Cognition and Instruction/Learning Mathematics
Mathematics contains many areas of study such as geometry, algebra, calculus, and probability; each requiring the mastery of specialized concepts and procedures. The challenges of teaching and learning mathematics can be understood and overcome through analysis of cognitive processes. In this chapter we examine cognitive theories and research that inform the practice of mathematics education. We discuss the relevant aspects of Piaget’s theory of cognitive development and the criticism that it has received. We explain the factors that influence individual students' abilities to learn mathematics and how teachers can account for these factors when designing lessons.
What is Mathematics?[edit]
Mathematics is the study of numbers, quantities, geometry and space, as well as their relationships and functions. It utilizes a combination of conceptual, procedural, and declarative knowledge.^{[1]} In order to successfully solve mathematical problems, students need to acquire this set of knowledge. To fully engage in their learning of mathematics, students must first gain a conceptual understanding, which requires utilizing background knowledge of learned concepts. Conceptual understanding of mathematics leads to the acquisition of more mathematical knowledge, helping to construct the other strands of mathematical proficiency: productive disposition, procedural fluency, strategic competence and adaptive reasoning. Growth in each proficiency leads to growth in the other proficiencies and leads to more knowledge. That is, conceptual knowledge enhances procedural knowledge and so on.^{[2]} For example, there are many different algorithms in mathematics that students need to be familiar with. When students have a clear understanding of mathematical principles and concepts, they will be able to select and recreate the appropriate algorithm for any mathematical problem. This demonstrates the connection between conceptual knowledge and procedural knowledge because students can have many learned strategies but they have to select the correct one and build upon it.^{[3]} In addition, when there are successes or failures while using certain procedures to solve complex mathematical problems, students can often learn more. Students can learn from their failure by selfquestioning their mistakes and can reconstruct their existing knowledge. As a result, this increases their conceptual knowledge. Declarative knowledge is definitely related to both conceptual and procedural knowledge because it requires students to retrieve mathematical concepts (i.e., conceptual knowledge) and specific mathematical algorithms (i.e., procedural knowledge) from the longterm memory. Deficiency in any one, or all, of these knowledge areas may cause learning difficulties in mathematics.^{[4]} Thus, this combination of conceptual, procedural, and declarative knowledge influences learning since they are all associated with one another.
Cognitive Theory and Mathematics[edit]
Piaget's theory of Cognitive Development[edit]
Jean Piaget has indicated four primary stages of cognitive development from birth to young adulthood, these includes sensorimotor (from birth to age 2), preoperational (about age 2 to age 7), concrete operational (about age 7 to age 11), and formal operational (about age 11 to age 15). Although everyone progresses through these stages differently, Piaget believed that every child would eventually experience every stage of thinking in the sequence and no one would miss a stage because one would not be able to develop to the next stage until they understand the previous one; it’s just a matter of time.
Piaget also pointed out that children’s learning is usually developed through movement and the five senses from birth to age 2. During the infants’ first few weeks, they start learning how to track objects and to get a hold of them by constantly practicing, which can help the parts of the brain that process and connect visual and motor behaviour to start developing. Once the infants recognize that learning follows by repetition, then they will start learning how to plan in advance and reach for the objects that they want by using a more efficient approach. Piaget claimed that infants are able to link numbers to objects at this stage^{[5]} and there is also evidence that children have already acquired some knowledge of the concepts of the numbers and counting^{[6]}. In order to develop the mathematical skills of infants at this stage, educators can offer activities that will integrate with numbers and counting. For example, educators can read books that have pictograms in them. This not only helps children to relate the pictures of objects to their corresponding numbers, but also helps build their reading and comprehension capabilities. During this period, Piaget has demonstrated that infants can already build their own ways of dealing with objects and knowledge about them, which can support gains reflective intelligence.^{[7]} Since Piaget believed that an individual needs to build upon knowledge that is acquired from the prior stage and therefore cannot move to the next stage until the current stage is mastered. Thus, in order to enhance infants' understanding of numbers, educators can provide a general foundation of mathematics by engaging activities that incorporate counting.
Children start acquiring language ability, symbolic thought, egocentric perspective and some degree of logic at around age 2 to age 7. During this period, children learn how to employ problemsolving skills that integrate with objects, such as numbers, blocks, etc. Although children have already gained some knowledge of concepts of numbers, they only have limited logic association, and cannot process operations in a reverse order. For example, children who understand that 5+3=8 may not have the mindset that 3+5=8 also. According to Piaget, this is because children can only identify one aspect or dimension of an object with the loss of other aspects. In order to enhance the children’s mathematical capabilities in this period, educators can ask them to build a specific object by using building blocks. While they are building it, they can learn how to group them based on their identical features, and also help them understanding that there are always multiple methods of combining them together.^{[8]}
According to Piaget, children’s cognitive development accelerates between ages 7 and 11. They can start using their five senses to distinguish objects, which can help them to identify two or three aspects of dimensions at once. For instance, Piaget used an experiment of pouring the same amount of liquid into different size containers. Children at this stage are able to notice the levels of liquid will be different based on the dimension of the container. Another major cognitive growth that occurs during this period is the ability of classification and seriation to separate objects.^{[9]}Children learn classification by grouping objects based on similar features, and acquire the ability of seriation by categorizing objects based on their increased or decreased value, such as length, width, volume, etc. Even though they may have already acquired some basic arithmetical operations at this stage, they do not know how to apply these concepts into solving math problems. For example, when they are being asked to count the pieces that are made of 3 rows of 5 building blocks, they do not know how to apply multiplication while counting. In other words, the abstract concepts of arithmetic must be directly related to physically available elements and operations. This also implies that they are still not capable of setting up a consistent system based on measurement at this stage.
The final stage of cognitive development often occurs at around age 11 to age 15. At this stage, children are able to form their own theories and construct their own mathematics concepts. They can also relate abstract concepts to concrete situations now. For example, when they encounter an algebra problem, they are now able to solve it by themselves instead of having a teacher to refer to a concrete condition. The reason that they can now develop abstract thought patterns into concrete situations is that that they start building their reasoning skills, which includes clarification, inference, evaluation, and application. In order to make students comfortable with these concepts, teachers can teach students on how to separate the word problems and understand the differences between related and unrelated information in the problem.
Piaget believed that if a child has a difficulty understanding a concept, it is because of the toorapid progress from the qualitative structure of the problem to the mathematical formulation. According to Piaget, in order to help the children to understand the concept, teachers should find an active approach that allows children to explore spontaneously, so they can learn and reconstruct their own concept, instead of having the teachers to give them the answers directly. ^{[10]}
Critiques of Piaget's Theory[edit]
Piaget’s belief of cognitive development  Criticism 

1) Children start developing an understanding of the object permanence 

2) Children’s sensory abilities and cognitive development occur in their first 6 months of birth 

3) Every child will experience the four stages in a specific order 

4) Piaget separated the cognitive development into definite stages 

Even though Piaget’s theory is widely used by teachers to monitor their students’ cognitive development in the classroom nowadays, his theory is controversial. Lots of educators rely on Piaget’s theory to measure students’ readiness for learning math. On the other hand, Hiebert and carpenter advised that Piaget's theory is not a useful guide, as lots of researches have proved that children who fail to follow Piaget’s theory are still able to learn the math concepts and skills.^{[11]} While Piaget focused on children’s internal exploration for knowledge, and believed that children start developing an understanding of object permanence (such as how to track for a hidden object) from birth to age 2, other researchers argue that Piaget neglected the children's need for motivation. Berger believes that external motivations and teachings play an important impact also.^{[12]} Kagan believes that the reason why an infant is able to reach for objects even with displacement is because their memory capacity has increased, not, as Piaget pointed out, in terms of the new cognitive structure.^{[13]} Piaget has also been criticized for broadly speaking of children’s abilities. He deduced that children’s sensory abilities and cognitive development occur in their first six months of birth. While Piaget believed that each child must go through those stages in a particular order, HeuvelPanhuize argued that Piaget’s theory underestimates young children’s abilities. For example, he found that since early childhood teachers’ belief of stages of cognitive development deeply relied on Piaget’s theory, they may have lower expectations for children’s knowledge of symbols, the counting sequence, and arithmetic operations than what the children are actually capable of.^{[14]} Beger also argued that their perceptual learning might actually develope before birth.^{[15]} Even though a child is supposed to be in a certain stage based on his or her age, not all learners are the same. They might be placed in a higher or lower stage based on their unique abilities. For instance, Gelman and Gallistel have pointed out that children in their preoperational stage are capable of thinking abstractly in terms of counting objects. In addition, Piaget fails to demonstrate the aspects of emotional and personality development of children. Even though Piaget’s theory explains an effective approach that can measure children’s intelligence and memory development, he neglects the remarkable aspects of creativity and social interaction of individuals.^{[16]} Christina Erneling argues that the pattern of development can be established only if the children are put in the right conditions. She believes that any concepts of learning require an expansive theory of education, and the fundamental part of cognitive development is to acknowledge the differences of an individual’s social and cultural backgrounds. In other words, Piaget seemed to be overlooking cultural effects. Since his research was done in a Western country, his theory of cognitive development may only represent Western society and culture. According to Piaget, scientific thinking and formal operations can only be reached at a certain stage. On the other hand, Edwards et. al argued that Piaget’s research was unreliable due to the lack of controls and small samples. He believed that there could a higher regard for the basic level of concrete operations in other cultures.^{[17]} Beger also argues against Piaget’s definite stages, he judges that Piaget had explicitly explained the children’s internal search for knowledge, but he tended to overlook the external factors, such as heredity, culture, and education. He suggested that Piaget’s stages of cognitive development should rather be seen as a gradual and continuous progress instead of separating into definite stages.^{[18]} Piaget’s theory has also been criticized for not offering a sufficient description of cognitive development in his last stage. He supposed that everyone will be able to develop abstract reasoning between age 11 to age 15. On the other hand, Paplia et.al believes that not everyone can acquire the skills of formal operations at this stage. And even though they may not attain this ability, it does not mean they are immature. We can only conclude that they have different phases of mature thought.^{[19]} Hence, a more persuasive belief of cognitive development should be perceived as an irregular process as children attain new skills and different behaviors individually at each stage.^{[20]}
Cognitive Domains[edit]
Cognitive theory and its relevance to learning mathematics has come a long way since Piaget. Numerous studies have been done which demonstrate the relationships between different cognitive abilities and mathematical abilities. As early as 1978, researchers were studying the relationship between academic abilities and patterns of brain related behaviour. In 1978, Rourke and Finlayson studied 914 year old children with learning disabilities and found that children lacking abilities in arithmetic performed as would be expected if their right cerebral hemisphere was not functioning correctly.^{[21]} More recent studies have been able to identify repeating patterns of even more specific relationships for cognitive abilities and functional deficiencies in math.
