Vector - BC Association of Math Teachers Helping Teacher K-12 School District ... learning outcomes...

Ve c t o rThe Official Journal of the BC Association of Mathematics Teachers

Summer 2011 • Volume 52 • Issue 2

Special Issue on Elementary Mathematics

Vector • Elementary Issue is published by the BC Association of Mathematics TeachersArticles and Letters to the Editors should be sent to:Peter Liljedahl, Vector [email protected]

Sean Chorney, Vector [email protected]

Membership Rates for 2010 - 2011$40 + GST BCTF Member$20 + GST Student (full time university only)$58.50 + GST Subscription fee (non-BCTF )

Notice to ContributorsWe invite contributions to Vector from all members of the mathematics education community in British Columbia. We will give priority to suitable mate¬rials written by BC authors on BC curriculum items. In some instances, we may publish articles written by persons outside the province if the materials are of particular interest in BC.

Articles can be submitted by email to the editors listed above. Authors should also include a short biographical statement of 40 words or less.

Articles should be in a common word processing format such as Apple Works, Microsoft Works, Microsoft Word (Mac or Windows), etc.All diagrams should be in TIFF, GIF, JPEG, BMP, or PICT formats. Photographs should be of high quality to facilitate scanning.

The editors reserve the right to edit for clarity, brevity, and grammar.

Summer 2011

The views expressed in each Vector article are those of its author(s), and not necessarily those of the editors or of the British Columbia Association of Mathematics Teachers.Articles appearing in Vector should not be reprinted without the permission of the editors. Once written permission is obtained, credit should be given to the author(s) and to Vector, citing the year, volume number, issue number, and page numbers.

Membership EnquiriesIf you have any questions about your membership status or have a change of address, please contact the BCAMT Membership Chair:Dave Ellis ([email protected])Notice to AdvertisersVector is published 3 times a year: spring, summer, and fall. Circulation is approximately 1400 members in BC, across Canada, and in other countries around the world.

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Technical InformationThe layouts and editing of this issue of Vector Elementary were done on a Dell using the following software packages: Adobe Acrobat Professional, Adobe InDesign, and Microsoft Word.

Summer 2011 2

3 2011-2012 BCAMT Executive5 Letter to the Editors

Carole Fullerton and Sandra Ball 6

What do they know? Assessing Mathematical Thinking in Kindergarten...

Carollee Norris 11Helping Students Reason About Fractions by Using Benchmarks

Janice Novakowski 18 Linear Measurements in Primary Classrooms

Carole Fullerton, Andrea Helmer, Selina Millar, Janice

Novakowski23 Investigating Pattern: The Big

Ideas Across the Grades

Lorill Vining 27 Marvellous Math:A Mathematical Reflection

Janice Novakowski on behalf of the Richmond Primary

Teachers’ Study Group31 Using Children’s Literature to

Inspire Mathematical Thinking

Ray Appel 36 “Math Again?”

Penny Morgan, Marthe Sivecki, and Cheryl Schwarz 41

Meaningful Collaboration Sparks a Successful Numeracy Action Plan

Janice Novakowski 47 Mathematics Apps for use with iPod Touches and iPads

Jeannie DeBoice 49

Book Review: Making Mathematics Meaningful for Students in the Intermediate Grades

52 Summer 2011 • Problem Set54 Interactive Math Websites

ON THE COVER: The cover is an interpretation of Kandinsky’s “Farbstudie”. It was created by Jesssica Rose Sutton who is in Grade 2 at Old Yale Road Elementary in Surrey, B.C. Jessica took into account the way that Kandinsky used colour to communicate. She also noticed that Kandinsky’s circles were not symmetrical and made hers in a similar manner.

3 Vector • Elementary Issue

The 2011–2012 BCAMT Executive

PresidentChris BeckerPrincess Margaret Secondary School (Penticton)Work: [email protected]

Past PresidentDave van BergeykSalmon Arm Secondary SchoolWork: [email protected]

Vice PresidentBrad EppSouth Kamloops SecondaryWork: [email protected]

SecretaryColin McLellanMcNair Secondary School (Richmond)Work: [email protected]

TreasurerKatie Wagner Robert A. McMath Secondary School (Richmond)[email protected]

Membership ChairDave Ellis Home: [email protected]

Elementary Representatives Sandra BallNumeracy Helping Teacher K-12School District 36 (Surrey)Work: [email protected]

Jennifer GriffinStride Avenue Community School (Burnaby)Work: [email protected]

Selina MillarNumeracy Helping Teacher K-12School District 36 (Surrey)Work: [email protected]

Carollee NorrisNumeracy Support Teacher School District 60 (Peace River North)Work: 250-262-6028 [email protected]

Donna WrightEcole Sandy Hill Elementary (Abbotsford)Work: 604-850-7131 [email protected]

Summer 2011 4

The 2011–2012 BCAMT Executive

Middle School RepresentativeDawn DriverH.D. Stafford Middle School (Langley)Work: [email protected]

Secondary RepresentativesMichael FinniganYale Secondary School (Abbotsford)Work: [email protected]

Sam MuracaLangley Secondary School (Langley)Work: 604-534-4171 [email protected]

Michèle RoblinHowe Sound Secondary (Squamish)Work: [email protected]

Danny YoungMoscrop Secondary SchoolWork: [email protected]

Christine YounghusbandHome: [email protected]

Bryn WilliamsProgram Consultant: Mathematics and Science (Burnaby)Work: [email protected]

Independent School RepresentativeRichard DeMerchantSt. Michaels University School (Victoria)Work: [email protected]

Post-Secondary RepresentativePeter LiljedahlAssistant Professor, Faculty of EducationSimon Fraser University (Burnaby)Work: [email protected]

Listserv ManagerColin McLennanMcNair Secondary School (Richmond)

NCTM RepresentativeMarc GarneauNumeracy Helping Teacher K-12School District 36 (Surrey)Work: [email protected]

Vector EditorsPeter LiljedahlAssistant Professor, Faculty of EducationSimon Fraser University (Burnaby)

Sean ChorneyMagee Secondary School (Vancouver)Work: [email protected]


Welcome to the BCAMT special Elementary Edition of VECTOR! As the BCAMT continues to promote excellence in mathematics teaching, we are pleased to offer this

issue in support of the elementary teachers in our province.

With the elementary math curriculum in full implementation, the challenges presented in the math classroom seem that much greater than in the past. In this edition of VECTOR, our hopes are that the elementary teacher will feel greater support, develop a broader repertoire of resources and gain a deeper understanding of the possibilities in their math classrooms.

All of the articles are contributions from teachers that are in the field, whether in the classroom or in supportive district roles. The topics include assessment, use of technology, a resource review and lesson ideas. The problem set revolves around the topics of patterning and equations and provides a way to differentiate instruction in the math classroom.

A heartfelt thank you and sincerest appreciation to the authors of the articles. Their passion for sharing and learning never cease to amaze us!

We hope this edition will encourage other elementary teachers to share ideas, experiences and developments in teaching and understanding the students and their mathematical “discoveries”. We gain so much from each other … we look forward to hearing from you!

Enjoy and please share!

Selina Millar Sandra Ball

Vector: Special Elementary Edition

Summer 2011 6

What do they know? Assessing Mathematical Thinking in Kindergarten By Carole Fullerton and Sandra BallCarole Fullerton and Sandra Ball work as district support staff in the Richmond and Surrey school districts respectively. They are excited at the learning and teaching potential of the “What Do They Know?” Assessment tool and welcome professional conversations about it. Please do not hesitate to contact us at: [email protected]; [email protected].

Kindergarten is a time for exploring – for play and talk and investigation. Motivated by

what’s important to them, our youngest learners come to understand key concepts through their play. This is true in mathematics more than other areas; little children spontaneously explore mathematical concepts through the activities they themselves choose. Kindergarten teachers know that their students quite naturally count and sort, order and compare, pattern and predict while at play in their classrooms.

We see so much capacity in our learners when we know what to look and listen for! Capturing and synthesizing this information in order to make instructional decisions is the fine art of teaching.

So what exactly is important to know – mathematically speaking – in kindergarten? What key skills and understandings are predictors of success in numeracy? And what tools – authentic, whole class and embedded in play-fullness – can we use to assess them?

This last question is harder to answer than one might assume. There are few quality numeracy assessments available for kindergarten-aged students, with the notable exception of the BC Early Numeracy Project Assessment. The Early Numeracy Project (ENP) was a collaboration between respected UBC teacher educators Dr. Heather Kelleher and Dr. Cynthia Nicol and teams of primary teachers throughout the province. Modeled on the Australian Early Numeracy Research Project and

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supported by its developer, Dr. Doug Clarke, the BC ENP is made up of an assessment and instructional component for K and K/1 classrooms. The assessment tool itself is rich and varied, and explores ideas within and well beyond the current K math curriculum, since it is developmentally framed, rather than tied to a curriculum. The downside to this formal research-based assessment tool is that it must be administered one-on-one, which makes it cumbersome to use with an entire class.

Looking for a tool that was less formal and more embedded in classroom practice, Sandra Ball and I came together in May of this year to design our own early numeracy assessment. Drawing on the ENP, we sought to create a tool that could be done in small groups or with the whole class; a tool that could be used to gather information in the fall and again in the spring; and a tool that might help to promote student-centered numeracy programs in early primary. A tall order!

An Overview of the Assessment

We started with a premise: that in order to be successful at the end of kindergarten, students needed to demonstrate capacity in the following areas of numeracy:

• Subitizing

○ The instant recognition of a quantity

• Partitioning or decomposition

○ The ability to break apart a number and put it back together again

• Patterning

○ The ability to recognize, represent and describe repeating patterns with different attributes

Much of the above list is explicit in the prescribed learning outcomes in the K and grade 1 curricula. Likewise, the research we have read and the wise teachings of John Van de Walle, Marian Small and the ENP developers all support our choice of these three predictors of numeracy success.

After all, we know that the capacity to subitize is important. When children can “see” quantities at a glance without counting, they understand that 5-ness has a shape – or rather a series of shapes. Regardless of its arrangement, though, 5 is still five, and young learners who have grasped this

We see so much

capacity in our

learners when

we know what to

look and listen

for!

Summer 2011 8

concept are more likely to be able to “count on” from a set, to “hold” a set in their mind, and to think more flexibly around number.

Subitizing is connected to the mathematical idea of partitioning. When students know that there are many ways to break up a set of 6 objects – into 5 and 1, 3 and 3, 4 and 2, etc – and that these smaller groupings still make 6 when they are put back together, they are demonstrating an understanding of partitioning. The ability to partition is critical to developing conceptual understanding of the operations: addition and subtraction as well as division and multiplication.

