mathematics summative assignmentkayladoris.weebly.com/uploads/1/3/6/6/13663342/...NCTM(Critical(Reviews:(Fractions,(Decimals(&Percents(BuildingUnderstandingof(Decimal(Fractions:(UsingGridscan(help((

Annotated Bibliography: Estimation Computational Estimation for Numeracy

Edwards, A. (1984). Computational estimation for numeracy. Educational Studies in

Mathematics, 15, 1, 59-‐73.

In his article Computational Estimation for Numeracy, Edwards explores how

the importance of computational estimation is generally acknowledged and the

widespread introduction of the calculator has great increased its significance in

teaching number sense. Edwards argues that the reason of failure to teach number

sense effectively has stemmed from the multiplicity of methods used in estimations.

It is said that because everyone has resources to hand=held calculators that there

will no longer be a need for computational estimation and that as a result, people

will rely to heavily on receiving their answers from a machine rather than being able

to estimate or solve the answers to problems themselves. Edwards explores in his

article some procedures that will not result in students relying on calculators to

receive their answers and that are developed in unusual situations but are problem

solving skills that are appropriate elsewhere. Suggestions are made for estimating

sums, differences, means, products, quotients and percentages that teachers should

use when teaching Mathematical concepts.

Computational Estimation

Fung, M. G., & Latulippe, C. L. (2010). Computational estimation. Teaching Children

Mathematics, 17, 3, 170-‐176.

This article explores the importance of children receiving a solid

understanding of number sense during the early elementary years. Elementary

teachers are ultimately responsible for constructing this understanding and

therefore it is crucial that they receive teacher-‐training problems that include an

emphasis on number sense to ensure that the teachers receive a sound

understanding for themselves. These programs should be based around integrating

the development of productive computation and estimation skills in the Elementary

level students. The article’s authors Fung and Latulippe argue that to better prepare

Elementary school teachers, the goal while teaching standard number systems

concepts is to implement a strong emphasis on computational estimation. A

comprehensive understanding of computational estimation in Elementary students

significantly improves awareness of and proficiency in estimation. They achieve this

understanding through careful selection of materials, manipulatives an activities

focused around estimation and its importance in everyday life. When teachers and

students understand the connection between Mathematical concepts and their

connections to day-‐to-‐day life, the understanding becomes reinforced. Fung and

Latulippe agree that continuing teacher education in estimation, mental math and

number systems will benefit the classroom because the more the teacher

understands a flexible way to work with numbers, the better the understanding of

the students will become.

Estimation’s Role in Calculations with Fractions

Johanning, D. I. (2011). Estimation's role in calculations with fractions. Mathematics

Teaching In The Middle School, 17(2), 96-‐102.

Johanning’s article entitled Estimation’s Role in Calculations with Fractions,

focuses on how estimation is more than a skill or an isolated topic and more of a

thinking tool that needs to be emphasized during instruction. By learning estimation

as a thinking tool, students will learn to develop algorithmic procedures and

meaning for fraction operations. Johanning uses the example that for students to

realize when fractions should be added, subtracted, multiplied or divided, they need

to develop a sense of size for various fraction quantities. In other words, for

students to successfully differentiate between when to apply a certain strategy,

students need to have a strong number sense. Number sense, in Johanning’s view

will enable students to use estimation as a tool with fractions and thus they will be

able to develop a sense of size and quantity for individual fractions and also in

relation to each other. Johanning argues that this experience of estimation as a

thinking tool in relation to fractions is necessary if students are to develop meaning

for fraction operations. Estimation is a useful thinking tool when exploring how to

add, subtract, multiply and divide fractions.

Benchmarks, Estimation Skills, and the “Real World.” Teaching Math

May, L. J. (May 01, 1994). Benchmarks, estimation skills, and the “real world.”

Teaching Math. Teaching Pre K-‐8, 24, 8, 24-‐25.

