**Math is fun!**

What are the issues?

Why are primary students disengaged in math?

Games as Tools for Learning

How do we combat these problems with math?

Q:

If Brajit or other students are disinterested or confused in math because of it's abstract quality or because they feel it's too boring, how can we change their minds?

Issues in primary school mathematics (Burns, 2004):

Therefore, we conclude primary school mathematics to be:

too abstract

not engaging or motivating enough for students

Math in Children's Literature

Motivation and Efficacy

Motivation: The process through which goal-directed behaviour is started and maintained.

Manipulatives

Games

A game is an activity that has these elements:

**What is wrong?**

**Making it concrete!**

**Making it fun!**

Conclusion for Case 2

Annotated Bibliography

Table of Contents

Glossary

Introduction

A:

We need to make math more concrete for them, by relating it to the real world, so they can visualize what is actually happening in a math concept vs. writing down numbers and memorizing. What tools are available to help teachers do this?

Source: Pixabay

Jean Piaget

Source: Encyclopaedia Britannica

Source: Pixabay

Source: Didax

Source: iQ Camp

Source: ClipArtHub

Source: Books Kids Love

Tyrannosaurus Math by Michelle Markel

In Case 2, Brajit doesn’t understand math! She can say that 10 minus 4 is 6, but when asked how many loonies are left if starting with 7 and given 3, she doesn’t know.

Why doesn’t she understand? Why isn’t her math education working? Could she be disinterested and unmotivated to learn math?

How do we engage primary students more meaningfully in the mathematics curriculum?

1. Glossary

2. Introduction

3. Our Own Stories

4. What are the issues?

Issues in mathematics curriculum

Issues in understanding how children learn math

5. How to make math more concrete?

What we can do

What is concrete knowledge?

Manipulatives

Math in children's literature

6. How to make math more fun?

Motivation and efficacy

Engagement

Games

Games as learning tools

7. Conclusion

8. Annotated Bibliography

Sensory-concrete

: when students use sensory materials to make sense of an idea; in early primary, students may need to physically touch objects (Clements & McMillen, 1996)

Integrated concrete

: combinations of separate ideas that are interconnected --> children combine physical objects, actions performed on those objects, as well as abstractions made, all into one interconnected structure of knowledge (Clements & McMillen, 1996)

There are 2 types of concrete knowledge: Sensory-concrete and Integrated concrete (Clements & McMillen, 1996)

What is Concrete Knowledge?

Manipulatives should be used to teach informal understandings of math concepts

Manipulatives can be used to actively engage student thinking about math problems, under the teacher's guidance. Just providing manipulatives is not enough, and does not lead straight to academic success (Puchner et al., n.d.)

Students' knowledge peaks when real-world situations, pictures, manipulatives, and spoken and written symbols are all interconnected (Clements & McMillen, 1996)

Engaging in interactive math lessons when using manipulatives and participating in group work leads to higher math achievement (Albertoni, 2014)

Students who use manipulatives during math usually outperform those who don't on retention and problem-solving tests (Clements & McMillen, 1996)

Students' attitudes towards math are improved when using concrete materials and guided by teachers (Clements & McMillen, 1996)

Piaget theorized that primary level children learn about the world through their sensorimotor capabilities, and can use these skills to perform operations on concrete objects, consequently informing their own knowledge (Piaget, via Nishida, 2007)

Children think concretely and cannot easily reason about abstract concepts. Manipulatives help make abstract concepts concrete and fosters children’s learning, allowing them to think about abstract concepts at an earlier stage of development (Piaget, via Nishida, 2007)

Maria Montessori

Yes! Using manipulatives is a great way to bypass the language barrier in a diverse classroom. With manipulatives, children are able to physically touch and visually see math concepts come to life, without needing a strong grasp of English. They can use manipulatives as a form of symbolism to help interpret math!

In fact, diverse students who actively used manipulatives correctly answered more test questions immediately following a math lesson compared to non-diverse students (Nishida, 2007)

Source: Pixabay

Engagement

Efficacy: The belief in one's capacity to control aspects of one's own life.