In 2001, Hanich, Jordan, Kaplan and Dick studied the mathematical performance of grade 2 students.^{[22]}. Children were divided into four groups, consisting of normal achieving students, children with math deficiencies, children with reading deficiencies, and children with both math and reading deficiencies. Children in each of the four groups were given seven mathematics tests in the same order, to assess performance in: a.) exact calculation in arithmetic combinations, b.) story problems, c.) approximate arithmetic, d.) place value, e.) calculation principles, f.) forced retrieval of number facts, and g.) written computation. They found that children with math and reading deficiencies struggled with both word problems and with standard computation (such as number facts, number combination and procedural computations); whereas children deficient in just math struggled only with standard computational skills. This, and subsequent studies, have led researchers to conclude that there is more than one cognitive domain for math, with each domain using different processes of the brain.
Fuchs, Fuchs, Stuebing, Fletcher, Hamlett, and Lambert noted that a number of studies have consistently found that predictors for computational success include: a.) working memory, b.) visualspatial working memory, c.) attention ratings, d.) phonological processing (detecting and discriminating differences in speech sounds), and e.) vocabulary knowledge (2008)^{[23]}. During a longterm, large scale study of students who were randomly sampled, the authors undertook to determine whether or not problem solving and computation were distinct aspects of mathematics. The authors assessed students for computational and wordproblem solving abilities, phonological skills, nonverbal problem solving, working memory, attentive behaviour, processing speed, and reading skills. They found that attentive behaviour and processing speed played dominant roles for computational difficulty.
Further, Fuchs et al also noted that working memory, short term memory, nonverbal problem solving (ability to complete patterns presented visually), concept formation, and language ability (including reading) were all predictors of problem solving ability. They also noted that deficiencies in language skills was a discerning factor for students who exhibited problem solving difficulties.
Processes of the brain for each cognitive math domain[edit]
Computation Cognitive abilities  Problem Solving Cognitive abilities 

Predictors for computational success:  Predictors for problem solving success: 
• Working memory
• Attention ratings • Processing speed • Language ability

• Working memory
• Short term memory • Nonverbal problem solving (ability to complete patterns presented visually) • Concept formation • Language ability

The Importance of Working Memory in Learning Mathematics[edit]
Working memory is the system responsible for temporarily holding new or previouslystored information which is being used for the completion of a current task. Its capacity is limited. There are two types of working memory: auditory memory and visualspatial memory. Visualspatial memory has been found to be important for solving computational problems. Auditory memory has been found to be important for all mathematical domains. The variation of an individual's capacity for working memory may be due to how fast information is processed, one's knowledge, or one's ability to ignore irrelevant knowledge.^{[24]} Executive processing activities, such as planning, organization and flexible thinking, may also affect working memory. ^{[25]}
On the other hand, short term memory is responsible for temporarily storing information which must be used, but not necessarily manipulated. Again, the capacity for short term memory is limited, maybe only a few seconds. This is where we store information such as a telephone number we need to remember for only a few seconds while we dial it.
In their study, The Relationship Between Working Memory and Mathematical Problem Solving in Children at Risk and Not at Risk for Serious Math Difficulties (2004), Swanson and BeebeFrankenberger concluded that working memory plays a critical role in integrating information during problem solving. They argue that working memory is highly important to integrating information during problem solving because "(a) it holds recently processed information to make connections to the latest input and (b) it maintains the gist of information for the construction of an overall representation of the problem."^{[26]}
A new study by H. Lee Swanson suggests that the capacity of working memory moderates the influence of cognitive strategies on problem solving accuracy.^{[27]} The author conducted an intervention study to ascertain what role working memory capacity played in strategy intervention outcomes and the role of strategy instruction on word problem solving accuracy.
Both verbal and visualspatial working memory were measured for all children in the study group. Children, both with and without math disabilities, were were then divided into three treatment groups for a randomized control trial. Group 1 was given verbal strategies for problem solving; Group 2 was given visualspatial strategies for problem solving; and Group 3 was given a combination of both verbal and visualspatial strategies. Each of the groups was also provided with lesson plans that regularly increased irrelevant information within the word problems. The author's strategy of adding irrelevant information was meant to teach the children to attend to relevant information only. This strategy was prompted by a number of other studies which showed that learning to differentiate between relevant and irrelevant information is significantly correlated with problem solving accuracy for students at risk for math disabilities.
The results of the study support the view that strategy instruction facilitates solution accuracy. However, it must be noted that the effects of strategy instruction were moderated by individual differences in working memory capacity. Those children with low working memory capacity did not benefit as much as expected. It was the children with higher working memory capacity, both with and without math disabilities, who were most likely to benefit from the learning strategies. All children with math disabilities, whether possessing high or low working memory capacity, did benefit from strategies that used visual information, however children with low working memory capacity needed the combination of both verbal and visual strategies. Lastly, the results suggest, academic tasks that train processes related to working memory for controlled attention may, in fact, influence later working memory performance.
Implications of this study would suggest that students with math disabilities be evaluated for working memory capacity and then strategies for addressing their individual concerns be determined based on their working memory capacity.
Factors that Affect Learning and Teaching Mathematics[edit]
Individual Differences[edit]
Every learner has their own distinct skills, background knowledge, culture, and interests. These aspects can affect learning and teaching mathematics because instructional strategies should be modified accordingly.
Differences in Skills[edit]
All learners have their own strengths and weaknesses. They may be skilled in some aspects in mathematics but may be incompetent in another area. It is important for teachers to know what skills the students have because they can utilize these skills to help improve the students’ weaknesses. If teachers do not recognize the students’ strengths and weaknesses, they might give students challenges. Students will face difficulty in the given task because they do not have the required skills. As a consequence, it may even influence the students' selfefficacy and create learned helplessness when students cannot accomplish the task. Hence, if teachers know what students are proficient in, then students will not have problems in learning new knowledge of mathematics. Mathematical problems require a set of preskills such as simple arithmetic, algebra and logic reasoning. For instance, solving word problems require mental representation of the problem and simple arithmetic to transform the word problem into a mathematical equation. As a result, students who are not skilled at formulating a mathematical equation will not be able to solve the word problem.^{[28]} Teachers should adjust their instructional practices according to the different preskills that the students have because these preskills play a big part in solving mathematical problems. When students gain more conceptual and procedural skills in mathematics, they become more competent and efficient in learning mathematics.^{[29]} In modern high schools, there are different levels in the course of mathematics such as beginner, principle, and advance level. Students are placed accordingly to their set of mathematical skills level. Otherwise, they can choose which level they want to be in. In this case, it is important that teachers support and evaluate the students' performance to see whether or not if they are suitable in the chosen level. Students do not want to be in a math class that is too difficult or else it would be too overwhelming, neither should it be too easy or else it would be too boring. Hence, by knowing what skills the students have, students can achieve new mathematical knowledge.