Benchmarks, Estimation Skills, and the “Real World.” Teaching Math. Teaching

Pre K-‐8, explores activities designed to help Elementary students with their

estimation understanding and skills. May reveals that without strong estimation

skills, it is difficult to function in the real world and answer day-‐to-‐day questions

such as: How high is it? How much does it weigh? or How long will it take? Although

students could carry around tools to figure out the precise answer, this is not always

a reasonable or realistic resolution. It is much easier to estimate your answer and

more often than not, a good estimate is the only answer you truly need. May then

goes on to explore different activities that are designed to help Elementary school

teachers to teach students to estimate distance, weight and time through

benchmarks. Benchmarks serve as a guide for making good estimates in any area of

measurement. These benchmark activities include using length of stride to pace-‐off

distances, using coins to estimate weight and counting out loud to estimate time.

After these benchmarks are taught and as estimation is taught in any Mathematical

concept, students can then explore their own benchmarks that help them estimate

in the real world.

One Fish, Two Fish, Pretzel Fish: Learning Estimation and Other Advanced

Mathematics Concepts in an Inclusive Class

Mittag, K. C., & Van, R. A. K. (1999). One fish, two fish, pretzel fish: learning

estimation and other advanced mathematics concepts in an inclusive class.

Teaching Exceptional Children, 31, 6, 66-‐72.

One Fish, Two Fish, Pretzel Fish: Learning Estimation and Other Advanced

Mathematics Concepts in an Inclusive Class, explores how a team of teachers

successfully taught a group of grade five students in an inclusive classroom to use

different strategies to learn Mathematical concepts and skills. This article shows

how teachers can work collectively to help students in inclusive classrooms to learn

Mathematics and how teamwork, researched-‐based strategies, student engagement

and ownership can are fundamental keys to success. The authors Mittag and Van

agree that when teachers can work collectively and empower students in an

inclusive classroom to take ownership of their learning, fundamental

comprehension of concepts occurs. The team of teachers used cooperative learning,

estimation techniques, calculators, graphic organizers, links to prior knowledge

(what they have learned up until this point), real-‐life problems and strong review

sessions to empower their students and get them engaged in the classroom

material. At the end of the year the teachers had noticed that the students had

gained an average of at least 10 to 20 percentage points over previous test scores.

The team of teachers concluded that the students learning and performance over

the past year had improved not only because they were taught how to complete

mathematical problems but because they were shown how to learn and how to

complete tasks, which are skills that are important in any subject, in any classroom.

Estimation and Number Sense

Sowder, Judith T. Grouws, Douglas A. (Ed), (1992). Handbook of research on

mathematics teaching and learning: A project of the National Council of

Teachers of Mathematics., 371-‐389.

Estimation and Number Sense by Judith T. Sowder focuses on topics in

estimation and related Mathematical areas that have proved to be of areas of

interest to researchers. Sowder found that computational estimation has received

the most research attenetion and the majority of her chapter focuses on studies of

how people estimate computations and what personal abilities are related to

estimation ability. Sowder then explores how computational estimation concepts

develop and how instruction of computational estimation occurs. Sowder argues

that it is important to emphasize mental computation in students because mental

computation is closely linked with computational estimation. In this article, Sowder

also has a brief discussion about recent thinking about number sense and its

importance for estimation. Sowder models that for students to be able to be able to

effectively estimate in completing Mathematical problems or in everyday life, they

must have a strong sense of numbers and their meaning. If students have a strong

sense of what numbers represent, their value and significance, they will be able to

make effective and accurate estimations.

NCTM Critical Reviews: Fractions, Decimals & Percents Building Understanding of Decimal Fractions: Using Grids can help

Students overcome Confusion about Place Value

D’Ambrosio, B.S. Kastberg, S.E. (2012) Building Understanding of Decimal Fractions:

Using Grids can help Students overcome Confusion about Place Value. NCTM: Teaching

Children Mathematics. 558-‐564.

