a good algorithm?

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e) Informal Multiplication by Non-Standard Partitioning

28 × 27 = (30 × 27) − (2 × 27)= (3 × 270) − (54)= 810 − 54= 756

If your students find some of these methods confusing please remember that this does not mean that they are bad algorithms; they are just not “right” for those students.

In my role as math coach for the Board, I have shown these different methods to students and teachers: not only do they find the different algorithms fascinating but in most cases, they often prefer one of the “new” methods. Indeed, the reason why these are often preferred is because they make sense to students. I would perhaps go so far as to suggest that the array method should be the first multiplication algorithm that students learn. It is efficient (certainly for up to three-digit by two-digit multiplication), it is understood by the students (providing that they truly understand the operation of multiplication), it connects with students’ previous knowledge (as it is a direct generalization of multiplication that has been modelled with base ten blocks) and it can connect with their future learning of the multiplication of decimals, fractions and binomials. I am certain that once students fully understand the array method, they will be ready to understand the other algorithms.

Having developed, learned and mastered a variety of algorithms (up to four-digit by two-digit as required in the Ontario Curriculum), our students need our help to realize when and where to use them. In years gone by, algorithms were extremely important as they were the best way of quickly finding the answer to lengthy calculations. Nowadays, we would mainly use technology for such questions and we would check the validity of our result mentally by using rounding techniques. Most situations in the real world can be solved if students have a good repertoire of mental math skills and informal pencil-and-paper techniques. Therefore we should perhaps be honing these “street math” skills in our students.

References

Devlin, K. (2005). The math instinct. New York: Thunder’s Mouth Press.

Fosnot, C. & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division. Portsmouth, NH: Heinemann

Kamii, C. & Dominick, A. (1998). The harmful effects of algorithms in Grades 1-4. In L.J. Morrow (Ed.), The teaching and learning of algorithms in school mathematics (pp 130-140). Reston, VA: National Council of Teachers of Mathematics.

Northcote, M. & McIntosh, A. (1998). What mathematics do adults really do in everyday life? Australian Primary Mathematics Teacher Vol. 4 No. 1, pp 19-21.

Nunes, T., Schliemann, A., & Carraher, D. (1993). Street mathematics and school mathematics. New York: Cambridge University Press.

van de Walle, J. (2001). Elementary and middle school mathematics: Teaching developmentally. New York: Pearson Education.

Willis, S. et al. (2006). First steps in mathematics: Number sense (Canadian Edition). Don Mills, ON.: Pearson Professional Learning.

Fall 200730

Transformations – An Exploration

Gary Tupper is a retired mathematics teacher living in Terrace.

In our curriculum we examine various transformations: translations, reflections and scalings. This paper addresses particularly the relationship between the algebraic (symbolic) and the geometric (visual) representations of 2-D relations.

The student will have learned that replacing x with a function of x results in a horizontal effect and replacing y with a function of y results in a vertical effect. The summative skill is exemplified when a student can describe the graph of, for example, y = 3sin4(x + π) – 2 in terms of the graph of y = sinx : “period quartered; amplitude tripled; translated down 2 and π to the left.” The student realizes that every term in the algebraic representation has a corresponding attribute in the graphical representation.

The student may have been perplexed upon noting that such replacements seem to have the opposite effect from what might be expected: replacing x with x – 3 results in a translation in the positive direction; replacing y with y/2 results in y-coordinates being multiplied by 2.

The following question arises naturally from the study of this topic. “Can we anticipate the effect on a given graph if we replace a variable in the original equation with an arbitrary function of that variable?” For example: what is the effect if we replace x with x2 or with x3 or with sin x?

Our method of addressing this question is essentially “scientific/experimental”: we will experiment by making various replacements and note the effects. Although the graphing could be done manually the author has greatly expedited matters by using computer software1 to produce all graphs.

Experiment 1:

Initial relation A y x: = 3

Replace x with |x| A y xτ : = 3

Figure 1.

Gary Tupper

A y x: = 3

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Figure 2. y x= 3

It appears that the quadrant 3 portion of A has reflected over the x-axis! But this contradicts our assumption that x-replacements imply horizontal effects.

Let’s try the same replacement on B: y = x3.

Figure 3. y = x3

Figure 4. y x= 3

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It appears that the quadrant 3 portion of C has been “discarded” and the quadrant 1 portion has been both left alone and reflected over the y-axis.

Let us next consider D y x x: .= − +3 3 1 5 , which has the property that none of the quadrantal portions are similar.

Figure 5 C y x : = −2 1

Figure 6 y x= −2 1

Figure 7 y x x= − +3 3 1 5.

This example seems to confirm the same aberration. Maybe our problem stems from the fact that both original graphs share the property that the quadrant 1 and quadrant 3 portions are point symmetric.

Let us examine an example like C y x : = −2 1 .

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Figure 8 y x x= − +3 3 1 5.

Our observation can be expanded to the whole plane: the quadrants 1 and 4 portions have remained and been reflected over the y-axis. The quadrant 2 and 3 portions are lost. Or, put another way, if (a, b) is a point on y = f(x) and if a is positive then there will be 2 corresponding points on y f x= ( ) : (a,b) and (–a,b). But if a is negative, there are no corresponding points on f x( ) .

Can we anticipate what might be the effect if we “absolute” both x and y variables? Using the basic relation in figure 7 we can examine the result of the two replacements. The relation of figure 9 is particularly useful because each quadrantal portion is distinct.

Figure 9 y x x= − +3 3 1 5.

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Figure 10 y x x= − +3 3 1 5.

When will (c, d) be a point of y f x= ( )? If and only if (|c|, |d|) is a point of y = f(x).

Now, let us examine the effect of replacing x with x2.

Experiment 2

Initial relation D y x x: . = − +3 3 1 5

Replace x with x2 D y x xτ : .= ( ) − +2 3 23 1 5

Figure 11 y x x= − +3 3 1 5.

It appears that the graph of y f x= ( )can be determined by simply examining the quadrant 1 portion of y f x= ( ) : that portion is replicated by reflection over both axes. And the Q3 portion is simply a subsequent reflection. For (a, b) in Q1, there will be three additional points generated: (–a, b), (–a, –b) and (a, –b). Points in the other quadrants will be discarded.

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Figure 12 y x x= ( ) − +2 3 23 1 5.

We do note that the y-intercept appears unchanged, and the y-axis is an axis of symmetry of Dt, which is an even function. Let’s examine further examples.

Figure 13 y = –x + 1 and y = –x2 + 1

Figure 14 y = x – 1 and y = x2 – 1

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3

4

x

y

(1,2)

(0,1)

(3,4)

(0,-3)

(1,-4)(-1,-4)

(-1,2)

(1.7,4)(-1.7,4)

Figure 15 R x y R x y: : –= + − = +1 2 1 22 and τ

We note that y = f(x) and y = f(x2) do share the y-intercept and that there is symmetry over the y-axis. But a complete understanding may be elusive, so we examine further.

Examining the coordinates of specific points we might deduce that (a, b) is a point of R R x yτ : ,2( ) if (a2, b) is a point of R(x, y) where R is used to denote a relation. Or, put another way, if (a, b) is a point of R(x, y) then ±( )a b, is a point of R x y2 ,( ) .

This clarifies why those points of R(x, y) in Quadrants 2 and 3 are discarded: their negative x-coordinates cannot be squares.

The same conclusion may be expressed with respect to R(x, y) and R x y,( ) : for (a,b) on R(x, y) to generate a point (c, b) on R x y,( ) then a must equal the absolute value of c, implying that only Quadrant 1 and 4 points of R(x, y) qualify.

We are now at the stage of hypothesizing about such substitutions in general terms. Suppose we wish to graph R R f x yτ : ,( )( ) , given the graph of R(x, y).

(a,b) is a point of R R f x yτ : ,( )( )

iff (f(a),b) is a point of R : R(x, y).

Suppose we start with E : y = 3x – 1 and wish to graph E y xτ : sin= −3 1.

Now (a,b) will be on Eτ iff (sin a, b) is on E. This implies that b, the y-coordinate, is 1 less than triple the sine of the first coordinate: i.e. , b = 3sin a – 1. Since –1 ≤ sin a ≤ 1, then b must lie between 3(–1) – 1 and 3(1) – 1 : that is, between –4 and 2.

