University of Ottawa

Assignment 3 – Annotated Bibliography & Summary

Daryn Caruso

EDU 5160 – Mathematical Thinking Across The Curriculum

Professor Yasmine Abtahi

April 11th, 2018

The topic of focus for the readings and summary was number sense. Number sense was

of particular interest because it is integral in the development of mathematical

understanding, particularly in the early years of mathematical development. The readings

were examined to see if they addressed the following questions:

1. From a teacher’s perspective, what are the challenges faced when trying to link

number sense concepts, such as: fractions, decimals, and percent?

relating fractions, decimals, and percent?

can implement?

Annotated Bibliography

Göbel, S. M., Watson, S. E., Lervåg, A., & Hulme, C. (2014). Children ’ s Arithmetic
Development : It Is Number Knowledge, Not the Approximate Number Sense,
That Counts. https://doi.org/10.1177/0956797613516471

Gobell, Watson, Lervag, and Hulme (2014) conducted an 11 month study of a group of

six year old students that fixated on the development of approximate number sense and

understanding of the Arabic numeral system as possible influences on arithmetic

development. It was interesting that learning and the ability to translate between Arabic

numerals and verbal codes were important for arithmetic development. It was concluded

that being able to match Arabic numerals with their verbal labels is a predictor of further

mathematical development. The reading reinforced that student numeracy needs can be

developed and integrated into the daily classroom routine early so students can have a

foundation for number sense prior to entering grade one.

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Moeller, K., Pixner, S., Zuber, J., Kaufmann, L., & Nuerk, H. (2011). Research in
Developmental Disabilities Early place-value understanding as a precursor for
later arithmetic performance — A longitudinal study on numerical
development. Research in Developmental Disabilities, 32(5), 1837–1851.
https://doi.org/10.1016/j.ridd.2011.03.012

Moeller, Pixner, Zuber, Kaufmann, and Neurk (2011) conducted a study that analyzed if

how grade one students learned basic numerical tasks (place value focused) could predict

success in specific arithmetic performances (addition). This reading demonstrated that if

an emphasis was placed on multi-digit number comprehension (place value) that it could

lead to easier comprehension of multi-digit problems in grade three. “the current data

suggest that early deficits in place-value understanding may still exert their influence on

later more complex arithmetic processes.” (Moeller, Pixner, Zuber, Kaufmann, and

Neurk, 2011) The concept of developmental trajectories was examined in that if certain

aspects are not developed in early grades when the students’ advance they will have to

absorb more information than many will entirely comprehend.

Murata, A., Siker, J., Kang, B., Scott, M., Lanouette, K., Baldinger, E. M., & Kim,
H. (n.d.). Math Talk and Student Strategy Trajectories : The Case of Two First.
Cognition and Instruction, 35(4), 290–316.
https://doi.org/10.1080/07370008.2017.1362408

Murata, Siker, Kang, Baldinger, Kim, Scott, and Lanouette (2017) focused on the

development of constructive student mathematics discussions in grade one. The teachers,

Carla and Mia, used different instructional strategies. Carla provided her students with a

detailed teacher-centric lesson. Mia incorporated student ideas in an organic way into her

lessons. When comparing the lessons it was concluded “the incorporation of a wider

variety of ideas and strategies during instruction will support individual students’ varying

learning trajectories to gradually converge toward the goals of classroom instruction.”

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(Murata, Siker, Kang, Baldinger, Kim, Scott, and Lanouette, 2017) An interesting facet

was how Mia had to be adaptable when incorporating ideas raised by her students.

“Effective use of talk moves requires inviting student ideas, which creates a certain level

of uncertainty as to how the instruction may play out.” (Murata, Siker, Kang, Baldinger,

Kim, Scott, and Lanouette, 2017) The study concluded that Mia’s lessons were not only

more engaging for the students but supported deeper student learning.