Beatriz S. D’Ambrosio teaches mathematics to preservice teachers at Miami

University. She is interested in place-‐value understanding and methods of

enhancing student reasoning and sense making about place value. Signe E. Kastberg

teaches at Purdue University in West Lafayette, Indiana. She is interested in building

models of the mathematics of students and using those models to guide instruction.

Building Understanding of Decimal Fractions: Using Grids can help Students

can help Students overcome Confusion about Place Value is intended for upper

Elementary School Teachers, Early Middle School Teachers and pre-‐service

Teachers. Although this article does not include any biographical information about

the authors, it explains that they intend to discuss the solution to the challenges of

ordering a set of decimals. The authors build on past research by including evidence

from 2003 where pre-‐service teachers were unable to arrange the values from

smallest to largest. There was also a significant amount of pre-‐service teachers who

could solve this task correctly but could not justify their solution by representing

each decimal in an area model using a decimal grid. The pre-‐service teachers

committed all the errors familiar to any educator working with students in the

upper elementary or middle school grades (Martinie and Bay-‐Williams 2003).

The objective of this article is to describe the challenges the adult learners

faced when they used grids to represent decimals and what the authors of this

article learned about their understanding of decimals from analyzing their work.

The authors’ language throughout the article was educational, professional and easy

to understanding. This article is includes a lot of Mathematical language and the

authors did a good job of making it understandable to their reader. A bibliography

is given at the end of the article and it is an appropriate length for this article, as it is

not very lengthy. The authors incorporate figures and exemplars of the work

completed to give the reader insight into what the question looked like. This is

beneficial to the reader because there are images to reinforce the information that is

being presented.

The authors’ major findings and conclusions are that the work with pre-service

teachers allowed them to determine activities that would push their understanding of

decimal numbers to a deeper level of understanding. D’Ambrosio and Kastberg found

that the teacher’s successful solution of a decimal-ordering task was often masking a

fragile understanding of important ideas regarding the use of decimal numerals to

represent fractional quantities. Using decimal grids, they were able to assess the

conceptual understanding of students. D’Ambrosio and Kastberg now believe that

students’ misunderstanding of how to represent decimals can be avoided if teachers begin

instruction of decimals with a vision of what makes understanding difficult and use this

vision to help students build understanding.

Fractions are Foundational

Fennell, F. (2007). Fractions are Foundational. NCTM News Bulletin.

The NCTM President (2006-‐2008), Francis Fennell, composed the article

Fractions are Foundational in 2007. The intended audience of this article is

Elementary School Teachers and pre-‐service Teachers. Francis Fennel’s intension

for this article is to discuss how Pre-‐K–8 mathematics instructions should provide

students with a strong sense of number without limiting their expectations for

student’s proficiency with whole numbers. Fennell agrees that such proficiency and

deep understanding are absolutely essential however; he argues that work with

fractions is equally important.

Fennell builds on past research as he explains that: Virtually every time he

asks teachers of algebra what they wish their incoming students knew, their

response is "fractions." The results of this informal polling were recently validated

in the National Survey of Algebra Teachers compiled by the National Opinion

Research Center at the University of Chicago for the National Mathematics Advisory

Panel of the U.S. Department of Education. Also, he recently asked fifth-‐grade

students to tell me where to place the fraction 9/5 on a number line. One student

informed that I couldn’t do that because the "top number" was more than 5, and the

number line went only to 1.

The objective of the article Fractions are Foundational is to show Fennels

main concern that we recognize the importance of curricular expectations that focus

on whole numbers but do not always acknowledge that a similar conceptual base is

necessary for fractions, decimals, and percents. Students need opportunities to work

with a variety of representations of fractions and to develop realizations of a

fraction. Similar to how students use counters to help anchor a mental image of a

whole number, they can use number lines to show how a fraction (or decimal or

percent) can be inserted between any two fractions. Number lines allow students to

compare fractions, decimals, and percentages.

The author’s language throughout the article is professional and easy to

understand. This is important because the article was released as a newsletter from

NCTM and their goal would be to have a general audience be able to understand the

meaning of the newsletter.