A child’s ability to pattern lays the foundation for algebraic reasoning – being able to predict what will come next (or even “before”) in a pattern is an essential skill. The more complex the pattern, the better! Students who can develop and extend a pattern with a complex core truly understand what a pattern is – and can generalize their learning to more challenging contexts.

Having identified these three predictors of success, Sandra and I set out to design tasks to match each one. Ultimately we

created a set of assessment tasks for the fall and another set of parallel tasks for the spring. The tasks themselves are very much like what a kindergarten teacher would do naturally in her classroom. This was important for Sandra and I to incorporate into the assessment. In fact, to make the assessment more classroom based, we included the use of a storybook for both the fall and spring assessments. Even the name for the assessment was consciously chosen. We call it the “What Do They Know?” numeracy assessment, since it focuses on students’ capacity, and how to build it…

The Tool at a Glance

Teachers begin by asking their students to perform some subitizing activities and making notes about their comfort level and flexibility in “seeing” number. Next, the teacher reads a story aloud to the class and poses an open-ended story problem of the children, who are then invited to model and solve the problem independently. Lastly, students create patterns to match characters on the stories and tell about what they have created.

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The spring task mirrors the fall task exactly, but includes more complex numbers and pattern cores.

From the description, above, it is clear that these tasks are informal – or rather they are structured like regular classroom lessons, with intentional prompts and focused observations. The assessment forms allow teachers to focus their observations on the key concepts, and to note interesting extensions or comments on the part of the children.

The “What Do They Know?” assessment package includes all the instructions, the line masters for data gathering and all necessary lesson props in full colour for both the fall and spring assessments. A rubric for use in the fall and spring is appended to the end of each term’s assessment; instructions on how to analyze and consolidate the observational data into the rubric are also included. To complement the assessment and to address the results, an instructional resource is included with suggestions for subitizing, partitioning and patterning lessons. The WDTK assessment also includes adaptations for grade 1, so that K/1 teachers can administer the assessment to all of their students.

The intention is that teachers assess their students in small groups or as a whole class in the fall, use the sample lessons included in the resource (among others) and assess again in the spring to judge the level of growth over time. Like with a “school-wide write”, we recommend that teachers use one rubric per child for both the fall and spring assessments, changing the colour of highlighter to show growth. This could ultimately be used for reporting to parents, should a teacher want to include the results in their more formal evaluation of students’ learning.

Initial Feedback

Sandra field-tested the “What Do They Know?” assessment items in the Surrey school district this June. The students enjoyed the stories and seemed comfortable with the tasks they were asked to complete. Sandra reported that she and the classroom teachers were able to learn a great deal about the students – and were even surprised at what some children knew and could do mathematically.

A child’s ability

to pattern lays

the foundation

for algebraic

reasoning –

being able to

predict what will

come next (or

even “before”) in

a pattern is an

essential skill.

Summer 2011 10

Feedback from the teachers has been overwhelmingly positive. They see both the value of the assessment and its fit within a play-full numeracy program at the Kindergarten level.

Future Plans – Where to From Here?

Teachers in the Surrey school district will have the opportunity to explore the WDTK Assessment in the fall of 2011. Each school will be provided with a copy of the assessment, the teaching props and a copy of the storybooks used. In Richmond, after school sessions will be offered in the fall of 2011, with participating K and K/1 teachers receiving professional development, the tool and a copy of each of the stories. In both districts, the participating teachers will have the chance to use the materials and then come together again to discuss what they are learning about their children – and together make plans for instruction based on the results.

Kindergarten is a time of joy – in learning, in play and in making sense of the world around. As early primary teachers we have the opportunity to experience that joy first hard – and to support our youngest learners along their path. The “What Do They Know?” assessment seeks to find out – in an authentic and classroom-familiar way – just what sense students are making out of their mathematical explorations, and to conscientiously build upon those understandings.

What do they know? Plenty.

Feedback from

the teachers

has been

overwhelmingly

positive. They

see both the

value of the

assessment

and its fit

within a play-

full numeracy

program at the

Kindergarten

level.

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Helping Students Reason About Fractions by Using BenchmarksBy Carollee NorrisCarollee Norris, [email protected], is the Numeracy Support Teacher for School District #60, Fort St John, BC, Canada. She is a sessional instructor for Simon Fraser University, teaching courses in mathematics education. She worked as a writer on the BC Ministry of Education’s 2007 Mathematics K-7 curricular document, and is presently on the executive of the BCAMT. You can read her blog at focusonmath.wordpress.com.

Fractions are usually the first experience students have where a number represents something other than a count. Fractions involve

parts of a whole, and generally lie on the number line between whole numbers. Many students find it surprising that the spaces on the number line contain numerical values. In fact, a single space, such as between 0 and 1, not only has many points, it has an infinite number of points. Students find this rather mind-boggling!

When they are first introduced, fractions present a paradox for children. In their early experiences with numbers, children learn to compare numbers to each other: 8 > 5, 3 < 4, etc. With this knowledge clearly in place, some students find it difficult to understand and remember that 1/5 <1/4. Since 5 > 4, students often reason falsely that 1/5 > 1/4.

Another thing that can present difficulty for children is the fractional representation. A fractional part may be represented in a variety of ways. It can be a portion of something, like a piece of pizza (area model), a length of something, like a string or rope (linear model), or it ca be pieces or objects that are part of a group (set model). In the case of the area model, 1/5 of the whole is a piece smaller than the original whole. For the linear model, 1/5 of a string is a shorter piece than the original. In the case the set model, 1/5 of a group, say of 15, is actually 3 pieces, not a single piece. The set representation is often the most difficult for students to make sense of. But no matter which model id used, a fractional part is only understood relative to its particular whole.

If students are to develop number sense around fraction concepts, then they must make meaning of the part-to-whole relationship that

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If students are

to develop

number sense

around fraction

concepts, then

they must make

meaning of the

part-to-whole

relationship that

is inherent in

fractions.

Summer 2011 12

is inherent in fractions. There is also value in understanding how any proper fraction is related to important reference points or benchmarks. Having such reference points allows students to quickly estimate the value of an unfamiliar fraction. Howden highlights the importance of understanding the relationships between numbers: “Number sense can be described as a good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are limited by traditional algorithms”. (Howden, Hilde. Arithmetic Teacher, February 1989, p.11.)

I have worked with a number of classes to benchmark fractions. The teacher pocket chart and the student mini pocket charts have been a particular success, with some students begging to take their pocket charts home to show their parents what they learned about fractions. What teacher would not be delighted about that kind of student enthusiasm about fractions?

All children will have had some experience with fractions. Even if they have had little or no formal instruction on them, children will have had the experience of splitting things in half, in quarters or other such fair sharing. Students need to understand that in writing a fraction, the denominator names the number of parts in the whole, and the numerator names the number of parts that are counted.

Beginning with Benchmarks:

Begin by showing some fractions one at a time to the students. Use fractions such as 1/8, 6/10, 13/14, 4/9, 93/100, 2/11, etc. For each fraction, ask the question, “Is this fraction greater than half or less than half?” Have students talk to a partner to determine the answer and then share their thinking with the class. Cards can be placed in the pocket chart to show each fraction’s value related to ½.

Show a fraction such as 7/8. Ask students to talk to their partners and decide together if this fraction is closer to 0, ½, or 1 [1]. It may be helpful to have students first determine if the fraction is greater than ½ or less than ½ [greater]. If a fraction is less than half, only 0 and ½ need now be considered further. If the fraction is greater than 1/2, only ½ and 1 need now be considered. Students write 7/8 on a card and place it at the appropriate spot in their pocket chart.

This same procedure is repeated with a number of different fractions such as these: 8/9, 49/100, 6/10, 2/9, 11/12, 8/15, 1/7, 5/12, 11/50.

[2/9, 1/7, and 11/50 are close to 1; 49/100, 6/10, 8/15 and 5/12 are close to ½; 8/9 and 11/12 are close to 1]

You may wish to ask students to remove the previous card before placing a new one. If students keep adding new fraction cards into the pocket


chart, it is much harder to glance around the room and do a quick evaluation of student answers for a particular question. If there is only a single card in each pocket chart, you will not lose time trying to read particular fractions in each student’s pocket chart.

It is important to fully investigate the reasoning behind student benchmark choices. When a student is sharing his ideas, it may be helpful to provide stems such as, “I believe that fraction is closest to ____ (name the benchmark number) because…” Once a student shares his answer and thinking with the class, others can build on those ideas. Encourage subsequent students to connect their thinking to the first student’s by starting out, “ I agree/disagree with _____ because…”

Allow non-traditional reasoning. If a fraction such as 4/9 is put forward to students, they may reason that half of 9 is 4 ½, so 4/9 must be close to the benchmark of half. Such thinking, that one can have 4 ½ pieces of things called ninths, is sound reasoning and should be encouraged. Although we do not usually write simple fractions with other fractions in the numerator, such ideas can be helpful in estimating the value of a fraction. It also gives students a glimpse of “fractions within fractions” which often happens in functions they will encounter in older grades.

Students may share reasoning that seems sound for a particular fraction but does not hold true for all fractions. The misconception needs to be probed and discussed. For instance, if a student says that 17/20 is close to a whole “because it is only missing three parts” that makes sense for that fraction. However, 2/5, which is also missing only three parts, is not close to a whole. The important idea for students to develop is that the number of parts a fraction has (or that are missing) must be considered relative to the whole. Stating that a fraction in a certain number of pieces away from a benchmark leaves that critical piece out. Students need to “muddle” over which statements can be generalized and which cannot. If students need additional support, representations of fractions with area/region linear, or set models of may be constructed to help students visualize specific quantities. Keep in mind, however, that when comparing two fractions, the wholes must be of equal size. Otherwise it is like comparing half of a large pizza with half of a small pizza, and clearly those halves are not equal.

Possible Questions for Students:

• How can a fraction be missing only a few parts and not be close to a whole? [If the whole has relatively few parts, missing a few makes a big difference.]

• How far away from being a whole (or a half) is this fraction? How do you know?

• Will fractions that are both missing the same number of parts both be near the same benchmark? [They may or may not be close to

Stating that a

fraction in a

certain number

of pieces

away from a

benchmark

leaves that

critical piece out.

Students need

to “muddle”

over which

statements can

be generalized

and which

cannot.