Differences in Background Knowledge[edit]
Students’ knowledge of mathematics can be affected from their background knowledge. Indeed, all students have different background knowledge because they all have different experiences in the social world. These reallife experiences are crucial because they learn about the functionality of mathematics symbols from these observations. For example, students can learn simple arithmetic from grocery shopping which involves dealing with money. Students can learn how to estimate the total cost of goods and how much change they should received back. Therefore, when mathematical concepts are taught in a way that is related to their background knowledge, students will be able to interpret these concepts more easily.^{[30]} In addition, students are more motivated and engaged when their learnings of mathematics are related to their realworld situations. This is because they find the acquired learnings very meaningful and important as they are applicable in their daily living.^{[31]} For instance, many students might find learning mathematics from a textbook boring or difficult. However, if mathematics are taught to solve reallife problems such as calculating the interest gained in the bank, the total cost of living expenses, or the probability of winning in a poker game. As a result, students will have a better understanding of mathematical symbols and concepts when these learnings are related to their prior experiences. In addition, challenging mathematical problems not only require background knowledge of mathematics, but also some knowledge of other subject areas such as physics terms or chemistry terms.^{[32]} Mathematical word problems require a good understanding of the text meaning before it can be solved which means that students need to be able to utilize their language knowledge to comprehend the text. As a result, students' background knowledge can impact their learning in mathematics. For instance, many math courses in University require prerequisite courses because the advance level math courses require understanding of some basic mathematical knowledge. Without these background knowledge, students will have difficulty comprehending the new math materials.
Differences in Interests[edit]
Everyone has different interests. Some students might enjoy mathematics because they were born or taught at a young age with strong mathematical skills, while other students might hate mathematics because they always face failure with mathematics which discourages them to continue to learn. Having interests in mathematics can increase students’ motivation to learn mathematics. This concept is an intrinsic motivation because students want to study mathematics out of their own interests.^{[33]} As a result, they are more engaged in the tasks and would try their best to solve the challenge. Students' interests are related with their beliefs on their selfperceptions, their ability, and their academic achievement.^{[34]} Thus, it is important to develop interests in mathematics for students in order to increase their academic performance. Indeed, there are many ways to increase interest in mathematics such as family, classmates, and teachers.^{[35]} Family can show support and encouragement to students in mathematics at home which can increases students’ value on mathematics. Students usually have social comparisons and like to follow what other classmates are doing. Hence, classmate influences play a big role in students. When students see their classmates enjoying a mathematics problem or game such as sudoku or a puzzle, students will also be interested in solving. Most importantly, teachers can organize fun and interactive games in a classroom setting while showing enthusiasm in their teaching.^{[36]} This will enhance students’ interests in learning a subject they do not enjoy. As a result, it is important that teachers create an enjoyable setting for students to learn in order to promote interests in mathematics. It would be very difficult to teach students mathematics if the learners hate mathematics. They will not want to learn the materials and only study because they have to.
Cultural Differences[edit]
Students with different cultural background have different academic achievement levels and different goals.^{[37]} Also, their values on mathematics might be different depending on their culture. When a culture values a particular subject such as mathematics, these children tend to be trained at a young age at school and at home. Hence, these students will have a higher efficiency of mathematics performance. Students who study mathematics regularly are likely to have a high level of automaticity because they have sufficient practices of the mathematical problems. They will be able to select the appropriate strategy and solve the mathematical problem more efficiently.^{[38]} Vice versa, when a culture does not believe that mathematics is important, these children might not be taught vigorously and will performed at lower competence levels. In order to excel in a subject area, it is important to have practices both at school and at home. Students who only practice their mathematics skills at school by the teachers' support do not have enough training because they are not encouraged to study actively and intensively at home. In addition, cultures that hold positive beliefs on performance such as high standards, effort, and positive attitudes can lead to high academic proficiency levels.^{[39]} Different cultures have different languages. By all means, their way of wording a mathematical problem may also differ. Research shows that the structure of Chinese number languages (e.g., 15 is ten five) is easier to learn than IndoEuropean number languages which is English (e.g., 12 is twelve and teens words are often inconsistent).^{[40]} It is often to faster to pronounce Chinese number languages than in English which affects students’ mathematics efficiency. Hence, Chinese has the ability to retain these numbers in shortterm memory longer especially in complex mathematical problems with multidigit numbers.^{[41]} Cultural differences should be taken into consideration when designing instructional practices since different students have different cultures that can affect how they approach mathematical problems.
SelfEfficacy in Mathematics[edit]
Students' selfefficacy in math is their belief in their ability to solve math questions. Students with a higher level of selfefficacy believe that they are capable in solving math questions, which they are more likely to engage in mathrelated tasks and have higher academic performance in math. On the other hand, students with low selfefficacy believe that they are not capable in solving math questions, which they will feel more anxious in solving math questions and have lower academic performance in math. Therefore, students' selfefficacy in math has strong connections with their engagement and academic performance in math.
SelfEfficacy's Impact in Math[edit]
Selfefficacy can influence the way students think, understand, and feel about their learning in math. Students with high selfefficacy believe that they have the ability and skill to perform well in math.^{[42]} Having the thought that they are capable in solving math, students will be more motivated to learn and study math. By doing so, students will encounter selffulfilling prophecy which fits their belief of their ability in math when their math improved after they studied. On the other hands, students with low selfefficacy in math will believe that they do not have the ability to perform well in math. ^{[43]}With this belief, students might have the thought that they cannot achieve math even if they tried very hard. Therefore, they are less motivated in doing math questions. Also, students with low selfefficacy in math might give up easily after a few trials of questions by thinking that they do not have the ability to get the right answer. When they do so, it reinforces their belief of their disability in math. The student will encounter selffulfilling prophecy which they act in a way that fulfill their belief in their low ability in math.
Assessing Students' SelfEfficacy[edit]
It is important to assess students' selfefficacy and know whether or not if they are confident in learning a particular topic in math because it may affect their performance. One of the ways to assess students' selfefficacy is to construct a list of firstperson statement and have students to rate their selfefficacy for each statement. ^{[44]} First, teachers have to identify the topic that they would like to assess their students' selfefficacy on. For instance, if the topic is on finding surface area, teachers then construct a list of firstperson statements on that topic. Then teachers can have students to rate the statement using a scale range from 0100 (0 which the statement is false and 100 which the statement is true).^{[45]} The following chart is an example of a student's rating his selfefficacy on the topic of surface area.
Rate (0100)  Statement 

80  I know what information do I need in order to find the surface area for a parallelogram. 
100  I can find the surface area of a rectangle when given the length and width. 
60  I can write the equation for the surface area of a trapezoid. 
50  I can explain to my classmate why the equation for the surface area of a triangle is bxh ÷2. 
90  I can calculate the area for a square which have the length of 4cm. 
After the student rated the statement, the teacher can estimate how confident the student is on that topic by adding up the scores. For the above example would be 80+100+60+50+90. From the scores, the teacher will have an idea on student's selfefficacy on that topic. Furthermore, the teacher can compare student's selfefficacy for a particular topic to their general efficacy in math. Also, when assessing students' selfefficacy, teachers should keep in mind that students' selfefficacy may impact their learning motivation and learning behavior. Therefore, teachers should adjust their teaching instructions to increase students' selfefficacy and match their level respectively.
Development of Students' SelfEfficacy[edit]
Bandura has proposed four major influences on the development of selfefficacy.^{[46]} The first influence is students' mastery experiences.^{[47]} For instance, when students succeed in a math test, their level of confidence in that area of math will go up. This will have a positive effect on students future performance, which students will be more confident that they have to ability to solve it when facing similar questions. The second influence is students' various experience.^{[48]} By observing others, especially peers with similar ability, students selfefficacy in doing a particular task will increase. When the teacher introduced a new topic in math, which students are uncertain about the level of difficult for that topic, by observing their peers completing the questions, their level of confident in understanding and completing the questions in the new topic will go up. Moreover, even watching a documentary on mathematicians doing math improves students' math selfefficacy.^{[49]} The third influence is social persuasion.^{[50]} This could be a positive phrase from the people which the students interact with, such as their parents, peers or teachers. Positive feedback from the teacher, such as "you are getting better in solving algebra questions" will increase students confident in solving algebra questions. The fourth influence is students psychological state.^{[51]} This refers to students emotional reaction toward a situation. For example, a student might feel that her failure of a math test is due to her inability of math, which in reality is a result of her anxiety. In this case, student misjudged her ability and lowered her confident in math. Another case might be student seeing her successful performance in a math test as luck, instead of her ability in performing well. In this case, the student lost a chance of building her confident in math. Therefore, students perception toward both positive and negative situations have an effect on building their selfefficacy. The way to increase students selfefficacy in this route is to have them to recognize their true ability in math and increase their positive feelings of their ability.