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-4 4

-5

6

x

y

y=3x-1

y=2

y=-4

Figure 16. E y xτ : sin= −3 1 lies between the lines y = 2 and y = –4

-8 8

-5

3

x

y(1,2) (arcsin1,2) (arcsin1,2)

(-1,-4)

(0,-1)

(arcsin-1,-4)

(arcsin0,-1)

Figure 17. Graph of y = 3sin x - 1 derived from the graph of y = 3x – 1

Now we can examine points of the segment of y = 3x – 1 between (1, 2) and (–1, –4) to generate the corresponding points of y = 3sin x – 1.

E (0, –1) will generate sin , : , , ,− ( ) −( ) −( ) −( )1 0 1 0 1 1 on E nτ π where n is an integer.

E(1, 2) will generate sin , : ,− ( )( ) +( )1 1 2 2 2 2 on E nτ π π where n is an integer.

We can further generalize as follows.

a b R x y

f a b R f x

, ,

,

( ) ( )( )( ) (

on

will yield on -1 ))( ), y

This concluding property will have a parallel equivalent version for y-replacements.

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We can make further observations.

• Although the computer generally allows a direct access to the desired graph without the need for any analysis, it may also be used as a tool to gain analytic insights.

• The “scientific/experimental” approach may be somewhat messy and not exactly “efficient”: there may be dead ends or ambiguous conclusions. There may be digressions.

• There may be an inclination on the teacher’s part to spare the students by simply stating the concluding property and to reinforce it by examining sample cases. This will regrettably reinforce the notion that mathematics is simply a collection of procedures to be practised.

Can we offer a decent example as to why the effect is the opposite of what might be expected? Consider the statement “The number of children we have is equal to the number of my month of birth (March = 3)”. If we replace my month of birth with my wife’s month of birth (June : 3 more than mine), then our statement becomes “The number of children we have is 3 less than the number of my wife’s month of birth.”

Or, in mathematics parlance: given a function f:y = f(x) such that (a, b) is a solution, we can say that f maps a to b, or f(a) = b. If we replace x with g(x), then f will map g(x) to b provided g(x) = a. If g maps c to a [i.e. c = g-1(a)] then we can conclude that (c, b) will be a solution of the composite function F:y = f(g(x)).

Conclusion:

The foregoing presentation is intended to give some insight into an experimental approach to mathematical discovery: including its false leads, incorrect conclusions and general inefficiency. With the obvious exception of the study of Euclidian geometry, most students are presented with mathematics as a finished product. They spend most of their time mastering taught techniques to solve a well-defined set of problems. The skill required boils down to two steps: first identify the type of problem, then recall and apply the appropriate solution procedure. In contrast, an experimental approach requires that the student attempt to figure out what is happening, and thereby gain some understanding.

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1. GrafEq V1.12 produced all graphs, which were pasted into Word®.

-10 10

-10

10

x

y

Figure 18

Exercise: Given the graph of x2y2 = 64, using the methods above, sketch the graph of (1/x)2y2 = 64.

Fall 200740

Musimathics or Mathemusic: Investigating the Effects of Music on Spatial-Temporal Reasoning

Sarah Jimenez is enrolled in the Technology and Critical Thinking (TACT) elementary education program at the University of British Columbia.

East High—a school in Albuquerque, New Mexico—is brimming with jocks, science geeks, and drama freaks. East High’s cafeteria tables distinguish the typical high school social hierarchy. The tether that holds the labelled students to their cafeteria placements will only lengthen with time. Fortunately for a chance meeting of the basketball star with the president of the academic club, and a successful musical audition, the star-crossed lovers reconcile their differences, and the differences of all the other cliques in East High. In Hollywood, and in Disney’s TV hit High School Musical, it is possible for music to have the power to bridge the gaps between two worlds; but is that all the power that music has?

What appeals to researchers and educators is the effect that those hours upon hours of listening to the High School Musical soundtrack has had on students’ brains. In particular, researchers and educators are interested in the effect of music on spatial-temporal reasoning, and mathematical ability. A great deal has been written about the relationship between math and music since the time of ancient Greece. Math is engraved in music, from harmony and number theory to musical patterns and group theory (Benson, 2007). Mathematics has also been related to music in the areas of tuning, temperament, and acoustics (Johnson, 2003). Contemporary science explores the beneficial relationship between music and math, and as it is a relatively recent contribution to the body of knowledge, it is still open to doubt.

Rauscher, Shaw, and Ky (1993) sparked discussion among researchers about math and music that has lead to relevancy of the topic in the education field. Rauscher et al. investigated the connection between music and spatial task performance. The experiment tested three listening conditions—music condition, relaxation condition, and silent condition— and the effects on the brain. In the music condition, participants listened to 10 minutes of Mozart’s Sonata K488; in the relaxation condition, participants listened to 10 minutes of relaxation instruction; in the silent condition, participants sat in silence for 10 minutes. After the ten minutes, participants were given one of three abstract reasoning tests of equal significance for measuring abstract ability. The tests were taken from the Stanford-Binet Intelligence Scale. The results of the experiment showed that participants who listened to Mozart’s sonata scored 8-9 IQ points higher than the participants of the other conditions (Rauscher et al., 1993). Participants performed better on abstract/spatial reasoning tests after listening to Mozart. Since then, the connection between music and spatial-temporal reasoning has been dubbed “the Mozart effect.”

Norman Weinberger (1998) has also explored the beneficial aspects of making music versus listening to music. He cites a follow-up study done by Rauscher et al. (1993) that compared the results of three age-calibrated tests of preschoolers aged 3-4 before and after they were given keyboard training or computer lessons. The first two tests assessed spatial and temporal reasoning and the third test assessed spatial recognition. The preschoolers

Sarah Jimenez

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who were given keyboard training showed significant improvements in the first two tests (Weinberger, 1998). Since the children who were given computer lessons had no improvement in spatial/temporal assessment, it is hypothesized that musical training can benefit math where spatial and temporal reasoning is important.

Spatial and temporal reasoning that connects music to math is defined as “belonging to both space and time” (OED). It is the ability to visualize spatial patterns and manipulate them mentally. An example of this is the paper-folding test where participants are shown a sequence of folds in a piece of paper. A set of holes is then punched through the paper, and the participants must choose which of a set of unfolded papers with holes corresponds to what they have just witnessed (Wikipedia). Spatial-temporal reasoning allows a person to transform images and relate them mentally in space and time. Mathematical principles of number patterns, measurements, 3-D objects and 2-D shapes, and transformations require spatial and temporal reasoning.

The brain and the nervous system filter and process information in different ways that are relevant to both music and math (Campbell, 1983). A musician uses the cerebral cortex to make decisions and formulate logic to read and interpret the notes on a musical piece the same way that a person engaged in a math question will visualize a problem in order to solve it. In Emblems of the Mind: The Inner Life of Music and Mathematics, Edward Rothstein finds that musical composition and mathematical formula are both “genius in the very notation that has developed for giving representation to ideas that seem to lie beyond ordinary language” (Rothstein, 1995, p 17).

Physical evidence of brain differences in musicians and non-musicians is available from studies using fMRI: functional magnetic resonance imaging (Schmithorst & Holland, 1994). These neuro-imaging studies attempt to settle the ongoing debate about the Mozart effect by the physical differences between people with musical and non-musical background. The imaging shows that musicians have increased corpus callosum size, increased white matter organization in corpus callosum and decreased white matter organization in the corticospinal tract (Schmithorst & Holland, 1994). While performing mathematical operations, musicians showed significantly greater brain activation than non-musicians in the parts of the brain that were associated with processing “shape information and visual-perceptual semantic processing” (Schmithorst & Holland, 1994, pp. 194-195). This increased brain activity in musicians can be attributed to their increased proficiency of processing shape information, and their years of reading and interpreting musical notation.

Music and mathematics are two of the oldest disciplines in our history. The Mozart effect is a relatively new debate in academia. Many musical theories and structures have been supported by mathematical equations and rationales derived from Pythagoras, Galileo to the recent scholars like Babbitt and Clough (Johnson, 2003). There definitely exists an interrelationship with music and math, which requires further exploration. The implications have already affected mathematics education in the form of available resources such as lesson plans, commercialized teaching tools, and a student demographic that come with musical background because of the intent of the Mozart effect. While I doubt that High School Musical will have any effect on the future generation of math students, the high interest in the musical aspect of the show may prompt children to make music, and train and practise in the musical knowledge that explains many of the studies on spatial-temporal reasoning.

