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Vector �

The Official Journal of the British Columbia

Association of Mathematics Teachers

Vector

Spring 2008�

Vector is published by the BC Association of Mathematics Teachers.

Articles and Letters to the Editors should be sent to:

David Tambellini, Vector Editor John Kamimura, Vector EditorBox 445 Kwantlen Park Secondary SchoolChristina Lake, BC 10441 132nd StreetV0H 1E0 Surrey, BC V3T [email protected] [email protected]

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Spring �008 • Volume 49 • Number �

Vector �

Inside this issue…

Vector

Official Journal of the BC Association of Mathematics Teachers

Spring 2008 Volume 49 Issue Number 1

6. Letters to the Editors

7. Spring �008 Puzzles

8. Teaching Math Effectively Rick Wunderlich

�0. Connecting Art and Mathematics Cristina Casado

�9. Teacher Math Anxiety and Lack of Conceptual Understanding Denise Flick

�5. Zero’s Lament Jim Vance

Werner Liedtke�7. Take the Calendar and “Stuff it” – At Least in the Early Grades

Marlow Eidiger

Al Sarna

�5. Improving the Math Curriculum

4�. Mathemagic V: The Voyage Home

55. Methods of Integration Duncan McDougall

4. The BCAMT Executive �007-�008

7�. Solutions to the Fall �007 Puzzles

Spring 20084

BCAMT Executive

Elementary School Representatives

Selina MillarNumeracy/Science Helping Teacher (Surrey)Work: (604) 592-4322Fax: (604) 590-2588Email: [email protected]

Sandra BallPrimary Numeracy Helping Teacher (Surrey)Work: (604) 590-2255E-mail: [email protected]

Lorill ViningNumeracy Support Teacher (Campbell River)E-mail: [email protected]

BCAMT President & Newsletter Editor

Past President

Dave van BergeykSalmon Arm Secondary SchoolSchool: (250) 832-2188Fax: (250) 832-6112E-mail: [email protected]

Vice-President

Robert SidleyH. J. Cambie Secondary School (Richmond)School: (604) 668-6430Fax: (604) 668-6132E-mail: [email protected]

Secretary

Rupi Samra-GynanePrincess Margaret Secondary School (Surrey)School: (604) 594-5488Fax: (604) 594-4689E-mail: [email protected]

Membership Chair

Dave EllisE-mail: [email protected]

Secondary School Representatives

Marc GarneauMathematics Helping Teacher (Surrey)Work: (604) 590-2588E-mail: [email protected]

Sam MuracaDistrict Coordinator - Numeracy (Langley)Office: (604) 534-9285Fax: (604) 530 2906E-mail: [email protected]

Phil LeeMagee Secondary School (Vancouver)School: (604) 713-8200Fax: (604) 713-8209E-mail: [email protected]

Michèle RoblinHowe Sound Secondary School (Squamish)BCAMT Hotline: (877) 888-MATHSchool: (604) 892-5261Fax: (604) 892-5618E-mail: [email protected]

Treasurer

Kathleen WagnerHugh Boyd Secondary School (Richmond)School: (604) 668-6615Fax: (604) 668-6569E-mail: [email protected]

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BCAMT Executive

Ministry of Education RepresentativeRichard DeMerchantMinistry of Education Office: (250) 387-4416Fax: (250) 356-2316E-mail: [email protected]

Peter LiljedahlFaculty of EducationSimon Fraser UniversityWork: (604) 291-5643E-mail: [email protected]

Post-Secondary Representative

Marc Garneau(see above)

NCTM Representative

Canadian NCTM Regional Representative (Zone 2)

Carol MatsumotoEmail: [email protected]

Regional Representatives

Chris BeckerPrincess Margaret Secondary School (Penticton)School: (250) 770-7620Fax: (250) 492-7649E-mail: [email protected]

Brad EppSouth Kamloops Secondary School School: (250) 374-1405Fax: (250) 374-9928E-mail: [email protected]

Independent Schools Representative

Chris StroudWest Point Grey AcademySchool: (604) 222-8750E-mail: [email protected]

2007 - 2008

Spring 20086

Letters to the Editors

Dear Editors:

Although test anxiety is always an issue with some students in every course, I currently have two students in my PMath11 class for whom this is a major issue. Both students have averaged around 30% on their first two tests of the semester. In one case in particular, I know the student works extremely hard, works with a good tutor, understands and completes homework correctly and contributes intelligently in class. However, put a test or quiz in front of her and she produces almost nonsense. Based on what I see (not including tests), I’d expect her to be a 65 - 75% student.

I am wondering what research has been done on extreme test anxiety, and on its ability to make that form of evaluation invalid. What are some of the things that are recommended to do by researchers? What are some of the things other teachers are doing?

In the past, I’ve had students write tests in more comfortable environments. I’ve also used hand-written tests and called them “work sheets” for students to complete. I’ve tried a few other things with good success in some cases, not so good in others.

Ultimately, I feel that the low marks some of these students have received for years have sent a very negative message to them, and severely affected their self-images and even their future abilities to learn new material. The frustrating part is that it’s sometimes a false message. The students may have good mathematical intelligence, but receive feedback that says they are “not as smart” as their peers. Worst of all, it is the fault of the education model we are using.

I find this to be one of the hardest parts of my job to deal with.

Michael Bruins Maple Ridge Christian School

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Spring 2008 Puzzles

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

David Tambellini, Vector Editor Box 445Christina Lake, BC V0H [email protected]

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.

Triangle ABC is isosceles with AB = AC. Point D on AB is such that ∠ BCD = 15° and BC = AD. Find, with proof, the measure of ∠ CAB.

(Reprinted with the permission of the author, who will be revealed with the solution in the summer 2008 issue of Vector.)

Four people, A, B, C, and D are on one side of a river. To get across the river they have a rowboat, but it can fit only two people at a time. A, B, C, and D could each row across the river in the boat individually in 1, 2, 5, and 10 min-utes respectively. However, when two people are on the boat, the time it takes them to row across the river is the same as the time necessary to row across for the slower of the two people. Assuming that no one can cross without the boat, and that everyone is to get across, what is the minimum time for all four people to get across the river? Show that your answer is the minimum.

(Reprinted with permission of the Canadian Mathematical Society)

Spring 20088

Teaching Math Effectively

I believe math teachers have the best, and certainly the most important job in the school system. Grandiose? Hardly.

Think about the world’s profound problems. What images come to mind? Is it Al Gore riding the scissor lift showing the graph of greenhouse gas carbon dioxide? It is the photos of African or inner city poverty? Is it images of pollution? Think next of each of these problems in terms of the mathematics inherent in them. Without mathematics, climate change is unnoticed, the economics of poverty nonexistent and the causes of pollution not quantified.

Unfortunately, it’s not simply mathematics that will provide us with solutions to the world’s problems, it is people. And at that point, fellow math teachers, is where we come in. I believe we no longer have the luxury of producing citizens who are less than the best they can be mathematically. Even more importantly, we cannot afford to produce citizens who believe they “can’t do math.” We must believe every student is capable of thinking mathematically, because we know all citizens must participate in the solutions to world-shaping issues such as climate change, poverty and pollution.

So, how do we get there? We begin by being more intentional about accessing the affective domain. The affective domain is the attitude or emotional content the student carries into learning. I love the descriptor that says “affect is the fuel that students bring to the classroom, connecting them to the ‘why’ of learning.” Our job is not only to explain the math, but to enable students to own the learning and to feel the “why.”

Ownership of learning is enhanced when teachers create an environment where students want to learn the math because it meets their needs. This occurs when students are given a problem to solve in such a way that they sense they own both the problem and the resultant solution. Some ways of creating this type of learning environment include providing rich, multi-solution problems that strike at the very nature of the individual student. I love geometry projects that have students designing jewelry and needing to find surface area for gold plating purposes. I love spreadsheet projects where students create their own virtual store and must do everything from stocking the store, pricing items, estimating

Rick Wunderlich is a full time teacher in Salmon Arm, BC.

Rick Wunderlich

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sales and profit, to providing staffing solutions and costs. I love it when students are asked how the school they attend might become more “green,” and then use algebra to defend their projections.

I also love the conversations I get to listen to when students of all abilities are truly engaged in something they believe in, and are using mathematical language and skills to express themselves. Nothing is sweeter than overhearing “I’d use a pie chart if I were you. It shows stuff better. I’ll show you how.” Or being asked, “Are we allowed to use graph paper and make everything to scale? It’ll make the condo layout easier to see.” Or, perhaps best of all, “this can’t be math, I like it.”

If we can get all students to emotionally own their mathematical learning, we will have done a great thing. And frankly, I don’t think we earthlings have any other option.

Teachers of mathematics, you have the best and most important job in education. Be proud.

Contact Rick: [email protected]

Spring 2008�0

Connecting Art and Mathematics

Cristina Casado is enrolled in the University of BC elementary teacher education program, as a member of the Technology and Critical Thinking (TACT) cohort.

When teaching art to children I’ve often observed myself and others both explain and model some part of the activity being taught. When I reflect on the language used, I can distinctly remember words such as half, triangle, pattern, parallel lines and equal occurring naturally as a way of describing specific features of the activity. As part of an artistic project, the vocabulary fits in context naturally, and requires almost no explanation. Learning happens implicitly and eventually that interiorization comes to the surface when children start to talk about what they have done, using the vocabulary that they have been exposed to.

Math and visual arts have been closely intertwined throughout history: from the intricacy of the tile patterns of the Alhambra in Spain and the designs in stained glass windows in Gothic cathedrals, to the perfect lines of Renaissance buildings and the drawings of Escher. In fact, “mathematics is the source of a virtually limitless number of rich and beautiful images and structures, provided one has the inclination and some skills to ‘read’ and understand their aesthetic” (Kent & Sharp, 2005). The connections between art and mathematics have served as models for artists’ endeavours, and have been recreated and transformed to give place to a rich history of art.

In education, there is a constant struggle between a tendency to compartmentalize subjects (in an attempt to simplify their teaching and learning), and an awareness of the need to integrate them (since few things in life are so compartmentalized). In mathematics, the situation is no different since it is considered that “mathematics must be made accessible to all students, that it must be presented as a connected discipline rather than a set of discrete topics, and that it must be learned in meaningful contexts” (House, 1995, p. vii).

In the past few years, we’ve seen a move towards teaching through making connections between subjects. Mathematics is increasingly being integrated with other subjects for the purpose of achieving a more holistic understanding. Mathematics learning is a language-mediated activity and is increasingly being approached from new angles, such as using children’s literature in a math class or keeping a math journal in which students “examine their ideas and reflect

Cristina Casado

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on what they have learnt. When students write about mathematics they are actively involved in thinking and learning mathematics” (Burns, 2003, p. 13).

Young children who cannot yet write express themselves through pictures. Even in the older grades, in cases where drawing and writing have been combined, evidence has suggested that “drawing may be one way to reveal what students know but cannot put into words” (DeJarnette, 1997, p. 151). In The Primary Program: A Framework for Teaching we read the following.

Enhancing children’s awareness of elements such as pattern, line, and symmetry in various art forms also helps children appreciate the aesthetic dimension of mathematics and fosters their numeracy development. Students who are visual or kinaesthetic learners, who are learning in a second language, or who have special learning needs, are supported in language-mediated activities when they have access to the fine arts.

Ministry of Education, 2000, p. 22

There is an increasing awareness of the need to communicate mathematical understanding and individual thinking beyond the unique use of the language of numbers. Visualization is essential for this to take place and its use “in the study of mathematics provides students with the opportunity to understand mathematical concepts and make connections among them” (Ministry of Education, 2007, p. 19). Therefore visual arts provide a unique tool in order to increase students’ reflection and understanding of mathematics.