Usher has conducted a research on measuring the four different sources of middle school students selfefficacy's development in math, by interviewing the students, parents, and teachers. ^{[52]}The result of the research is consistent with Bandura's proposed idea on the development of selfefficacy, which mastery performance, vicarious experiences, social persuasion and physiological states all have a connection with students confidence in math. For mastery performance, it showed a strong relationship with students development of selfefficacy. A strategy that usher suggested which math teacher can use to increase students confident through mastery performance is to "deliver instruction in a way that maximize the opportunity for mastery experiences, however small."^{[53]} For instance, a teacher could teach the students the correction strategy on math topic like algorithm and algebra. An example question is 18 ÷ 6 =?. The teacher could teacher the students to selfcheck the answer by multiplying the quotient by the divisor (3 x 6= 18) and if the answer is the same as the dividend, then is correct. Students who have been taught and used the correction strategy had increased their mastery performance in math.^{[54]} Assign challenging assignments for students which are within their ability to complete it will also increase students' mastery experience.
In addition, some evidence in the Usher's research has shown that the four sources have a connection with each others too. For vicarious experiences, the finding has shown that both of the parents and the teachers' experience with math have a connection with students math confidence. One of the compelling findings in the research is which a student interpreted his parents' failure in math as evidence that he could be different.^{[55]} This shows that not only successful experiences, unsuccessful experiences with math could have a connection with have students math confident. The finding also shows students' physiological states would have an effect on how they interpret others' experience. For social persuasion, the finding has shown that the messages both parents and teachers have sent to the children could largely impact students' belief in their ability.^{[56]} For instance, a message that belief math is a fixed ability would result in student's lack of motivation. So, if parents tell their children that with math ability they either have it or not, their children might end up believing that they do not have the ability to perform well and lower their confidence in math. In this case, social persuasion could have an effect on students' physiological states.
Teachers Efficacy[edit]
Teachers' teaching efficacy refers to the belief that they can make a significant change in their students,^{[57]}such as students' academic performance, selfefficacy, motivation, attitude and interest in learning. In order for teachers to establish a high level of teaching efficacy, they need to have a positive attitude, rich pedagogical knowledge and content knowledge toward their teaching subject. Teachers' attitude towards math may have a strong influence on students' attitudes and academic performance. A study has examined teachers' attitudes toward math in four different groups through interviewing the teachers and having them to complete a teacher attitude scale.^{[58]} The four different groups are K4 teachers, middle school teachers, other educators (Principals, other administrators) and special education teachers. The result indicated that among the four groups, middle school teachers have the strongest positive attitude toward math (60% strongly positive, 30% neutral, 10% strongly negative), whereas K4 teachers have the strongest negative attitude toward math (43% strongly positive, 23% neutral, 34% strongly negative). ^{[59]} The result shows that math is less emphasize and valued in elementary level then in middle school level. By having a negative attitude towards math, teachers are less likely belief that they can make a change in their students' learning, which is correlated to their teaching efficacy. Teachers' pedagogical knowledge and content knowledge in math are also factors that affect their teaching efficacy. A current research has studied teachers' math pedagogical knowledge and math content knowledge in relation to teachers' teaching efficacy and students' achievement in the topic of algebra i.^{[60]} The result have found that they are strong correlation between teacher's teaching efficacy with their pedagogical knowledge and content knowledge, which indicating that teachers with a rich pedagogical knowledge and content knowledge are more confidence with their teaching and more likely to believe that they can make a significant change in their students' learning.^{[61]}
Teachers' teaching efficacy can affect students' learning in many different ways. One of the more observable factors is students' academic achievement. A study had conducted K12 school teachers' selfefficacy beliefs and found that their selfefficacy beliefs are positively associated with students' achievement. ^{[62]} Besides students' achievement, teachers' teaching efficacy could also affect student motivation, interest and strategies use in learning. This is because teachers with higher teaching efficacy are more likely to use praise instead criticism, to be more accepting and more task oriented. ^{[63]} Another research has found that teachers with higher efficacy will teach their students more learning strategies and have more focused academic learning time, which will increase students' performance.^{[64]}
SelfRegulated Learning[edit]
People might think that students' low mathematics achievement is due to their low ability in math or the consequences of not studying. But that may not be the case in all situations. Sometimes, students' low mathematics achievement might be a result of not using the most appropriate strategies to study due to their lack of selfregulated learning skills. SelfRegulated Learning is students' ability to control all aspect of their learning, from advance planning to how they evaluate their own performance afterward ^{[65]}. There are three core components for selfRegulated Learning. The first one is metacognitive awareness, which refer to how students' set their goal and their plan of reaching that goal. ^{[66]}The second one is strategies use, which refer to a list of selfregulated strategies that students could apply to their studying. Skilled learners use more effective strategies when they are learning.^{[67]} The last one is motivation control, which is students' ability to set goals and their positive belief on their academic skills and performance.^{[68]} The ability of selfregulated learning has a big impact on students' mathematic achievement. Students will use better strategies and have a better understanding on how to study mathematics, when their selfregulated learning skills improved, which will increase this mathematic achievement.
Mathematics SelfRegulated Learning Program Study[edit]
A research in Southeast Asia had established a mathematics selfregulation learning program and the result had shown that when students are being taught with selfregulated learning skills, their mathematic achievement increases. The research involved with 60 lower mathematic achieving students in elementary level. 30 students are being placed in the experimental group, which they have to attend a mathematics selfregulated learning program.
This program contains 30 sessions, which serve a purpose of increasing students' selfregulated learning skill by increasing their motivational control and teaching them the selfregulation strategies. (Sessions 15) The program started with developing students' selfregulation belief system. They introduced students' to the value of personal responsibility, selfefficacy, learning goal and attribution to effort by lecturing students with storytelling and having them to share their ideas in a group.^{[69]} (Sessions 611) Then, they introduced students the 14 selfregulated learning strategies that were proposed by Zimmerman.^{[70]} Each strategy was explained by emphasizing its usage and important in learning mathematics. Afterward, students are given the opportunity to practice each strategy on their own. (Sessions 1230) Lastly, students are guided to apply selfregulated learning strategies in their regular mathematic lessons. Also, they have to evaluate their own progress by completing the goal setting, selfevaluation and selfconsequating forms. After the students completed the 30 sessions in mathematics selfregulated learning program, they will take a mathematic achievement test and a selfregulated learning test. The results have shown that students who attended the program scored higher in both tests compared to those who did not attend the program.
Applying SelfRegulated Learning strategies in Mathematic
Strategies  Application in Mathematic 

Selfevaluation  Students do so by making sure that they have the right answers for the questions with the appropriate steps. 
Organizing and transforming  Students' ability to organize math questions. Some of the ways are using graphs, equations, and diagrams. 
Goalsetting and planning  Students setting goals and their plan of achieving those goals. 
Keeping records and monitoring  Taking notes in class. Organizing the equations. 
Environment structuring  Studying in an environment which benefits their study. 
Selfconsequences  Students' own punishment or reward on its own success or failure in mathematic. 
Rehearsing and memorizing  Student learned by doing a lot of different forms of math questions. 
Seeking information  Student seek information from the nonsocial source. 
Seeking social assistance  Students seek help from their peers, teacher or other adults. 
Reviewing records  Students reread textbook, notes or their homework questions. 
After attending 30 sessions of a mathematics selfregulated learning program, students showed significant improvement in their mathematical achievement and selfregulated learning test. ^{[71]} This shows that it is possible to teach lowerachieving math students with selfregulated learning skills. When they were equipped with these skills and taught to focus on the processes and strategies, their math solving skills improved. With improvement, students will gradually recognize their ability to do better in math. Praising and rewarding themselves for their improvement will provide students will have even greater improvement in math. As a result, their selfefficacy and their interest in math will rise. This creates a positive cycle: when students believe that they have the ability to achieve math, they will work even harder in math with the appropriate selfregulated learning skills.