Motivation and efficacy are related to academic achievement, and to each other (Schunk, 1990).

Motivation is related to aspects of aptitude and efficacy in the subject, perceived importance of the subject, and the student's future goals. (Schunk, 1990)

Efficacy is dependent upon using learning strategies, such as practicing, planning, monitoring, and effort management. Motivation plays a key role in sustaining these strategies (Schunk, 1990).

In other words, the more a student believes a subject is useful to their life and future, and the stronger their confidence in the subject area is, the higher their motivation in that area will be. The higher their motivation is, the more they will be able to sustain the strategies necessary for higher efficacy, which then feeds into their confidence. It is a self-sustaining feedback loop.

1. Issues with the mathematics curriculum itself.

lessons involve following a set of procedures instead of making sense of questions

leading to lack of relation to real world situations

teachers may lack confidence in teaching the curriculum themselves

teaching of addition and subtraction at the same time

Kamii and Lewis (2003) stress emphasizing addition computation skills before subtraction to help increase mathematical acheivement

elementary students are skilled at grasping concrete operations but mathematics may be too abstract (Piaget, via Rivera Vega, 1996)

2. Issues in understanding how children learn mathematical concepts.

Burns (2004) lists two ways in which children can learn math:

via

logic:

developing skills and knowledge through reasoning and understanding

e.g. what does 6 divided by 2

mean

?

via

social convention:

learning socially accepted symbols and terminology

e.g. the social convention for representing "plus" is "+"

Three forms of engagement:

Therefore: If the lesson is about a

social convention

, then memorization and math worksheets will suffice.

If the lesson is dependent on

logic

, then students must internalize and come to their own understandings.

However... this is not always followed, thus leading to a lack of motivation.

In relation to the case...

Brajit is able to compute numerical equations, yet has difficulty applying math concepts to the real world. We then assume she is not fully internalizing and coming to her own understandings with these concepts.

Behavioural Engagement: the display of positive behaviours such as rule-following, participation, contribution to school and extracurricular activities, and involvement in learning.

Cognitive Engagement: having the psychological investment in learning, which includes a desire for challenge and the putting forth of effort towards understanding and mastery of the lessons taught. Involves metacognition and the use of learning strategies.

Emotional Engagement: having positive feelings towards school, teachers, and friends. This includes interest, happiness, and believing that one belongs and that school is valuable.

Engagement encompasses how a student behaves, thinks, and feels.

Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School Engagement: Potential of the Concept, State of the Evidence. Review of Educational Research, 74(1), 59–109. http://doi.org/10.3102/00346543074001059

This article reviews the relevant research that has gone into developing a model of student engagement. Breaks the broad term "engagement" into three interrelated components: behavioural engagement, cognitive engagement, and emotional engagement. It describes what each of these are, and the state of research on them. Includes a discussion on the importance of engagement and its relationship to academic success, as well as an exploration of why the levels of engagements may be low and how educators can raise them. This was very useful for the package in that it helped develop and define our model of engagement, and strengthened the claim that games can be educationally useful.

Garris, R., Ahlers, R., & Driskell, J. E. (2002). Games, Motivation, and Learning: A Research and Practice Model. Simulation & Gaming, 33(4), 441–467. http://doi.org/10.1177/1046878102238607

Extensively researched, this paper was vital to the "Making it Fun!" section. They make a compelling argument for the use of games in education, as they lay out the various studies that demonstrate the strength of games and link them to educational philosophies. The article provides an in-depth definition of what a game is and what games entail, and it creates a model to describe how student learning through play is achieved. They link play to motivation and engagement, and even demonstrate how and why games can succeed as educational tools through a game used by the US Navy for educational purposes.

Heuvel-Panhuizen M.V.D., and Elia I., (2013). The role of picture books in young children’s mathematics learning. Advances in mathematics education. 227-230.