The author’s major finding and conclusions are: comprehension with

fractions is an important foundation for learning more advanced mathematics.

Fractions provide the best introduction to algebra in the elementary and middle

school years. It is necessary to spend a significant amount of time and emphasis on

developing the links among fractions, decimals and percents and solve problems

involving their use.

Masterpieces to Mathematics: Using Art to Teach Fraction, Decimal and

Percent Equivalents

Scaptura, C. Suh, J. Mahaffey, G. (2007) Masterpieces to Mathematics: Using Art to

Teach Fraction, Decimal and Percent Equivalents. NCTM: Mathematics Teaching in

the Middle School. 13, 1, 24-‐28.

Christopher Scaptura teaches sixth grade at Garfield Elementary School in

Springfield, VA. He is currently pursuing his master’s degree in elementary

education at George Mason University, Fairfax, Virginia. Jennifer Suh is an assistant

professor of Mathematics Education at George Mason University in Fairfax, Virginia.

Suh’s research interests focus on developing students’ mathematical proficiency

through problem solving and building fluency and teachers’ pedagogical content

knowledge in mathematics. Greg Mahaffey taught sixth-‐grade mathematics at

Westlawn Elementary School for the Fairfax County Public Schools in Virginia. He is

interested in broadening and increasing students’ interest in mathematics though

curricular and real-‐life connections.

The intended audience of this article is Middle School Mathematics Teachers

who may be interested in using Art to facilitate the understanding of Fractions,

Decimals and Percents. The authors build on past research and state that

historically fractions and decimals are taught separately without providing students

with the opportunity to make the connection between the two, which stunts their

ability to fully understand rational numbers. Scaptura, Suh and Mahaffey also argue

that past research shows that students are not taught these concepts in a relevant

way, meaning that teachers need to play a more active and direct role in providing

relevant experiences to enhance student understanding. This article shares how

students created their own Optical Art, and how they connected that work of art to

rational numbers. Students identified colored portions of a grid and recognized

fraction, decimal, and percent breakdowns of their own designs. Through visual and

mathematical representations of rational numbers, they learned mathematics

through artistry.

The authors language throughout the article is particularly effective because

it is educational yet easy to understand and therefore easy to read. The bibliography

is a substantial length for this short article and reflects that the authors used recent

research when writing the article. References are used to support the articles claims

and underline the importance of teaching students in a manner that engages them in

the classroom. Pictures and tables are used throughout the article to show the

reader what this lesson looked like in the classroom and what tables the students

had to complete before moving on to integrating Optical Art.

The authors found that this activity helped build students’ understanding of

the relationships among rational numbers by seeing how fractions, decimals, and

percents are related. It also stimulated their interest in Optical Art and allowed them

to express themselves artistically, while learning the Mathematical concept.

Students have fewer out-‐of-‐school experiences with rational numbers, which makes

it necessary for teachers to provide relevant experiences to engage students into

learning about fractions, decimals and percents.

Math Manipulatives In my classroom, manipulatives will be available to the students whenever the need

them. Rather than having students come get them (which may lead to them not

wanting to admit to their peers that they need to use them), I will have

manipulatives on their desks during the entire Math lesson so that no one has to feel

uncomfortable.

Base-‐Ten Blocks

The base-‐ten blocks are a very useful manipulative to use in the Mathematics

classroom. They allow students to visually understand basic mathematical concepts

including addition, subtraction, number sense, place value and counting. Base Ten

Blocks allow the student to manipulate the blocks in different ways to express

numbers and patterns. Interlocking Base Ten blocks help to clarify place value

concepts because they allow students to manipulate and visualize varying

quantities. They are frequently used in the classroom by teachers to model concepts,

as well as by students to reinforce their own understanding of the mathematical

concepts. Physically manipulating objects is an important technique used in learning

basic mathematic principles, particularly at the early stages of mathematical

learning.