In the traditional classroom, math is viewed as an answercentred subject rather than a processcentred subject. By emphasizing speed and accuracy, students will develop skills in copying and memorizing mathematical facts instead of understanding math. Also, the learning only flows one way, from teacher to students. In this kind of classroom setting it would be hard for students to apply selfregulated learning strategies because when the students are not allowed to have choice and control over their study, they are not likely to learn strategies for selfregulation, nor willingly selfinitiate and control the use of various strategies. ^{[72]}Therefore, in order for students to apply selfregulation skills, the classroom environment is very important. One of the best ways to develop selfregulated learning skills is to give a certain degree of control to students for their own learning. Math teachers should promote the sharing of knowledge and decision making. When students have a voice in setting goals, planning activities and evaluating their own performance, they have a chance to practice their selfregulated learning skills, which will have a positive impact on their math achievement.
Uppergrade students can apply selfregulated learning skills better than lower grade students. ^{[73]}This is because older students are more capable of understanding concepts and ideas that are presented in selfregulated learning theory. Also, some of the selfregulated learning strategies require prior knowledge and skills, such as writing a plan or organizing learning materials. Therefore, it is easier for uppergrade students to learn some of the selfregulated strategies. As a result, uppergrade students show more improvement in mathematical achievement than lower grade students when they are taught selfregulated learning skills.
Implications for Teaching[edit]
MathematicsLearning Disabilities[edit]
Recent studies into cognition, working memory and mathematics learning disabilities all point to a need to distinguish between computation and problem solving learning disabilities in math. To this point, mathematics assessments have been generic and have not given appropriate consideration to the different features of each domain. Professionals must consider these two skills separately when diagnosing students. Teachers should also take into account the different domains when instructing children with mathematical learning disabilities. Some suggestions and tools that may help students with their mathematical learning are:
External Representation[edit]
External representation is a helpful tool in mathematics because mathematical problems can be complicated to solve mentally at times. By using external representation, it provides a clear understanding on the concept of mathematics by which students can develop knowledge acquisition. Some external representations are workedout examples, animations, and diagrams.
Workedout Examples[edit]
Workedout examples are a useful instructional method that teachers use to facilitate students in learning mathematics. Research shows that using workedout examples can increase the students who have low mathematics performance level. One reason is when students are given a problem to solve, their optimal goal is to solve the problem rather than to learn mathematics. In contrast, when students are given workedout examples, they actually learn and try to interpret the materials on their own. ^{[74]} Thus, workedout examples focus more on intentional learning for students. Students usually do not understand the mathematical theory or proof because they are complicated to comprehend. However, workedout examples are easier for students to acquire learning and understand the concept of mathematics. Without giving explicit instruction, teachers simply show the steps of how to solve the mathematical problem as an example for the students to refer to. There are detailed explanations on the steps required to solve the mathematical problem. Then, students have the autonomy to selfexplain similar types of mathematical problem on their own. Thus, they can use the workedout examples as references to solve many mathematical problems. ^{[75]} They can explicitly reflect their thinking on how to solve the problem by referring to the workedout examples that the teachers provide. Hence, this can also enhance the students in selfregulated learning as they are practicing their critical thinking in solving the problem. This metacognitive strategy can help students improve their problemsolving skills especially on mathematical word problems. Metacognitive strategies include selfquestioning, selfevaluating, summarizing, and illustrating the problem. ^{[76]} These strategies are believed to acquire knowledge for students while constructing a deeper understanding from the workedout examples. Research shows that students who can selfexplain the problem and solve them have higher mathematics achievement. When students explain the steps of how to solve the mathematical problem on their own, they are exercising their reflective thinking which can construct a greater understanding beyond what the information was given. Indeed, students can develop new and sophisticated knowledge of mathematics because they consolidate the newly learned materials with their prior knowledge. ^{[77]} In addition, workedout examples can also be used in group settings where students can discuss with their classmates in solving mathematical problems. Research found two ways that students can use workedout examples in classrooms. ^{[78]} One way is students who understand the workedout examples can explain to those who do not understand. The other way is students interpret the workedout examples altogether by using their logic and reasoning skills. Both ways engage students in learning in a social interactive setting by discussing the details of the workedout examples. Learning in a social setting can strengthen the understanding of the materials because students are elaborating the examples more in depth. They can also ask any questions that they have with the workedout examples in order to get a clear comprehension. ^{[79]} Therefore, it is important that students should discuss further on the workedout examples in small groups to reflect on the problem procedure and to generate knowledge acquisition beyond their existing knowledge.
Animations[edit]
To increase the students’ interest in learning mathematics, animation is a great instructional tool to use to teach students. Since mathematics can be quite boring and uninteresting at times, animations can attract students’ interest in learning mathematics. Most importantly, animations claim to facilitate students’ problemsolving skills in mathematics. ^{[80]} Before students solve any mathematical problem, it is important that students identify the problem and know what to solve. Henceforth, when students find the problem hard to translate, animation becomes most effective because it consists of visual representation that makes it easier for students to interpret the question. In contrast, when students just take notes on the problem, they do not have a clear understanding of what the problem means because they are just simply copying the text. By having a pictorial representation along with the verbal explanation of the problem, students can visualize what is happening in the problem fully. For instance, the concept of addition and subtraction is hard to explain through text to an elementary school student. However, when using animations to display a before and after frame of what happened in the problem can construct a clear comprehension. In the case of addition or subtraction mathematical problems, animations can demonstrate an increase or decrease of objects to represent the solution. In addition, animations can illustrate abstract math theories by showing visible objects, concrete results, and specific instances. Thus, animations can be used to convey the abstract concepts of mathematics with reference to distinct examples. ^{[81]} Animations can facilitate the acquisition of abstract principles and the comprehension of workedout examples due to the visual representation of the problems. Although workedout examples are known as a effective instructional practice, animations can be used to effectively improve these examples. ^{[82]} Workedout examples may not always have a pictorial representation but only have written texts. Therefore, when each of the steps of the solution procedure of the workedout examples have a visual representation, students can imagine what is going on in the problem. Students can also interpret the workedout examples better with the explanations and the pictures given. As a result, it is recommended that teachers should use animations as an instructional tool in their practices to fully consolidate the students’ learning in mathematics.
Diagrams[edit]
To produce an informational diagram can be a very difficult procedure, because students do not only required to interpret the verbal information into the visual information, but also needed to identify and integrate the related information together before associating to their prior knowledge.^{[83]} Larkin and Simon believed that diagrammatic representation is easier and more efficient than sentential representation because of three aspects in regards to searching, matching, and inference. First, it clearly retains all the information about the topographical and geometric relations between the elements of the word problems. Therefore, students can search for particular information easily. Second, since all the related elements are grouped together, it shows the connections between the concrete representations and the pictograms. Hence, it can simplify the process of identifying the related information. Besides that, the memory load is lower if the problem is produced by drawing a diagram, as the students can clearly see the essential inference between the related information.^{[84]} Many studies have suggested that the use of diagrams can improve the efficiency in problem solving.
Banerjee has conducted a research on the effects of using diagramming as a representational technique on high school students’ achievements in solving math word problems. The result has proved that the diagramming method (such as focusing on the creation and labels of diagrams to represent the mathematics) can significantly improve their achievements in solving math word problems.^{[85]}
In a study with the use of diagrams in solving the math word problems, Uesaka, Manalo, and Ichikawa made a comparison of students in Japan and New Zealand. ^{[86]} The diagram that was drawn by a Japanese student was using a oneobject problem, and the one that was produced by a New Zealand student was using a two dimensional object to solve the math word problems. Results indicated that the percentages of correct answers by the New Zealand students were significantly higher than the Japanese students. The reason is that producing a diagrammatic representation can index the sentences by location, so students can observe the details at a specific location explicitly, which ease them on understanding the problem.^{[87]}
In order to promote students on using diagrams to solve math problems, teachers should first teach them on 1) what diagrams are, 2) the importance of using diagrams to solve problems, 3) when to apply the diagrams in solving problems, 4) which type of diagram should be using for the math problems, 5) how to generate a diagram, and 6) how to use a diagram effectively. The reason that students should know the fundamental concepts of diagrams is that diagrams may not apply on all the math problems. Uesaka and Manalo pointed out that students tend to use diagrams when solving math word problems in regards of length and distance instead of spatial problems, because it usually involves a concrete relationships and known quantities.^{[88]} After teaching them the important concepts of diagrams, teachers can then instruct them the 3 step procedure – Ask, Do, and Check.^{[89]} Van Garderen and Scheuermann suggested that students should first concentrate on what needs to be solved; then they should produce a diagram. Finally, they can solve the problem by using the diagrams. For example, in order to focus on what needs to be solved, students can use the key word method to search for the information, and place the information that is given from the problem.^{[90]} In conclusion, diagrams can be an effective strategy when solving math problems; it does not only help students to think critically, but also aid them in solving problems by using a different approach.