This chapter explains that children’s literature provides a context where children can make certain meaning; this process can be achieved via relating the reading material to one’s own personal experience. Specifically, the author explains that children will easily come to learn meaning-making via reading children’s picture books, which always provides a leeway into getting at the concept of mathematics. Picture books contain mathematical concepts such as the concept of growth, fairness, order, cause/effect, etc.. Through exposure to picture books, children can learn to think on mathematical grounds, asking their questions, searching for answers, taking into account different points of view, etc. Lastly, the explicit visual images provide a mode through which children can use this representation to get at the mathematical concepts. I chose this article for case 2 as a way for the teachers to be able to help the ELL learners. Via the use of picture books, which includes big visual images and which incorporates some mathematical concepts, I believe both ELL’s and non-ELL’s will acquire a deep mathematical understanding of some concepts: big vs. small, sequence, order, combination, shapes, etc.

Kamii, C., & Lewis, B.A. (2003). Single-digit subtraction with fluency. Teaching Children Mathematics, 10(4), 230-236.

This article addresses the idea that subtraction is a much harder concept for young learners to grasp than is addition. Although primary students learn about fact families, they often have a difficult time translating this information into the world of subtraction. Kamii and Lewis explore the idea that educators must first ensure that their students understand addition before introducing a negative action that is a secondary construct to addition. I like how this study recognizes the challenges that are produced in mathematics when language is introduced. The article gives some good insight into why children have encounter challenges like we see in case two when it comes to math, especially subtraction, and it outlines ways in which we as teachers can work to eliminate these challenges.

From: Fredricks, Blumenfeld, & Paris (2004)

Kim, S., & Chang, M. (2010). Computer games for the math achievement of diverse students. Journal of Educational Technology & Society, 13(3), 224.

This article tests the effectiveness of using computer games to improve the math achievement of language minority (ELL) and gender-specific students. The authors found that non-ELLs who played computer math games everyday in school displayed significantly lower math achievement than those who never played. On the other hand, there were positive effects for ELLs who played everyday. I picked this article as it provides an alternative method of learning math for Brajit, because it seems that the current method the teacher is using does not resonate with her. Using games will create a fun learning experience that can at the same time deepen Brajit's understanding of the applications of math.

Nishida, T. K. (2007). The use of manipulatives to support children's acquisition of abstract math concepts.

This article investigated whether actively manipulating objects influences children's acquisition of math concepts. The research found that diverse children that actively used manipulatives correctly answered more posttest questions immediately following the lesson than children in the other conditions. The article shows how ELLs can benefit from manipulatives, allowing us to help ELLs from our Case 2 class with more abstract math concepts.

Puchner, L., Taylor, A., O'donnell, B., & Fick, K. (n.d.). Teacher Learning and Mathematics Manipulatives: A Collective Case Study About Teacher Use of Manipulatives in Elementary and Middle School Mathematics Lessons. School Science and Mathematics, 313-325.

The evidence from this study shows that manipulatives are important in learning math for young children. Specifically, the author analyzes how the teachers are using the manipulatives and if they are effective towards children’s learning. The author stresses that it is necessary for pre-service teachers to know that providing manipulatives does not lead straight to students’ academic success; it is important to recognize the relationship between pedagogical approaches, content, and type of manipulatives. I believe this is a valuable article because it provides examples of what is appropriate use of manipulatives and how to support content learning.

Randel, J. M., Morris, B. A., Wetzel, C. D., & Whitehill, B. V. (1992). The Effectiveness of Games for Educational Purposes: A Review of Recent Research. Simulation & Gaming, 23(3), 261–276.

http://doi.org/10.1177/1046878192233001

An early paper reviewing the extensive research already done on educational video games by that point. The results were mixed, showing that games helped in some areas (such as math and language literacy) but not in others (such as social studies). Given the early nature of computer games at the time, and the fact that teachers probably did not know how to effectively implement them yet at that time, plus the inherent diversity of games in general and the difficulty in objectively comparing one to the other in a controlled manner, I (Jonathan) don't really let these findings dampen my interest and belief in games as educational tools. In any case, it was relevant and useful to this package as it demonstrated that, even at this early point, math was one of the subjects that benefited from educational games.