Utley, J., & Reeder, S. (2012). Prospective Elementary Teachers’ Development of

Utley and Reeder (2012) reported on prospective elementary teachers comprehension of

fractions and followed them as they “unlearned” some teaching practices and reinforced

their comprehension. The reading took a curious perspective when it acknowledged

fractions, as an area of mathematics that is challenged not only students. “Research

shows that the challenge of learning fractions was not limited to elementary students but

to adults as well.” (Utley and Reeder, 2012) The conclusion of the study was pessimistic

as even after prospective teachers have been retaught fractions many still struggled. Utley

and Reeder (2012) concluded that: “If the strength of these prospective elementary

teachers' understandings likes only in a part-whole understanding of fractions, then as

mathematics educators we should be concerned.” (Utley and Reeder, 2012)

Sasanguie, D., Göbel, S. M., Moll, K., Smets, K., & Reynvoet, B. (2013). Journal of
Experimental Child Approximate number sense, symbolic number processing,
or number – space mappings: What underlies mathematics achievement ?
Journal of Experimental Child Psychology, 114(3), 418–431.
https://doi.org/10.1016/j.jecp.2012.10.012

Sasanguie, Gobel, Moll, Smets, and Reynvoet (2013) focused on students aged: 6-8 with

the hopes that by having them complete different mathematical tasks they would be able

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to predict future mathematical comprehension. They found that the recognition of

performance on symbolic number line estimation best predicted success on math

assessments. When the two different tests were administered to the students the study

concluded that the type of assessment (a timed athematic test and a general curriculum

based test) did not vary the results by a significant number. The idea that the way that

students are taught is more important than how they are assessed is an idea that I have

believed in for the last several years. If students have a strong numerical sense and are

comfortable working with the numbers and mathematical symbols the method of

assessing them should not make a large difference in their assessment results.

Sweeney, E. S., Quinn, R. J., & Quinn, R. J. (2018). Connecting Fractions, Decimals,
& Percents. National Council of Teachers of Mathematics 5(5), 324–328.

Sweeny and Quinn (2000) analyzed student experiences when learning about the

relationship between decimals, fractions, and percent. The reading offers a five-phase

lesson concept that, through observations in several classrooms, is successful at linking

the three concepts. These activities can be completed over three days. I found this reading

to be helpful as it presents the reader with methods that could be used in the classroom in

a way that is straightforward for teachers. The five phases are: pre-assessment (quiz),

instruction, a demonstration game, playing the games in groups, and lesson conclusion

(post-assessment quiz). The Sweeny and Quinn article was the shortest of the readings

analyzed but contained useful information for the classroom that can be easily applied.

Wagner, D., & Davis, B. (2010). Feeling number : grounding number sense in a sense
of quantity, 39–51. https://doi.org/10.1007/s10649-009-9226-9

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Wagner and Davis (2010) concentrated on the development of number sense quantity.

Their research was separated into three sections. The first section focused on

developing a strong number sense. The second section focused on cultural, including

symbolic and verbal, number representations. The third section offers teaching

strategies for developing quantity and number sense. It was interesting to be exposed

to how some First Nations groups in Canada have specific words to represent

quantities of certain objects. The teaching examples in the third section also contained

quantity activities that could further students understandings of quantities of numbers,

making them less abstract. “typical applications of mathematics, which appear mostly

in word problems, may not connect sufficiently to students’ personal

contexts.”(Wagner & Davis, 2010) The concept of developing a “feel” for numbers,

especially the quantity of numbers was a innovative concept.

Whitacre, I., & Nickerson, S. D. (2016). Investigating the improvement of

Whitacre and Nickerson (2014) recounted a number and operations course for

prospective elementary school teachers that focused on the development of framework

for reasoning about fraction specific guided instruction. The objective was for

prospective teachers to become aware of different methods of instruction in hopes that it

would result in improved guided instruction. Specifically, reasoning on fraction

comparison tasks. At the conclusion, the prospective teachers “used a wider variety of

valid strategies to perform those comparisons, demonstrating improved flexibility in

reasoning about fraction magnitude.” (Whitacre and Nickerson, 2014) Whitacre and

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Nickerson (2014) fixated on an aspect of mathematics education that, from first hand

experience, causes students difficulty in demonstrating comprehension of questions and

defining terms. As the mathematics taught in elementary schools is not as complex as to

what students encounter in middle and high school, some teachers may not have the

comprehension to properly prepare their students

Summary

The readings corroborated that mathematical confusion can start as early as the

beginning of elementary school and can continue into adulthood, where teachers have

difficulty grasping some of the concepts they are required to teach. This was evident in

the Utley and Reeder (2012) and the Whitacre and Nickerson (2014) readings that

focused on teaching strategies that helped improve teacher comprehension and number

sense lesson strategies. The research indicated it was difficult for some prospective

teachers to view mathematical concepts in a transformed way. “Although these

prospective teachers had many years of school mathematics…their reasoning about

fraction concepts was often incorrect and largely based on limited understandings or

misconceptions they had previously developed.” Wagner and Davis (2010) also

referenced this and they included a section on retraining teachers about number sense,

in hopes they will be able to explain the concepts more thoroughly to their students.