Algorithms[edit]
An algorithm is a series of steps to help students solving math problems. If they follow these procedures, they will always be able to compute a correct answer every time. Algorithm involves with repeating sequences, it applies to addition, subtraction, multiplication, and division. By using algorithms, students can learn how to explain what is happening in each step, and able to track their mistakes if they yield an incorrect answer in the end. It requires them be attention to details when they are problem solving, that is, when they are working through a multiple step solutions, they are required to recall the algorithms from their long term memory and have a set of steps in their mind already. Also, teachers should instruct students that algorithms must be solved in a sequential order, none of the steps can be jumped over. For example, when students are learning basic arithmetic operations, they have to learn that there is a specific order to solve a problem like 5+8×6. Students need to understand that they have to do the multiplication first, then the addition part. If they can follow the correct order, they can always yield to a correct answer. However, Paul Cobb has conducted a study in regards of Grade 1 and 2 students solving doubledigit addition problems. He noticed that all of the students were managed to give a correct answer for 16+9 by using various methods. Conversely, if they were asked to use the traditional school algorithm with carrying to solve the same problem but with a vertical context, many of them tend to yield an incorrect answer. He concluded that the reason of causing the students to have a higher possibility of making errors with a traditional school algorithm is that they were only forcing themselves to follow the rules instead of fully understanding how the algorithms work.^{[91]} J.S. Brown and Burton found out that there is a significant amount of students are using one or more wrong versions of algorithm consistently to solve their math problems. Even though lots of incorrect algorithms yield to a correct answer, yet it may not apply to all cases.^{[92]} For example, some children had a preconception that the subtraction algorithm means taking the smaller number from the larger in every single column, regardless of which number was on the top. The diagram on the left can explain why incorrect algorithms may not produce a correct solution all the time.
Brown and Burton pointed out that even though the children who have the wrong perception of the subtraction algorithm may seem to understand the arithmetic operations of subtraction, as this can guide them to yield the correct solution on part a) and part c). However, they will yield an incorrect answer on part b) and part d), as the numbers on top in the second columns are smaller than the numbers in the bottom.
Nagel and Swingen believed that the traditional algorithms with carrying or borrowing can only increase their efficiency and accuracy, yet neglect the sensemaking for the students.^{[93]}
Therefore, in order to deal with the serial aspects of algorithms effectively, educators should teach students to use their spatial abilities when applying multiple steps to solve a problem. For example, they need to learn how to keep numbers aligned and spaced correctly to solve the problems successfully; especially when they are computing column subtraction, multiple digit multiplication, etc. Teachers should encourage students to develop and use their own algorithms to solve problems. They can encourage their students to incorporate mnemonics with algorithms; this approach can help them to remember things such as the procedures in solving problems.^{[94]} For example, PEDMAS can tell them the order when carrying out operations. Instead of simply solving an arithmetic operation from left to right, they now understand that they have to solve the brackets first.^{[95]} Moreover, teachers should ask the students to look over the entire problems first before trying to solve for an answer, then they should teach them how to break the problem into small parts and to determine which parts will require using the algorithms. They should also know which algorithms they should apply on for each parts; and finally, they should reflect on their answers for every steps. By showing steps, students can always track their mistakes and come to a correct solution ultimately.
Word Problem Strategies[edit]
Word problems present a special case for all children, but especially those with problem solving learning disabilities. The most significant difference between computational problems and word problems is the addition of linguistic information. In other words, children must first read written words and filter out the information in order to translate the written problem into a computational number sentence. Children must then identify the missing information, as well as the relevant information, before completing the actual math portion of the problem.
Word problems are challenging for many students to comprehend but the problem is compounded when the learner’s first language is not English. According to Jan, S. and Rodrigues, S. (2012)^{[96]}, children with English as a second language cannot comprehend problem statements due to language barriers. They tend to rely on key words or misinterpret the problem statement and so their resulting solution may be incorrect. Relying on key words can distract students from trying to understand the problem. “Key words can cause confusion in differentiating between everyday language and mathematical language.” ^{[97]}
Findings from this study suggest that class or small group discussions will provide students with an opportunity to clarify the nature of a problem so that they can understand what is being given and what is being asked. Providing students with opportunities to read, understand, share each other’s ideas, and to consider the problem and solution from a number of different tactics will provide the students with a greater understanding of the problem.
In taking a cognitive approach to teaching word problems, it is important for the teacher to provide ample opportunity for students to think about and discuss the meaning of the word problems, and then consider multiple solutions with their classmates. This approach is valuable for both those students who have language barriers and those students with math learning disabilities.
The Council for Learning Disabilities ^{[98]} recommends some of the following strategies for instructing students in problem solving:
FAST DRAW (Mercer & Miller, 1992) Find what you’re solving for. Ask yourself, “What are the parts of the problem?” Set up the numbers. Tie down the sign.
Discover the sign. Read the problem. Answer, or draw and check. Write the answer.
Questions and Actions (Rivera, 1994) Step a. Read the problem. Questions Are there words I don’t know? Do I know what each word means? Do I need to reread the problem? Are there number words? Actions Underline words. Find out definitions. Reread. Underline. b. Restate the problem. What information is important? What information isn’t needed? What is the question asking? Underline. Cross out. Put in own words. c. Develop a plan. What are the facts? How can they be organized? How many steps are there? What operations will I use? Make a list. Develop chart. Use manipulatives. Use smaller numbers. Select an operation. d. Compute the problem. Did I get the correct answer? Estimate. Check with partner. Verify with calculator. e. Examine the results. Have I answered the question? Does my answer seem reasonable? Can I restate question/answer? Reread question. Check question/answer. Write a number sentence.
3. TINS Strategy (Owen, 2003) Different steps used to analyze and solve word problems are represented with this acronym. Thought: Think about what you need to do to solve this problem and circle the key words. Information: Circle and write the information needed to solve this problem; draw a picture; cross out unneeded information. Number Sentence: Write a number sentence to represent the problem. Solution Sentence: Write a solution sentence that explains your answer. Example: Kyle bought 6 baseball cards. The next day, he added 11 more cards to his collection. How many cards does he have in all? Thought: + Information: 6 baseball cards, 11 baseball cards Number Sentence: 6 + 11 = Solution Sentence: Kyle has 17 baseball cards in his collection.
4. Problem Solving (Birsh, Lyon, Denckla, Adams, Moats, & Steeves, 1997) Read the problem first. Highlight the question. Circle the important information. Develop a plan. Use manipulatives to represent the numbers. Implement the plan. Check your work.
Cognitive Tutor for teaching algebra[edit]
In 1985, Anderson, Boyle, and Reigser added the discipline of cognitive psychology to the Intelligent Tutoring Systems. Since then, the intelligent tutoring system adopted this approach to construct cognitive models for students to gain knowledge was named Cognitive Tutors.^{[99]} The most widely used Cognitive Tutor is Cognitive Tutor® Algebra I.^{[100]} Carnegie Learning, Inc., the trademark owner, is developing fullscale Cognitive Tutor®, including Algebra I, II, Bridge to Algebra, Geometry, and Integrated Math I, II, III. Cognitive Tutor® now includes Spanish Modules, as well.
How to teach[edit]
Two builtin algorithms, model tracing and knowledge tracing can help monitor students' learning during using the software. Model tracing can provide justintime feedback, ondemand hints, and give contentspecific advice based on every step of the students’ performance trace.^{[99]} Knowledge tracing can individualize learning tasks for every user based on the prior knowledge.^{[99]}^{[100]}
You can go to the Chapter of Problem Solving, Critical Thinking, and Argumentation (2.5.2 The theoretical background of Cognitive Tutor) to get more detailed information of how Cognitive Tutor can facilitate algebra learning via justintime feedback, ondemand hints, contentspecific advice, and personalized tasks.
Mixed effects of Cognitive Tutor® Algebra I[edit]
Regarding the effectiveness of Cognitive Tutors, previous research evidence supports more effectiveness of Cognitive Tutors than classroom instruction.^{[99]}^{[101]}^{[102]}^{[103]} However, recent independent largescale study, What Works Clearinghouse,^{[104]} established by the U.S Department of Education's Institute of Education Sciences, reviewed 6 out of 22 studies on Cognitive Tutor® Algebra I which includes 12,840 students in grade 813 in 118 locations. The researchers found that Cognitive Tutor® Algebra I has mixed effects on algebra and no statistically significant or substantively important effect on general mathematics achievement for secondary students.