Source: Pixabay

Engaging students in their schoolwork is critical to their academic success, as engagement shares numerous similarities with motivation and efficacy. Emotional engagement, for instance, is very similar to motivation, as it includes elements of interest and value. Cognitive engagement is very similar to efficacy, as metacognition and the following of learning strategies is important to both (From: Fredricks et al., 2004). Thus if we can engage students emotionally and cognitively, we can help to motivate them and improve their efficacy.

So how can we engage students? Well one very useful way is through games! If they are fun and interesting, they will engage students emotionally. If they are challenging and learning-centric, they will engage the students cognitively.

Our Own Stories

Carol

Growing up, I do not remember being very good at math. To this day, that probably still applies. From this, I can see the importance of making math fun and meaningful as it can effect long term perceptions of the topic itself.

Vivian

When I was young, my dad would borrow all of the interactive math games from the library and bring it home for me to practice math with. I grew up being heavily engaged with math, and I was interested to see how this method of learning math would impact other children.

Jonathan

I'm a huge fan of games, and my lifelong passion for and involvement with gaming has made me pretty familiar with their potential. Honestly, I'm pretty sure video games helped me to learn to read. I was really interested in trying to apply math education to a fun game, as I firmly believe that showing kids that math can be useful in fun and games might help motivate them to pursue it further.

Montessori theorized that mental development is connected with and dependent on movement (Montessori, via Nishida, 2007)

Believed that children built on their physical experiences of the world through their senses (Montessori, via Nishida, 2007)

Could extend their learning and understanding through manipulating interesting materials (Montessori, via Nishida, 2007)

Source: Montessori Australia

Source: Pixabay

Adapted from: Garris, Ahlers, & Driskell (2002)

Source: Pixabay

The Game Cycle

There are many ways to make math fun and engaging for students like Brajit. These include the use of manipulatives, interactive activities, as well as many forms of games (digital, fantastical, etc.). Through these methods, children are able to transform abstract mathematical concepts into more concrete, relevant operations in connecting to their own lives and/or interests.

By engaging her in enjoyable math activities and solidifying what she learned in math class through tangible, visual methods, we hope that Brajit will have a clearer grasp on math concepts and can answer her father correctly next time!

Ultimately, it is hoped that mathematical achievement will improve and perceptions of the curriculum as an experience will prove more positive!

User Judgments: How the player feels about the game (ie, emotional engagement). These judgments include interest, enjoyment, task involvement (how immersed they are in the game) and confidence.

User Behaviour: These judgments determine player motivation, which in turn determines behaviours such as persistence, pursuit of goals, involvement in the task, and challenge-seeking (cognitive engagement).

System Feedback: Judgments (and thus behaviours) are regulated by comparisons of performance with the standards of the target goal. If performance consistently exceeds the goals, the game is demed too easy and motivation declines. However, if performance falls short, and the goal is clear and the difficulty is not too severe, motivation is increased.

Debriefing is critical for the use of games as learning tools. This is the role of the teacher: we must link the activities of the game to real world problems and lessons.

From: Garris et al. (2002)

From: Garris et al. (2002)

But keep in mind that...

Will this work for ELLs?

Source: Openclipart

A great deal of research has been done (and is being done) on games as learning tools. Games' strongest selling point is how engaging and enjoyable they are. Garris et al. (2002) is able to link the game cycle to Duffy and Jonassen's constructivist and Dewey's experiental learning models, stating that the player is actively constructing knowledge from experience. An effective learning environment can be established with the debriefing stage, and students can learn by doing.