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Summer 2011 14

the same benchmark; e.g., 5/11 and 95/100 are both 5 parts away from a whole but have different benchmarks; 7/14 and 8/15 are both 7 parts away from a whole but have the same benchmark.]

• If you know a fraction is less than half (more than half), how do you then decide if it is closer to 0 or to ½ (½ or 1)?

Possible Assessment Strategies:

• Ask students to review all of their fraction cards and sort them into three piles based on near benchmarks and record the three groups of fractions.

• Offer students a particular denominator, say /25. Ask students to write a fraction with that particular denominator which would be close to 0, one that would be close to ½, and one that would be close to 1. Have them explain in a sentence for each fraction why they believe it is close to the benchmark. (Answers will vary.)

Comparing Fractions:

Give students 49/100 and 1/7 at the same time and ask, “Which fraction is larger”. Have them discuss the fractions with their partners and place the fractions in their pocket chart. The placement will allow them to tell which is larger. Discuss reasoning with the whole class. [49/100 is larger since it is close to ½, while 1/7 is close to 0]. Repeat this process with other pairs: 2/9 and 47/50 [2/9 > 47/50].

Fractions that are both close to the same benchmark require a different kind of reasoning. Consider 6/10 and 5/12. Some students may quickly jump to the conclusion that they are equal because they are both near ½. Encourage students to consider if each fraction is a little more than ½ or a little less than ½ and discuss their reasoning with a partner. In so doing they should see that 6/10 is greater than 5/12.

The fractions 2/10 and 2/15 present another problem. They are both closest to the zero benchmark, so students cannot use the “little larger, little smaller” strategy they used for 5/10 and 5/12. Hopefully students will notice that since both fractions have the same number of parts being counted, they need only compare the size of each part in relation to each other. Fifteenths will be smaller than tenths, thus 2/15 must be smaller than 2/10.

Continue to give students pairs of fractions to compare and discuss. By giving specific pairs of fractions, students have the opportunity to develop a variety of ways fractions can be compared:

• Two fractions each closer to different benchmarks.

○ 3/8 and 9/10 [9/10 > 3/8]

Encourage

students to

consider if each

fraction is a little

more than ½ or

a little less than

½ and discuss

their reasoning

with a partner.

In so doing they

should see that

6/10 is greater

than 5/12.


○ 2/25 and 8/14 [8/14 > 2/25]

• Two fractions each closer to ½, but one a little larger than ½ and the other smaller than ½.

○ 11/20 and 3/8 [11/20 > 3/8]

○ 6/13 and 8/15 [8/15 > 6/13]

• Two fractions with the same numerator. If the number of parts is the same, the larger fraction always has larger pieces.

○ 3/5 and 3/8 [3/5 > 3/8]

○ 5/8 and 5/6 [5/6 > 5/8]

• Two fractions with the same denominator. [If the size of the parts is the same, the fraction with more parts is greater.]

○ 2/7 and 4/7 [4/7 > 2/7]

○ 9/10 and 8/10 [9/10 > 8/10]

• Two fractions that are the same number of parts away from a benchmark.

○ 7/8 and 8/9 [Both are both missing one part of a whole, but since ninths are smaller parts than eighths, 8/9 is missing the smaller part and thus is larger.]

○ 5/8 and 6/10 [Both are one part more than half, but the extra eighth is larger than the extra tenth, making 5/8 larger.]

Ordering Sets of Fractions:

Students can use the same understanding about comparing two fractions to order three or more fractions from least to greatest. It is important to choose sets of fractions that can be ordered solely by using the benchmarks and mathematical reasoning.

Have students work with their partners to order sets of fractions. They may do so using their fraction pocket chart, or they may wish to mark them on a sketched number line (0 to 1 with ½ added in the middle) to demonstrate their approximate placement.

Suggested sets:

• 5/8, 2/9, and 2/100 [2/100, 2/9, 5/8]

• 1/5, 11/12, and 6/10 [1/5, 6/10. 11/12]

• 9/10, 19/20, and 7/10 [7/10, 910, 19/20]

• 6/10, 3/8, and 11/20 [3/8, 11/20, 6/10]

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• 49/100, 4/8, and 1/9 [1/9, 49/100, 4/8]

You can also have students create sets of three fractions that can be put into order using the benchmarks and mathematical reasoning. Have them record the answers, and then exchange sets with another student and solve each other’s sets.

Extensions:

The comparing and ordering activities can also be done including decimals and/or percents. In such cases students need to understand that the equivalent benchmarks for decimals are 0, 0.5, and 1, and for percents they are 0%, 50%, and 100%. Mixing fractions, decimals, and percents allows students to develop fluidity between these different kinds of part-to-whole representations.

• Give students pairs of numbers to compare with one a fraction and the other a decimal (e.g., 3/8 and 0.9; 89% and 1/5).

• A mixture of fractions, decimals, percents can be given when ordering fractions (e.g. 47/50, 49%, 4/8, 1/10, and 0.53)

Conclusion:

If we want students to really understand fractions, we need to be mindful of what Howden said and give students many opportunities to explore fractions and visualize them in a variety of contexts. The pocket charts can be used in some “regular” lessons in class as students learn to reason about fractions, but I have also found them very helpful for regular review activities. When students have the opportunity each week to benchmark a few fractions, compare some pairs of fractions, and order sets of fractions, the students revisit in their minds many important understandings about fractions. In facilitating these activities we provide the important time factor needed to build number sense in this area of mathematics.


Instructions for Making a Fraction Benchmark Pocket Chart

Teacher’s Demonstration-Sized ChartMaterials:• Poster board cut to 28 inches by 11 inches (i.e., cut in half lengthwise)• Stapler and staples• Marker• Yard stick or other long straight edge

Fold up a “pocket”, approximately three inches deep, along the entire 28” length of the poster board. This gives an overall size of 28” by 8”. Secure the pocket at each end with several staples. (Note: if the pocket gapes open, staple the very bottom of the chart about 1/4-1/2 inch above the fold in several places. This will “tighten” the pocket.)

Using the long straight edge as a guide, draw a line across the pocket. Placement of the line should be about 1 to 1 ½ inches from the top of the pocket and run almost the full width of the pocket. Place a small vertical line at each end of the line segment and mark the middle with another small vertical line. Moving left to right, mark the benchmarks 0, ½, and 1 below the vertical lines. Write a title at the top of the pocket chart (e.g., “Fractions” or “Benchmarking Fractions”).

3” x 5” file cards work for writing fractions to be placed. Be sure to write in the upper portion of the card, as the lower portion will be hidden by the pocket.

Individual Fraction Pocket Charts for StudentsMaterials (for each pocket chart)• Manila tag or cardstock, approximately 11 inches x 4 1/4 inches (cut sheets in half

lengthwise)• Stapler and staples (or tape)• Pen or Marker• Ruler or other straight edge.

Fold up a “pocket”, approximately 1½ inches deep, along the entire 11” length of the card stock. This gives an overall size of 11” by 2 ¾ ”. Secure the pocket at each end with one or two staples. (Note: if the pocket gapes open, staple the very bottom of the chart by the fold near the middle. This will “tighten” the pocket.)

Using the ruler or straight edge as a guide, draw a line across the pocket. Placement of the line should be about ½ inch from the top of the pocket and run almost the full width of the pocket. Place a small vertical line at each end of the line segment and mark the middle with another small line. Moving left to right, mark the benchmarks 0, ½, and 1 below the vertical lines. Write a title at the top of the pocket chart (e.g., “Fractions” or “Benchmarking Fractions” to match the teacher’s version).

3” x 5” file cards cut into thirds work for writing fractions to be placed in the student packet charts. Be sure to have students write in the upper portion of each card, as the lower portion will be hidden by the pocket.

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Summer 2011 18

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IONLinear Measurement in

Primary ClassroomsBy Janice NovakowskiJanice Novakowski is a teacher in the Richmond School District where she is currently teaching at Blair Elementary.

Primary students are naturally curious and genuine inquiry about measurement often emerges in the classroom as students play, wonder

and compare the sizes of different objects, particularly in comparison to themselves. Young students are interested in finding out what is the “biggest” or “longest” or “tiniest” of something. Younger students snap together cubes to see how long of a train they can make winding through the classroom. Older students are fascinated by the Guinness Book of World Records and who is the tallest living human or how long the longest fingernails are! In developing linear measurement concepts and skills in the classroom, there are several “big ideas” to consider when planning opportunities for students to explore measurement.

Big Ideas in Linear Measurement• objects can be compared directly when they are placed along the

same baseline• we can use language like “longer than” and “shorter than” or “about

the same as” to describe our comparisons• when objects cannot be directly compared, we can use

another object like a piece of string to measure both objects and then compare them

• we can use all sorts of different units to measure with• units need to “iterate” from end to end of the linear

dimension being measured without any spaces between the units

• multiple copies of a unit can be used to measure or a unit can be used over and over multiple times to measure

• a measurement requires a number of units and the type of unit being used

Linear Measurement is measurement in one dimension. It is the number of units along the line or curve of an object.Dimensions that are measured are length, height, width, depth, distance and perimeter.


• standard units help us to communicate clearly about measurements• there are many tools that we can use to help us measure with

standard units (ruler, metre stick, measuring tape, trundle wheel)• the numbers on a measuring tool shows how many units there are

up to that point• referents (either personal or mathematical) can be used to both

estimate and measure

BC K-7 Mathematics IRP (2007)Prescribed Learning Outcomes (K-3)Linear Measurement

Kindergarten• Use direct comparison to compare two objects based on a single

attribute such as length

Grade One• Demonstrate an understanding of measurement as a process of

comparing by: - identifying attributes that can be compared - ordering objects - making statements of comparison -filling, covering or matching

Grade Two• Relate the size of a unit of measure to the number of units (limited

to non-standard units) used to measure length• Compare and order objects by length, height, distance around using

non-standard units and making statements of comparison• Measure length to the nearest non-standard unit by:

- using multiple copies of a unit - using a single copy of a unit (iteration process)

Grade Three• Demonstrate an understanding of measuring length (cm, m) by:

- selecting and justifying referents for the units cm and m - modelling and describing the relationship between the units cm and m - estmating length using referents -measuring and recording length, width and height• Demonstrate an understanding or perimeter of regular and irregular

shapes by: - estimating perimeter using referents for centimetre or metre - measuring and recording perimeter (cm, m) - constructing different shapes for a given perimeter (cm, m) to demonstrate that many shapes are possible for a perimeter

Primary students

are naturally

curious about

measurement

as students

play, wonder

and compare

the sizes of

different objects,

particularly in

comparison to

themselves.