Morgan and Ritter,^{[105]} conducted a within teacher experiment in grade nine algebra classes in five different schools in Moore, Oklahoma. In this study, each teacher was assigned at least one Cognitive Tutor® Algebra I integrated classroom and one traditional classroom. The findings suggested that students who learned with Cognitive Tutor® Algebra I performed better than their peers who did not use the software, as well as tending to have positive attitudes towards mathematics, such as greater confidence in math.
Cabalo, Jaciw, and Vu^{[106]} conducted a randomized experiment to examine the effectiveness of Cognitive Tutor® Algebra I in five secondary schools settings in Maui County, Hawaii. After six months implementation of Cognitive Tutor® Algebra I, the students were required to take the NWEA Algebra EndofCourse Achievement Level Test at the end of 200506 school year. The findings suggested students overall reported positive attitudes towards Cognitive Tutor® software, and most students, whether using the software or not, showed improvements on math tests. However, students who had low scores before using Cognitive Tutor® improved significantly compared to those students with high initial scores.
Campuzano, Dynarski, Agodini, and Rall^{[107]} conducted a 2year congressionallymandated study on the effectiveness of technologybased instruction, including employing Cognitive Tutor® Algebra I in the second year in nine highpoverty schools in four districts. The researchers adopted the methods of randomized controlled trial and randomly assigned the teachers to either use the software or keep using the existing school curriculum. All the students were taken ETS EndofCourse tests in fall and spring, and the students who used the software had significantly higher scores in the second year compared to which in the first year. However, the difference in exam scores between the intervention group and the comparison group is little (p<0.3).
Pane, Griffin, MaCaffrey, and Karam^{[108]} adopted randomized controlled trial to examine the effectiveness of the technology integrated algebra curriculum in America. The research lasted for two consecutive school years, and the Cognitive Tutor® Algebra I software was implemented both in the teacherdirected classroom instruction (3 days a week) and the computerguided instruction (2 days a week). The results in high schools showed a little difference of learning achievements between students in the intervention group and the comparison group in the first school year (p<0.46). However, the evidence firmly supported the benefits of integrating Cognitive Tutor® Algebra I in the second year (p<0.04), the lower achievement students in the intervention group had larger improvements compared to highperformance students in the same group.
Glossary[edit]
Algorithm is a procedure with a series of steps in mathematics that when used appropriately to solve a mathematical problem, it will yield a correct solution.
Application occurs when students are able to make associations between mathematical concepts and daily life situations.
Clarification occurs when students identify and analyze aspects of a problem, it allows them to interpret the information that they need in order to solve the problem.
Classification is the ability of grouping objects based on similar characteristics.
Conceptual knowledge is the mental structures that promote students' reasoning and understanding of mathematics.
Declarative knowledge is when mathematical concepts, that are factual knowledge, are being retrieved from the longterm memory; hence, using these concepts to solve other complex mathematical problems.
Evaluation occurs when students can use a particular rubric to determine the correctness of a problem solution.
Inference occurs when students are able to use general concepts to specific situations and distinguish the similarities and differences among objects.
Intrinsic motivation is when students want to perform mainly for their own personal interests.
Metacognitive is the knowledge used to control one's thinking and learning.
Procedural knowledge is the knowledge about how to solve mathematical problems using the sequence of strategy steps.
Seriation is the ability of ordering objects from small to large based on the sizes, such as length, weight, or volume.
Selfregulated learning is the ability to control one's learning, from planning to how one evaluate performance afterward.
Short term memory is responsible for temporarily storing information which must be used, but not necessarily manipulated.
Working memory is the system responsible for temporarily holding new or previouslystored information which is being used for the completion of a current task.
Suggested Reading[edit]
 A case study of novice teachers' mathematics problem solving beliefs and perceptions. Baker, C. K. (2015). A case study of novice teachers' mathematics problem solving beliefs and perceptions. Dissertation Abstracts International Section A, 75
 Piaget and Vygotsky: Many resemblances, and a crucial difference. Lourenço, O. (2012). Piaget and Vygotsky: Many resemblances, and a crucial difference. New Ideas In Psychology, 30(3), 281295. doi:10.1016/j.newideapsych.2011.12.006
References[edit]
Fuchs, L. S., Fuchs, D., Stuebing, K., Fletcher, J.M., Hamlett, C. L. , & Lambert, W. (2008). Problem solving and computational skill: Are they shared or distinct aspects of mathematical cognition? Journal of Educational Psychology 100 (1), 30
Hanich, L. B., Jordan, N. C., Kaplan, D., & Dick, J. (2001). Performance across different areas of mathematical cognition in children with learning disabilities. Journal of Educational Psychology, 93, 615–626.
Rourke, B. P., & Finlayson, M. A. J. (1978). Neuropsychological significance of variations in patterns of academic skills: Verbal and visualspatial abilities. Journal of Abnormal Child Psychology, 6, 121–133.
Swanson, H. L., & BeebeFrankenberger, M. (2004). The relationship between working memory and mathematical problemsolving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 96, 471–491.
Swanson, H. L. (2003). Agerelated differences in learning disabled and skilled readers’ working memory. Journal of Experimental Child Psychology, 85, 1–31.
 ↑ Wong, B., Graham, L., Hoskyn, M., & Berman, J. (Eds.). (2008). The ABCs of learning disabilities (2nd ed.). Boston: Elsevier Academic Press.
 ↑ Bruning, R., Schraw, G., & Norby, M. (2011). Cognitive Psychology and Instruction (5th ed.). Boston: Pearson Education.
 ↑ Bruning, R., Schraw, G., & Norby, M. (2011). Cognitive Psychology and Instruction (5th ed.). Boston: Pearson Education.
 ↑ Wong, B., Graham, L., Hoskyn, M., & Berman, J. (Eds.). (2008). The ABCs of learning disabilities (2nd ed.). Boston: Elsevier Academic Press.
 ↑ Piaget, J. (1977). Epistemology and psychology of functions. Dordrecht, Netherlands: D. Reidel Publishing Company.
 ↑ Fuson, K. C. (1988). Children’s counting and concepts of numbers. New York: Springer.
 ↑ Piaget, J. (1963). The origins of intelligence in children. New York, NY: W.W. Norton & Company, Inc.
 ↑ Thompson, C. S. (1990). Place value and larger numbers. In J. N. Payne (Ed.), Mathematics for young children (pp. 89–108). Reston, VA: National Council of Teachers of Mathematics.
 ↑ Piaget, J. (1977). Epistemology and psychology of functions. Dordrecht, Netherlands: D. Reidel Publishing Company.
 ↑ Piaget, J. (1968). Genetic epistemology. New York, NY: Columbia University Press Retrieved June 13, 2003 from the World Wide Web: http://www.marxists.org/reference/subject/philosophy/works/fr/piaget.htm
 ↑ Hiebert, J., and T. P. Carpenter. "Piagetian Tasks as Readiness Measures in Mathematics Instruction: A CriticalReview." EDUCATIONAL STUDIES IN MATHEMATICS 13: 329345, 1982
 ↑ Berger, K.S. (1988). The developing person through the life span (2nd ed.). New York: Worth Publishers Ltd.
 ↑ Berger, K.S. (1988). The developing person through the life span (2nd ed.). New York: Worth Publishers Ltd.
 ↑ Cross, C. (2009). The Early Childhood Workforce and Its Professional Development. In Mathematics learning in early childhood paths toward excellence and equity. Washington, DC: National Academies Press.
 ↑ Berger, K.S. (1988). The developing person through the life span (2nd ed.). New York: Worth Publishers Ltd.
 ↑ Papalia, D. E., Olds, S. W., and Feldman, R. D. (1998). Human development(7th ed.). Boston: McGrawHill.
 ↑ Edwards, L., Hopgood, J., Rosenberg, K., & Rush, K. (2000). Mental Development and Education. Retrieved April 25, 2009, from Flinders University
 ↑ Berger, K.S. (1988). The developing person through the life span (2nd ed.). New York: Worth Publishers Ltd.
 ↑ Papalia, D.E., Olds, S.W., & Feldman, R.D. (1998). Human development (7th ed.). Boston: McGrawHill.
 ↑ Berger, K.S. (1988). The developing person through the life span (2nd ed.). New York: Worth Publishers Ltd.
 ↑ Rourke, B. P., & Finlayson, M. A. J. (1978). Neuropsychological significance of variations in patterns of academic skills: Verbal and visualspatial abilities. Journal of Abnormal Child Psychology, 6, 121–133.
 ↑ Hanich, L. B., Jordan, N. C., Kaplan, D., & Dick, J. (2001) Performance across different areas of mathematical cognition in children with learning disabilities. Journal of Educational Psychology, 93, 615–626.