The efficacy of games as teaching tools has shown mixed results. My (Jonathan) personal belief is that the wrong games were being used, or that they were being used ineffectively. Regardless, an earlier study (Randel, Morris, Wetzel, & Whitehill, 1992) found that the effectiveness of games depended largely on the subject area. The subject that showed the best results from games was math, thus giving strong support for their use in this case. Furthermore, Kim and Chang (2010) found that ELL students had higher math achievement after playing math-based computer games daily. This shows that games can be especially useful in diverse classrooms.

Source: Pixabay

Source: Pixabay

Rivera Vega, L. M. (1996). Background for the study. In The effect of mental computation instruction on third grade mathematics students (2). Retrieved from: http://ponce.inter.edu/cai/tesis/lmrivera/cap2.htm

This website was useful in that it helped us understand the perspectives of many theorists in regards to children’s understanding of mathematical concepts. These theorists give us a framework in which we were able use in answering the question of how we can better engage students in the math curriculum.

Schunk, D. H. (1990). Introduction to the special section on motivation and efficacy. Journal of Educational Psychology, 82(1), 3–6. http://doi.org/10.1037/h0092681

Schunk summarizes the research on motivation and efficacy in the special edition of the journal focusing solely on that topic. Useful in giving an overview of the topic, the article goes through the major areas of study in motivation and efficacy, and explains the findings. Helpful in defining motivation and efficacy, as well as linking them together and supporting their importance to academic achievement.

Setati, M. (2005). Teaching mathematics in a primary multilingual classroom. Journal for research in Mathematics Education, 36, 447-466. Retrieved from http://www.nctm.org/publications/journal-for-research-in-mathematics-education/

This article looks at the relationship between math and language (i.e. knowing mathematical terms, concepts) in a South African classroom where the students’ and teacher’s home language differs from language of instruction. The author states that there are two types of discourses in math that students may engage in (i.e. procedural discourses and conceptual discourses) and found that the students and the teacher in the South African class spoke in English during procedural discourses, the home language during conceptual discourses, and that conceptual knowledge was somehow less valued than procedural knowledge. The author argues that language facilitates student’s communication of (and learning) of math and that teachers need to permit the use of children’s first language when learning math. Given that the class in case 2 has ELLs or linguistically diverse students, this article is extremely useful in understanding how ELLs might engage in mathematical learning and what kinds of challenges they may face while doing so. It can also help us think about what the teacher can do in order to engage all students in her math lessons.

Willis, J. (2010). Learning to love math: Teaching strategies that change student attitudes and get results. Alexandria, VA, USA: Association for Supervision & Curriculum Development (ASCD). Retrieved from http://www.ebrary.com

Willis provides suggestions on interventions that can help students overcome negative attitudes towards math, with the concept of achievable challenge as an important underlying concept throughout all her suggestions. This includes accommodating for different students' learning needs, and addressing “how to evaluate each student's achievable challenge level to each new unit so instruction can be applied appropriately to lower the barriers, not the bar”.

Does it work?

Children's literature provides context for children to make certain meanings

Children can relate the reading material to their own personal experiences, providing a leeway into concepts of mathematics (e.g. growth, fairness, order, etc.)

They learn to think on mathematical grounds, asking their questions, searching for answers, taking into account different points of view, etc

Explicit visual images provide a mode through which children can use this representation to get at the mathematical concepts (Heuvel-Panhuizen & Elia, 2013)

For example, in Tyrannosaurus Math by Michelle Markel, not only are there 6 dinosaurs, but they are all in different colors according to how many there are (1 orange, 2 blue, 3 purple), giving a visual clue to accompany the math problem of 1+2+3=6

Storybooks are a good way to engage students in math and provide context, but they can possibly be troubling for ELLs if chosen inappropriately for their level.

Many mathematical difficulties of ELLs are related to the language demands of math tasks. Many ELLs have high numerical proficiency that is masked when assessed through language-heavy means, such as complex word problems. Cognitive switching to translate technical math terms can lead to errors (Alt, Arizmendi, & Beal, 2014)

To counter this, teachers can lighten the language load by using manipulatives, or provide visuals that do not rely heavily on language. (Alt, Arizmendi, & Beal, 2014)

Concrete knowledge: Comes in 2 forms: sensory-concrete and integrated concrete.