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Summer 2011 20

Mathematical Tasks to Develop Linear Measurement Skills and Concepts

Whenever possible, create authentic, purposeful contexts for measuring. Opportunities for measuring may arise in the classroom or may be suggested by students.

Direct comparison• create a centre with tubs of materials for students

to compare/order from shortest to longest(ribbons, plastic snakes etc) or from shortest to tallest (stuffed animals, matroushka dolls)

• create a longer/shorter/same as chart for students to draw or write objects compared to a referent object

Indirect comparison

• using masking tape on the floor, create “crooked” or curved paths for students to measure (see what students come up with…they may ask for string, strips of paper, etc)

Nonstandard measurement• provide students with objects to measure with different

nonstandard units (ie. Measure the length of this book using three different units and record your findings)

• have students estimate and then measure an object with a nonstandard unit and then change the unit and re-estimate and re-measure the object, considering the relationship between the units

• have students share the different ways they measure with nonstandard units, discussing the difference between using multiple units and using a unit multiple times (iterating)

• have students make their own unit ruler (ie. 5 popsicle sticks long) and mark unit length in middle of unit to reinforce counting of units

Standard measurement • have students explore and compare a variety of

standard measuring tools (different types of rulers, metre sticks and measuring tapes) – how are they the same and how are they different?

• have students explore how a trundle wheel works• measure lengths that are longer than 30cm (one

standard ruler length) so students need to iterate and count on

• have students use their ruler to measure short lengths with a different start point than zero (ie. line up object at the 2 cm mark)


Estimation is important to develop alongside measurement skills. Estimating lengths, distances, etc can be embedded into all measurement tasks. Students should become comfortable using personal referents (ie. a student’s foot length) as well as standard referents (ie. a metre stick) to help them estimate linear measures.

Children’s Literature to support the development of linear measurement concepts and skills:

Actual Size By Steve Jenkins

Author and illustrator Steve Jenkins shows some amazing animals (or parts of them) in their actual size. All measurements are imperial but can easily be converted/measured in metric units.

I’m the Biggest Thing in the Ocean By Kevin Sherry

A giant squid travels through the ocean boasting that it is the biggest thing in the ocean as it encounters smaller animals. The squid eventually finds an animal bigger than it.

Just how long can a long string be?! By Keith Baker

Little bird and Little ant investigate all the different ways a string can be used and discover that a string’s length depends on its purpose.

How Deep is the Sea? By Anna Milbourne

Pipkin wonders how deep the sea is and goes off on an adventure to investigate. He shares his amazing findings with his mother. A scale diagram in both metres and feet is provided in a pocket at the back of the book.

Down Down Down By Steve Jenkins

The book begins above the ocean surface and travels down to the ocean floor, showing the animals that live at each depth. On each double-page spread there is a scale indicating the ocean’s depth and temperature in both imperial and metric units. Information about the animals, including their measurements is in a visual glossary at the back of the book.

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Summer 2011 22

Measuring Penny By Loreen Leedy

Lisa’s homework is to measure something in as many different ways as she can. She chooses to measure her dog Penny and measures her dog using standard and non-standard units.

Jim and the Beanstalk By Raymond Briggs

This classic variation has Jim needing to find ways to measure the giant’s head for a new wig, mouth for new teeth and eyeglass size.

How Big is a Foot? By Rolf Myller

This classic book is ideal for helping students understand the purpose of using standard measuring units. The King wants a bed made for his wife and uses his “foot” to count out the length and width of the bed.


Mathematics

is the science

of pattern.

The ability to

recognize,

represent and

extend pattern

is critical for all

students

Math Across the GradesInvestigating Patterns The Big Ideas Across the GradesBy Carole Fullerton, Andrea Helmer, Selina Millar, Janice Novakowski (BCAMT Executive; Elementary reps; 2004/2005)

Seeing how the math curriculum develops across the grades can be a challenge. The ideas and activities shared in this article help to support elementary teachers with meeting this challenge along with integrating literature and ways to assess the students’ learning. Adapted from the BC Association of Mathematics Teachers (BCAMT) Newsletter.

Mathematics is the science of pattern. The ability to recognize, represent and extend pattern is critical for all students. When we

have students create and think about repeating patterns and growing patterns in primary, these early experiences are developing algebraic thinking which lays the foundation for formal algebra in grades six and seven.

Early primary students work with blocks and materials to create, model and extend patterns. Intermediate students learn to represent

patterns with number, and later, algebraic expressions.

A big idea in mathematics is to be able to generalize relationships. Talking about our pattern rule (“My pattern goes red-blue-blue,

red-blue-blue.”) in kindergarten develops into the ability to generalize a function rule (“The total is doubling and increasing by 1 each time.”).

In the Classroom . . . Strategies for Elementary Teachers

K - 3Provide opportunities for students to explore manipulatives (pattern

blocks, buttons, shells, unifix cubes, tiles, etc). Talk with students about their patterns to develop descriptive language–the attributes of colour, shape, texture, and direction for example. Young children need

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Summer 2011 24

lots of experience exploring and describing patterns. Older students (grades 2-3) will need practice labelling and representing their patterns with

different materials. After these experiences, children may record their

patterns by drawing, labeling or even writing about them. Make sure children have the chance to explore growing, circular and surrounding patterns!

Why not try a pattern activity in the computer lab??

Use KidPix to stamp out a pattern!

A grade 2 sudent explores a growing pattern

A Kindergarten student works on a circular pattern

Literature connections K-3 The Cake that Mack AteBy: Rose Robart

Look out Bird!By: Marilyn Janovitz

If You Give a Mouse A CookieBy: Laura Joffe Numeroff

Beep-Beep Vroom-VroomBy: Stuart Murphy


3 - 5Here is a number machine pattern:

3 goes in, 15 comes out



What’s the function rule?

What is happening each time a number goes in?

What will happen if 10 goes in? What will come out? How do you know?

Show your thinking in at least 2 different ways. Can you build your solution?

Challenge: 45 came out. What went in?

Extra special challenge: 1/5 went in. What came out?

5 - 8The Handshake Problem

There are 6 people in a room. They all shake hands with each other. How many handshakes

will there be?

Show your thinking in at least 3 ways. How could you create a model to represent your solution?

How many handshakes would there be for 10 people in a room?

Challenge:

Describe the rule for the total number of handshakes for any number of people in a room.

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Literature connections 3-5Anno’s Mysterious Multiplying JarBy: Masaichiro and Mitsumasa Anno

Two of Everything By: Lily Toy Hong

Try This!

It’s a T=chart

Literature connections 5-8Spaghetti and Meatballs for All By: Marilyn Burns

One Grain of Rice: A Mathematical FolktaleBy: Demi

The King’s Chessboard By: David Birch

Anno’s Magic SeedsBy: Masaichiro and Mitsumasa Anno

Summer 2011 26

Assessment . . . Strategies for Elementary TeachersDid you know?

The new BC IRP will have more explicit outcomes around algebraic

thinking beginning in kindergarten. This kind of thinking can be developed through tasks like the ones we have included here, focusing on pattern rules and function rules.

Check www.wncp.ca for a draft of the new K-7 Western and Northern

Canadian Common Curriculum Framework. These outcomes are being adopted by our BC IRP document. Earliest optional implementation is September 2007.

What to look for while kids manipulate patterns?

Communication and Representation

Can students describe the pattern or pattern rule they are working with?

Can students represent their thinking about patterns using numbers, pictures and words?

Each of these tasks is a performance task that could be assessed using a Numeracy Performance Standard. They can be found at:

http://www.bced.gov.bc.ca/perf_stands/numeracy.htm


How can

teachers help

their students

become both

confident in

math and be

able to hold

onto concepts

learned

throughout the

year?

Marvellous MathA Mathematical ReflectionBy Lorill ViningLorill Vining has taught grades 1 to 6. She currently teaches on Quadra Island and lives in Campbell River

You want a challenge? Teach Math. A subject where 30 students are at 30 different levels. A subject where there are numerous

outcomes which build on the knowledge learned in the previous year. A subject where concepts that are often taught and mastered one year are completely forgotten by the time those concepts are revisited the following year. How can teachers help their students become both confident in math and be able to hold onto concepts learned throughout the year?

THE PROBLEM

It is September. You have just welcomed your new class and you are keen to find out what they know. You read with them. You write with them. All is well. And then you give them a Numeracy Assessment based on last year’s outcomes. If your class is anything like the ones I have taught in the past number of years, the marking of those “Assessments FOR Learning” will be done in just a few minutes as most answers are left blank, at which point a sense of panic will start to set in. How is it possible that, over the summer months of splishing and splashing, these students have forgotten everything they have ever learned in the world of math?

Let’s say your class is like mine, Grade 6. They claim they have never seen division before (one year I had students hope to learn about division and they described it as the line with the two dots…), fractions don’t exist in their world. The class average is 20% on the computation part of the test. The results show that the students are struggling with basics addition and subtraction facts to 18… and what is a decimal? There is no concept of multiplication, the 2 in 245 means 2 (instead of 200), and what is a sum anyways? OK, go ahead… hit the panic button!

There is a way to bring these students up to grade level. But first, let’s look at the biggest reason why students forget the concepts learned

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Summer 2011 28

throughout the year. Teachers follow the curriculum and teach the outcomes as they progress through the year. In September, for example, we teach basic number sense and computation. We then move into measurement, back to fractions, take a break from the hard stuff and do some geometry, then move into decimals and percents, ending the year with probability and data analysis. The following September we repeat, knowing that it has been 12 months since we did number and computation the previous Fall. So, we review it quickly and add more ideas on top of a foundation that barely exists. To start all over takes too long as we have a big curriculum to cover… maybe they’ll get it next year! And this goes on until the students land in my class in Grade 6. And this is where we have to start.

THE WILL…

I tell my students their learning is similar to a tall building in Vancouver. I tell them to imagine they are building a high rise and need to save some money in the budget. They decide that a concrete foundation will cost a lot of money and won’t be seen anyways, so they skip that part of the building. They start to build the structure and the first few floors are going great. But the sand they are building on starts to sink in one corner. What is going to happen before long? The building will fall over. It will come crumbling down. It won’t work. No matter how well they build that structure, if there is no foundation it will eventually crash. That is the same with our learning. If we try to learn ideas without the foundation behind it, the new ideas won’t make sense… they will crumble and be useless. So, we start at the beginning. Building numbers. Pulling numbers apart. Basic facts… doubles, doubles plus one. We build a foundation, a common language, a word wall. This is where it has to

start. This happens in our Math class. This is the time in our daily agenda entitled, “Math”. But this is not the only part of our Math learning. That other critical piece comes in the morning during our “settling in” routine. We call this routine Marvelous Math. This activity is the key to success with my students.