Another concept in some of the readings was that teachers could have

mathematical misconceptions, which they are unaware of, that they transfer to their

students. For example, Utley and Reeder (2012) observed teachers in the classroom

focusing on the procedure of solving math questions but lacked teaching their students a

fundamental comprehension of what they are solving. They state “teachers must include

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activities that force the conceptual understanding of fractions rather than merely content

procedures.” (Utley and Reeder, 2012) Wagner and Davis (2010) discussed their desire

for teachers to have stronger number sense comprehension that they can pass on to their

students. Wagner and Davis (2010) hope that further teacher training “will awaken them

to be less numb to their physical and social environment and more aware of the role of

mathematics in society.” (Wagner and Davis, 2010) Murata, Siker, Kang, Baldinger, Kim,

Scott, and Lanouette (2017) compared different types of lesson methods with one

compactly structured and one more spontaneous. It was observed that a teacher who had

more spontaneous lessons, which depended on their number sense comprehension,

produced students who had strong comprehension. They concluded “incorporating

different mathematical ideas in class discussions we simultaneously open doors for

diverse learners and increase academic rigor for all students.” (Murata, Siker, Kang,

Baldinger, Kim, Scott, and Lanouette 2017) However, the ability for this to be

accomplished depended on the teacher’s competence to teach a more complete number

sense understanding, not just simply how to solve the problems.

Two of the topic questions, developed prior to selecting the readings, were

addressed and provided auxiliary comprehension. The readings addressed number sense

challenges encountered by teachers and students. More emphasis was placed on the

teachers’ perspective; in hopes that improved number sense teaching strategies would

result in stronger student comprehension.

Several readings addressed classroom approaches that teachers can implement.

Sweeny and Quinn (2000) examined the use games in mathematics class with students

allowed to have influence in creating the games they play. “Playing a game increases the

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excitement of any mathematics lesson, but when creating the game is an opportunity to

learn in itself, the result is even better.” (Sweeny and Quinn, 2000) Letting the students

create their own games is a unique approach, which I have yet to implement. Gobell,

Watson, Lervag, and Hulme (2014) examine how teachers can focus their planning and

conclude that it benefits students when they are introduced to place value at an early age.

“In addition to knowledge of single Arabic digits, an understanding of multi-digit

numbers, and especially place-value understanding, is also crucial for arithmetic

development.” (Gobell, Watson, Lervag, and Hulme, 2014) Although Gobell, Watson,

Lervag, and Hulme (2014) focus more on lesson goals rather than lesson implementation,

the readings compliment each other as they focus on different aspects of the teaching

process.

A question that was not addressed in the readings is “Is demonstrating the ability

to estimate the most important aspect for students of relating fractions, decimals, and

percent?” A focal point of my teaching strategy is to train my students to be able to

accurately estimate their answers prior to solving the problem. This is in hopes that the

students will have some confidence that their answer is correct if it is similar to their

estimate. It is my belief that if students are not able to estimate that when they solve a

problem they may not be sure if it is correct or not.

A common theme in several of the readings is that when young students

(kindergarten to grade one) are exposed to further number sense concepts (place value

and symbolic translation) it can result in later success in mathematics. “focusing on

symbols and the translation to their corresponding representation is very important in

learning contexts because this process seems so crucial for later math achievement.”

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(Sasanguie, Gobel, Moll, Smets, and Reynvoet, 2013). Gobell, Watson, Lervag, and

Hulme (2014) concluded “number-identification ability assessed at 6 years of age was a

powerful independent predictor of growth in arithmetic skill over the next 11 months.”

(Gobell, Watson, Lervag, and Hulme, 2014)

In regards to assumptions made, the readings did not contain worthwhile lesson

differentiation information. Although the lesson content introduced in the readings

presents teachers with different perspectives in hopes of influencing their pedagogy there

seems to be an assumption made that these ideas can be applied to all students or that the

teacher already has a repertoire to assist students who are having difficulty.

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