“Examples of art from around the world can easily fit into class discussions of geometry, measurement, symmetry and patterns” (Shirley, 1995, p. 40). These notions set the bases for future learning; students need to interiorize them first, in order to grasp the depths of more complex concepts. For example, “learning to work with patterns in the early grades helps develop students’ algebraic thinking that is foundational for working with more abstract mathematics in higher grades (Ministry of Education, 2007, p. 13).

Art doesn’t have to be a static painting that one is to observe and not touch, as if we were in a museum. When brought into the classroom, art can be a source of inspiration: a model that can be altered, an image that can be photocopied, cut and turned into a jigsaw, or a model of a sculpture that can be observed, analysed, deconstructed and turned into a manipulative that students can work on, create and recreate. Specific works of art can be seen as artists’ own understandings of mathematical ideas, as interpretations of their own reality that are represented in the form of a sculpture, painting, mosaic or building.

This opens the door to art being a means of mathematical discovery and an opportunity to look into and search for hidden features. It also allows for the freedom to reinvent and create one’s own understanding of similar

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mathematical concepts. It even expands this discovery into “observing and recognizing patterns in nature [that] are fundamental concepts taught in art classes” (Chong, 1997, p.12). The nature of art itself allows for self-expression and communication of both perception and understanding. Art thus becomes both a beginning and an end in itself.

An approach to mathematics through art allows for connections between mathematics and its applications at a concrete and tangible level. Artists’ connections can be seen as sources of inspiration and are also used as firm ground through which more complex concepts can be exposed and studied. Understanding these concepts and seeing their applications in one’s immediate reality are goals that can be reached through several routes. However, art provides a clear trans-cultural and historical connection with mathematics that should not be wasted.

Following the bibliography are some ideas on how you could connect art and mathematics in the classroom. They are meant to be just brief guidelines of different ways in which a lesson could be oriented, not succinct explanations of each one.

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References

Ministry of Education. (2000). The primary program: A framework for learning. Victoria: Province of British Columbia.

Ministry of Education. (2007). Integrated Resource Package. Victoria: Province of British Columbia.

Burns, M. (2003). Writing in math class. California: Math Solutions Publications.

Chong, K. (1997). 2 – 4 – 6 – 8 Everybody tessellate. Vector, 38(1).

DeJarnette, K. (1997). The arts, language, and knowing: an experimental study of the potential of the visual arts for assessing academic learning by language minority students. In R. Deasy (Ed.), Critical links: learning in the arts and student academic and social development (p. 151). Retrieved October 2007 from http://www.aep-arts.org/files/publications/CriticalLinks.pdf

House, P. (1995). Preface. In P. House & A. Coxford (Eds.), Connecting mathematics across the curriculum (pp. vii –viii). Virginia: National Council of Teachers of Mathematics.

Kent P. & Sharp, J. (2005). Bridges between mathematics and art. Mathematics Teaching, 193, 23-26.

Shirley, L (1995). Using ethnomathematics to find multicultural mathematical connections. In P. House & A. Coxford (Eds.), Connecting mathematics across the curriculum (pp. 34-43). Virginia: National Council of Teachers of Mathematics.

Spring 2008�4

Patterns

Sunshine and Shadow – Amish quilt

Talk about quilts as part of the Amish tradition.For primary grades:- Initially, use quilts that have simple patterns, and increasingly move onto more

complex ones.- Identify the patterns in each of the rows. Compare the different patterns used.

Predict what the pattern of the next row will be. Explore how we can find a different pattern when we look at the quilt from one of the corners.

- Make an individual mini-quilt/or big classroom quilt using scraps of card, recycled materials, felt or fabric:

• Measure each of the shapes. • Predict how many shapes you will need to fit on a given surface. • Decide on a pattern and predict what it will look like. • Display them in the classroom.For intermediate grades:- Relate to the addition table: • Can you see a relationship? • How will the pattern change if we related it to the multiplication table?- How many squares are there in total? How many ways can you count them? Estimate the number of squares.For more information see:Ernie, K. (1995). Mathematics and Quilting in Connecting Mathematics across

the Curriculum (pp. 170 – 176) Virginia: National Council of Teachers of Mathematics.

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Geometry

Senecio by Paul Klee

- Talk about Paul Klee and his work.

- Look at the painting and talk about the shapes that students can see. Are the triangles “proper” triangles? How are they different from other triangles? How many squares are there? How many are in the neck? And in the head?

- Trace the main shapes of the painting, photocopy it and give copies to students for them to cut and try and reconstruct as if it were a jigsaw puzzle.

- Give students scissors and paper of different colours and get them to explore other ways of making the neck using more/less squares.

- Get students to reflect on the adjustments they have to make for the squares to fit in that space.

Spring 2008�6

Geometry

Untitled (Red, Blue and Purple) Elizabeth McIntosh, 2005-2006, oil on canvas

- Talk about Elizabeth McIntosh and her work.

- Once students are familiar with tangrams, show them the painting and get them to reproduce it as accurately as possible, using the tangrams.

- Points for discussion: • What are the main differences between both of them? • How can we make them the same?

• Estimate the length by which certain triangles would have to be modified in order for them to fit the tangram model.

- Get students to make a design that is even more removed from both the tangram and the painting models, using only triangles that touch each other. • Talk about the different types of triangles used to make designs.

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Geometry/Symmetry

Tiles in the Alhambra (Granada, Spain)

- Talk about Islamic design in Spain.

- Talk with the students about the shapes in the tile wall. What patterns can they see? How are the shapes arranged?

- Enlarge and make a photocopy of one of the bigger circles for each student. Talk about the shapes in that specific circle.

- Get students to fold the circle through the exact middle and explore how the shape has changed. Is it the same on both sides?

- Keep folding it and explore how the shape changes and yet both sides continue being exactly the same. If the students are old enough, introduce the concept of line of symmetry and find out how many lines of symmetry are in the circle.

- Cut out different shapes of paper and make doilies. Explore the different shapes created and the number of lines of symmetry that are created each time.

- A variation would be to use Mandalas and stained glass windows.

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Pattern/Symmetry

- Talk about Rangoli art and about the elements that constitute the design.

- Focus on the dots and lines and the symmetry. • Is there another way of making a symmetric design just by changing the

lines but without moving the dots? • Answers can be on geo-boards or drawn on paper.

- Provide the students with a sample of a part of the pattern. Using that as a model, get them to make a growing pattern.

Rangoli Art (India)

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Teacher Math Anxiety and Lack of Conceptual Understanding

Math Makes Sense. This is the title of a new elementary math series recently launched in Canada. The title was chosen to indicate to prospective users (teachers and districts) that here at last was a resource that would encourage and support teachers’ efforts to teach for deeper mathematical understanding. Educators would be persuaded and instructed to teach math, using pedagogy that would support the constructivist theory of learning and students would understand concept before procedure and algorithm. Quality would be valued above quantity. Engagement in mathematical activity would be the goal. Hallelujah!

The books were delivered. Handsome teacher guides complete with CDs accompanied the student texts. Colourful boxes of manipulatives beckoned the teacher to open, organize and store in equally colourful bins, the wonderful array of geoboards, pattern blocks, pentominoes, and tiles. One teacher was heard to remark, “I arrived in September and boxes of math books and supplies were waiting in my classroom. It was just like Christmas!”

It was not long before Christmas morning euphoria succumbed to Boxing Day disillusionment. Soon teachers were heard to remark and snicker, “Hah! Math Makes Sense? This Math Makes No Sense. There’s not enough practice. I can’t spend that much time teaching patterning. The classroom is too noisy. There is too much discussion and not enough drill. ”

What was the problem? How could a program that had promised so much hope so quickly become a source of ridicule?

Yes, the program was different. No longer could a teacher present a concept, demonstrate, assign practice questions and return to her seat to mark the daily spelling work. The teacher was now required to circulate and facilitate as students engaged in exploration of newly introduced concepts and then attempted to connect new learning to previous knowledge. The teacher had to deeply understand the concepts so that she might facilitate discussion. The teacher had to possess mathematical knowledge above and beyond the grade

Denise FlickDenise Flick teaches Grade 7, and is the K-7 Numeracy Coordinator with School District #20 (Kootenay- Columbia).

Spring 2008�0

level being taught. The teacher had to field questions that she did not always have the answers for and adjust the pace and content of the lesson to suit the individual needs of students.

Wait! This wasn’t right! Is not the teacher supposed to be all knowing? Is not the teacher supposed to be in charge of the lesson? Is not the teacher the sage on the stage and not the guide on the side? A review of all provincially sanctioned resources soon confirmed that all recommended resources were now enabling the educator to organise, present and deliver math instruction in a like manner. Oy-vey!

The sad reality was that this math did not make sense. It did not make sense to many of the teachers attempting to teach it.

Elementary school teachers are required to be all and know all. Language, science, art, physical education, music and yes, mathematics are subjects in which we are to be competent, knowledgeable, and proficient. Very few individuals are so multi-talented that they can easily and effectively provide for their students all the necessary components of a well-rounded education. I know that I feel inadequate when teaching French. My knowledge and ability are lacking. Others in my schoolhouse profess to be unable to carry a tune and as a result go to great lengths to avoid teaching music and devote limited time to musical endeavours in their classrooms.

Is it not just as likely that some individuals experience the same feelings of inadequacy or anxiety when teaching math? Is it not true that in some classrooms, math is the first class to be shortened or skipped when an assembly is called or when a poster contest needs to be completed?

As educators, we have always known that math anxiety is a very real problem for many students. Math anxiety is also a very real problem for some teachers. The problem of math anxiety “becomes acute when the person most afraid of numbers and equations is standing in front of the classroom trying to teach the subject” (Campbell, 2006).

The teaching and learning of mathematics has historically been concerned with rote memorization and the carrying out of procedure. Deep understanding of mathematical concepts has not been stressed. In the 1990s some effort to make math more meaningful was bandied about and engaged in. To connect math learning to real life, students were often working with pizza fractions or favourite food graphs. Word problems were highlighted as the way to encourage problem solving. Three birds plus two birds equals how many birds? Three fish plus two fish equal how many fish? It was not always realized that the task was only problem solving if there was a problem to be solved.

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Games were introduced to aid in the memorization of basic facts. Manipulatives were used in an attempt to help children explain why “you carry the one.” (Carry the one?) Teachers believed that simply asking students to explain was sufficient to promote deep understanding. Statements were confused with explanations. In elementary school, even high-performing math students often had a shallow grasp of mathematical concepts. These students carried on to secondary school and then on to university.

Is it not possible that many intelligent, yet math anxious individuals are drawn to post secondary studies? These individuals do not continue their mathematical studies at university and are therefore drawn to career study that does not require additional math courses. Elementary teaching is such a career. It is frequently the case that future elementary school teachers are required to take only one math course. The course has little if anything to do with conceptual understanding. The course is a methods course.

In recent years, the Western and Northern Canadian Protocol (WNCP) has been established. This protocol comprises the western provinces (British Columbia, Alberta, Manitoba, and Saskatchewan) as well as the territories (Yukon, Nunavut and Northwest Territories). A coordinated math curriculum and mandated pedagogy have been established. The number of concepts taught at each grade level has been decreased. This decrease is intended to provide teachers and students the opportunity for rich and full mathematical learning of each concept covered rather than the surface study that was available previously as teachers were pressed to cover far too many concepts in each school year.