THE WAY!

Marvelous Math. Each and every morning when my students come in the class, they take their duotangs out of the Marvelous Math tub and get started on the 10 questions that are on the board. These questions are review questions only. There is nothing new in Marvelous Math, only concepts that have been taught in the Math class. The first nine questions are always a mixture of concepts using language off the

There is no

concept of

multiplication,

the 2 in 245

means 2

(instead of 200),

and what is a

sum anyways?

OK, go ahead…

hit the panic

button!


We start at

the beginning.

Building

numbers.

Pulling numbers

apart. Basic

facts… doubles,

doubles plus

one. We build

a foundation,

a common

language, a

word wall.

word wall. Each time a new math term/concept is introduced, I put it on the word wall on a sentence strip with the word and a brief definition and sometimes an example.

When using these terms in Marvelous Math, I underline the vocabulary words that are on the word wall so they know where to look if they forget the meaning of a word. The tenth question is a word problem, again using concepts we have learned previously in class.

Time is given for all students to finish all 10 questions. This gives me time to work the room and see where gaps are, who needs concepts reviewed or re-taught and who needs extra challenge. Sometimes I need to send a small group who need extra help to finish with an Educational Assistant, volunteer parent, Learning Assistance Teacher, or whoever I can round up. Those who choose not to finish stay in with me at recess or lunch. This only happens a few times!

When all students are finished, we mark the questions together. This is our time to share our different strategies and thinking. Essentially, the kids reinforce and teach each other the concepts being reviewed. I always ask students to explain their thinking as to how they got their answer and also ask if anyone did the question another way. Students who are comfortable will exchange their books with a partner. I do ask students to be honest when marking as there is a reward for getting 10/10… an earned reward.

Each day I look through these duotangs. It is a quick check and takes about 20 seconds per book. I am looking for a date as well as a quick scan to see how the student understands the concepts reviewed that day. If they get 10/10, they earn the letter F. The next time they get 100%, they will see FR at the top of the page. The third time they get 100%, which might be a few days later, they will find FRE. And the fourth time they get all questions correct they will find FREE! When they open their books and find FREE, they are excused from Marvelous Math that day and can do a quiet centre or activity of their choice. This is a great motivator. They LOVE free days! On the day they did not do MM because of a FREE day, I initial above the word FREE so I know they have taken their free day and they won’t take 5 free days in a row! Believe me, they’ll try! In order to get a letter, they must have the date on their page which allows me to keep track of any days away as well as if they are skipping days. I find it helpful. When marking the MM together, all students in the class participate, including those who

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Math Word Wall at the end of the year.

Summer 2011 30

Same student in April

had a free day.

At the beginning of this year, I had a handful of students who needed more difficult questions in MM. For the first four months, I put up two sets of questions. Simple questions for the majority of my students, and more challenging questions for a select few. Only those who demonstrated a solid understanding of the questions on the simpler version were permitted to try the more difficult set. Some students tried the more difficult questions and then opted back to the easier set which was perfectly fine with me. By January, everyone was doing the same set as the gap was quickly narrowing. Those huge learning gaps that caused the alarm bells to go off were being filled in.

THE RESULTS!

Upon return from Christmas Holiday (the first day back), my students rewrote the computation section of the assessment for learning that they had written back in September. The class average jumped from 20% to 66%. We graphed the data and celebrated. Their hard work was paying off. Everyone was excited. When we returned from Spring Break (two weeks), the class average jumped to 96%. Concepts mastered. I have 22 students. All but one wrote the test. Five students have learning disabilities that put them into the 1701 category. This class was flying now!!! Every student felt successful.

We made a triple bar graph of the three different test results then brought in Timbits… a real celebration.

These students have done a x, +, -, and ÷ question at grade level every day since November when they were ready to learn these concepts (once the basic understanding and strategies for basic facts were solid). Every outcome taught has been reviewed almost daily as I rotate through the word wall. I fully believe that this exercise is the real reason

why my students have become math lovers like me! They own the math!

PS. When your students ask you why they have to do math twice a day, tell them the MM doesn’t count as it is a settling in activity and morning routine! They seem to buy it!

Sample from September


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Using Children’s Literature to Inspire Mathematical ThinkingSubmitted by Janice Novakowski on behalf of the Richmond Primary Teacher’s Study GroupThe Primary Teachers Study Group is a group of Richmond teachers that meets every six weeks or so to share ideas from their classrooms and engage in personally chosen professional development. For two years, our group chose to focus on non-fiction literacy and how children communicate their thinking during mathematical problem solving. We were fortunate to receive a grant from the BCAMT and purchased several children’s books from which we developed problem-based mathematics lessons. During these lessons, we focused on students communicating their thinking, both orally and with pictures, numbers and words. The use of non-fiction text features, such as diagrams, labels and symbols, were utilized as students communicated in their math journals. The tasks and problems we worked on were intended to be open-ended and create opportunities for all students to access and be engaged with mathematics. Much like Japanese lesson study, we developed lesson ideas together and then met to discuss how our students did with the lessons and then we discussed how we could improve the lessons even further and shared questions that we asked that got students thinking and how we encouraged to share how they figured out the problems. This process helped us be more intentional in our mathematics teaching and we learned so much from our students and each other. We have included two of the collaboratively produced lessons here.

Grouping PenguinsLesson Background: The students had been working on mental mathematics strategies, visualizing groups as well as different ways to communicate their thinking.

365 PenguinsBy Jean-Luc Fromenthal and Joelle Jolivet(available in English and French)

Summary: A family receives a mysterious delivery of one penguin a day for a whole year.

Summer 2011 32

Problem:How many different ways could you put the 60 penguins in equal groups?

Lesson Overview:• The students were read the first few pages of the book, stopping to

answer some mental mathematics questions as we went.• When we got to the page where the dad organizes the 60 penguins

into four groups of 15, we stopped and asked the students to think about how else the dad could have organized the penguins into groups.

• The students were asked to solve the problem in as many ways as they could, showing how they figured it out. A large (11x17) piece of paper was given to each student and manipulative materials were set out in tubs.

• As the students worked, they were encouraged to discuss their findings with each other and were asked questions such as:

“Is there another way to group the penguins?” “What could you use/do to help yourself with this?” “How are those two ways alike and how are they different?” “Does that way give you an idea for another way?”• After students worked on their groupings for about half an hour, the

students sat in a circle on the carpet and turned to a partner to share and compare their work.

Other Possibilities:• How many penguins will the family have

to take care of by the end of ____ (month)?• How many penguins will the family have

on your birthday?• On what date will the family have 100

penguins? (do after reading a few pages)

Teacher’s Thoughts:The book is written in a light, fun tone that held each child’s attention. The students were enthusiastic and engaged during the entire lesson. The ability for students to communicate their ideas was excellent to see and hear. Every student was able to explain their ideas and strategies in a way that made sense to them. Although each student was working at his/her level of skill, the problem offered each of them considerable challenges. With prompts, most students were able to extend their thinking. For my class, knowing that they are able to use any strategy they feel comfortable with and with any manipulatives they like, allows them to begin a problem with enthusiasm and a motivation to find as many solutions to the problem as possible. This book and problem was a great example of this.The pair share and the sharing circle, at the end of the lesson, were great ways to discuss the problem. Students were able to learn about different

The students

were

enthusiastic

and engaged

during the

entire lesson.

The ability for

students to

communicate

their ideas was

excellent to see

and hear.


strategies in “kid-friendly” language. After recess, students came back eager to finish the book and their level of mathematical engagement had increased.

Student ResponsesThis student was one of only a few students to use manipulatives. Instead of choosing one of the tubs of materials set out for the class, he searched for the base ten blocks. His comment was, “These are tens, so 6 of these make 60.” He continued to find other ways to group 60 penguins, using multiples of ten and he recorded his groupings with drawings of the blocks, number labels and equations.

This student also used multiples of ten in his groupings but used tally marks and equations to record his work. When asked by a teacher how he knew that five groups of 12 was 60, his response was, “Well, five 2s are 10 and five 10s are 50 so that’s 50 + 10 is 60.”

How Many Feet?Lesson Background:

The students had been working on decomposing five at the beginning of the school year. This lesson introduced the number 10 and different ways to make it, including different ways to count to 10.

How Many Feet in the Bed? by Diane Johnston Hamm

Summary: A family enjoys a special morning together as they all climb into mom and dad’s bed and then leave as they go off to get ready for their day.

Problem:If there were 10 feet in the bed, who/what could there be?

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Summer 2011 34

Lesson Overview:• The teacher read the story aloud, having students count the number

of feet in the bed as members of the family jumped on, then left the bed.

• We counted the feet by 2s, as per the text in the story.• After our first read, the teacher modeled the people going onto the

bed using a piece of paper and unifix cubes. A ten frame could also be used. The students suggested different ways to organize the cubes and different ways to count them.

• The students were then asked to solve the problem: If there were 10 feet in your bed, who/what could there be?

• The students worked in their math journals at tables or on the carpet.

• Most students chose some sort of manipulative material to work with and most students began with all “people” in their beds.

• Once one student included a pet in her bed, and that was shared with the whole class which led to many students trying interesting combinations of animals and people in their beds.

• After about half an hour, the students sat in a circle on the carpet and “read” their math stories to the class.

Other Possibilities:• Change the number of feet in the bed.

• If there were 20 (36) feet in the bed, what are the different ways you could count them? (the book models counting them by 2’s)

Teacher’s Thoughts:Some students changed how many feet a regular person would have

There was

high student

engagement in

the story, and

quite a few of

the students

grasped

the concept

because it was

a story that they

could relate to,

build with blocks

and create on

paper.


to make ten feet with only drawing 2 in the bed (the example where the Uncle has 8 feet). There was high student engagement in the story, and quite a few of the students grasped the concept because it was a story that they could relate to, build with blocks and create on paper. This problem was completed at the start of the year so it was hard for some students who didn’t have one-to-one correspondence - it would be interesting to see read the story again at the end of the year and notice the difference in their answers and understanding

~Jamie Lepore, Kindergarten, Errington Elementary

Student ResponsesKindergarten students at Errington used 10 unifix cubes to model the 10 feet in the bed. The students used the cubes to think of what person or

animal could be in the bed and then drew a picture showing the “ten feet” in the bed.