Beginner’s Balance

A Beginner’s Balance is important for the early grade-‐levels of Elementary to

introduce the concepts of mass and measurement. This balance would be

particularly useful in a kindergarten class because of the balancing bears; children

would be engaged and curious about how to make the bears balance. I think this is a

great way to introduce and get children thinking about mass and measurement.

Geometric Solids

Geometric solids are important in the classroom because students can explore

shape, size, pattern, volume and measurement in a hands-‐on visual way. By

exploring spheres, cubes, cylinders, pyramids, prisms, hemispheres and rectangles,

students can begin to think about where we see these shapes in everyday life which

will enhance the idea of math in the real world. I feel that these geometric solids are

a great manipulative for any classroom.

Wooden Pattern Blocks

Pattern Blocks are a wonderful manipulative for students in the Elementary

classroom. Children can create patterns and designs by matching the geometric

shapes and explore concepts of one-‐to-‐one correspondence, sorting, matching,

symmetry, fractions, measurement and problem solving. These Pattern Blocks are

made of colourful wood, which would be durable and engaging for the children

while working with them.

Plastic Coin Set

I feel that a plastic coin set is a great manipulative to have in any Elementary

classroom. Students can learn about the value of each coin, how to mix the coins to

efficiently create a certain sum of money, play money-‐themed games (cash register)

and how to make change. Children can easily connect to coins because they are

prominent in their daily lives such as: milk money, tooth fairy money, lunch money,

etc. A coin set can be beneficial to many lessons in the classroom and I would plan to

have a set for each student so they can explore them individually.

Math Technology SmartBoard Lesson: Fraction Review

This SmartBoard Lesson was designed as a Fraction Review for grade four students.

The students would complete this SmartBoard Fraction Review after learning the

concept of fractions. It is important to evaluate the comfort level and

comprehension of fractions for each individual child, before being assessed or

introducing the concept of decimals and percents in relation to fractions. The

purpose of this activity is to evaluate and reinforce representing and describing

fractions before assessment or introducing decimals and percents.

To download a copy of this SmartBoard Lesson, click on the link in the outline.

Podcast: Fractions

The Fractions’ Podcast Unit website is a great technological resource that I can see

myself using in the future. It outlines the whole unit of Fractions and then gives

examples of Podcasts that each student would make to demonstrate their

understanding of fractions and provide one example of a question to ask to the class.

I think this would be very effective in the classroom because students would be

engaged with creating their podcast but also would be motivated to fully understand

the concept so they could create a podcast to share with the class.

To view the website and listen to an example of the podcast, please visit:

https://sites.google.com/site/welcometosilvinafissoreclass/

Useful Math Website: www.math.com

Math.com is a great website for Mathematics students of any age. Visitors simply

have to click on their required grade level and they will instantly be directed to the

concepts taught during their grade. Visitors can look up concept summaries if they

need extra help at home or complete practice questions to enhance their learning of

the concept. Math.com also has each provincial curriculum explanation (if you sign

up for their website), where you can see the curriculum expectations as well as the

progress of each province in Mathematics.

To view this educational website please visit: www.math.com

Journal Entries

Kings Landing I feel that our field trip to Kings Landing Historical Settlement could be used

as a theme for lessons in Math very easily. The most prominent way that I think you

could use the field trip in an educational way would be the learning of shapes. There

are many shapes at Kings Landing and it would be a great way for Elementary

students to link a Mathematical concept with an everyday experience. Shapes such

as circles, rectangles, squares, triangles, etc. can get students thinking of the shapes

that surround them and how they are everywhere in the world. Shapes are

extremely important in basic and more advanced math. Basic Mathematical

understanding of shape patterns and spatial perception help to develop sequencing

and logic skills that will lead into more challenging Mathematical Concepts later in

students Educational Careers.

Another way that our field trip to Kings Landing Historical Settlement could

be used as a theme in Math is with Number Sense. At Kings Landing there are

multiples of everything, everywhere. There are usually more than one of each king

of animal, house, farm, etc. Students can again link numbers to everyday life and

could even create a short presentation their classmates about the numbers of

animals, objects, buildings, or gardens they found while they were on their field trip.