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Works Cited

Benson, D.J. (2007). Music: a mathematical offering. Cambridge, UK: Cambridge University Press.

Campbell, D.G. (1983). Introduction to the musical brain. St. Louis, Missouri: Magnamusic-Baton.

Johnson, T.A. (2003). Foundations of diatonic theory: a mathematically based approach to music fundamentals. Emeryville, CA: Key College Publishing.

Oxford English Dictionary Online. Spatio-temporal. Retrieved October 24, 2007.

Rauscher, F.H., Shaw, G.L., & Ky, C.N. (1993). Music and spatial task performance. Nature. 14 October 1993, 365, p. 611. Retrieved 21 October 2007 from http://www.nature.com/nature/journal/v365/n6447/pdf/365611a0.pdf

Rothstein, E. (1995). Emblems of the mind: the inner life of music and mathematics. New York, NY: Times Books.

Schmithorst, V.J. & Holland, S.K. (2004). The effects of musical training on the neural correlates of math processing: a functional magnetic resonance imaging study in humans. Neuroscience Letters. 16 January 2004, 354, 3, pp. 193-196.

Weinberger, N.M. (1998). The music in our minds. Educational Leadership, Nov 98, 56, 3, pp. 36-41. Retrieved 22 October 2007 from http://web.ebscohost.com/ehost/pdf?vid=3&hid=12&sid=f53e432b-dcae-4a48-8fab-744d50e2890a%40essionmgr7

Wikipedia. Spatial-temporal reasoning. Retrieved 21 October 2007 from http://en.wikipedia.org/wiki/Spatial_temporal_reasoning.

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A New Chess Set for Teaching Mathematical Chess

Frank Ho is a teacher at the Ho Math and Chess Learning Centre (http://www.math andchess.com).

If one thinks that chess consists of warriors or commanders and kings and queens battling in the field, then this notion does not really address the reason why chess pieces move in pattern-like directions. For example, a rook moves up and down or left and right and a bishop moves diagonally. Is chess a reflection of ancient war or is it an invention based on a mathematical principle? The author believes that chess was invented by using the concept of geometric symmetry. This conjecture is based on analyzing the moves of each chess piece and I thus conclude that chess was created from a mathematical point of view.

For a fair game, the positions of the chess pieces and the layout of the chessboard must be symmetric. Perhaps it is not coincidental that the playing field of chess is all about squares and that the Chinese character for “rice field” is also a 2 by 2 square. Given a fixed perimeter, the square has the largest area. The chessboard is an 8 by 8 tessellation of 64 squares. There are 4 lines of symmetry in each square and these 4 lines constitute the moves of the rook, queen, king, pawn and bishop. It makes sense that each chess piece moves along these lines.

How Chess Moves Originated

To play a symmetric game, the smallest board required is 5 by 5. I believe that the possible moves of each chess piece are originally intended to be 360 degrees of circular movement. For example, take a look at a 5 by 5 chessboard. (See figure 1.) If a chess piece is placed at c3, how many ways can this chess piece reach out to form the shape of a circle? In addition to a circle, a square shape could also be formed, depending on how the points are connected.

The first “easy” way would be to move down or up and left or right, from c3 so as to reach the limit of a circle. In this way, the moves of the rook are born. Its motion is called a translation or a slide in geometry. If you connect the 4 out-reached points with 4 straight lines, the shape is actually a square; however with contours it then forms a circle.

The second way of moving to form a circle is to move in the directions of two main diagonals. A circle is thus produced and the moves of a bishop are born. Arguably, the four points also make a square shape. This motion from c3 to each of the 4 diagonal points is also a double-slide. The bishop can move in 360 degrees.

Combining the above two ways of a rook and bishop moving, we have the most powerful of all chess pieces: the queen. This is the birth of a queen’s move. Finally, the king follows the moves of a queen, but can move only one square per turn.

In a 5 by 5 chessboard, we notice that all chess squares on each of 4 sides are covered by the moves of a rook and bishop except a2, a4, b1, b5, d1, d5, e2, and e4. So from an attacking or defending point of view, this is a problem: there are 8 squares that are not covered. This is the reason for the birth of another chess

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piece called the knight, which covers the 8 remaining squares by jumping. This perhaps is the reason why the knight jumps since it does not trace any squares in one straight line to reach any one of those eight unreachable squares.

By using up/down, left/right, diagonal, and diagonal jump moves, every square on a 5 by 5 chessboard is completely covered from c3, the central point.

Figure 1: A 5 by 5 Chessboard

This geometric view of chess moving in 360 degrees explains how all chess moves were originated. The move directions show clearly the relationship between geometric symmetry and chess.

Discovery the Key Linking Math and Chess

For me, the discovery that chess moves use the symmetry property of a square is the secret key that links math and chess. The difficult task when teaching children as young as four or five is that it understandably takes considerably more time for such young children to become familiar with how each piece should move. Without mastering chess moves, children cannot experience the joy of playing chess. Often it even becomes frustrating and discouraging for some children to pursue further. The main reason that children cannot master the moves of each chess piece quickly is that there is no clear relation between each chess figurine and its moves. For example, it does not seem to make sense to children why a castle-shaped rook should move left/right and up/down.

Based on my discovery, I created the Geometry Chess Symbol (GCS, patent pending) using the concept of geometry and also a new chess teaching set that is based on the Geometry Chess Symbol. The responses to using GCS to link math and chess and to the testing of this new chess set at Ho Math and Chess are very favorable. Children could start to play chess almost immediately after being shown that they just have to move each piece according the moves marked on that piece.

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Figure 2. Geometry Chess Symbol and the Ho Math and Chess Teaching Set

This incredible chess teaching set plays just like an ordinary chess set but offers the additional advantage that the move of each chess piece is clearly marked on its flat surface. This makes chess not only easier to learn but also fun for children. It is a “what you see is what you move” chess set. Also, no more spills or bumps for small hands when moving pieces.

The geometric concepts of lines, line segments, transformations, and intersections are used to design this revolutionary set. With its starting point in the middle, each chess piece is a great pattern tool to enhance children’s skills in observation, orientation, and decision-making. Children can picture themselves at the intersection of the lines and then move according to the directions provided by the arrows on the line segments. All move directions are consistent with the instructions of any typical chess book. For example, the knight moves in an L shape, starting with 2 squares like the rook and then one square perpendicular to that.

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An additional advantage of this pocket-sized, flat surfaced set is to play Blind Chess, which uses a very simple move rule and is fun to play.

Blind ChessSince the Ho Math and Chess Teaching pieces have flat surfaces and uniform square size, the pieces cannot be identified and are indistinguishable from each other when they are turned face down.

This special feature allows children to play a special game called Blind Chess or Half Chess (also called Banqi in Chinese). Blind Chess is very easy and fun to play.

The BoardBlind Chess is played by two players on a board (4 by 8) that is half the size of the normal chessboard.

Starting the GameThe 32 pieces are shuffled and then each of them is randomly placed face down on a square. The first player turns over a piece and the colour of the first piece uncovered will be the side of the first player.

Moving a Piece There are 3 kinds of moves. A player may turn a piece face-up, move a piece, or capture an opponent’s piece. A player may move only face-up pieces of his or her own colour. Unlike normal chess moving rules, there is only one rule to move pieces in Blind Chess: a piece can move only one square up, down, left, or right: i.e., all pieces move like a rook, but only one square at a time.

Capturing an Opponent’s PiecesTo capture, a face-up piece may move only to a square occupied by an opponent’s face-up piece. All pieces (black or white) are ranked according to the following hierarchy and the capturing rule is strictly according to this defined hierarchy.

• The king has the highest rank and can capture all the opponent’s pieces other than a pawn.

• The queen can capture all the opponent’s pieces other than the king.

• A rook can capture all the opponent’s pieces other than the king or queen.

• A bishop can capture only the opponent’s knight or pawn.

• A knight can capture only the opponent’s pawn.

• A pawn has the lowest rank but can capture the opponent’s king.

How the game ends The game ends when a player cannot make a move or until all pieces are captured. If the game is forced in an endless cycle of moves then it is a draw.