As an alternative to the story, the kindergarten students at Grauer Elementary read The Mitten by Jan Brett and thought about what 10 feet might be hiding in the mitten. (Thanks to the primary teachers from

Hamilton for this idea!)

This student realized she drew too many animals when she started to count their feet so she used a circle and an arrow in her drawing to show that those animals “left” the mitten.

Grades 3 and 4 students at Tomsett Elementary chose their own number of feet and drew pictures and showed how they grouped/counted the feet in the bed.

This student just shows the dinosaurs’ feet in the bed and includes word labels and a key/legend to record the number of feet.

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Summer 2011 36

“Math Again?”By Ray Appel Ray has taught grades 2-7, been a Faculty Associate at Simon Fraser University (with a K-12 Module), and has also been a District Math/Science Coordinator (K-12). For over six years, Ray stepped back from the classroom and has traveled across Canada (from Inuvik, NWT, to Hartley Bay BC, Regina, to beautiful New Brunswick) giving numerous workshops on topics ranging from math, assessment to science and fine arts, as well as teaching in hundreds of classrooms. Currently, Ray is once again a Faculty Associate at Simon Fraser University.

We hear that math is all around us and, yet, so many of our students struggle. Even if they do well on tests, it just seems that they still

don’t really understand the ideas within basic mathematical principles. How can we remedy such a dilemma?

Many years ago, I remember giving my students in grade 5 some long division questions. They were pretty typical textbook questions that I found in some old textbooks. One day Kyle, (who always got straight As) was frustrated with me because part of being an effective math learner meant also understanding what we were learning. He repeatedly got all his solutions perfect, without any errors, yet didn’t really understand what an answer of “34 R6” meant. His lack of understanding, while getting all the right fill-in-the blank answers taught me a lesson. The lesson was this: my approach would have to be more multi-faceted, since a superficial cursory approach merely gives one perspective. In order to get more into the processes of math, I would have to differentiate both the tasks and the assessment. Lesson learned.

The following year, I had a student teacher. I suggested (in late September), that we start with the concept of division. My student teacher gulped. Now? As a mentor to my student teacher, I was pretty nervous. Why was I scared? Well, to begin with, in the older textbooks, division is always placed in the last part of the text. Why? Because the only way taught is usually standard “long division”. Kids have to know how to multiply, divide, subtract, add, follow through with four or five basic steps, and repeat the process until they arrive at a solution. It’s not easy. (But give them a cookie and ask them to divide it evenly and you have a different story!)

Before starting the ‘division’ unit in late September, I gave the class two numbers (3 and 157) and asked them to show different strategies for showing how to divide 157 into 3 equal parts. I was amazed at

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My approach

would have to

be more multi-

faceted, since

a superficial

cursory

approach merely

gives one

perspective.


how many strategies the class found! After 45 minutes into the lesson, a student said, “Hey, we haven’t talked about long division yet!”. How true. The kids were finding other ways to arrive at the solution of how to divide 157 into 3 equal parts, in ways that made sense!

Probably the most commonly used strategy is what the kids called the ‘pizza method’. (It was called the ‘pizza method’ because to many kids it looked like 3 pizzas!) One solution to dividing 157 into 3 equal parts is shown below:

Each student would begin by drawing the number of circles as indicated by the divisor (in this case, 3). Then, they would begin estimating the quotient (how many equal units in each circle). For students that are good at estimating, they might not have to write down 10 in each group followed by 20, followed by another 5 in each group, as so on. They might be able to estimate “50” in each group with some left over. Since estimation is a major foundational part of the math curriculum, and since knowing how to estimate is vital in everyday life, we were actually helping each other in ways beyond just the correct answer.

However, the key when asking students to sometimes come up with their own strategies is to make sure that whatever the strategies that they use, the better ones are: accurate, flexible and efficient. Using tally marks to group might be accurate, and flexible, but it might not be efficient for all students. If we think about what we want for our students, it often boils down to wanting students to be more efficient (faster, more organized, and perhaps more productive). I learned as well, that this can be different for the variety of learners in my classroom. What’s efficient for one, may not be as efficient for others.

After students brainstorm strategies, introduce the criteria for powerful strategies (i.e., accurate, flexible and efficient). Then, focus on the generated strategies that meet ALL the criteria, and go from there. If a student can divide using a strategy similar to the ‘pizza’ method, but struggles with long division, does that mean they know how to divide? What is our collective bottom line? In the end, we want to build kid’s number sense, while they learn and use effective strategies. My approach has been to start with the real world, and allow kids to invent strategies, but at the same time make sure the invented strategies are accurate, efficient and flexible. If we start with an old textbook from the early eighties, and do that ‘math’, we run into the problem that that is what kids think math really is. It isn’t always.

Keith Devlin in his book, “The Math Instinct” argues, that babies as

Since knowing

how to estimate

is vital in

everyday life,

we were actually

helping each

other in ways

beyond just the

correct answer.

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young as a few days old can distinguish between one-ness, two-ness and three-ness! What does that mean for us? It means that “math” is much more than the textbook or worksheet. It’s how we talk in the classroom, at home ...and how we engage mathematically in our world. What this might mean in the classroom is that students come to school with a sense of ‘mathness’. At times, they can see the big picture. …But, it’s not always that easy.

One day, I remember being really frustrated when I heard, “I give up!” for the umpteenth time during math class. I had just started teaching. It was my first year, and I wondering why math was the one place where patience and perseverance didn’t seem to matter, or even exist. One thing that got me moving was something another teacher said to me. She asked me what it was like for me when I am learning something new. Is it easy? What happens with the inevitable struggle that comes with deep learning? At the time, I was trying to get into shape. I was just starting to run. I was trying to watch what I was eating. Was it easy? No. Did I want to give up? Yes...

That conversation got me thinking about some truths I’ve learned in my life when it comes to learning something new. Think about a new relationship you’ve had, exercising, losing weight, taking big risks, going back to school, saving for a vacation in Europe...

We all struggle, and we learn some important truths while we struggle:

1) Meaningful things take time. 2) It helps when others allow me to take risks. 3) It‘s good when I listen, and when others listen to me as I learn. 4) I need to try things in more than one way, even if it seems wrong

at first. 5) I need to make mistakes. If I feel free to make them, then I’ll really

learn.

One of the ways that we’ve heard at probably every math workshop we’ve ever gone to is to make math “fun” and “engaging”. In some ways, this helps with perseverance and patience, as our students get into the learning without always realizing they’re learning.

We want our kids to develop best practice and healthy, persevering attitudes, including those in math. If I can look at how I struggle and learn in my own life, I can be more empathic to students as they struggle with their learning. That got me thinking ... (This was the tough part) how was I approaching math? What were my attitudes? Where was I getting frustrated? This made a difference. Firstly, I noticed a pattern in the way I was speaking with the students.

Essentially, I used a pattern that Gordon Wells calls, “Initiation-Response- Evaluation”. I initiate a question, I get a response, then I evaluate. Notice that I’m speaking two out of the three times per unit of

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It‘s good when I

listen, and when

others listen to

me as I learn.


conversation in the math class, and doing most of the work. This struck me as interesting. If I was doing most of the talking and evaluating, then what was left for the students to do?

I decided to do a few things. Firstly, I listened more to the students. What were their ways to solve problems? How many different ways could they solve 48 + 52 without using pencil and paper? How could they estimate? What strategies could they use? Were they using strategies that were efficient, accurate and flexible? Secondly, I decided to allow mistakes to be made. I had been saying this in my class for years (“It’s okay to make mistakes”), but did I really believe that? If so, how far did I believe that? In allowing kids to make mistakes (and openly share those), they felt more comfortable to do persevere.

But, here’s the secret! Use peer sharing throughout the math class so students feel more comfortable taking risks and sharing with their peers first. Then, slowly build yourself into the picture. That’s not easy at first. I found that I wanted to do a lot of the hinting, guiding, and the talking students through it. Looking back, I guess this is somewhat understandable, and there are times when we simply have to model something as simple as “the steps” needed to work through a math problem. One of the things I realized one day is the need for language to help students who don’t know how to begin. One of the prompts we came up with was:

First I… Next I… Then I… Finally I…

To help the students feel more comfortable, I put this on the tops of their sheets, made graphic organizers, and posted these prompts on chart paper. This was one step to get the students talking more. Often, kids will listen to their peers more than they’ll listen to you. Use that to aid the learning process. You can start by asking them to show one example of where they learned “x” really well, and one example where they didn’t.

Thirdly, I realized that learning takes time. Imagine if you were learning a new math resource, textbook series, or a new provincial math curriculum. Is it easy? Are you struggling? Are you ready to throw down your pencil on your desk? Of course, but like anything meaningful, it takes time.

When I say that things take time, I don’t mean that we should spend all of September teaching place value and then doing this well into October until every student gets it. What I mean is that we should layer and repeat throughout the year (i.e., place value in context of everything else). As well, we need to continue to build number sense while we help kids to estimate.

What have I learned? I’m a learner too. Just as I struggle with keeping fit,

I decided to

allow mistakes

to be made. I

had been saying

this in my class

for years (“It’s

okay to make

mistakes”),

but did I really

believe that? If

so, how far did I

believe that?

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or going back to take courses, or learn a new math resource, textbook series, or a new provincial math curriculum, so too do my students struggle as they learn.

It’s part of life.

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Five Tips & Tricks Make life easier in the classroom. How? Follow these easy steps...

1. Ask the students how the topic fits in with everyday life.

2. Give the students the problem and the solution. Then, students use strategies to show how to get there.

3. Ask students to write a word problem instead of just answering them. This will show you to what extent they understand.

4. Ask yourself “What is the bottom line?”. If it’s to build numerate students with a deep sense of number, then do they always have to only show it in only one way?

5. Continue to listen to the students. It’s always fascinating how they arrive at solutions, use mental math or invent strategies.

Check out Ray’s website: www.zapple.ca


Meaningful Collaboration Sparks a Successful Numeracy Action PlanBy Penny Morgan, Marthe Sivecki, and Cheryl Schwarz

Penny Morgan, Marthe Sivecki, and Cheryl Schwarz are elementary teachers at Blue Mountain Elementary, in Maple Ridge. They recently completed a graduate diploma program in Exploring Numeracy through Simon Fraser University’s Field Programs. They are committed to improving their students’ attitudes towards mathematics, and to help them become competent problem solvers.

The staff at Blue Mountain Elementary, a small rural school, was interested in changing their teacher practice in numeracy education.