I would suggest that this noble effort to create the opportunity for improved student conceptual understanding will create a new wave of math anxious teachers. Many hesitant teachers of math had established some level of comfort in the how and what they were teaching. But now… Math Makes Sense, Math Everywhere, and Math Links are all commercially available and provincially accepted resources. All these programs reflect the change in content but more importantly, they reflect an important change in pedagogy. The WNCP approved resources are doomed to fail miserably if the new is taught in the old way and if our teacher-training facilities have not done what is needed to prepare the next generation of teachers to use the next generation of resources.

A key factor in the success of these many initiatives and instructional strategies in elementary school mathematics is the ability of the teacher to effectively implement the improved programs. As Krulick and Rudnick (1982) stated, “before elementary teachers can teach problem solving, they must themselves

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become problem solvers.” I would suggest that before elementary teachers can teach for deep conceptual understanding, they must themselves possess deep conceptual understanding.

Many researchers claim that the majority of elementary school teachers are “under prepared in mathematics, and that they also possess an ongoing case of mathematics anxiety” (Battista, 1986; Buhlman & Young, 1982; Kelly & Tomhave, 1985). These authors report that significant portions of elementary school teachers possess levels of mathematics anxiety. The levels of mathematics anxiety are enough to negatively affect their classroom teaching practices. “The disproportionately large number of mathematically anxious teachers at the elementary school level is often said to influence not only the effectiveness of instruction, but may promote the early onset of mathematics anxiety among students” (Hackworth, 1985, p. 8; Oberlin, 1982). These authors state “that if elementary teachers are to make instruction more relevant and exciting to their students, they must first overcome any fears or negative attitudes that may have a negative influence on their planning and teaching.”

It is not this article’s purpose to criticize math anxious and/or concept deficient teachers. The purpose is to emphasize the importance that needs to be placed on the diagnosing and addressing of deficits in teachers’ mathematical content knowledge, and to identify the mathematical beliefs and attitudes of those who will or do work with students. It cannot be assumed that teachers are trained to teach mathematics for deep understanding and are then able to do so with confidence. Many may leave university lacking the skills needed to be good teachers of mathematics. Classrooms may be home to teachers who lack the pedagogy, content knowledge and disposition necessary to do the best job possible for their students.

Several problems exist. One is the often inappropriate and insufficient method and content courses offered to teachers in training. Another problem is the scarcity of continued education in pedagogy and continued opportunity to improve mathematical understanding and disposition for teachers already entrusted with the mathematical education of children. At the beginning of one’s teacher training, negative disposition in math needs to be recognized and addressed. Within the teaching profession, a climate of trust and opportunity needs to be developed to ensure that in-service teacher mathematics anxiety can be admitted and effectively and respectfully addressed. Quinn, (1998) proposed that teachers who possess poor attitudes to math are more likely to pass those feelings on to their students. Those students may then become teachers and the cycle continues.

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The teaching community must not be afraid to acknowledge that the greatest deterrent to “good” elementary math instruction cannot be “fixed” only with improvement in curriculum, resources and/or pedagogy. Teacher disposition, knowledge of pedagogy and deep understanding of the subject matter must be addressed.

Professional development is too often ineffective. Professional development is costly and yet for this great expenditure it is infrequent, superficial, and noncumulative. The professional development opportunities presented are not connected to deeper learning but are more often delivered in the form of a “tune up” by well meaning individuals in educationally themed sweaters and jewelry.

It is my wish to inspire and validate discussion among teachers, teachers in training, school districts and teacher educators. The education profession must first acknowledge the existence of math anxiety and ill preparedness and then must endeavour to provide appropriate and successful methods by which this problem can be eliminated.

Summary

North American children continue to lag behind other countries in mathematical achievement (Trends in International Mathematics and Science Study [TIMSS], 1999). Increased testing, variation in instruction and use of math games and manipulatives have not improved learning. Our children’s mathematical learning will not improve until their teachers’ mathematical learning is provided for. Universities and colleges must recognize the mathematical needs of pre-service teachers. School boards and district personnel must recognize the mathematical needs of the teachers that they now have in their employ. If pre-service and in-service elementary teachers are to make instruction less anxious for their students, they must first overcome any fears or negative attitudes that may have a negative influence on planning and teaching.

I believe it is through the development of conceptual and pedagogical understanding that teachers can improve their instruction of mathematics and improve the ability of their students to think and work mathematically. I believe that it is through personal reflection and with the support of teacher educators, colleagues and district personnel that teachers can address their personnel difficulties with math anxiety and can then create an atmosphere in which all students can investigate all things mathematical in a safe, supportive, and effective environment.

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Bibliography

Battista, M. (1986). The relationship of mathematics anxiety and mathematical knowledge to the learning of mathematical pedagogy by pre-service elementary teachers. School Science and Mathematics, 86, 10-19.

Buhlman, B. J., & Young, D. M. (1982). On the transmission of mathematics anxiety. Arithmetic Teacher, 30, 55-56.

Campbell, G. (2006). Popping math anxiety. Retrieved October 8, 2006, from ASU Research http://researchmag.asu.edu/stories/cresmet.html

Hackworth, R. D. (1985). Math anxiety reduction. Clearwater, FL: H & H Publishing.

Kelly, W., & Tomhave, W. (1985). A study of math anxiety and math avoidance in pre-service elementary teachers. Arithmetic Teacher, 32, 51-53.

Krulick, S. & Rudnick, J.A. (1982). Teaching problem solving to pre-service teachers. Arithmetic Teacher, 29, 42-45.

Oberlin, L. (1982). How to teach children to hate mathematics. School Science and Mathematics, 82, 261.

Quinn, R. J. (1998). The influence of mathematics methods courses on pre-service teachers’ pedagogical beliefs concerning manipulatives. Clearing House, 71, pp. 235-289.

Vector �5

Zero’s Lament

Jim Vance is a Professor Emeritus at the University of Victoria.

Hello. My name is Zero. I’m here to say that I feel hurt, as you would, when people call me a “nothing.” I am a Number and although I’m different from other numbers and am known to be a source of difficulty for some learners, I am needed in everyday life and have many characteristics that make me unique and indispensable in the world of mathematics. Let me tell you a little about myself.

As numbers go, I am Even tempered, Wholesome, and approachable. Although Complex in nature, I am a Real and a Rational entity. Obviously, I’m not Perfect (such numbers are few and far between) and I can’t claim to be Prime, Positive or even Natural. On the other hand, I am not thought of as unlucky, nor do I belong to the Odd, Negative, Irrational or Imaginary number families–not that there would be anything wrong with that.

Sure, I have my faults, like everyone else. You can’t really count on me and I can be impossible at times, like when you try to divide by me. I always get my own way under multiplication, but when you add me to other numbers, I don’t try to steal their identity or change them in any way.

Although my roots go back to far earlier times, my invention in India around the ninth century is recognized as one of the most important developments in the history of mathematics. I serve two essential functions. As a front extension of the Counting numbers (which begin 1, 2, 3), I am the Cardinal number that tells how many things there are in a group when there aren’t any. As a digit in the base-ten place value numeration system, I appear in the numerals representing powers of ten and as a place holder in other multi-digit numerals, making it possible to name decimal fractions so small and whole numbers so large as to be conceived only in the mind. (e.g., a googol is 1 followed by a hundred zeros.)

My creation led to other major advancements beginning with negative integers and algebra. Today, I alone stand between the positive numbers and their negative counterparts on the real number line, and I name the point at the center of the planes for complex numbers and coordinate geometry. In

Jim Vance

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the general form of linear, quadratic and other polynomial equations, you will find me on the right of the equals sign. The roots of a polynomial equation are the “zeros” of the corresponding function. Calculus involves the notion of limits and the enigmatic idea of an infinite sequence of numbers getting closer and closer to me, but never quite touching me. And may I say that I am honored to be a part of the most beautiful and famous equation in mathematics – Euler’s Identity – which links five fundamental and remarkable constants: 0, 1, π, i, e.

On measurement instruments, such as rulers and thermometers, I occupy the benchmark position. Even though in physical measurement I can never represent the size of an object, I am used to describe abstract situations, such as the width of a line and the number of dimensions a point has. Water freezes at zero degrees on the Celsius scale, and the lowest possible temperature is called absolute zero. The probability of an event that can never occur is zero. Common expressions that capitalize on my essence include zero gravity, zero down and zero tolerance.

Since I do sometimes behave strangely and in confusing ways with the operations, let me review my properties.

(1) The sum of zero and any number is that number.

(2) Any number minus zero is that number; zero minus any number is the inverse of that number.

(3) The product of zero and any number is zero.

(4) Zero divided by any non-zero number is zero; any non-zero number divided by zero is undefined (because there is no solution); zero divided by zero is undefined (because there are infinite solutions).

(5) Zero to any non-zero power is zero; any non-zero number to the zero power is one; zero to the zero power is either undefined or defined to be one (the value most consistent with the theory of exponents).

Now that you know more about me, I hope you will remember that although I may not be large in size, I play an important role in mathematics and society, and am held in high repute among my fellow numbers (each of which is also special in his or her own way). Do not feel sorry for me or think of me as somehow incomplete or lacking, and please never again refer to me as “nothing.”

Vector �7

Take the Calendar and “Stuff it” – At Least in the Early Grades

Werner Liedtke is a Professor Emeritus at the University of Victoria.

A few years ago, at the beginning of the last session of the course Diagnosis and Intervention in Mathematics two students, Dale and Jennifer, made a request. When the wish was granted, a ukulele appeared, song sheets were distributed, and to the tune of Baa Baa Black Sheep all twenty-seven students sang ten verses that referred to different ideas that had been part of the course. One verse went as follows.

Always use the best aids

Ifyoureallydon’twanttofluffit.

Forget the abacus and number line

Take the pocket chart and stuff it.

These ideas are based on the fact that many responses from students having some difficulties with mathematics learning indicate lack of number sense. These students are unable to visualize the numbers behind the symbols or number names they are trying to manipulate. The use of inappropriate manipulatives, aids that do not clearly illustrate existing relationships, may be to blame. Hence the conclusion to “stuff” such aids for intervention IEPs. (I have seen attempts to overcome the inappropriateness of a pocket chart that I think make matters worse for students who experience difficulties. These attempts included using different colours for different pocket positions, or even worse, changing the size of the pocket pieces that are labeled ONES, TENS, HUNDREDS, etc.)

I believe that the use of a calendar is detrimental to any early learning about numbers and for attempts to foster the development of number sense. Therefore, I will discuss reasons why I think it should be included in the list of “aids” that should be “stuffed,” at least as part of learning about numbers in the early years.

It was satisfying for me to see that during the recent updating of The Common Curriculum Framework for K – 9 Mathematics (Western and Northern Canadian

Werner Liedtke

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Protocol for Mathematics), which will become the BC curriculum, the writing team members were unanimous in not referring to “calendar mathematics.” I have heard that there are teachers/educators who disagree with this decision. While supervising student teachers, I have observed many calendar activities that took up quite a bit of time. These teachers would probably agree with the statements, or parts thereof, that appear in the Ministry’s Response Draft of the Classroom Assessment Model – Grade 1. Under the heading “Planning for Assessment,” it states as follows.

The calendar is a valuable tool in mathematics. It provides opportunities for multiple practice and creates rich, meaningful ideas for developing rote counting abilities and number sense. It also allows maximum benefit in the building of conceptual understanding.

(2006, p.223)

“Assessment Strategies” include the following suggestions.

Students need to practice and gain confidence in their counting skills in a meaningful and purposeful way. After students make a connection between rote counting and how many are in a set, they become more able to work with larger quantities as they can count in groups rather than counting by ones.