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The Predictive Ability of the FSA on Provincial Mathematics 10 Examinations

Brad Epp teaches at South Kamloops Secondary School. This article is an extract from his recent Master of Arts thesis at the University of Victoria. Leslee Francis-Pelton is an associate professor at the University of Victoria, in the Department of Education.

Student performance on the recently implemented Provincial Mathematics 10 examinations concerns mathematics teachers. Could results from the Grade 7 Foundation Skills Assessment (FSA) be used to implement early strategies for students’ improvement in secondary mathematics as well as to place students into appropriate secondary curriculum pathways?

Design

My research attempted to use prior performance on Grade 7 FSA tests to predict student performance on standardized Grade 10 math examinations. Aside from student scores on the numeracy and reading comprehension portions of the Grade 7 FSA, I considered other factors as independent variables: gender, aboriginal status, English as a second language status, and school size. The dependent variables for this study were the student’s best score on the Grade 10 provincial mathematics examination and the curriculum pathway in which the student was enrolled for Grade 10: Applications of Mathematics; Essentials of Mathematics; or Principles of Mathematics.

I used scatter plots, descriptive statistics, analysis of variance, and linear regression to describe and quantify the relationships between the independent and dependent variables.

The primary data was collected from the Ministry of Education through Edudata Canada (http://www.edudata.educ.ubc.ca/); thus, student personal information remained anonymous. I obtained the following information for each student.

• FSA scores at Grade 7 for two domains (reading and numeracy)• Grade 10 mathematics best examination score• the mathematics course enrolled in at Grade 10• gender• whether an ESL student or not• whether an aboriginal student or not• the total number of Grade 10 students enrolled in the student’s school • the identification number of the student’s school

Project Participants

Eighty three percent (48391) of all Grade 10 students wrote a provincial mathematics 10 examination in the 2004/05 school year. Six percent (3440) wrote Applications of Mathematics 10 (AMA 10), 13% (7740) wrote Essentials of Mathematics 10 (EMA 10) and 64% (37207) wrote Principles of Mathematics 10 (PMA 10). The

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remaining 17% (9911) did not write the examination. This may be because these students did not complete the course objectives, were not in a Grade 10 mathematics course, or were on an individualized education program (IEP).

The data provided by the Ministry of Education had some anomalies and missing values. First, since my research concerned the ability of the FSA subtests to predict performance on the Grade 10 examinations, students who did not participate in either portion of the FSA were not considered in this study. Second, since a low number of students results in unstable estimates of student performance, students in schools with less than five Grade 10 students were eliminated from consideration. With these restrictions, the project participants therefore included 27 292 students who wrote the PMA 10 examination, 5052 students who wrote the EMA 10 examination, and 2662 students who wrote the AMA 10 examination for a total of 35 006 students. This total represented 84% of the initial data set.

Instruments

The FSA at Grade 7 takes approximately 4.5 hours to complete. It is comprised of reading comprehension, numeracy, and writing skills. The numeracy portions of the assessment are written as two subtests, each taking approximately 45 minutes to complete. In the Grade 7 numeracy subtests, students are asked to answer 40 multiple-choice questions and two written response questions. The reading comprehension portions of the assessment are written as two subtests, each taking approximately 30 minutes to complete. In the Grade 7 reading comprehension subtests, students are asked to answer 40 multiple choice and 3 written response questions. The questions are based upon a student’s interpretation of three pieces of writing. The assessment was written in May 2002. Students wrote each section of the FSA only once; there is no opportunity for rewrites. The examination has strong content validity because questions are created from a specific table of specifications that is a subset of the provincial learning outcomes from Grades 4 to 7. The questions are also developed by teams of teachers who are experienced in teaching the grade levels being assessed. The examinations are then field-tested and revisions made.

Each Grade 10 examination covers the respective curriculum of its pathway. There are some curriculum organizers omitted from the examination because it is felt that these organizers are better assessed by classroom teachers. Even though the content of each of the three Grade 10 examinations differs, the methods of questioning do not. Four types of questions are asked: traditional multiple-choice (one correct answer, and three distracters); numerical answer (students write in the numerical answer and a machine scores it); matching questions (several different stems covering a broad curricular area are matched with several distracters); and true/false questions. Students complete 60 questions over a period of two and one-half hours. Students have the opportunity to write the examination at four different times during the school year: January, April, June, or August within a three-year period upon completion of the course. If a student dislikes her or his results on the examination, she or he has the opportunity to rewrite the examination once. The higher mark of the two sessions becomes the student’s best examination mark. If a student dislikes the second mark, the student has the option of repeating the course. This examination has content validity because the examinations are based upon the provincial curriculum developed through a specifications document. Teams of teachers create questions, proof questions, and the exams are then trial written by experienced mathematics 10 teachers.

Data Collection and Analysis

Data were requested from Edudata Canada in January 2006. Scatter plots were created and correlations calculated to help visualize the relationship, if any, between students’ scores on the Grade 10 mathematics assessments and their FSA results.

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Results

Three sections present the results of the data analysis. First, I examined the relationship between student performance on the reading comprehension portion of the FSA and the numeracy portion. Second, I conducted a preliminary view of the Grade 10 examination performance on all three examinations versus the two FSA subtests. Third, regression analysis was used to predict performance on the Grade 10 examinations.

1) The Relationship between FSA SubsetsBefore analyzing the data with the independent and dependent variables, I calculated the correlation between students’ results on the reading comprehension portion of the FSA and the numeracy portion. If there was no significant relationship on the Grade 7 results, it was likely that the reading comprehension would not be significant on the Grade 10 exam. The correlation between students’ results on the FSA reading comprehension and the FSA numeracy was 0.592 and was significant at the 0.001 level.

Since there was a correlation between the reading comprehension and numeracy portions of the FSA, it was likely that a relationship between the Grade 10 mathematics examination results and the reading comprehension portion of the FSA existed.

2) Correlation Coefficients Obtained from ScatterplotsA cursory view needs to be taken for all three mathematics examinations versus the respective FSA subtests. In all cases a significant relationship existed between students’ FSA results and best exam performances. However, the degree of the relationships appeared to be small and have a positive slope. In all cases, the variation accounted for between the FSA subtests and the best exam score was a maximum of 27.0%. Following are the correlation coefficients.

• Principles of Mathematics 10 best examination performance vs. numeracy performance on the FSA (n = 27 292).

The correlation between these two variables was 0.516 and was significant at the 0.001 level.

• Principles of Mathematics 10 best examination performance vs. reading performance on the FSA (n = 27 292).

When comparing PMA 10 examination performance and the reading subtest, there appears to be a small positive linear relationship. The correlation between these two variables was 0.368 and was significant at the 0.001 level.

• Essentials of Mathematics 10 best examination performance vs. numeracy performance on the FSA (n = 5 052).

The correlation coefficient was 0.329, significant at the 0.001 level.

• Essentials of Mathematics 10 best examination performance vs. reading performance on the FSA (n = 5 052).

The correlation coefficient was 0.341, significant at the 0.001 level.

• Applications of Mathematics 10 best examination performance vs. numeracy performance on the FSA (n = 2 662).

The correlation coefficient was 0.405, significant at the 0.001 level.

• Applications of Mathematics 10 best examination performance vs. reading performance on the FSA (n = 2 662).

The correlation coefficient was 0.379, significant at the 0.001 level.

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In summary, the student FSA numeracy results had a small, positive and significant correlation with performance on the provincial mathematics examinations. There were indeed small, positive, and significant correlations between the FSA reading results and the three provincial mathematics 10 examinations. The correlations were greater for the numeracy results. This was to be expected, since both assessments (FSA numeracy and provincial mathematics 10 examinations) were measuring the students’ ability to answer questions on a mathematics assessment, albeit at different grade levels. The most pronounced linear trend appeared in the scatter plot of the best examination results for the PMA 10 students versus the FSA numeracy results. The rest of the scatter plots showed a linear trend, but not as clearly as the PMA 10 versus numeracy.

3) Linear Regression

Linear regression is another method of predicting best examination performance for each of the three mathematics pathways. The students’ best examination was regressed versus FSA reading, FSA numeracy, ESL status, aboriginal status, and school size. In all of the cases, the models were significant. (See tables 1, 2, and 3.)