We believed the best way to achieve this would be through a ‘teachers helping teachers’ model. So we became mentors for our staff and proposed the implementation of a professional learning community in mathematics. We would like to share what we have learned through this collaborative process.

In the spring of 2010, the teachers at Blue Mountain Elementary decided to develop an action plan to improve the effectiveness of math instruction in the classroom. Together we created an intent statement for our action plan. The backbone for this action plan was our School Improvement Plan goal: To improve student understanding in numeracy through problem-based teaching that focuses on the big math idea. With this in view, we developed the following action plan intent statement:

To improve teacher understanding and change teacher practice in numeracy education through exposure to:

a. Problem-based Teaching, which is modeled on the constructivist approach, where children learn best through discovery and active

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a. participation in the learning process. This approach requires that the teacher becomes the mediator between the child and the math, rather than someone who imparts knowledge to the student. At Blue Mountain we accept and expect that the children will struggle in order to learn how to become effective problem-solvers. Through this process the children learn the value of perseverance, knowing that their teacher will not immediately ‘rescue’ them by simply telling them how to solve the problem.

b. The Big Math Idea, which focuses on the enduring math understandings, rather than only teaching to the learning outcomes in the IRPs. When the BMI is made explicit, the children have a greater understanding of the important connections between math concepts, as well as connections to the real world. For example, a grade one math learning outcome is being able to state the number ‘one more’ or ‘one less’. The big math idea is that numbers are related to each other in a variety of ways. This same BMI extends throughout the grade levels. When children have these enduring understandings, it allows them to connect new learning to prior knowledge in a more meaningful way.

c. The math processes, as outlined in the BC Math IRPs, are: making connections, reasoning, visualization, mental math and estimation, problem-solving, use of technology, and communication. Using these processes helps the children to think and learn more effectively. It encourages teachers to focus on conceptual understanding rather than procedural knowledge. When students use the processes to learn and solve problems, teachers are more able to assess their learning and the degree of their understanding.

We felt that there were several key factors needed to make a school action plan successful:

1. Commitment and support from all staff members 2. A well-designed intent statement, crafted collaboratively by all

team members3. Time to focus on the goal4. Money 5. Expertise 6. Reflection and Evaluation 7. Celebration

1. Commitment and support from staff members. Without the full support from all staff members, true change is hampered. According to McLaughlin and Yee (1988), change within a school community is more likely when staff members share collegiality, have a unity of purpose, a positive approach to change and are

Children learn

the value of

perseverance,

knowing that

their teacher will

not immediately

‘rescue’ them

by simply telling

them how

to solve the

problem.


1. invested in the well-being of the school as a whole. The foundation of our mentoring teams was based on mutual respect and trust, and the ability to be honest with each other about what worked and what was not successful. As well, we openly shared our doubts and trepidations about changing teacher practice without fear of judgment from each other.

2. A well-designed intent statement crafted collaboratively by all team members. An intent statement must be specific, realistic and attainable in terms of its outcome, with a clear focus for measurable evaluation. Student growth and learning should always be kept in view as the ultimate goal. It also should be broad enough in its scope to allow for fluidity and change throughout the year based on the team’s needs. The staff decided together to focus on a very specific aspect of math teaching each term (i.e. the big math idea, the math processes, the problem-solving approach) in order to narrow focus and allow for maximum development and understanding of the concept. Consideration for timing was important (i.e. a new focus each term, evaluation at year end) in order to move the action plan forward.

3. Time to focus on the goal. We feel that this was one of the most important factors in making a school action plan successful. We devoted all non-instructional days in 2010/2011 to discuss, learn, explore, and reflect on the most current theories and practices in math education. We built in protected time to our school day to meet in mentor/mentee groups. During this time, similar grade teachers met, with each group deciding on the topics of discussion and how the mentor could best support the mentees. Collaborative planning, demonstration lessons as well as team teaching were some of the activities resulting from these meetings. Using a ‘buddy class’ model, as well as utilizing our noon-hour supervisors, we allotted 45 minutes every two weeks for teachers to meet. We also established a protected math learning time throughout the school (11-12am every day) and made it a priority to avoid any unnecessary interruptions during that time (i.e. fire drills, assemblies, PA announcements).

4. Money. In this age of cutbacks, little money is provided by the government to support teacher professional development. It is essential to find any money that is available through grants. We were able to secure three grants from several sources (our school district, BCAMT, and our local teacher’s association) which provided release time for meeting and planning, allowed us to bring in expert math educators to provide demonstration lessons and discussion time, and to purchase resources, such as texts for book studies.

It encourages

teachers

to focus on

conceptual

understanding

rather than

procedural

knowledge.

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1. Expertise. We felt that it was essential to use funding to secure people with expertise to support us in reaching our goal. Math is a very complicated and difficult subject to teach, and little training is given to pre-service or to practicing teachers in this area. We feel that the current model of ‘one-hit’ pro-d is not working, as there is no follow-up and no chance to practice what has been learned in a structured, supportive setting. We used our grant money to secure a leading expert in math education, Carole Fullerton, to spend a day at our school delivering demonstration lessons to different grade groups (primary, early intermediate, late intermediate). Release time was used to ensure all teachers were able to watch these lessons, and later discuss and reflect upon the experience. This came after a district-wide pro-d session where Carole Fullerton spoke to a large group on teaching the math processes. Following this, fellow colleagues from our district led break-out sessions in relevant numeracy topics. Another facet to this topic was the establishment of a ‘book study’ to read and discuss current research in math education. Four resources that we considered for our book study were:

a. What if Your ABCs Were Your 1 2 3s by Leslie Mintonb. Comprehending Math By Arthur Hyde; c. Teaching Student-Centered Math by John Van de Walled. The Teaching Gap by James W. Stigler and James Hiebert

We chose What if Your ABCs Were Your 123s by Leslie Minton in order to highlight the connection between literacy and numeracy.

To help support our Special Education Assistants, we compiled a resource binder with a variety of meaningful, multi-grade math activities and games to be used with small groups of children. This was introduced to our SEAs during a professional development day, giving them a chance to explore several of the activities and ask any questions.

6. ReflectionandEvaluation. It is very important to follow-up on experiences through individual and group reflection. Meaningful reflection provided valuable data for evaluation in terms of obtaining our goal, as well as providing future direction and to help individuals recognize their own growth. Throughout the year, we asked our colleagues questions such as: “What was the big idea you remember teaching today?” or “What did you find meaningful/useful from this session/experience?” We also asked the staff to rate their understanding of teaching a concept (i.e. The Big Math Idea) on a scale of 1 to 5. We did this several times to show evidence of growth over time. More formal evaluation of student improvement was done using district numeracy assessment

Student growth

and learning

should always

be kept in view

as the ultimate

goal.


6. kits (Island Net) that were administered in September and again in May to gauge student’s level of understanding and growth of knowledge and skill. Another assessment resource that we used was The Problem-Solving assessment kit (Gr. K-7) from Surrey School District, also administered in September and the following May.

7. Celebration. We felt that celebrating our successes and growth was key to building confidence and maintaining interest as the year unfolded. We set up a Math Brag Board in our staff room where staff members were encouraged to share student work samples, lesson plans, etc... We found that this encouraged informal discussions in the staff room at recess and lunch. We also set up a Math wall in the main foyer of the school so that the children could see that math is an important and valued subject at our school. Quick presentations at monthly assemblies allowed classes to celebrate and highlight the math learning in their rooms, focusing on the big math idea.

FinalReflections–June2011

To gauge the effectiveness of our learning community, we held a year end lunch meeting in order to reflect on our experiences in learning and teaching math over the past year. On the whole, our staff felt positive about belonging to the learning community, feeling supported and acknowledging that they had grown in their personal journey in teaching numeracy. We concluded that the mentor/mentee relationships seemed to have been the most instrumental way to effect change in teacher practice. One teacher felt that being involved in a mentoring relationship helped to keep her accountable to her professional growth plan. Another teacher found that observing demonstration lessons by her mentor inspired her to find ways to teach numeracy more effectively in her classroom. Yet another teacher found that she was able to get support from her mentor that was relevant to what she was teaching at that specific time. Participation in the learning community continues to grow, as one teacher who did not have a mentor this year, requested to have one next year. For next year, we will continue to use the ‘buddy class’ model to access in school collaborative time. Many mentor relationships will continue and others will be newly formed.

Where we will go next:

For next year our staff has chosen to focus on technology in the classroom as one of our school improvement goals. Most teachers feel that, as a result of our numeracy focus this past year, they have a better ‘handle’ and understanding of how to deliver an effective math program. They are on their way! However, we believe that we need to continue to support and encourage our colleagues in numeracy education as next year unfolds. We have learned this past year that

Our staff felt

positive about

belonging to

the learning

community,

feeling

supported and

acknowledging

that they had

grown in their

personal journey

in teaching

numeracy. M

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there needs to be a core group of leaders taking ownership over the direction of our ongoing learning community and keeping the goals of numeracy education at the forefront. An intent as important as ‘improving teaching understanding and skill in teaching numeracy’ can not be completed successfully in one year. We will continue to be teacher leaders in numeracy, by sharing current research, resources, and professional opportunities.

Some of the activities we are considering for next year include increasing parent awareness of current numeracy educational practices through guest speakers, ‘math’ nights, informal conversations and written information in newsletters. We are also planning a school-wide ‘Math Does Make Sense’ Day, where students rotate through different stations in classrooms. We will offer more book study opportunities for staff as well as provide support for colleagues in developing their personalized professional growth plans. In addition, we will link next year’s focus on ‘technology in the classroom’ with our ongoing numeracy goal.

We were excited to see how this professional learning community supported our colleagues by increasing their confidence and skill in teaching numeracy. Coupled with this, our confidence and enthusiasm in being teacher leaders has mirrored that of our colleagues and we are dedicated to continuing our journey!

What was the

big idea you

remember

teaching today?


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Mathematics Apps for use with iPod Touches and iPadsCompiled By Janice Novakowski

Blair Elementary of the Richmond School District took part in an iPad Inquiry Project sponsored by CUEBC (Computer Using

Educators of British Columbia) in the spring of 2011. A focus of this inquiry was seeing how students responded to a variety of apps and how the iPad could be used to enhance learning in mathematics and science. The students worked in pairs and this allowed for an oral language focus on communication of mathematical thinking and use of mathematics vocabulary. Some of the apps (available through the iTunes App Store) that the students enjoyed and were used to practice mathematical skills and concepts include:

PopMath Maths PlusIn this multi-leveled app, students choose an operation or a combination of operations to practice matching facts to answers by popping bubbles. This was particularly popular with the primary students.