A major part of the courses I teach is devoted to an attempt to have students “watch their language” (Liedtke, 2002) as they try to identify and make statements about learning outcomes and assessment/diagnosis that are specific and non-subjective (Liedtke, 2007a). This attempt also involves identifying examples of “edu-speak”: impressive sounding statements that may leave a listener or reader puzzled. With such statements, it is difficult, or impossible, to know intent because they are too general, too subjective or open to more than one interpretation. The following quote, which I heard during a Northwest Mathematics Conference presentation, can be used to illustrate my attempts. “Without a focus or without specific outcomes, nothing specific will come out.”

The following initial questions that came to mind illustrate why I would label the entries on p. 223 of the Response Draft as “edu-speak.” (I realize these entries will likely be different in the final edition. At least I hope they will.)

• What are examples of ‘multiple practice’? Would this practice be appropriate (Charles and Lobato, 1998) and foster numerical power?

• How are rich and meaningful defined? What types of examples would meet this condition and what are some that would not?

• What are “rote counting abilities”? How many are there? Which of these are “developed” and how? (I am somewhat embarrassed to ask because I have no idea what these abilities are.)

• Which aspects or components of number sense are “developed”?

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• How is conceptual understanding defined and how is it “built”? “Conceptual understanding” of what? How is maximum defined and how is it possible to assess “maximum benefit”?

• What are some possible benchmarks for determining “gain in confidence”? What are the “counting skills”? Are these skills the same the “counting abilities” referred to under Planning Assessment?

• What are possible indicators of being “more able to work with larger quantities”? How is larger quantity defined?

I believe that both teachers and students will benefit from statements that are as specific as possible. Not only that, the mathematical language has to be correct and consistent. As far as discussions about a calendar are concerned, I think it is important to make distinctions between number names (numerals), number (cardinal number or numerousness) and ordinals. Without these distinctions, it is impossible to interpret a statement like, “identify the number before and after a given number” (p. 224).

In an article entitled Calendar Reading: A Tradition That Begs Remodeling, Schwartz (1994) points out that telling young children things like, “Whenever you have a 2 and a 3, you have 23. Today is Wednesday, March twenty-third.” is meaningless. According to Schwartz, many early childhood professionals resisted calendar reading tasks in Kindergarten because children:

• often do not accurately count to 31, much less read double-digit numerals;

• are unable to deal with a five-row, seven column matrix;

• cannot interchange ordinal and cardinal numbers which occur in calendar reading; and

• have little or no use in their lives for reading or using day-date information. (p. 104)

Peretti (1995) agrees with Schwartz. She calls calendarspeak an “unassessed tradition” that is developmentally inappropriate. Since I agree with Peretti’s conclusion that “the tragedy is not only that children are exposed to largely irrelevant material but also that they lose so much valuable time in the process,” I will present reasons for reaching the same conclusions that he did.

In his list of “criteria for mathematical thinking,” Greenwood’s (1993) first entry reads, “Everything you do in mathematics should make sense to you” (p. 144). As far as mathematics learning is concerned, making sense of number is the key part of this “sense making”; it is the foundation for numeracy (The Primary Program, 2000, p. 146). The calendar-mathematics activities I have read about and observed in classrooms while supervising students do not and did

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not contribute to any aspects of making sense of numbers. In facts, I think the majority of these activities are detrimental in that they do not and did not clarify any ideas about number.

Visualization is a key component of number sense. An important learning outcome deals with fostering this visualization. When children hear a name for a number (four, seven, ten, twenty-one, …) or when they see a numeral (4, 7, 10, 21, …) we want them to “see” the numbers represented by these names or symbols. The phrase used by one student, “I see it in my brain,” illustrates the goal of visualization. I think it is impossible to reach this goal by examining the ordinals on a calendar.

Fostering the development of number sense requires a growth plan: specific goals and learning outcomes as well as appropriate materials, questions and activities. The foundational activities in the early years need to be based on understanding, not rote learning. The latter is, according to some (Flewelling and Higginson, 2000) and I include myself in this group, an oxymoron. According to the authors, it is a major source of anxiety and “rote-learning-plus-practice techniques train problem solvers as well as paint-by-numbers techniques train artists” (p.27). “Rote activities” do not contribute to any aspects of conceptual understanding and do not foster the development of number sense. Research results show that “instruction can emphasize conceptual understanding without sacrificing skill proficiency” (Hiebert, 2000).

During one classroom observation, the numerals that were pointed to on the calendar were 21, 20 and 22 for today’s, yesterday’s, and tomorrow’s date, respectively. The children were told how to read these numerals as in the example quoted from Schwartz: “When we see 2 and 1, we say twenty-one,” etc. No reference was made to ordinals or to the meaning of these symbols. I found it a little contradictory and unusual to see that the remainder of the lesson dealt with numbers less than ten.

In one Grade 1 classroom I observed students generate complex addition equations consisting of several addends, some made up of two-digit numerals, based on the date on the calendar. During a recent conversation with a fellow educator at the airport I was told about one teacher he knew who has her students in Grade 1 create amazingly complex equations for this age. Both, the student and the educator, were looking for my reaction to such “amazing” feats.

I was and I am reminded of a film we watched as graduate students that showed how the mathematician Gattegno had a group of children in Kindergarten write addition equations involving two- and three-digit numerals that covered the length of a wall chalkboard. The following questions come to mind when I see or hear about this type of activity.

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• Why?

• How does “doing” this type of task connect to previous learning, to ongoing learning, or to meaningful experiences outside the classroom?

• Can children talk about this in their own words and mention possible reasons for doing it?

• Do these tasks contribute to aspects of the development of number sense?

• Are children able to visualize the numerals they are recording and manipulating?

I know my answers to these questions, and to me these tasks are nothing more than some sort of meaningless “mental gymnastics” that do not transfer to other ideas or procedures. There exist many different types of problems and activities that would be of value as far as fostering the development of number sense is concerned (Liedtke, 2004; Liedtke, 2007b).

I have seen suggestions for lessons that try to make the 100th day of school a ‘special mathematical’ occasion. The Classroom Assessment Model – Grade 1 includes the comment that this day “is an opportunity to motivate students to use mathematics in a meaningful way” (p. 225). The following questions illustrate some of my concerns for some of these suggestions.

• Is a separate calendar to be kept to record ordinals from first to 100th ?

If that is the case, how are the ordinals to be grouped: in sets of seven as on the calendar, in sets of ten, or in a linear sequence from first to 100th?

• Should the ordinals for October, November and part of December be covered and a new sequence of ordinals be entered until the 100th day is reached on the calendar?

• What about a possible confusion about the 100th day not being the 100th day of the year, or about the fact that it is unlikely, for various reasons, that it is not the 100th day for anyone in the classroom?

• What about the fact that when ordinals are used in a linear sequence, we are not counting?

• What about the fact that cardinality or numerousness cannot be abstracted from ordinals and it is impossible to perform an operation on ordinals?

• How is it possible to go from the 100th member of a linear sequence and arrive at 100, a special name for a special collection of discrete objects, since it tells us that there is 1 hundred 0 tens and 0 ones?

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• What mathematical abilities do children have and what indicators of number sense do we expect from these children on the 100th day of school that can make any kind of “sense” of activities related to 100th?

• On p. 225 it is suggested that a number line be made. How is the number line with numerals to be explained and how is a comparison between this number line and the linear sequence of ordinals to 100th to be made?

On the same page “Assessment Strategies” are presented. They are listed below in italics and followed by my reactions, concerns or questions.

Follow the counting sequence

The meaning of this is not clear to me.

Count forwards by 1s from any number / Count backwards by 1s from any number

I hope the appearance of the word number in these two statements is not a mistake, but I am afraid that might be the case. The reason I am happy to see “number” is because it implies an attempt to foster visualization, which is a requisite for rational counting. As they stand, the statements mean that a discrete set of objects is presented to children. They are told how many there are and then they are asked to keep repeating the placement or removal of one object, and after each move announce the number they are looking at. Since it is likely that these statements are meant to refer to a recital of number names on the calendar, they do not involve tasks that contribute to an understanding of number and do not foster the development of number sense.

Count by 5’s

When I tell my students that I think this is a meaningless task for children in Grade 1, some will tell me that it makes them “faster” when they count objects. I suggest that they place objects in front of children and have some find the answer to “How many?” by counting in groups of five while others just count by 1’s. Then see which group of students would be the first to complete this task. Since rote skip-counting by fives is not needed during the early stages of learning about numbers and is not of any use in students’ lives at this stage, I think it is a useless skill that is easy to learn at a later stage. If someone deems the skill of counting by 5’s important, hands or groups of fives should be used to aid with the visualization of the numbers in the counting sequence. (The use of nickels and skip-counting by fives to determine the amount of money does not foster visualization or the ability to count rationally.)

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Count by 10’s

I have observed lessons where children were asked to say, in a very loud voice, every fifth numeral and every tenth numeral, and these were identified by being underlined and/or coloured. Not only that, in one classroom, a puppet was made to jump out of a cardboard tube for emphasis every time one of these number names was stated! Since no attempt to foster visualization is involved, I think this is a meaningless ritual. The least that should be done for tasks like this is to use objects or groups of fingers to show the numbers for the numerals that are the focus of this activity. I think counting by tens should be part of representing two-digit numerals, finding the answer to “How many?” by first counting tens (“ten, twenty, thirty, …”), then ones. And to further foster visualization, read each numeral in two ways: e.g., 43 as “four tens three ones” or as “forty-three.”

Read the numerals 0 to 100

What are the possible advantages of learning to read numerals that are not part of the present mathematics learning? Are there any?

Write the numerals 0 to 100

No doubt the “write” should say “print” but that aside, how important is this printing: especially the printing of names for numbers that have not been examined? Printing is a difficult and time consuming task, especially for many boys. Should time be set aside for such an endeavor, as part of teaching mathematics? I am sad to say that I have seen it become a focus of mathematics activities.

My conclusion is that “calendar mathematics” does not contribute to any conceptual understanding nor does it foster the development of any aspect of number sense in the early grades. “Calendar mathematics” is conducive to the presentation of developmentally inappropriate activities and to the use of incorrect mathematical ideas and language. Since that is the case, I would hope that if students were to rewrite the verse quoted at the beginning, it would recommend teachers to “take the calendar and stuff it.”

Schwartz (1994), who begged that calendar readings be remodeled, includes quite a few suggestions for activities that he considers to be appropriate for young children. The words, numerals and recording method used for the numerals on a calendar can all be made meaningful to students in Grade 3 as they learn about the specific outcomes related to measurement and time.

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References

Charles R. and Lobato J. (1998.) Future basics: Developing numerical power. Monograph. Golden, Colo.: National Council of Supervisors of Mathematics.

Flewelling, G. and Higginson W. (2000.) A handbook on rich learning tasks: Realizing a vision of tomorrow’s mathematics classroom. Kingston, Ontario: Queen’s University.

Greenwood, J. (1993.) On the nature of teaching and assessing “mathematical Power” and “mathematical thinking.” Arithmetic Teacher, 31(3), 42-49.

Hiebert, J. (2000.) What can we expect from research? Teaching Children Mathematics, 6(7), 436-437.

Liedtke, W. (2002.) Watch your (our) language. Vector, 43(2), 15-20.

Liedtke, W. (2004.) A focus on number sense makes a lot of sense. delta-K, 41(2), 15-17.

Liedtke, W. (2006a.) Why should number sense be the focus for any and all students? (Let me try to count the ways.) Vector, 47(3),13-22.

Liedtke, W. (2006b.) Specificity and correctness: Necessities for reaching/assessingthegoalsoftheWNCP. Submitted to delta-K.

Ministry of Education. (2000.) The primary program – a framework for teaching. Victoria, BC: Ministry of Education.

Ministry of Education. (2006.) The common curriculum framework for K to 9 mathematics(WNCP). Victoria, BC: Ministry of Education.