Each student’s reading score and numeracy score were related with the student’s best examination performance. In all models, the numeracy score had the strongest (greatest) effect on examination performance. Gender had a small but negative effect on the examination results for AMA 10 and EMA 10 students, as well as a small positive effect on PMA 10 examination results. The coefficients of the aboriginal status variable were negative in all three models. The coefficient of the ESL variable in the EMA 10 regression was negative, but was positive in the PMA 10 models. The school size coefficient was small and negative in the EMA 10 regression; it was similarly small and positive in the PMA 10 regression.

Table 1.

Regression Analysis for Predicting Best Examination Performance for PMA 10 Students

Unstandardized Coefficients

Standardized Coefficients T Sig.

Independent Variables B Std.

Error Beta

(Constant) 54.515 .843 64.692 .000

FSA Reading 2.794 .104 .165 26.877 .000

FSA Numeracy 7.253 .102 .430 70.960 .000

Gender 1.822 .144 .064 12.673 .000

Grade 10 school size .006 .001 .040 7.879 .000

Aboriginal status -2.869 .337 -.043 -8.519 .000

ESL status 6.081 .175 .181 34.739 .000

The model accounted for 32.0% of the variation in the data (F(6,27285) = 2142.345, p<0.01).

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Table 2.

Regression Analysis for Predicting Best Examination Performance for EMA 10 Students.

Unstandardized Coefficients

Standardized Coefficients T Sig.

Independent Variables B Std.

Error Beta

(Constant) 77.513 1.428 54.286 .000

FSA Reading 3.608 .234 .230 15.403 .000

FSA Numeracy 4.097 .295 .206 13.892 .000

Gender -1.709 .317 -.070 -5.387 .000

Aboriginal status -1.314 .455 -.037 -2.887 .004

ESL status -3.135 .440 -.093 -7.133 .000

Grade 10 school size -.008 .002 -.062 -4.724 .000

The model accounted for 16.9% of the variation in the data (F(6, 5045)=170.706, p<0.01).

Table 3.

Regression Analysis for Predicting Best Examination Performance for AMA 10 Students.

Unstandardized Coefficients

Standardized Coefficients T Sig.

Independent Variables B Std.

Error Beta

(Constant) 70.107 1.599 43.832 .000

FSA Reading 4.411 .317 .266 13.903 .000

FSA Numeracy 5.293 .375 .271 14.110 .000

Gender -3.305 .445 -.128 -7.422 .000

Aboriginal status -1.566 .754 -.035 -2.076 .038

The model accounted for 23.1% of the variation in the data (F(4, 2659)=199.652, p<0.01).

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Recommendations

Recommendations are made to the following audiences: education professionals (teachers, principals, counselors, district officials), parents, Ministry of Education officials, and researchers. Education professionals and parents need to be aware of the implications of this research. In addition, Ministry of Education officials need to be made aware of students’ reading ability possibly affecting student performance, and a suggested change to reporting student performance on the Grade 10 examinations.

The results of this study have implications for education professionals and parents. The coefficients of the variables determine student performance on Grade 10 examinations and predict placement into the three Grade 10 courses. Each of the independent variables predicts student performance on the Grade 10 examinations to differing degrees. All models show the importance of the FSA results in predicting future examination performance in PMA 10, AMA 10, and EMA 10. The magnitude of the coefficients of these variables illustrates that these results cannot be discounted. Students who score poorly on this assessment should receive interventions in both mathematics and reading comprehension in their Grade 8 and 9 years to help correct deficiencies in their prior learning. By investigating their students’ performance on the FSA, teachers can use the results to help plan for their students’ learning by discovering the likelihood of their students’ success in Grade 10. Parents can be more informed and advocate for their children to receive support to make the likelihood of success greater. Learning assistance programs can be developed to intervene to help students. School counselors and principals can help students by creating, supporting and placing students into the appropriate recovery program.

In addition to the recommendations to education professionals and parents, Ministry of Education officials have some considerations that are exclusive to their domain. When constructing future provincial examinations for mathematics, the Ministry of Education needs to be aware of students’ proficiency in English. It might be the case that students in the EMA 10 and AMA 10 programs do not read as proficiently as their peers in PMA 10. If the purpose of the mathematics examination is to measure students’ proficiency in that subject area and not their reading ability, care must be taken so that the examinations are actually measuring students’ mathematics performance and not their reading ability. When reporting student performance to researchers, the ministry should consider providing student scores as a raw score or to include some questions that will discriminate between the students who score above 80%. This might improve the correlation between the student results from the numeracy subtest and the PMA 10 examination. The negative coefficient of the aboriginal status variable should be a systemic concern. Education officials need to determine the causes of this and look for possible solutions.

Limitations and Suggestions for Further Research

Attempting to predict student performance two years into the future is fraught with difficulty. There are initial concerns with the low correlation between the two FSA sub-tests. While not measuring the same concepts, “strong” students will generally perform well on both assessments and “weaker” students will not.

Each of the regression models is significant in predicting student performance on the Grade 10 examinations; however, the models do not explain a great deal of the variance in the results. When examining results from the regression analysis, the variance explained by the models ranges from a low of 16.9% for the EMA 10 model to 23.1% for the AMA 10 model and to a high of 32% for PMA 10 model.

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Why You Should Choose Math in Secondary School

Dr. Espen Andersen is an Associate Professor at the Norwegian School of Management and Associate Editor of Ubiquity.

[The following article was written for Aftenposten, a large Norwegian newspaper. The article encourages students to choose mathematics as a major subject in secondary school: not just in preparation for higher education but because having math up to maximum secondary school level is important in all walks of life. Note: This translation is slightly changed to have meaning outside a Norwegian context.]

A recurring problem in most rich societies is that students in general do not take enough math – despite high availability of relatively well-paid jobs in fields that demand math, such as engineering, statistics, teaching and technology. Students see math as hard, boring and irrelevant, and do not respond (at least not sufficiently) to motivational factors such as easier admission to higher education or interesting and important work.

It seems to me we need to be much more direct in our attempts to get students to learn hard sciences in general and math in particular. Hence, addressed to current and future secondary school students, here are 12 reasons to choose lots of mathematics in secondary school.

1) Choose math because it makes you smarter. Math is to learning what endurance and strength training is to sports: the basis that enables you to excel in

the specialty of your choice. You cannot become a major sports star without being strong and having good cardiovascular ability. You cannot become a star within your job or excel in your profession unless you can think smart and critically – and math will help you do that.

2) Choose math because you will make more money. Winners of American Idol and other “celebrities” may make money, but only a tiny number of people

have enough celebrity to make money, and most of them get stale after a few years. Then it is back to school, or to less rewarding careers (“Would you like fries with that?”). If you skip auditions and the sports channels and instead do your homework – especially math – you can go on to an education that will get you a well-paid job. You can earn much more than what most pop singers and sports stars make: perhaps not right away, but certainly if you look at averages and calculate it over a lifetime.

3) Choose math because you will lose less money. When hordes of idiots throw their money at pyramid schemes, it is partially because they don’t know

enough math. Specifically, if you know a little bit about statistics and interest calculations, you can look through economic lies and wishful thinking. With some knowledge of hard sciences you will probably feel better, too, because you will avoid spending your money and your hopes on crystals, magnets and other swindles – simply because you know they don’t work.

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4) Choose math to get an easier time at college and university. Yes, it is hard work to learn math properly while in high school. But when it is time for college or

university, you can skip reading pages and pages of boring, over-explaining college texts. Instead, you can look at a chart or a formula, and understand how things relate to each other. Math is a language, shorter and more effective than other languages. If you know math, you can work smarter, not harder.

5) Choose math because you will live in a global world. In a global world, you will compete for the interesting jobs against people from the whole world. The

smart kids in Eastern Europe, India and China regard math and other “hard” sciences as a ticket out of poverty and social degradation. Why not do as they do: get knowledge that makes you viable all over the world, not just in your home country?

6) Choose math because you will live in a world of constant change.

More and more, new technology and new ways of doing things are changing daily life. If you have learned math, you can learn how and why things work, and avoid scraping by through your career, supported by Post-It Notes and Help files – scared to death of accidentally pressing the wrong key and running into something unfamiliar.