Number Sense is extremely important for Elementary Students to grasp because

literally every other concept in Math builds onto Number Sense.

Although the Field Trip could be used in many different ways across many

different subjects, I feel that Mathematics would be a great subject. I feel that any

time Mathematics can be linked to everyday life and in personal ways it should be.

This is a great way to get students engaged in the Mathematics classroom, as it is

something that they can relate to.

Fractions, Fractions and more Fractions! Upon starting the Education Programme at St. Thomas University I was very

apprehensive about the Elementary School Math Methods Course. I feel that I have

always struggled in the field of Mathematics and one of my least favourite parts of

the subject was Fractions! Every time someone would mention the word I would get

a sweaty-‐palms, anxious feeling because I would think of how I was going to

respond to the question they were about to ask me. To my relief, the Elementary

School Math Methods Course has taken this anxiety away from me! I believe now

that the source of my anxiety had come from building this fraction concept into way

more than it needed to be. This course has helped me to realize that fractions are all

around us in our everyday life and they do not simply end in the classroom once you

finish the unit. Fractions relate to many Mathematical concepts and should not

simply be forgotten once the unit is over, which is something I was under the

impression of during my academic career.

My Math Methods course has given me the confidence boost that I have been

waiting for, for a very long time. I feel that when I am teaching and the unit of

fractions begins, I will not build them up into this big, scary concept that many of my

teachers did for me. I will now tell my students that fractions are a part of everyday

life, they are all around us and that it is extremely important to learn about them.

When “scary concepts” such as fractions are introduced, I believe that teachers

should relate it to as many personal, everyday situations for the students as

possible. When students can link Mathematical Concepts to something in their own

experiences, I feel that they become more engaged and able to take control of their

own learning. It is very important that instead of using difficult and formal

manipulatives when instructing about fractions, we use things that relate to

students such as: chocolate bars, pizza or cookies. When fractions are represented

by something that means something to the students I feel that they will be able to

interact with the concept much easier.

Relating Mathematics to Students I feel that it is sometimes difficult for students to appreciate the importance

of Mathematics. Students often find the subject boring or hard to understand

because they do not understand that Math is all around them. The way that Math

has been taught throughout Elementary, Middle and High school is so far

disconnected from real life examples that students have a hard time engaging and

connecting with Mathematical Concepts. Throughout the years, and probably

throughout the centuries, teachers have struggled to make math meaningful by

providing students with problems and examples demonstrating its applications in

everyday life. Although I feel that teachers are doing a much better job at connecting

Math with the real world, I feel that we constantly need a reminder that students

engage best with what they can relate to. I personally had been tutoring a boy in

grade 1 that was having a hard time with anything related to Math. After speaking

with him and realizing that he liked anything to do with cars, and integrating cars

into his Mathematical problems, the young boy had fewer problems in the subject

area. Although he still struggled with some of the concepts, he was much more

engaged and open to trying new things in the subject because he could relate to

what the questions were asking him.

Today teachers can also use technology to let students experience the value

of Math, instead of simply reading about it. Mathematical games have been made

online, online self-‐help websites are created and review websites are also available

for students. I believe that technology is something that Math teachers should use to

their advantage because it is something that students relate so easily to. They are

also technology-‐natives so it is relevant to them, to be able to learn, practice and

complete tasks using technology. However, I believe that no matter how teachers

connect to their students, it is important that students are aware that Math is all

around them and teachers must find new innovative ways to engage students on a

level that is interesting to them.

When I was in Public School, Math was taught in a way that was so

disconnected to everyday life that I had an extremely hard time connecting with

what my teachers were trying to teach me. I feel that this is the case for many of my

fellow classmates which is why I believe that it is crucial that Mathematics are

taught in a real life, engaging and interesting way so that students can connect Math

to everyday situations which in turn will facilitate their learning.

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