Ministry of Education. (2006.) Classroom assessment model - Grade 1, response draft. Victoria, BC: Ministry of Education.

Peretti, D. (1995.) A timely piece. (readers’ exchange). Teaching Children Mathematics, 1(8), 51.

Schwartz, S. (1994.) Calendar reading: a tradition that begs remodeling. Teaching Children Mathematics, 1(8), 104-109.

Vector �5

There are many criticisms pertaining to a lack of student achievement in public schools. Much of the criticism comes from low test scores. Test results from the Trends in International Mathematics and Science Study (TIMSS), and The National Assessment of Educational Progress (NAEP), have caused furor from selected observers in terms of student achievement. Mathematics, as one academic discipline, has been tested in each of these two tests.

Those who justify student achievement in mathematics state that the following create differences among international student achievement in the TIMSS.

• Cultural differences exist among nations whereby academic achievement has extreme value in some nations.

• Compared to the United States, some nations have aligned their mathematics curriculum more with the objectives of TIMSS.

• When making these comparisons, statistical sampling errors of students within nations have occurred.

• TIMSS test items favour particular nations in validity. (See Ediger and Rao, 2000.)

According to some educators, dissatisfaction with NAEP test results may be due to the following.

• Test items were written on too complex a level, especially for secondary students.

• Test items favour those states that have highly qualified mathematics teachers.

• There is the possibility of sampling errors among students being tested. (Kennedy and Tipps, 1991)

Whatever the reasons, there are ways of improving mathematics instruction in the public schools. The balance of this paper will present means to strengthen teaching and learning situations.

Improving the Mathematics Curriculum

Marlow Ediger is a Professor Emeritus at Cambridge University.

Marlow Ediger

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Evaluating Mathematics Teachers

Teachers need to possess much knowledge in the area they are teaching. For example, an algebra teacher must be highly competent in the subject matter being taught. He/she cannot just be one or two lessons ahead of students. Specialization is a key concept. Algebra teachers who are highly qualified have increased opportunities to assist learners as needed. In remedial instruction, the algebra teacher who possesses essential mathematical wisdom can diagnose student difficulties more easily.

Specializing not only in knowledge, but also in methods of teaching is needed to optimize learner progress. If one method has not worked in assisting students, a different procedure may. To strengthen mathematics teaching then, the following are offered as suggestions.

• Encourage teachers to take senior mathematics courses online or at a university.

• Guide teachers, in an in-service activity, to diagnose the kinds of errors students make and to assist students in overcoming these problem areas.

• Provide time for mathematics teachers to share ideas on what works in teaching/learning situations.

• Conduct workshops on improving mathematics instruction, using the assistance of highly qualified instructors.

• Establish a professional library for the teaching of mathematics. Introduce and encourage its use.

• Establish personal goals in writing at the beginning of a new school year to improve mathematics instruction and follow through with student achievement. The goals of the project need to be shared with other mathematics instructors. (See Science and Children, October 2006, on Measurement.)

Improving the quality of mathematics instruction is always possible. Innovative ways must be found to provide for the growth and development of mathematical instruction. Time should be given to teachers to observe each other’s teaching and diagnose the quality of instruction, based on recommended standards. If direct observation is not possible, then video recording of a lesson also provides opportunities for observation, as well as for critiquing.

Lesson plan evaluation has become quite popular in Japan and has been introduced in selected schools in the United States. A committee of mathematics teachers assesses in depth a lesson plan, which results in modification and change. The lesson is then taught, evaluated and required changes made. If changes are made, then re-teaching occurs.

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It is important that mathematics teachers work together in devising the best curriculum possible.

Summative and Formative Evaluation

A good mathematics teacher is a good evaluator. He/she observes rather continuously if students are actively engaged in each lesson. The following learner traits are also observed.

• kinds and types of errors made in daily work

• remediation for mistakes made

• responsibility of the student for task completion

• interest shown in mathematics

• behavior exhibited making for a learner-centred classroom (Ediger, 1997).

New topics to be presented are built upon what has been learned previously. Thus teachers furnish background information for future objectives. Readiness for achieving the new concepts is then in evidence.

Quality sequencing is important in student learning. With improved sequence or order in learning activities, the learner is more likely to experience success. The teacher must encourage careful listening to each learning step.

Summative evaluation occurs within an ongoing unit of study, and may well involve the following.

• teacher written tests such as multiple choice and essay

• discussions within the class as well as in small groups

• teacher and student self evaluation in terms of desired criteria

• student attentiveness in class

(National Council of Teachers of Mathematics, 1989).

Using formative evaluation results to identify specific errors, the teacher is better able to assist students where needed. Adequate teacher attention must be given to formative evaluation results to notice the kinds of errors made and to overcome the making of each. Following diagnosis and remediation, each student should be ready to attain the next objective.

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Palanivasan (2006) lists the following common causes of errors made by students in mathematics.

• computation

• lack of reasoning ability

• poor procedure or complete absence of a systematic approach

• difficulty in selecting a procedure

• failure to comprehend the problem

• insufficient and ineffective reading skills

• vocabulary difficulties

• short attention span

• inability to select essential data

• carelessness in reading the data

• lack of interest

• prevalence of vague guesswork in an attempt to secure a quick answer.

It is necessary for mathematics teachers to study the kinds of errors students make in order to strengthen teaching and learning situations. Such formative evaluation provides opportunities for teachers to make changes in an ongoing unit of study.

Summative evaluation is used to observe the end-of-unit progress of each learner. The teacher needs to notice how much progress each student has made since the beginning of the now completed unit of study.

Which essential changes might the teacher make in teaching a unit to a subsequent group? These entail objectives stressed, the sequence of mathematical learning activities, explanations provided by the teacher in diagnosis and remediation, grouping procedures, and the securing of learner attention, among others. The purpose is to get the mathematics unit ready for the next set of students to be taught. (See Peressini, 1997.)

It is important for the teacher to use appropriate psychological principles of learning in teaching and learning situations. Thus, each student must perceive the following.

• purpose in each sequential step of learning

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Acceptable reasons for working on each problem need to be present. It is difficult for learners to be serious about purposeless work.

• interest in mathematics problems

If student interest is lacking, the teacher must devise approaches to stimulate learner achievement.

Conclusion

The mathematics teacher must be well prepared to teach mathematical subject matter. He/she needs to grow continually as a professional teacher. Relevant objectives and quality learning activities for students to achieve the stated ends of instruction must be present. Teachers need to use evaluation procedures that are diagnostic. It is important to use the tenets of educational psychology in teaching and learning situations.

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References

Ediger, M. (1997). Teaching mathematics in the elementary school. Kirksville: Simpson Publishing.

Ediger, M., and Rao D. (2000). Teaching mathematics successfully. New Delhi: Discovery Publishing House.

Kennedy, L., & Tipps, S. (1991). Guiding children’s learning of mathematics. Belmont, California: Wadsworth Publishing Company.

National Council Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, Virginia: NCTM.

Palvanivassan, M. (2006). Analysis of errors committed by higher secondary students in calculus. Ph.D. thesis evaluated by the author for the University of Madras, Chennai, India.

Peressini, D. (1997). Parental reform of mathematics education. The Mathematics Teacher, 90 (6).

Science and Children, (2006). 44 (2). Measurement. Arlington, Virginia: National Science Teachers’ Association.

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Mathemagic V: The Voyage Home

Al Sarna teaches mathematics at Kitsilano Secondary School in Vancouver.

Here are a few more mathemagical “tricks” that you can use to spice up a math class or introduce a new lesson.

A Magic Sum and Product

• The spectator starts with a random list/column of integers, selects any two numbers (say A and B) from the column and replaces them with the number AB + A + B at the top of the next column. The process ends when only one number remains. The mathemagician reveals the final number in advance.

Example: The spectator chooses a column of numbers {2, 4, 5, 6}. The mathemagician reveals that the final number remaining will be 629. The spectator chooses the 2 and the 5, crosses them out, and replaces them with the number AB + A + B = (2)(5) + 2 + 5 = 17 at the top of a new column. The process continues until only the number 629 remains.

2 4 17 5 4 34 6 6 17 629

SolutionThe mathemagician forms the product of one more than each of the numbers and then subtracts 1. Our example would be

(2 + 1)(4 + 1)(5 + 1)(6 + 1) – 1 = 629.

When the two numbers A and B are replaced with AB + A + B, the result equals (A + 1)(B + 1) – 1.

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Let the set of numbers be S = {A1, A

2, A

3, … , A

n}. The mathemagician forms

the product P = (A1 + 1)(A

2 + 1)(A

2 + 1) … (A

n +1) – 1.

Another Baffling Prediction• A spectator shuffles the deck and places it on the table. The mathemagician writes the name of a card and hands it to another spectator to hold. Twelve cards are dealt face down. The spectator touches any 4 and they are turned face up. The remaining cards are returned to the bottom of the deck. A count is established and the card is found!

1. Suppose the 4 face up cards are 4, 7, 9, and King. For now we will let face cards have a value of 10, but you could let the spectator assign any value less than or equal to 10 to face cards and the trick will still work.

2. On top of the 4 cards, deal enough cards to bring the total up to 10. For instance, on the four you would count “five, six, seven, eight, nine, ten.” On the seven you would add 3 more cards, on the nine you would add 1 more card, and on the King no more cards would be added.

3. The values of the four cards are now added: 4, 7, 9, and King would equal 30. The spectator takes the deck and counts down to the 30th card and turns it over.

4. The mathemagician’s prediction is now read and it is, of course, the name of the card turned over.

5. The prediction is the bottom card of the deck after the spectator has shuffled it. When you put the 8 left over cards on the bottom in the beginning this makes the seen card the 44th card from the top (or ninth card from the bottom).

SolutionLet the value of the four face up cards be A, B, C, and D. If you are counting up to 10, then the number of cards in each pile is 11 minus the value of the face up card as shown:

11 – A

A

11 – B 11 – C 11 – D

B C D

The number of cards in the four piles is equal to

(11 – A) + (11 – B) + (11 – C) + (11 – D) = 44 – (A + B + C + D).

Vector 4�

The number of cards left in the deck is

52 – [44 – (A + B + C + D) = 8 + (A + B + C + D).

You now deal down to the (A + B + C + D)th card and are now at the 9th card from the bottom (or the 44th card from the top). Remember, you can let the spectator assign any value less than or equal to 10 to the face up cards and it still works.

The Amazing Twin• This is a nice variation on the last trick and lends itself to various extensions. The spectator chooses 3 cards from a full deck, and the mathemagician chooses one from the deck to put in his pocket. Then the spectator chooses a card from the deck that the 3 previous cards determine. It turns out that the mathemagician’s card has the same color and the same face value as the one chosen by the spectator.

1. Have the spectator choose any 3 cards, without looking, and place them face down. Have the spectator shuffle the deck.

2. The mathemagician announces that he will now choose a card that will have the same color and same face value as the one the spectator will choose.

3. The spectator now turns over his 3 cards and counts from the face up value of the card (Jacks are 11, Queens are 12, and Kings are 13) to 15. Place these face down in front of the face up cards. For example, if the face up card is an 8, count “9, 10, 11, 12, 13, 14, 15.”

4. The spectator now counts down, from the remainder of the deck, to the sum of the face up cards. For example, if his cards were an 8, 5, and a Queen, he would count down to the 25th card. When he turns it over it will be the “twin” of the one the mathemagician has.

SolutionThe mathemagician chooses a card that matches the color and face value of the 4th card from the bottom and puts it in his pocket. (The 4th card from the bottom might have to be changed if the spectator has already chosen its “match.”)