Glow Burst LiteWith this app, students touch moving glowing circles with numbers in them and need to touch them in numerical sequence (ie. 2, 14, 34, 42) to proceed to the next level. Later levels include negative integers mixed with positive integers.

Rocket MathStudents complete computational questions in order to acquire enough points to launch their rocket. As the move up through the levels they acquire points to add features to their rockets such as colourful designs or extra boosters.

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TanZen HD LiteThis app allows you to move tangram pieces into an outline puzzle. Pieces can be slid, flipped and rotated.

Fractions AppThis app begins with a great little tutorial about fractions with lots of supporting visuals. It also includes practice exercises and a “test”.

Pizza Fractions: Comparing Simple FractionsStudents are presented with two fractions and they need to tap on the correct symbol to make the statement true….greater than, less than or equal to. If students are unsure, they can tap to see the two fractions represented in visuals of pizzas to help them with their answer.

My First Weighing ExercisesThere are three “modes” (ways of playing) and three levels of play for each mode that makes this app usable for a wide variety of grades/levels. The different ways of playing all focus on algebraic thinking and balancing both sides of a scale using fruits of different values and weights in grams.

Number LineSeveral circles pop up with a percentage, fraction or decimal fraction in each. The player drags each circle on to a number line, putting the circles in order from least to greatest. This app encourages great mental math work and practice of conversions between fractions, percentages and decimals.


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Making Mathematics Meaningful for Students in the Intermediate GradesReviewed By Jeannie DeBoiceAuthor: Werner LiedtkeISBN: 978-1425184186© 2008 Trafford PublishingPaperback, 174 pages

With the advent of our 2007 BC math curriculum (K-7), the idea that students need to make sense of the math they are learning has

never been more prevalent. Teachers are always looking for new ways to help their students to create personal meaning for the math ideas they are learning, and Werner Liedke has published a book to help teachers do just that. Making Mathematics Meaningful: for Students in the Intermediate Grades (Trafford publishing, 2010) consists of 8 chapters covering the 4 strands – number sense and operations, measurement sense, spatial sense and patterns. These chapters emphasize teaching strategies, classroom activities and diagnostic tasks and is BC specific, referring to our BC curriculum throughout. Suggestions for lessons and activities are all based on experience, observations, action research and interviews, so they are very practical. Theory is kept to a minimum, but referred to when it is pertinent to the concept or lesson.

Liedtke emphasizes the importance of student conversations to develop conceptual understanding. He uses many examples throughout of good questions, and prompts teachers to provoke student learning through problem solving, making connections and communicating, such as: “Try to think of a different way or a way nobody in this classroom has thought of” (p. 22) and “If you were to teach younger students how to add decimals what would you tell them that you think they should know?” (p. 112). One suggestion I have is that it would be helpful to include some student responses. For example, under ‘fractions’, “How can you tell whether a fraction is closer to 0 or closer to 1 [on a number line] by looking at the fraction?”(p. 49) is a good question; however, some teachers may find examples of student responses helpful, too, such

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as, “⅞ is closer to 1 than ¾ is because it’s only ⅛ away from 1 and ¾ is ¼ away from one.”

Liedtke refers to using manipulatives often in math lessons, and makes the important point that visualization doesn’t come from manipulatives or visuals (technology) alone:

The right questions and proper problems need to be posed or presented as materials are handled. Without high-order thinking questions and open-ended problems little, if anything related to the desired mathematics learning outcomes will ‘travel’ from the hand or the hands, up the arm to the brain. (p. 15)

He describes how to use manipulatives to aid in ‘seeing’ such abstract concepts as decimals, integers and fractions, emphasizing that a variety of tools is important to give students several ways into a mathematical idea. One suggestion I’d make is to include many more illustrations of these materials, perhaps even being used by students, to give new teachers a picture of what these materials look like.

Liedtke views number sense as “…the key foundation for numeracy” (p. 31) and devotes 4 of the 8 chapters (half the book) to developing number sense, basic facts and computational procedures. The chapters focus on sense-making, visualization, mental math and the development of personal strategies. This matches up nicely with our Elementary mathematics curriculum that also has a heavy emphasis on number sense. (If teachers skim through the K-6 “Assessment Overview Tables” they will see that the suggested weighting for Number Sense is never any less than 40-50%!) The author includes a good discussion of the role and importance of estimation with activities chosen to foster the use of benchmarks. He emphasizes a move away from ‘rounding’ to estimating and instead suggests students develop intuition about number through many activities fostering awareness of benchmarks.

Assessment for and of learning is referred to in every chapter and Liedtke offers many useful and insightful suggestions to make assessment an integral part of mathematics instruction. He emphasizes the importance of interviews for diagnostic work over written tests. “Probing thinking is research at a high level; the teacher takes the role of a cognitive diagnostician” (p. 26) Again, well-phrased questions are key here. He also makes the point that an interview can be adjusted on the spot unlike a written test, such as through illustration (having the child use objects or pictures to further explain their thinking) and redirection (changing the question depending on the response) for example.

In the section on Assessment, Liedtke emphasizes the need to focus on much more than rote, procedural learning to accurately report on a child’s under-standing. He provides many tasks and activities

He uses many

examples

throughout of

good questions,

and prompts

teachers to

provoke student

learning through

problem solving


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throughout the book that allow the teacher to collect assessment data on a child’s mathematical understanding that is far more meaningful than a percentage or a grade. He also reminds us how important it is that we are specific and non-subjective in our reporting to parents. He explains that it is common to read statements in report cards that sound impressive but are actually nebulous: “Has good number sense”, “Lacks number sense” or “Does not understand fractions”. One example he cites is a Special Education teacher who was asked to work with a child who “…does not understand chapter 4”!

Making Mathematics Meaningful for Students in the Intermediate Grades is a helpful addition to any intermediate teacher’s professional resource collection. It also offers great suggestions for Learning Assistance teachers who are trying to help students make sense of math and foster their numeracy. Well done, Werner!

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TSummer 2011 • Problem SetThis following set of problems are from “Good Questions: Great Ways to Differentiate Mathematics Instruction” (Marian Small 2009). The samples of open questions are one example of implementing differentiated math instruction into the classroom.

Guiding principles for Open Questions invite:

• correct responses at the variety of levels of abilities• students to participate together as a class in follow-up discussions.

When assessing the student’s work, it is helpful to keep the mathematical Big Ideas in mind. They can reveal what your students genuinely know and understand.

Here are some Patterns and Algebraic Thinking big ideas to guide the following tasks:

• Patterns can be represented in a variety of ways.• Some ways of displaying data actually highlight patterns.• Much of mathematics of the other strands is built on a pattern foundation.• Algebra is a way to represent and explain mathematical relationships and to describe

and analyze change.• Relationships between quantities can be described using variables.

Marion Small

GRADESK-2

BIG IDEA. Any pattern, algebraic expression, relationship, or equation can be represented in many ways.

Sabrina made the pattern below.

Make a pattern that you think is like this. Tell why the patterns are alike.

REFERENCESmall, Marian. Good Questions: Great Ways to Differentiate Mathematics Instruction. New York: Teachers College Press, 2009.


GRADES3-5

BIG IDEA. A group of items form a pattern only if there is an element of repetition, or regularity, that can be described with a pattern rule.

Use shapes like these to make a pattern that changes in two ways.

GRADES6-8

BIG IDEA. A pattern, algebraic expression, relationship, or equation can be represented in many ways.

A number pattern can be described by a table of values. The first number tells the position of the number in the pattern, and the second number is the pattern value. For example, the pattern 2, 4, 6, 8, … is described by:

1 2 3 4 5 62 4 6 8 10 12

Then the table of values can be graphed; the x-coordinate is the position number and the y-coordinate is the term value. Suppose the graph for a pattern goes 2,3). What could the pattern be?

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Game-based learning is best used to supplement or support classroom instruction. When used effectively, web based games and activities can increase motivation and engagement for some students and provide further practice and reinforcement. Compiled by Deanna Lightbody, Deanna is a District Teacher - Instructional Service for the Langley School District

BitesizeBritish site for Primary to Secondary studentsNote: Key Stage 1 (age 6 to 7), and Key Stage 2 (age 8 to 11)

http://www.bbc.co.uk/schools/bitesize

Math PlaygroundAn action-packed site for elementary and middle school students which includes math games, logic games math tutorial videos.

http://www.mathplayground.com/

MrNussbaumHighly engaging math games which includes Canadian money games

http://www.mrnussbaum.com/

LearnNowBC: Youth Learning CentreFun, interactive games and activities for young learners

http://www.learnnowbc.ca/learningcentre/

CTK Math Games for KidsA collection of games and puzzles with mathematical contents and a few practice exercises disguised as games. Gr. 3-6.

http://www.ctkmathgamesforkids.com/

Count Us InGames to practice early number and counting skills which can also be used on an interactive whiteboard (SMART Board).

http://www.abc.net.au/countusin/default.htm

Interactive Math Websites for Elementary Students and Teachers


British Columbia Association of Mathematics Teachers

What does the BCAMT do? Our goals and objectives include:

1. to promote excellence in math education throughout the province, including all areas from K-122. to promote development and implementation of sound curriculum3. promoting good communication with members, other educators, Ministry of Education, parents, students and the community

How do we do this?

1. Conferences • Fall Math Conference every October – see image for this year’s conference • New Teachers Math Conference every spring • Interior Math Conference every spring • Partner with North Central Zone Conference every spring • Regional professional development meetings throughout the province for elementary, middle, and secondary teachers2. Publications • Vector – math education journal published since the 1960s • Newsletter 3 times a year • Various other initiatives, such as recent “Questions Worth Asking” publication3. Communication • Thriving math education listserve with over 800 members • Website www.bcamt.ca4. Grants & Awards • Various teachers awards (new teacher, elementary, secondary, service, lifetime membership) • Grants to teachers for initiatives that support our goals

What you get when you join?

• Three Vector journals per year• Three Newsletters from the President of the BCAMT, updating you on current happenings in BC math education• Other publications from the BCAMT, such as the recently completed “Questions Worth Asking” book• Opportunity to apply for a grant to help fund a math initiative that your are involved in• Access to BCAMT mentorship program• Ministry updates related to math education

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Sign up as a member, tick appropriate math checkbox, detach sheet and mail to address shown below.

Peter Liljedahl Faculty of Education Simon Fraser University 8888 University Drive Burnaby, BC V5A 1S6

Vector - BC Association of Math Teachers Helping Teacher K-12 School District ... learning outcomes...

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