7) Choose math because it doesn’t close any doors. If you don’t choose math in high school, you close the door to interesting studies and careers. You might

not think those options interesting now, but what if you change your mind? Besides, math is most easily learned as a young person, whereas social sciences, history, art and philosophy benefit from a little maturing – and some math.

8) Choose math because it is interesting in itself. Too many people (including teachers) will tell you that math is hard and boring. But what do they know?

You don’t ask your grandmother what kind of game-playing machine you should get, and you don’t ask your parents for help in sending a text message. Why ask a teacher – who perhaps got a C in basic math and still made it through to his or her teaching certificate – whether math is hard? If you do the work and stick it out, you will find that mathematics is fun, exciting, and intellectually elegant.

9) Choose math because you will meet it more and more in the future.

Mathematics becomes more and more important in all areas of work and scholarship. Future journalists and politicians will talk less and analyze more. Future police officers and military personnel will use more and more complicated technology. Future nurses and teachers will have to relate to numbers and technology every day. Future car mechanics and carpenters will use chip-optimization and stress analysis as much as monkey wrenches and hammers. There will be more math at work, so you will need more math at school.

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10) Choose math so you can get through, not just into college. If you cherry-pick the easy stuff in high school, you might come through with a certificate that makes

you eligible for a college education. Having a piece of paper is nice, but don’t for a second think this makes you ready for college. You will notice this as soon as you enter college and have to take remedial math programs, with ensuing stress and difficulty, just to have any kind of idea what the professor is talking about.

11) Choose math because it is creative.* Many think that mathematics has to do with only logical deduction and that somehow it is in opposition

to creativity. The truth is that math can be a supremely creative force if only the knowledge is used right, not least as a tool for problem solving during your career. A good knowledge of math in combination with other knowledge makes you more creative than others.

12) Choose math because it is cool. You have permission to be smart, you have permission to do what your peers do not. Choose math so

you don’t have to, for the rest of your life, talk about how math is “hard” or “cold.” Choose math so you don’t have to joke away your inability to do simple calculations or your lack of understanding of what you are doing. Besides, math will get you a job in the cool companies, those that need brains.

You don’t have to become a mathematician (or an engineer) because you choose math in secondary school. But choosing math helps if you want to be smart, think critically, understand how and why things relate to each other, and to argue effectively and convincingly.

Math is a sharp knife for cutting through thorny problems. If you want a sharp knife in your mental tool chest - choose mathematics!

* This point was added by Jon Holtan, a mathematician who works with the insurance company If.

Source: Ubiquity Volume 7, Issue 11 (March 21 - March 27, 2006) http://www.acm.org/ubiquity

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Metacognition in Mathematics Education

David Tambellini is a retired mathematics teacher living at Christina Lake.

Metacognition may be defined as the critical contemplation of one’s own cognition. To be metacognitive is to think about one’s own thinking, to know about one’s own knowledge, or as Gattegno (1974) puts it, to be “aware of one’s own awareness” (p. 80). Schoenfeld outlines the following three aspects of metacognition.

a) Your knowledge about your own thought processes. How accurate are you in describing your own thinking?

b) Control, or self-regulation. How well do you keep track of what you’re doing when (for example) you’re solving problems, and how well (if at all) do you use the input from these observations to guide your problem-solving actions?

c) Beliefs and intuitions. What ideas about mathematics do you bring to your work in mathematics, and how does that shape the way that you do mathematics?

(1987, p.190)

Thus metacognition refers to one’s self-knowledge, self-orchestration, and beliefs as one engages in cognitive activity. Finally, a further aspect of the concept that transcends all the above (and which is a theme of this article) is one’s ability to reflect on and to understand the utility of self-awareness itself.

Metacognitive strategies are processes, tactics or rules that one uses to plan and to monitor one’s own cognitive progress in a task. As Flavell (1976) succinctly writes, “cognitive strategies are invoked to make cognitive progress, metacognitive strategies are invoked to monitor it” (p. 232).

In mathematics education, metacognition involves one’s self-awareness as a learner of mathematics. Such awareness could include for example, one’s mathematical strengths or weaknesses, or one’s preferences for various mathematical topics and learning modes. As mathematical behaviour, metacognition refers to monitoring one’s progress in a task, and to making strategic decisions based on one’s self-observations. It involves the learner’s beliefs about mathematics and mathematics learning, and the effects those beliefs have upon mathematical achievement. Last, metacognition in math entails the learner’s appreciation not only of the value of heuristics and cognitive strategies, but also of the value of metacognitive strategies themselves.

Many pupils fail to use cognitive and metacognitive strategies as spontaneously, frequently, and efficiently as their peers. For some students, the absence of metacognitive behaviours can mean failure just as the presence of such processes may promote good problem solving (Schoenfeld, 1983). Some students possess the necessary talent to solve a problem, but nonetheless fail in their attempts. In these cases, their failure can stem from attitudinal problems or from metacognitive ineptitude (Lester, Garofalo & Kroll, 1989).

David Tambellini

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To explain this lack of metacognitive behaviours, some researchers posit actual strategic disabilities or deficits. However, others also consider certain students, especially those with learning disabilities, as “passive learners” who are unwilling or unmotivated to behave in a metacognitive (or cognitive) way (Cherkes-Julkowski, 1985; Torgesen, 1977). This trait stands in direct contrast to the constructivist viewpoint that “implies a way of teaching that acknowledges learners as active knowers” (Noddings, 1990, p. 10).

Some underachieving students become inactive problem solvers quite easily. With relatively few academic successes of any kind, they show little or no faith that any strategic action on their part will result in a solution to a problem. These pupils may develop a “learned helplessness” that causes them to rely on others for help or else to give up. From their point of view, heuristics and metacognitive plans are “for the brainy guys in the class.”

There is abundant support for the recognition of metacognition as an integral part of mathematics education (Crosswhite, 1987). For example, Surtaam (2004) maintains that “the attainment of problem-solving ability is dependent on five inter-related components: attitudes, skills, concepts, processes and metacognition” (p. 8). Gattegno (1974) posits that educating students means striving to make them aware of certain powers they already possess that they can use in the same way as mathematicians do.

Among the many different functionings of man’s mind we can see two which go to make the mathematician and these are the awarenesses, first, of relationships as such, and second, of the dynamics of the mind itself as it is involved in any functioning. Knowing this, teachers can serve their students best by bringing them to the state of watchfulness in which they perceive how one becomes aware of relationships and of the dynamics of the mind.

(p. 81)

Likewise, Dawson (1992) states that

… being aware of what one is doing is the only way in which learning will occur. The additional step of being aware of oneself as one is learning is a challenge that requires working with students to help them develop tools for accomplishing this task.

(p. 24)

Garofalo (1987) maintains that if we want our students to become active learners and doers of mathematics rather than mere knowers of mathematical facts and procedures, we must design our instruction to help develop their metacognition. Rohwer and Thomas (1989) call for instruction that will explicitly aid pupils to acquire the metacognitive strategies necessary for planning and monitoring during problem solving.

Research literature from the learning disabilities field provides ample evidence that metacognitive strategies in mathematics can and should be taught to special students. First, pupils can be trained to use and maintain metacognitive skills (Borkowski et al., 1989; Bos, 1988; Garofalo & Lester, 1985). Second, there is a positive relationship between success in mathematics and metacognitive strategy instruction (Garofalo & Lester, 1985; Loper, Hallahan, & Ianna, 1982; Montague, 1992). Training in metacognitive skills leads to a decrease in impulsivity and distractibility in students with behavioural problems (Davis & Hajicek, 1985; Frith & Armstrong, 1985) and to increases in on-task behaviour (Hallahan, 1987; Hallahan & Lloyd, 1987; Snider, 1987). Using metacognitive strategies can have positive effects on the self-esteem of students (Borkowski, 1992; Labercane & Battle, 1987). Indeed, the positive value of metacognitive training for students with learning disabilities appears to be unequivocal.

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In regard to these successes, Garofalo and Lester (1985) offer the following plea.

As mathematics educators and teachers, not only should we incorporate such metacognitive supplements in our efforts to train students to be proficient in applying algorithms and heuristics, but we should also help our students adopt a metacognitive posture toward mathematical performance in general.