Let the three chosen cards be A, B, and C as follows:

A B C

Spring 200844

If you are counting up to 15, then the number of cards in each pile (counting the original card) is equal to (16 – A), (16 – B), and (16 – C) respectively. The number of cards left in the remainder of the deck is equal to

51 – [(16 – A) + (16 – B) + (16 – C)] = 51 – 48 + (A + B + C) = 3 + (A + B + C).

This means that the (A + B + C)th card is the 4th from the bottom.

The following could be some extensions.• What card should we look for if we count up to 13?• What card should we look for if we use 4 cards and count up to 12?• What change would be necessary if we also let the spectator define

face cards differently? Perhaps letting them have a value of 5?

Revealing the Remainder• The mathemagician reveals the remainder after an arbitrary number

is divided by 9, and then multiplied by a number chosen by the mathemagician!

1. Choose any arbitrary number.2. Divide by 9 and tell the mathemagician the remainder.3. Multiply the original number by a number the mathemagician

chooses.4. Divide this result by 9.5. The mathemagician reveals this remainder. This remainder is equal

to the product of the sum of the digits of the original remainder times the number the mathemagician chose. The mathemagician simply keeps adding the digits.

Example:Suppose the number chosen is 5837. The remainder when it’s divided by 9 will be equal to 5 + 8 + 3 + 7 = 23. Now suppose the mathemagician chooses 7. The remainder when 5837 • 7 is divided by 9 will be equal to 7 • 5 = 35 and 3 + 5 = 8. In other words, 5837 • 7 = 40859, and the remainder when 40859 is divided by 9 is equal to 8.

SolutionThe remainder when any number is divided by 9 is equal to the sum of its digits. (Keep reducing to a single digit.)

Consider a general 4–digit number: N = 1000x + 100h + 10t + u.

Rewrite this number as N = 999x + 99h + 9t + (x + h + t + u).

Vector 45

Now it is seen that upon division by 9, the remainder is equal to the sum of the digits.

Consider any number N = 9q + r. Multiplying by any number, d, gives Nd = d(9q + r) + (dr). Clearly the remainder is dr.

Calendar Magic• A spectator circles any 4 by 4 block of dates on a calendar. He selects

4 numbers from the block according to the mathemagician’s directions and the mathemagician reveals the sum.

Example:1. The spectator circles any number in the 4 by 4 block selected.

2. He/she then eliminates all the numbers remaining in that row and that column.

3. Repeat the process until there are no more numbers to circle: as seen below, the spectator will have circled 4 numbers.

4. The magic sum of the 4 circled numbers will be equal to twice the sum of the endpoints of either diagonal.

Suppose 27 is selected first. Cross out everything else in the same row and same column. Suppose 15 is selected next. Again cross out everything else in the same row and column. If 23 is selected next, the only remaining number will be 7. Notice that the sum 27 + 7 + 15 + 23 is equal to twice the value of 27 plus 9 (or twice the value of 6 plus 30).

2007 March 2007Sunday Monday Tuesday Wednesday Thursday Friday Saturday

1 2 34 5 6 7 8 9 1011 12 13 14 15 16 1718 19 20 21 22 23 2425 26 27 28 29 30 31

Spring 200846

Solution

Consider the general 4 x 4 grid as it would appear in a calendar. x x + 1 x + 2 x + 3x + 7 x + 8 x + 9 x + 10x + 14 x + 15 x + 16 x + 17x + 21 x + 22 x + 23 x + 24

No matter which 4 numbers you pick, you will have only 1 number per row. So to add them, you will have at least (x + 0) + (x + 7) + (x + 14) + (x + 21). Also, you will have only 1 number per column. Thus your sum of the 4 numbers will also include (0 + 1 + 2 + 3). In other words, you will have 4x + 48 = 2(2x + 24). The sum of the diagonal endpoints is 2x +24.

Here are two challenging extensions you might want to use with your class. • What would the magic sum be if you used an arbitrary 10 by 10 grid (i.e.

the numbers from 1 to 100 filled in) and selected a 5 by 5 block?• What would the magic sum be if we used an n by n grid and selected

any k by k square, k < n?

Revealing the Crossed Out Digit• This is another variation on the many ways of revealing a crossed

out digit. The spectator writes down any number, with no zeroes, (at least 4 digits is nice) and forms a new number by reversing its digits. They subtract one number from the other and cross out any non-zero digit in this difference. They reveal the sum of the remaining digits in the difference to the mathemagician and he reveals the crossed out digit.

Example:Suppose the number is 2598. The number with the digits reversed is 8952. The difference between the two numbers is 6354. Suppose the 3 is crossed out. The sum of the remaining digits is 15. The mathemagician immediately reveals that the crossed out digit was a 3.

He does this by subtracting 15 from the next multiple of 9; 18 – 15 = 3. A faster way is to simply add the digits, 1 + 5 = 6, and subtract this sum from 9.

Vector 47

Solution

Without loss of generality, consider an arbitrary 4–digit number and the difference between it and the number formed when its digits are reversed:

N = 1000x + 100h + 10t + u – (1000u + 100t + 10h + x).

This equals 999x + 90h – 90t – 999u. Clearly this is divisible by 9. If the number is divisible by 9 then the sum of the digits of the number must be divisible by 9. This is seen by rewriting N = 1000x + 100h + 10t + u as N = 999x + 99h + 9t + (x + h + t + u). Now it is seen that if N is divisible by 9, then the sum of the digits must also be divisible by 9.

After a digit is crossed out and the sum of the remaining digits is revealed, all the mathemagician has to do is find the difference between this sum and the next multiple of 9. For those familiar with “casting out nines,” the easiest way is to simply reduce everything to a single digit, by adding the digits together as often as necessary to get to a single digit. Incidentally, the reason I restrict the crossed out number to one not containing any zeroes is that I do reduce to a single digit, and in this approach (casting out nines) you can’t distinguish between a 9 and a 0.

The Wizard’s Magic NumberThe following is from the Internet. The URL is

http://www.geocities.com/Baja/4954/blend1.html

• You pick a card and perform some calculations. The wizard picks his card but keeps it face down. In the end he reveals the units digit of your number.

The screens are shown below.Suppose your number is 7. The result of the calculations above will give the result 73. Now you click on the ten’s digit of your number on the following screen. (The directions are missing from the screen on the actual site.)

Spring 200848

Pick a card, any card. Don’t click on it, memorize it. The next step is the important step: Watch as the wizard selects his card!

The wizard says he doesn’t want to reveal his card until the end.

In the meantime, let’s get our calculations done.

Continue.

Multiply the value of your card by two.

Add two to this number.

Multiply that number by five.

Subtract seven from that number.

Remember the result. It’s a “magic number.”

Vector 49

When you move your mouse over the Wizard’s card it will reveal a 3.

SolutionWhen you follow the calculations given you will arrive at

2x –> 2x + 2 –>10x + 10 –> 10x + 3.

The calculations simply move your original number into the ten’s location and the unit’s digit will always be a 3.

The Lightning Calculator• The mathemagician multiplies a 3–digit number by two different 3–digit

numbers and adds the results faster than anyone with a calculator.

1. Have the spectator write down a 3–digit number twice and choose an arbitrary 3–digit number to multiply the first number written down by.

2. Under the second number the mathemagician writes down the 9 complement (the number whose digits will add to 9 with the original number, as the example shows).

3. To get the answer, the mathemagician subtracts 1 from the spectator’s number and then appends the 9 complement to this result.

Example: Let the spectator’s number be 437 and suppose they want the first one to be multiplied by 316. The mathemagician will write 683 (the 9 complement)

The wizard’s card is the blank one.

Spring 200850

under the second 437 as the example shows. The answer will be 436563. The 9 complement of 436 is 563.

Solution

We are really multiplying by 999, which is the same as multiplying by 1000 – 1. In other words, what we have is 437(1000 – 1) = 437 000 – 437.

Consider

It should be obvious why subtracting 1 from the number and appending the 9 complement works. Clearly there is nothing special about using 3–digit numbers.

A Simply Impossible Card Revelation• A spectator shuffles a deck of cards and deals two equal piles of any

number of his choice (unknown to the mathemagician). He notes the top card of one pile and places the second pile on top, giving the remaining cards in the deck to another spectator. The second spectator guesses where he thinks the chosen card will wind up. He counts this many cards (without telling the mathemagician the number) from the deck and places them on the original packet. After a simple deal, the mathemagician reveals the card.

1. The spectator deals some of the deck into two equal piles. The number is not important, just that the piles be equal and that some of the deck be left over. It will speed things up if the piles contain no more than 15 cards each.

2. He notes the top card of one face down pile and then places the other pile face down on top of it.

3. A second spectator guesses where the chosen card will appear. Suppose he selects 8.

4. He deals 8 cards from the remainder of the deck and places them face down on the pile. The number 8 is not revealed yet.

5. The mathemagician now takes the pile, grasping it face down from above and in the right hand. Now use the left thumb to draw off the top card while the fingers of the left hand simultaneously draw off the bottom card. This

+ 437316 437

6× × 883

000 437437

−

Vector 5�

pair of cards is placed faced down. Continue this process putting the pairs on top of one another until all the cards have been drawn off and placed in a pile onto the table.

6. In plain view, the mathemagician takes the top card and places it at the bottom of the pile.

7. Now ask the spectator his number, 8 in our example. Deal 8 cards off the top of the pile. The next card will be the chosen card.

Example/ Solution:Consider the movement of the cards in the first example where the spectator chooses piles of 3, and in the second example where the spectator chooses piles of 4. The card marked (*) is the chosen card. The number of cards in each initial pile is x, and the number chosen by the second spectator is n. The second column is after the top/bottom cards are dealt off. The third column simply moves the top card to the bottom.

Example 1: The first spectator chooses two piles of 3 and the second spectator guesses the chosen card will appear after 2 cards.

Example 2: The first spectator chooses two piles of 4 and the second spectator guesses the chosen card will appear after 5 cards.

AB

n

x

=

=

2

123456

2 6*

2314

5

6

*B

A

314

5

62

*B

A

Notice the chosen card is originally at

( ) ( )2 3 1n x

+ + .

In the last step it is at the position

( )2 1n

+ .

The deal moves x (3) cards beneath the number of cards (2) already underneath the selected card.

Spring 20085�

The interested reader can now duplicate the above examples using only n and x.

The College Bet• The spectator shuffles the deck as much as he likes and then

cuts the deck into 3 face down packets. The mathemagician now states that she will bet even money that an ace, king, or a five (or any 3 cards you like) will be on top of one of the piles. About 55% of the time you will win. This is a nice intro to probability with combinations (or the fundamental counting principle) for Principles of Math 12.

Solution

The probability that you will win with the first card is 12 out of 52 (the number of aces, kings, and fives out of the total number of cards). The probability of winning with the second card is equal to 40/52 (the 40 remaining cards) times 12/51 (the probability of winning with the second card). This equals 40/221. The probability of winning with the third card is equal to 40/52 times 39/51 times 12/50 (a different card on the first pile, a different card on the second pile, and one of the 12 winning cards on the third pile). This product is 12/85. Since we are only interested in winning on the first, or the second, or the third card

ABCDE

n

= 5

12345678

2 8*

=x

Notice the chosen card is originally at

( ) ( )5 4 1n x

+ + .

In the last step it is at the position

( )5 1n

+ .

The deal moves x (4) cards beneath the number of cards (3) already underneath the selected card.

13E4D5*C6B7A82

213E4D5*C6B7A8

Vector 5�

only, the results are mutually exclusive and we can add the probabilities. The sum of the probabilities is

313

40221

1285

4785

0 55+ + = . .

Another way of looking at it would be to think of the number of 3–card combinations from a 52-card deck: 52 3 22100C = . The number of possibilities excluding the aces, kings, and fives is calculated as follows:

40 39 383 2 1

9880× ×× ×

= .

