mathematics summative...
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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.