(p. 173)

To promote metacognition, teachers might adhere to the following precepts derived from research in both mathematics education and learning disabilities. Initially, the instructor could introduce tasks and strategies that result in success for the students, thereby inducing “the belief in the general utility of being strategic” (Borkowski et al., 1989, p. 59). It is also beneficial to give praise (or marks) for all attempted solutions (rather than just for correct answers) so the students will come to appreciate that even a failed or dead-end cognitive strategy can add positively to his/her metacognitive repertoire. Self-expression can be encouraged by having each learner complete questionnaires or written response logs about both negative and positive mathematical experiences. In sum, as Wilson and Clarke (2004) comment, “the promotion of metacognition within the curriculum must start with its legitimization as a topic of classroom conversation” (p. 44).

The teacher can also be a co-problem solver with students. This “team effort” approach will help actively involve socially withdrawn students and help to eliminate low attainers’ negative view of “teacher as omnipotent possessor of all knowledge.” Then, the teacher can demonstrate metacognitive problem solving as she/he teaches, posing questions such as “Have we seen this situation before?” and “How well do you feel this approach working so far?” as examples of metacognitive behaviour.

The effective promotion of metacognition might require attitudinal changes on the teacher’s part. Thirty-five years later, I still recall my beginner’s trepidation whenever the superintendent or principal visited my classroom. My main fears focused on the possibility that this particular class of all classes would become unruly and/or confused about the lesson of the day. However, in contrast to the “avoid bewilderment at all costs” maxim of years past, there is today a new perspective on students’ befuddlement in mathematical tasks. John Mason (1992) speaks of how a “beautiful confusion” in a student’s mathematical thought can provide opportunities for group learning. In other words, there is relatively little that the mathematics class can extract from the neat situation when a student gives a correct answer, or professes to understand a concept. But there is pedagogical value in an individual’s ambiguity or consternation. As Mason and Davis (1991) suggest, something can be learned from being stuck.

If and when you get stuck, acknowledge that fact by writing STUCK!, (and if you recognise negative feelings such as anger or frustration arising inside you, put that into the writing of the word STUCK!). Then accept the fact that being stuck is an honourable state, a state from which much can be learned about yourself, and ultimately, about helping pupils. Only you can decide when to take a break, when to ponder a question as you go to sleep and upon waking, and when to conclude that you have made as much progress as you are likely to… .

(p. 5)

The above comments are examples of metacognitive strategies that teachers can introduce to failure-prone students who are attempting to do mathematics.

For the development of self-regulatory skills, Schoenfeld (1987) advocates that students learn mathematics in small groups. He maintains that such cooperative learning emulates the metacognitive activity that occurs in actual collaborative communities of mathematicians. Schoenfeld talks of a “society of mind” in which metacognition comes alive naturally through the dialogues between active learners.

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Finally, metacognitive expertise develops through extensive practice and experience. After all, one learns to read well by reading a lot, and one learns to write expertly by writing a lot. The difficult task for teachers is not to provide extensive metacognitive experience. Rather, it is to provide the meaningful mathematical context for such experiences to occur. Students cannot be expected to use metacognitive strategies about, for example, long division or irrelevant word problems. As Lester, Garofalo and Kroll (1989) aptly put it, “teachers who expect their students to be metacognitive must insure that their students have something to be metacognitive about” (p. 86).

Beliefs and mathematics education

Beliefs, the third aspect of metacognition, have as strong an influence on mathematical behaviour as do self-awareness and self-orchestration. This point was driven home to me after hearing an alternate school student (Jag) reveal his beliefs about the nature of mathematics learning:

The teacher teaches you at the beginning of the class, and gives you a bunch of questions to practice on, and at the end of the chapter gives you a test. That’s the way I’ve been doing it since grade school. That’s the way you do math.

He goes on to explain how he began to fail mathematics.

About half way through Grade 10, I started looking in the back of the book and copying the answers from the back, ’cause it would take too long, and I would just copy the answers. And that was why I didn’t understand what I was doing. I just started failing tests… . Sometimes I wouldn’t even go to math class because I didn’t understand. I thought there’s no point in going to math. So I’d just walk out to the store or something. I don’t know...I didn’t understand it. If didn’t copy the answers out of the back of my book I’d get in trouble, so I’d just, I’d say forget it and go.

Jag believed that school mathematics meant listening to a teacher, practising questions from the text, and writing chapter tests. For him, Mathematics 10 involved getting the correct answers to homework exercises. This last belief was so strong, he considered it sufficient to placate the teacher with a semblance of completed homework. However, copying from the answer key precluded conceptual understanding, which led to test failures, frustration, and to the inevitable truancies for which he was eventually suspended. Although aware that he lacked the understanding necessary to pass exams, he nonetheless puts his beliefs into action, or rather inaction.

It is not difficult to ascertain the origins of Jag’s beliefs: he was merely following the rules of the game called Mathematics 10. In other words, his beliefs about mathematics were derived from the current curricular structure that gives individual homework and seatwork such prominence.

There is a strong consensus amongst mathematics educators about the nature of many students’ beliefs towards mathematics. (Baroody & Ginsberg, 1990; Borasi, 1992; Lester, Garofalo & Kroll, 1989; Schoenfeld, 1992; Schoenfeld, 1989; Silver, 1987; Silver, 1982). For instance, students can have mistaken beliefs about mathematics such as the following.

• Mathematics is mostly memorization.

• There is usually only one way to solve any mathematics problem.

• If one really understands the mathematics, then problems should only take a few minutes.

• Mathematics is based entirely on rules.

• Being stuck in mathematics is ignominious and means one is stupid.

Unfortunately, as Schoenfeld (1989) comments, “they practice what they claim to believe” (p. 349).

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Many students also have conflicting beliefs about mathematics and mathematics learning. For instance, using a questionnaire adapted from Borasi (1992), I asked my students to rate different statements as either DEFINITELY TRUE, SORT OF TRUE, NOT VERY TRUE, or NOT TRUE AT ALL. One student, Derek, qualified the statement “There is always a rule to follow in solving mathematical problems” as definitely true, while he rated the declaration “Every mathematical question has only one right answer” as not true at all. When I pointed out how his thinking seemed contradictory to me, Derek replied, “I don’t personally see any conflict.” Likewise, Schoenfeld (1989) describes students with apparently contradictory beliefs.

Despite their assertions that mathematics helps one to think logically and that one can be creative in mathematics, they claim that mathematics is best learned by memorization. (p. 348)

For some students, lack of mathematical sophistication may account for their inability to see such conflicts in their beliefs. In other words, they may not have the cognitive maturity or competence to discuss the complex philosophical underpinnings of mathematics. For others, claims about creativity and discovery in mathematics might simply be regurgitations of the favorable rhetoric they hear from “progressive” mathematics teachers. In Derek’s case, I suspect this latter reason.

There is, however, virtual agreement amongst educators as to the main cause of beliefs such as those listed above (Schoenfeld, 1992). As McLeod (1989) comments, “There is nothing wrong with the students’ mechanism for developing beliefs about mathematics” (p. 247). Indeed, Borasi (1992) points out that such beliefs are “quite justified by the mathematical experiences students are likely to have had during their years of schooling” (p. 208). In other words, while they are not true reflections of mathematics, such beliefs are often accurate descriptions of the mathematics offered in schools. Schoenfeld (1987) sums up the situation well.

As a result of their instruction, many students develop some beliefs about what math is all about that are just plain wrong – and those beliefs have a strong negative effect on their mathematical behaviour. (p. 195)

The effects of such erroneous convictions are legion. For instance, students will not attempt to understand mathematics that they believe is only to be memorized. They will forego strategy monitoring and self-regulation if they believe that there is only one way to solve a problem. They will quickly give up on problems that they believe are typically solvable in a few minutes. Students’ mathematical inquiries will be limited to the search for rules that they believe govern all mathematical situations. Given such negative repercussions, educators cannot disregard such pre-existing beliefs.

Research findings by Lee and Wheeler (1986) provide a good example of the effects of beliefs on the behaviour of algebra students. These researchers discovered that many students rarely turn spontaneously to algebra to solve problems, even when judged capable of performing the necessary algebraic manipulations. It was found that these students attributed an extremely low status to algebra; in particular, they saw algebra “as an irrelevant activity that for some mysterious reason seems to please teachers and researchers” (p. 101). Thus, faulty belief structures can inhibit desirable metacognitive behaviours (e.g., strategy choice) as well as cognitive ones.