This means that the number of ways of winning is 22100 – 9880 = 12220. In other words, you can win 12220 – 9880 = 2340 more ways than you can lose.

The probability of winning is 1222022100

0 55. .

FinallyMany more mathemagical “tricks” can be found in the books listed in the bibliography or by doing a search on the Internet for topics like “Math Magic,” etc. Relax, enjoy, and may the force stay with you.

≅

≅

Spring 200854

Bibliography

Fulves, K. (1983). Self-working number magic. New York: Dover Publications.

Fulves, K. (1984). More self-working card tricks. New York: Dover Publications.

Gardner, M. (1956). Mathematics magic and mystery. New York: Dover Publications.

Gardner, M. (1975). Mathematical puzzles and diversions. Great Britain: Pelican Books.

Mira, J. (1971). Mathematical teasers. New York: Barnes and Noble.

Simon, W. (1964). Mathematical magic. New York: Charles Scribner’s Sons.

Vector 55

Methods of Integration

Duncan McDougall teaches at TutorFind Learning Centre in Victoria.

It has been my experience that students continue to struggle with methods of integration despite our best intentions, instructions and coaching. Let us all agree that not everyone learns the same amount at the same rate. Let us also understand that there are some teaching styles that don’t transmit all the necessary concepts and/or details to all the various learning styles that exist. In my mind, it is necessary to make the instruction as clear and simple as possible without being condescending or over simplified; however, we do have to assume some basic knowledge.

I have seen second and third year students studying differential equations who are frustrated because they couldn’t complete the resulting integral. These are students who should be able to calculate most integrals with little or no difficulty but who haven’t yet mastered the techniques. This shows up at the worst of times: final exams. It is quite possible that the subtleties of the particular techniques weren’t grasped properly or practised enough. There are more reasons of course, but these are some of them.

In this text, we will attempt to rectify this situation by plainly putting into words those tricks and techniques that may alleviate some of the difficulties still experienced. All four analytical methods have little quirks that need to be clarified so that mastery can begin early and permanently.

For example, how would you integrate ln xx

dx ∫ versus ln x dx∫ ? And how would you integrate 1

92xdx

+∫ versus 192x

dx−∫ ? The principle concern

whenever integrating is to know the best approach. Using the most efficient method saves a lot of work and frustration. Being able to spot the subtleties of each approach permits us to make better decisions. Therefore, I would suggest that the most difficult part of integration is deciding on the approach.

As there are so many types of integrals, it is difficult at first to discern the various types from one another. In terms of pedagogy, my thinking is that we classify integrals fundamentally, get experience with these, and then branch out to the more intricate questions. Hence, a strategy for beginners is to learn and master substitution, parts, trig substitution and partial fractions. We will look at each method individually, noting its subtleties as examples are demonstrated.

Duncan McDougall

Spring 200856

Integration by Substitution

This method is best used when the derivative of part of the integrand already appears in the integrand. This does appear confusing when we first read it; let us clarify with a few examples.

1. 3 22 3 2x x x x dx−( ) −∫

At the onset, this looks frightening. The length of the integrand alone is enough to scare most! However, we do notice that the derivative of x3 – x2 (namely 3x2 – 2x) appears as part of the integrand. We now know two things: we can use substitution and we know what to let u equal.

Let u x x= −3 2 (the function whose derivative also appears in the integrand).

Therefore, dudx

x x= −3 22 and du x x dx= −( )3 22 As you see, we multiply by dx in order to see clearly what portion will be replaced by du.

It is very important to remember that all portions of the given integral are to be put in terms of u and du. Now let us substitute u and du into the original integral. We get

Now this is an easier equation to integrate! Integrating it we get

.

Now remember that the original is not in terms of u but in terms of x, so don’t forget to substitute back. Our final form is

23

3 232( )x x C− + .

As substitution is the easiest and most efficient method, we don’t need to try another method if this one works. Let us examine this technique applied to other functions.

23

32u C+

Vector 57

2. ln xx

dx∫

The derivative of ln x is 1x

so we let u = ln x where dudx x

=1 . Therefore, du

xdx=

1 .

Substituting in du and u we haveln x

ux

dx du ∫ ∫= .

This integrates to uC

2

2+ .

Now, putting the equation back in terms of x we get

(ln )xC

2

2+ or ln2

2x

C+ .

Let’s try another example.

3. xe dxx2

∫Let u = x2 so du = 2x dx.

Don’t worry that the constant factor 2 isn’t in the original integral. As long as the variable is there (in this case the x), we can introduce the factor 2 after the integral sign provided we compensate for it by a factor of 1/2 in front of the integral sign.

Therefore, we are really only multiplying the integral by 1.

xe dx xe dxx x2 212

2∫ ∫=

So now our du is present precisely in the integrand, and we can substitute in u and du.

12

12

22

x dx due ex u∫ ∫= .

When we integrate this through we end up with

12

e Cu + .

Now simply substitute for u to make the equation in terms of x and we get

12

2

e Cx + .

Spring 200858

Here’s another one to try.

4. sin cos4 x x dx ∫Let u = sin x so du = cos x dx

Now substitute u and du into the original equation and we get

sin cos4 4x ux dx du ∫ ∫=

Doesn’t this look much easier to integrate? When we do, we have

15

5u C+ .

Now substitute back into terms of x and we get

15

5(sin )x C+ or 1

55sin x C+ .

5. x x dx−∫ 1

Now here’s a tricky one. We’ll let u = x – 1 so du = 1 dx.

Now we still have an x left unaccounted for so we use the equation u = x – 1 and solve for x. We get u = x + 1.

Now let’s substitute in u and du and the value we found for x so that the entire equation is in terms of u.

x x u udx du− = + =∫ ∫1 1 ( )

This still looks scary so multiply out and we have

u u du u du + ∫∫ .

We can simplify further and write it as

u du u du32

12 + ∫∫ .

Integrating we get 25

23

52

32u u C+ +

= 25

1 23

152

32( ) ( )x x C− + − +

.

Vector 59

Integration by Parts

This method is best used when we cannot use the substitution method. We tend to look for an integral made up of functions that seem to be unrelated. If we do not see the derivative of part of the integrand already appearing in the integrand then we look for algebraic and/or transcendental and/or trig functions, or a combination thereof.

Recall our definition for integration by parts: u dv uv v du ∫ ∫= − .

Integration by parts breaks down into one of the following two situations.

1. a single portion 2. a double portion

Let us start with an example of situation 1.

1. ln x dx∫

Here we have situation 1 because we have a single function whose derivative does not appear anywhere else in the integral. In terms of method for a situation 1 type integral, always let u equal the given function and always let dv equal the dx.

Let u = ln x, so dudx x

=1 and du

xdx=

1 .

Let dv = dx so v = x.

Now using the definition and substituting in our values, we get

u dv uv v du = − ∫∫

ln lnx dx x x xx

dx = − ∫∫1

= − ∫x x dxln

= − +x x x Cln .

Here is another example of a situation 1 type integral.

Spring 200860

2. arctan x dx ∫Once again, we have a single function whose derivative does not appear anywhere else in the integral.

Let u = arctan x, so u dv uv v du = − ∫∫ .

Let dv = dx so v = x.

This gives

.

arctan arctanx dx x x xx

dx = −+∫∫1

1 2

= −+∫x xxx

dxarctan1 2

= − + +x x x Carctan ln( )12

1 2 .

Now let’s try an example of situation 2.

3. xe dxx∫This is a situation 2 integral because x and ex aren’t related as functions per se. Since we have more than one given function, it becomes a matter of best choices for both u and dv.Let u = x, so du = 1 dx.

Let dv = ex dx so v = ex.

Now, substituting back using the definition we get

xe dx xe e dxx x x= − ∫∫

Choices for u and dv: We tend to let u equal any function that would eventually disappear under repeated differentiation. These usually are positive powers of x. As for dv, we tend to let this equal any function whose derivatives go in cycles such as trig functions or eax.


= − +xe e Cx x

= − +e x Cx ( )1


Vector 6�

Unfortunately, we have an exception to all this and that is any integrand containing ln x. Whenever this happens, we immediately let u = ln x and dv be whatever is left. The reasoning for this is that the beginner doesn’t necessarily know that the anti-derivative of ln x is x ln x – x Even if the beginner did know this fact, letting du = ln x makes the process of solving the integral much longer. Here is an example of this.

4. x x dxln ∫We would not use substitution because the derivative of ln x does not appear in the given integrand. Instead we try integration by parts (situation 2).

Let u = ln x, so dux

dx=1

Let dv = x dx, so v x=

2

2.

We have u dv uv v du = − ∫∫ ,

x x dxx

xx

xdxln ln = − •∫ ∫

2 2

2 21

= − ∫x

x x dx2

212

ln

The Vicious Cycle: What do we do for a situation 2 integral whereby neither of the given functions disappears under repeated differentiation: i.e. both functions have cyclical derivatives?

Here is an example of this type of integral.

5. e x dxx cos ∫As we can see, neither of the derivatives disappears so our choices for u and dv are arbitrary. Since we will have to do this process more than once for this integral, we should be consistent with our choices. That is, if we let u = ex the first time around, then we should let u = ex the second time around too.

Let u = ex, so du = ex dx.

Let dv = cos x dx, so v = sin x.

= −

+x

xx

C2 2

212 2

ln

= − +x

xx

C2 2

2 4ln

Spring 20086�

Using our definition, we get:

u dv uv v du = − ∫∫e x dx e x e x dxx x xcos sin sin = −∫ ∫ .

Golden Rule of Parts:

If, after our initial choices of u and dv, the resulting integral is more difficult then the original, then stop! Instead, make different choices for both u and dv, and begin again.

Here, however, e x dxx sin ∫ is not more complicated but is as complex as the original, so begin again by solving e x dxx sin ∫ .

Let u = ex, so du = ex dx.

Let dv = sin x dx, so v = –cos x.

By definition we have

u dv uv v du = − ∫∫e x dx e x e x dxx x xsin cos cos ∫ ∫= − + .

Now substituting this new equation into the original

( e x dx e x e x dxx x xcos sin sin = −∫ ∫ ) we get:

e x dx e x e x e x dxx x x xcos sin cos cos ∫ ∫= − − +

= + − ∫e x e x e x dxx x xsin cos cos .

In order to break the cycle, we simply bring −∫ e x dxx cos over to the left-hand side: in other words, add e x dxx cos ∫ to both sides of the equation.

Now we have

2 e x dx e x e x Cx x xcos sin cos ∫ = + + .

This simplifies to

.e x dx e x x Cx xcos sin cos ∫ = +( ) +12

Vector 6�

Integration by Trig Substitution:

This method is best used when we cannot use Substitution or Parts as seen previously. The key factors are the sum or difference of two perfect squares, usually under a square root sign.

Here is a table of the identities that you will encounter:

Identity Algebraic Form

Integral Examples

Substitution

(1)

sin coscos sin

2 2

2 2

11

θ θθ θ

= −

= −

xax a

=

=

sin

sin

θ

θ

or

We could use xa

= cosθ but the derivative

involves a minus sign.(2)

sec tancsc cot

2 2

2 2

11

θ θθ θ

= +

= −

xax a

=

=

tan

tan

θ

θ

or

We could use xa

= cotθ but the derivative

involves a minus sign.

(3)

tan seccot csc

2 2

2 2

11

θ θθ θ

= −

= −

xax a

=

=

sec

sec

θ

θ

or

We could use xa

= cscθ but the derivative

involves a minus sign.

Following is an example that must be solved by using trigonometric substitution.

1. 1

1 2−∫ xdx

Looking at the table above, we see that we should use identity(1).