Sensory-concrete involves manipulation of sensory materials to make sense of an idea.

Integrated concrete is more advanced and is built through learning. For children, the physical objects, the actions they perform on the objects, and the abstractions they make are all interrelated in a strong mental structure supporting understanding.

Efficacy: The belief in one's capacity to control aspects of one's own life.

Engagement: The state of being committed and invested in something. In terms of school, there are three interrelated parts:

Behavioural: Coming to school, following rules, etc. Easily observable

Cognitive: Psychological investment in learning, understanding, and metacognition

Emotional: The feelings a student has about school, relationships with various aspects of the school, and their opinion on the value of learning

Fantasy, Endogenic: When the fantasy is intrinsically related to the underlying game system (ie, learning about physics by piloting a spaceship into orbit)

Fantasy, Exogenic: When, in a game, the fantasy is draped around the underlying game system (ie: solve fractions to slay a dragon)

Game: An activity that involves the elements of fantasy, rules and goals, sensory stimuli, challenge, mystery, and control

Albertoni, M. J. (2014). Promoting engagement in math through interactive lessons.

This article found that students were more engaged in interactive math lessons when using manipulatives and participating in group work. It also found that student achievement increased in math when using manipulatives. This article is useful as it makes interaction and manipulatives key to helping children do better in math and have higher participation in the classroom.

Alt, M., Arizmendi, G. D., & Beal, C. R. (2014). The relationship between mathematics and language: Academic implications for children with specific language impairment and English language learners. Language, Speech, and Hearing Services in Schools, 45(3), 220-233.

This article examines relationship between language and math performance of students with specific language impairments (SLI) and students who are ELLs, as well as looks at different theories as to why ELLs have more difficulty with math compared to English monolingual students. It offers some ideas on multimodal approaches that may decrease the performance gap between ELLs and English monolingual students. Tying back to the case, this article is significant because it emphasizes that educators must recognize that language-heavy mathematical problems do not honestly reflect ELLs’ mathematical abilities. It highlights the importance of using different modalities, other than text, so that ELLs are able to make meaningful connections between the math they learn in class and the world they live in.

Burns, M. (2004). A can of coke leads to a piece of pi: A professional development exercise for educators is an adaptable math lesson for many grades. Journal of Staff Development, 25(4), 16.

This article brings up issues in teaching math that teachers face: they think that math does not involve sense-making, just a following of procedures, and that teachers lack security in their own mathematical knowledge, so it is difficult for them to transmit knowledge to their students. It also explains how students learn math through logic and social convention, and gives classroom strategies for teaching. This article is relevant as it shows us a major barrier in teaching math and how it affects the students. We can apply the classroom strategies for more effective teaching.

Clements, D. H., & McMillen, S. (1996). Rethinking "concrete" manipulatives. Teaching Children Mathematics,2(5), 270-279.

This article agrees that the common problem with children learning math is the abstract quality of it. Clements defines 2 different types of concrete knowledge, and explains how they tie in to manipulative usage. He also goes over different mediums of manipulatives including computer generated manipulatives. This article in incredibly helpful has it gives us a solid understanding of what "concrete" actually means, so that we can build our ideas around it.

Internalize: to understand a concept with one's own mental framework and relate it to one's own experiences in order to make meaning

Logic: the use of rational reasoning when making sense of mathematical concepts.

Manipulative: Any physical object that a student can use to support their hands-on learning and connect mathematical ideas and symbols that they learn in the classroom with those objects. Some examples are unifix cubes, and pattern blocks.

Motivation: The process through which goal-directed behaviour is started and maintained.

Real-world: framing math problems in the context of probable life situations

Social convention: that which is accepted by society as a norm

**By: Carol, Jonathan, Vivian**

Source: Carson-Dellosa Publishing Group

Source: Youtube

IT'S MATHEMATICS SONG