What steps can be taken to promote more accurate and productive belief systems? First, it should be appreciated that beliefs, unlike emotions, are mainly cognitive in nature, and build up slowly over long periods of time (McLeod, 1989). They are perhaps more difficult to alter than students’ faulty cognitions or weak mathematical understandings. As well, beliefs have a strong affective component, and thus rooted in deep emotional layers, they may be extra resilient to modification. Finally, as Borasi (1992) warns, shifts in beliefs are dependent upon an institution’s paying more than mere lip service to new educational approaches.

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As an important first step, Borasi (1992) suggests that teachers encourage students “to reflect on their beliefs... to engage them in activities that require them to state what they think and feel about mathematics” (p. 209). In this way, what Silver (1987) calls the “hidden curriculum” of deleterious beliefs can be brought out in the open.

Finally, true credal change will come only after modifications to the source of those student beliefs: the mathematical content and methodologies of our present curricula. In the words of Silver (1987),

[i]n designing mathematics curricula or planning instructional activities, we need to be mindful that students will integrate their experience with the activity, unit, or course that we are preparing with their prior experiences to form or to modify attitudes toward and beliefs about mathematics and mathematical problem solving. Let us teach toward this hidden curriculum to allow our students to develop attitudes and beliefs that reflect a view of mathematics as vibrant, challenging, creative, interesting, and constructive. (p. 57)

Indeed, let’s give our students something to believe in!

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Bibliography

Baroody, A., & Ginsburg, H. (1990). Children’s mathematical learning: A cognitive view. In R. Davis, C. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics. (pp. 51-64.). Reston: National Council of Teachers of Mathematics.

Borasi, R. (1992). Learning mathematics through inquiry. Portsmouth NH: Heinemann.

Borkowski, J. (1992). Metacognitive theory: a framework for teaching literacy, writing, and math skills. Journal of Learning Disabilities, 25(4), 253-257.

Borkowski, J., Estrada, M., Milstead, M., & Hale, C. (1989). General problem solving skills: relations between metacognition and strategic processing. Learning Disability Quarterly, 12, 57-70.

Bos, C. (1988). Academic interventions for learning disabilities. In K. Kavale (Ed.), Learning disabilities: State of the art and practice (pp. 98-122). Boston: College Hill.

Cherkes-Julkowski, M. (1985). Metacognitive considerations in mathematics instruction for the learning disabled. In J. Cawley (Ed.), Cognitive strategies and mathematics for the learning disabled (pp. 99-116). Rockville, Md: Aspen.

Crosswhite, J. (1987). Cognitive science and mathematics education: A mathematics educator’s perspective. In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 265-277). London: Lawrence Erlbaum.

Davis, R., & Hajicek, J. (1985). Effects of self-instructional training and strategy training on a mathematics task with severely behaviorally disordered students. Behavioral Disorders, 10, 275-282.

Dawson, A. (1992). We’ve come a long way, folks! Logo Exchange,11(1), 22-25.

Flavell, J. (1976). Metacognitive aspects of problem solving. In L. Resnick (Ed.), The nature of intelligence (pp.231-235). New York: Wiley.

Frith, G. & Armstrong, S. (1985). Self-monitoring for behavior disordered students. Teaching Exceptional Children, 18(2), 144-148.

Garofalo, J. (1987). Metacognition and school mathematics. Arithmetic Teacher, 34(8), 22-23.

Garofalo, J., & Lester, F. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16, 163-176.

Gattegno, C. (1974). The common sense of teaching mathematics. New York: Educational Solutions.

Hallahan, D. (1987). University of Virginia Learning Disabilities Research Institute. Learning Disability Quarterly, 1, 77-78.

Hallahan, D., & Lloyd, J. (1987). A reply to Snider. Learning Disability Quarterly, 10, 153-156.

Labercane, G., & Battle, J. (1987). Cognitive processing strategies, self-esteem, and reading comprehension of learning disabled students. BC Journal of Special Education, 11(2), 167-185.

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Lee, L. & Wheeler, D. (1986). High school students’ conception of justification in algebra. In G. Lappan & R. Evans (Eds.), Proceedings of the eighth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 94-101). East Lansing, MI: PME-NA.

Lester, F., Garofalo, J., & Kroll, D. (1989). Self-confidence, interest, beliefs, and metacognition: key influences on problem solving behavior. In D. McLeod & V. Adams (Eds.), Affect and mathematical problem solving: A new perspective (pp. 73-88). New York: Springer-Verlag.

Loper, A., Hallahan, D., & Ianna, S. (1982). Meta-attention in learning disabled and normal students. Learning Disability Quarterly, 5, 29-36.

McLeod, D.(1989). Beliefs, attitudes, and emotions: New views of affect in mathematics education. In D. McLeod & V. Adams (Eds.), Affect and mathematical problem solving: A new perspective (pp. 245-258). New York: Springer-Verlag.

Mason, J. (1992). Thinking mathematically. Address given at the Summer Institute in Teacher Education: New Perspectives in the Teaching and Learning of Mathematics. Burnaby BC: Simon Fraser University.

Mason, J. & Davis, J. (1991). Fostering and sustaining mathematical thinking through problem solving. Deakin University.

Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25(4), 230-248.

Rohwer, W., & Thomas, J. (1989). Domain-specific knowledge, metacognition, and the promise of instructional reform. In C. McCormick, G. Miller, & M. Pressley (Eds.), Cognitive strategy research: From basic research to educational applications. New York: Springer-Verlag.

Schoenfeld, A. (1983). Beyond the purely cognitive: Belief systems, social cognitions, and meta- cognitions as driving forces in intellectual performance. Cognitive Science, 7, 329-363.

Schoenfeld, A. (1987). What’s all the fuss about metacognition? In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189-215). London: Lawrence Erlbaum.

Schoenfeld, A. (1989). Explorations of students’ mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20(4), 338-355.

Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.)., Handbook for research on mathematics teaching and learning. New York: Macmillan.

Silver, E. (1982). Knowledge organization and mathematical problem solving. In F. Lester & J Garofalo (Eds.), Mathematical problem solving: issues in research (pp. 15-25). Philadelphia: Franklin Institute Press.

Silver, E. (1987). Foundations of cognitive theory and research for mathematics problem-solving instruction. In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 33-60). London: Lawrence Erlbaum.

Snider, V. (1987). Use of self-monitoring of attention with LD students: Research and application. Learning Disability Quarterly, 10, 139-151.

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Surtamm, C. (2004). Assessing mathematical processes – A complex process. Ontario Mathematics Gazette, 43(1), 7- 9.

Torgesen, J. (1977). The role of nonspecific factors in the task performance of learning disabled children: A theoretical assessment. Journal of Learning Disabilities, 10, 27-34.

Wilson, J. & Clarke, D. (2004). Towards the modelling of mathematical metacognition. Mathematics Education Research Journal, 16(2), 25-48.

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Solutions to the Summer 2007 Puzzles

Puzzle 1

Solutio

nThe

As the result of our draw, Stephen Cox of Shawnigan Lake wins the BCAMT design-er T-shirt for his correct solutions.

(Source: The Golden Ratio by Mario Livio, p. 212)

Find the next 3 numbers in the following sequence.

1, 10, 101, 10110, _____, ______, ______

Explain your answer.

There could probably be many patterns. One interesting pattern is as follows. Start with the number 1, and then for the second term, replace 1 by 10. Subsequently, replace each 1 by 10 and each 0 by 1. This will yield the following sequence.

1, 10, 101, 10110, 10110101, 1011010110110, 101101011011010110101

Note that the number of 1s starting with the first number form a Fibonacci sequence, as do the number of zeros starting with the second number.

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Puzzle 2

Solutio

nThe

What is the final digit of the number 31001?

(Source: Lore of Large Numbers by Philip J Davis, p. 129)

Notice the pattern of the last digits in successive powers of 3.

30 = 131 = 332 = 933 = 2734 = 8135 = 24336 = 72937 = 2187

The odd powers 31, 35 , 39 , 313 , 317 ,... all end in 3.

The odd powers 33 , 37 , 311, 315 , 319 ,... all end in 7.

Since 1001 is a member of the arithmetic series 1, 5, 9, 13, 17, … then the power 31001 ends in a 3.

The repeating pattern of the last digits is 1, 3, 9, 7, 1, 3, 9, 7, …