We let x = sinθ so dx d= cosθ θ . When we substitute in we get

11 2−∫ x

dx

a x2 2− 12 2a x

dx−∫

a x2 2+ 12 2a x

dx+∫

x a2 2− 12 2x x a

dx−∫

Spring 200864

=

−∫1

1 2sincos

θθ θ d

= ∫1

2coscos

θθ θ d [identity (1)]

= ∫1

coscos

θθ θ d

= = + = +∫ d C x Cθ θ Arcsin .

Here is another example.

2. 11 2+∫ x

dx


We let x = tanθ so dx d= sec2 θ θ . When we substitute in we get

1

1 2+∫ xdx

=+∫

11 2

2

tansec

θθ θ d

= ∫

12

2

secsec

θθ θd

= = + = +∫ d C x Cθ θ Arctan .

Try this example.

3.1

12x xdx

−∫


We let x = secθ so dx d= tan secθ θ θ . When we substitute in we get

112x x

dx−∫

=−∫

112sec sec

tan secθ θ

θ θ θ d

=

−∫1

12sectan

θθ θ d

Vector 65

= ∫

12tan

tanθ

θ θ d

= ∫1

tantan

θθ θ d

= = + = +∫ d C x Cθ θ Arcsec .

Now here is a tricky one.

4. 59 16 2−∫ x

dx

First we want to change the 9 to a 1 so we take out a 9 from inside the brackets. We also take the 5 out for simplicity. Last, the 9 comes out from under the square root sign as a 3.

=

−

∫5 1

3 1 43

2

x

dx

=−

∫5 1

9 1 169

2xdx

59 16 2−∫ x

dx

Now this looks more familiar! We recognize this from the table above as identity(1).

We let 43

x = sinθ . Thus x =34

sinθ and dx d=34

cosθ θ .

Now go ahead and solve the problem as in example 1.

Here’s another tricky one.

5. 14 52x x

dx+ +∫

First we need to complete the square, so we split up 5 into a 4 and a 1.

14 52x x

dx+ +∫

Spring 200866

=+ + +∫

14 4 12x x

dx .

=+( ) +

∫1

2 12xdx .

Now this looks familiar! We recognize this from the table above as identity(2).

We let x + =2 tanθ and substitute tanθ into the equation. Now you can integrate.

Partial Fractions:

This method is best used when we cannot use substitution, parts, or trig

substitution. Here, we will look at rational expressions of the form P xQ x

( )( )

whose

denominator is factorable. For example, 111 302x x

dx+ +∫ has a factorable

denominator that becomes 1

6 5( )( )x xdx

+ +∫ .

Note: If the denominator is not factorable, consider completing the square and using trig substitution.

Let’s go through an example.

Example 1 1

11 302x xdx

+ +∫

We factor the denominator and we get

1

11 302x xdx

+ +∫ = 1

6 5( )( )x xdx

+ +∫ .

Working with the rational expression, we let

1

6 5 6 5( )( ) ( ) ( )x xA

xB

x+ +=

++

+.

Now to find constants A and B, we multiply both sides of this identity by the common denominator (x + 6)(x + 5). The result is

1 = A(x + 5) + B(x + 6) = Ax + 5A + Bx + 6B

Vector 67

Now we’ll group the x’s together and the numbers together and we get

1 = x(A + B) + (5A + 6B).

Since we have no x term, we write 1 as 0x + 1 and then equate the components.

0x +1 = x(A + B) + (5A + 6B)

We now have 0x = x(A + B) and 1 = (5A + 6B).

0x = x(A + B)

0x = A + B

–A = B

Now we can put everything in terms of A. We substitute –A in for B and the result is

1 = 5A + 6B

1 = 5A – 6A

1 = –1A

A = –1 .

And since –A = B, we have B = 1. Now we substitute in the values for A and B in the original integral and we get

1

6 516

15( )( ) ( ) ( )x x

dxx

dxx

dx+ +

=−+

++∫ ∫∫ .

Therefore, we have 1

6 56 5

( )( )ln ln

x xdx x x C

+ += − + + + +∫

=++

+ln xx

C56 .

Spring 200868

Four Golden Rules of Partial Fractions

To decompose P xQ x

( )( )

into partial fractions, do the following.

1. If the degree of Q(x) is greater than the degree of P(x), proceed to step 2 below.

If the degree of Q(x) is not greater than the degree of P(x), then divide P(x) by

Q(x) to obtain P xQ x

h xR xQ x

( )( )

( ) ( )( )

= + [where h(x) is a polynomial]. Then apply

rules 2, 3, and 4 below to the denominator of the rational function R xQ x

( )( )

.

For instance, consider P xQ x

x x xx x x

( )( )

=− + +

− − +

4 2

3 2

2 4 11

. We note that the degree of

Q(x) is 3, but the degree of P(x) is 4. Since the degree of Q(x) is not greater than the degree of P(x), we perform long division obtain

x x xx x x

xx

x x x

4 2

3 2 3 2

2 4 11

1 41

− + +− − +

= + +− − +

.

Then, we would apply rules 2, 3, and 4 below to decompose the new denominator x3 – x2 – x + 1 into partial fractions. (We will revisit this later as example #4.)

2. Factor the denominator Q(x) completely. The factors will be of the form (px + q)m and/or (ax2 + bx + c)n where (ax2 + bx + c) is not factorable. (If the denominator is not factorable, other methods must be used.)

3. For each factor of the form (px + q)m the following m partial fractions must be

included.

D

px qD

px qD

px qD

px qn

m1 2

23

3( ) ( ) ( ) ( )++

++

++

+…+

4. For each factor of the form (ax2 + bx + c)n, the following n partial fractions must be included.

E x Fax bx c

E x Fax bx c

E x Fa

1 12

2 22 2

3 3++ +

++

+ ++

+( ) ( ) ( xx bx c

E x Fax bx c

n nn2 3 2+ +

++

+ +) ( )…+

Vector 69

Example 2

252 32( ) ( )x x

dx− +∫

As per rules 3 and 4 above, we will decompose the integrand into the following partial fractions.

25

2 3 2 2 32 2( ) ( ) ( ) ( ) ( )x xA

xB

xC

x− +=

−+

−+

+

Now multiply each side by the common denominator ( x2 – 2)2(x + 3). The result is

25 = A(x – 2)(x + 3) + B(x + 3) + C(x – 2)2

25 = A(x2 + x – 6) + B(x + 3) + C(x – 2)2

Rewriting the 25, and collecting terms, we have

25 = Ax2 + Ax – 6A + Bx + 3B + Cx2 – 4Cx + Cx

0x2 + 0x + 25 = (A + C)x2 + (A + B – 4C)x + (4C+ 3B – 6)

Therefore, 0x2 = (A + C)x2, so 0 = A + C and –A = C.

We also have 0x = (A + B – 4C)x , so 0 = A + B – 4C.

Substituting –A for C, we get 0 = A + B – 4(–A) and 0 = 5A + B and –5A = B.

Last, 25 = –6A + 3B + 4C so 25 = –6A + 3(–5A) + 4(–A).

Thus, 25 = –6A –15A – 4A and 25 = –25A.

Our results are A = –1, B = 5 and C = 1.

Now substituting these values back into the original we get

252 3

12

52

132 2( ) ( ) ( ) ( ) ( )x x x x x− +

=−−

+−

++

= − − − − + + +−ln ( ) lnx x x C2 5 2 31

= −

+−

−−

+ln( )

xx x

C32

52

252 3

12

52

132 2( ) ( ) ( ) ( ) ( )x x

dxx x x− +

=−−

+−

++

∫

∫ dx

Spring 200870

Example 3 32 12( )( )x x

dx+ −∫ .

By rules 3 and 4 above, we decompose the rational expression as follows.

32 1 2 12 2( )( ) ( ) ( )x x

Ax Bx

Cx+ −

=++

+−

To find the constants A, B, and C, we multiply both sides of this identity by the common denominator (x2 + 2)(x – 1). The result is

3 = (Ax + B)(x – 1) + C(x2 + 2) 3 = Ax2 – Ax + Bx – B + Cx2 + 2C 3 = (A + C)x2 + (B – A)x + (2C – B).

Since we have no x2 or x terms, we write 3 as 0x2 + 0x + 3. This implies

0x2 = (A + C)x2

0 = (A + C) G = – AAlso, we have 0x = (B – A)x

0 = (B – A)

A = B

Finally, we can write 3 = –B + 2C.

Putting everything in terms of A we get

3 = –A + 2(–A) = –3A A = –1This means B = –1 and C = 1. Substituting in the values for A, B, and C into the original fraction we get

32 1

1 12

112 2( )( ) ( ) ( )x x

xx x+ −

=− −

++

−

32 1

1 12

112 2( )( ) ( ) ( )x x

dxx

xdx

xdx

+ −=

− −+

+−∫ ∫ ∫

= −

+−

++

−∫ ∫ ∫x

xdx

xdx

xdx

( ) ( ) ( )2 221

21

1

= − + − + − +12

2 22

22

12ln lnx x x CArctan

Vector 7�

As promised, here is the example mentioned in Rule 1 above.

Example 4x x x

x x xdx

4 2

3 2

2 4 11

− + +− − +∫

Since the degree of the numerator (4) is greater than the degree of the denominator (3), we perform long division on the rational expression to obtain

x x xx x x

xx

x x x

4 2

3 2 3 2

2 4 11

1 41

− + +− − +

= + +− − +

Next we decompose the expression 413 2

xx x x− − +

into partial fractions.

Factoring the denominator we get

41

41 13 2 2

xx x x

xx x− − +

=− +( ) ( )

.

By Rule 3, we obtain4

1 1 1 1 12 2

xx x

Ax

Bx

Cx( ) ( ) ( ) ( )− +

=−

+−

++

.

Now multiply through by the LCM (x – 1)2(x + 1) and collecting like terms we have

4x = A(x – 1)(x + 1) + B(x + 1) + C(x – 1)2

= (A + C)x2 + (B – 2C)x + (B + C – A).

Rewriting 4x as 0x2 + 4x + 0 and equating coefficients gives us the following three equations.

A + C = 0

B – 2C = 4

B + C – A = 0

Solving this system, we get A = 1, B = 2, and C = –1.

Substituting these values in gives us the following solution.

x x x

x x xdx

4 2

3 2

2 4 11

− + +− − +∫

= + +

− − +

∫ x

xx x x

dx1 413 2

Spring 20087�

SummaryThe purpose of this article was to demonstrate these four methods of integration. We do recognize that there are methods that cannot be solved by using any one of the four methods mentioned. One could use numerical methods such as Riemann sums, the trapezoidal rule, and/or Simpson’s rule. Another method involves the use of series.

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= + +−+

−−

+x

xxx x

C2

211

21

ln

= + + − −−

− + +x

x xx

x C2

21 2

11ln ln

= + +−

+−

++

∫ x

Ax

Bx

Cx

dx11 1 12( ) ( )

= + +−

+−

+−+

∫ x

x x xdx1 1

12

1112( ) ( )

= + +− +

∫ x

xx x

dx1 41 12( ) ( )

Vector 7�

Solutions to the Fall 2007 Puzzles

Puzzle 1

Solutio

nThe

Source: Michael Shackleford. Used with permission.

Using only the numbers 1, 3, 4, and 6, together with the operations +, −, ×, & ÷, 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.

24 =6

1− 34

The solution is unique.

Spring 200874

Puzzle 2

Solutio

nThe

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

(Source: http:www.freepuzzles.com)

The key is the number 12. The number 12 holds the following properties.

52 + 122 = 132

92 + 122 = 152

It just happens that 5 + 9 = 14 .

13

5 9

15

12

14

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