 ↑ Fuchs, L. S., Fuchs, D., Stuebing, K., Fletcher, J.M., Hamlett, C. L. , & Lambert, W. (2008). Problem solving and computational skill: Are they shared or distinct aspects of mathematical cognition? Journal of Educational Psychology 100 (1), 30
 ↑ Swanson, H. L., & BeebeFrankenberger, M. (2004). The relationship between working memory and mathematical problemsolving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 96, 471–491
 ↑ Swanson, H. L. (2003). Agerelated differences in learning disabled and skilled readers’ working memory. Journal of Experimental Child Psychology, 85, 1–31.
 ↑ Swanson, H. L., & BeebeFrankenberger, M. (2004). The relationship between working memory and mathematical problemsolving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 96, 471–491.
 ↑ Swanson, H. L. (August 4,2015). Cognitive strategy interventions improve word problem solving and working memory in children with math disabilities. Front Psychol., 2015; 6: 1099. Retrieved from doi: 10.3389/fpsyg.2015.01099
 ↑ Wong, B., Graham, L., Hoskyn, M., & Berman, J. (Eds.). (2008). The ABCs of learning disabilities (2nd ed.). Boston: Elsevier Academic Press.
 ↑ Bruning, R., Schraw, G., & Norby, M. (2011). Cognitive Psychology and Instruction (5th ed.). Boston: Pearson Education.
 ↑ Wong, B., Graham, L., Hoskyn, M., & Berman, J. (Eds.). (2008). The ABCs of learning disabilities (2nd ed.). Boston: Elsevier Academic Press.
 ↑ Wong, B., Graham, L., Hoskyn, M., & Berman, J. (Eds.). (2008). The ABCs of learning disabilities (2nd ed.). Boston: Elsevier Academic Press.
 ↑ Bruning, R., Schraw, G., & Norby, M. (2011). Cognitive Psychology and Instruction (5th ed.). Boston: Pearson Education.
 ↑ Bruning, R., Schraw, G., & Norby, M. (2011). Cognitive Psychology and Instruction (5th ed.). Boston: Pearson Education.
 ↑ Upadyaya, K., & Jacquelynne, S. E. (2014). How do Teachers’ Beliefs predict Children’s Interests in Math from Kindergarten to Sixth Grade? MerrillPalmer Quarterly, 60(4). Retrieved from http://web.b.ebscohost.com.proxy.lib.sfu.ca/ehost/pdfviewer/pdfviewer?sid=aaa92fa222f84cd79d9449896f602e89%40sessionmgr120&vid=0&hid=109
 ↑ Frenzel, A.C., Goets T., Pekrun R., & Watt, H.M.G. (2010). Development of Mathematics Interest in Adolescence: Influences of Gender, Family and School Context. Journal of Research on Adolescence, 20(2), 507537. doi:10.1111/j.15327795.2010.00645.x
 ↑ Frenzel, A.C., Goets T., Pekrun R., & Watt, H.M.G. (2010). Development of Mathematics Interest in Adolescence: Influences of Gender, Family and School Context. Journal of Research on Adolescence, 20(2), 507537. doi:10.1111/j.15327795.2010.00645.x
 ↑ Tsao, YL. (2004). A Comparison of American and Taiwanese Students: Their Math Perception. Journal of Instructional Psychology, 31(3). Retrieved from http://web.b.ebscohost.com.proxy.lib.sfu.ca/ehost/pdfviewer/pdfviewer?sid=97c9159942224cc88717703e6b774b06%40sessionmgr110&vid=19&hid=109
 ↑ Imbo, I., & LeFevre, JoAnne. (2009). Cultural differences in Complex Addition: Efficient Chinese versus Adaptive Belgians and Canadians. Journal of Experimental Psychology: Learning, Memory, and Cognition, 35(6),14651476. doi:10.1037/a0017022
 ↑ Imbo, I., & LeFevre, JoAnne. (2009). Cultural differences in Complex Addition: Efficient Chinese versus Adaptive Belgians and Canadians. Journal of Experimental Psychology: Learning, Memory, and Cognition, 35(6),14651476. doi:10.1037/a0017022
 ↑ Imbo, I., & LeFevre, JoAnne. (2009). Cultural differences in Complex Addition: Efficient Chinese versus Adaptive Belgians and Canadians. Journal of Experimental Psychology: Learning, Memory, and Cognition, 35(6),14651476. doi:10.1037/a0017022
 ↑ Imbo, I., & LeFevre, JoAnne. (2009). Cultural differences in Complex Addition: Efficient Chinese versus Adaptive Belgians and Canadians. Journal of Experimental Psychology: Learning, Memory, and Cognition, 35(6),14651476. doi:10.1037/a0017022
 ↑ Schunk, D. H., & Zimmerman, B.J. (2006). Competence and control beliefs: Distinguishing the means and ends. In P.A. Alexander & P.H. Winne (Eds.), Handbook of educational psychology (2nd ed., pp.349367). Yahweh, NJ: Erlbaum.
 ↑ Schunk, D. H., & Zimmerman, B.J. (2006). Competence and control beliefs: Distinguishing the means and ends. In P.A. Alexander & P.H. Winne (Eds.), Handbook of educational psychology (2nd ed., pp.349367). Yahweh, NJ: Erlbaum.
 ↑ Bandura, A. (1997). Selfefficacy: The exercise of control.New York, NY:Freeman.
 ↑ Bandura, A. (1997). Selfefficacy: The exercise of control.New York, NY:Freeman.
 ↑ Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs, NJ, US: PrenticeHall, Inc.
 ↑ Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs, NJ, US: PrenticeHall, Inc.
 ↑ Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs, NJ, US: PrenticeHall, Inc.
 ↑ Hekimoglu, S., & Kittrell, E. (2010). Challenging students' beliefs about mathematics: The use of documentary to alter perceptions of efficacy. PRIMUS: Problems, Resources, And Issues In Mathematics Undergraduate Studies, 20(4), 299331. doi:10.1080/10511970802293956
 ↑ Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs, NJ, US: PrenticeHall, Inc.
 ↑ Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. Englewood Cliffs, NJ, US: PrenticeHall, Inc.
 ↑ Usher, E. L. (2009). Sources of middle school students’ selfefficacy in mathematics: A qualitative investigation. American Educational Research Journal, 46(1), 275314. doi:10.3102/0002831208324517
 ↑ Usher, E. L. (2009). Sources of middle school students’ selfefficacy in mathematics: A qualitative investigation. American Educational Research Journal, 46(1), 275314. doi:10.3102/0002831208324517
 ↑ Ramdass, D., & Zimmerman, B. J. (2008). Effects of selfcorrection strategy training on middle school students' selfefficacy, selfevaluation, and mathematics division learning. Journal Of Advanced Academics, 20(1), 1841.
 ↑ Usher, E. L. (2009). Sources of middle school students’ selfefficacy in mathematics: A qualitative investigation. American Educational Research Journal, 46(1), 275314. doi:10.3102/0002831208324517
 ↑ Usher, E. L. (2009). Sources of middle school students’ selfefficacy in mathematics: A qualitative investigation. American Educational Research Journal, 46(1), 275314. doi:10.3102/0002831208324517
 ↑ Woolfolk, A. E., & Hoy, W. K. (1990). Prospective teachers' sense of efficacy and beliefs about control. Journal of Educational Psychology, 82(1), 8191. doi:10.1037/00220663.82.1.81
 ↑ Kolstad, R. K., Hughes, S., & Briggs, L. D. (1994). Teacher attitudes toward mathematics. Journal Of Instructional Psychology, 21(1), 4448.
 ↑ Kolstad, R. K., Hughes, S., & Briggs, L. D. (1994). Teacher attitudes toward mathematics. Journal Of Instructional Psychology, 21(1), 4448.
 ↑ Fox, A. M. (2015). Teacher selfefficacy, content and pedagogical knowledge, and their relationship to student achievement in algebra i. Dissertation Abstracts International Section A, 75,
 ↑ Fox, A. M. (2015). Teacher selfefficacy, content and pedagogical knowledge, and their relationship to student achievement in algebra i. Dissertation Abstracts International Section A, 75,
 ↑ ChearsYoung, J. B. (2015). The association between math teachers' moral judgment development and selfefficacy beliefs, and their relationship with student achievement. Dissertation Abstracts International Section A, 76,
 ↑ Kagan, D.M. (1992). Implications of research on teacher belief. Educational psychologists, 27, 6590.
 ↑ Ghaith, G., & Yaghi, H. (1997). Relationships among experience, teacher efficacy, and attitudes toward the implementation of instructional innovation. Teaching and Teacher Education, 13, 451458.
 ↑ Zimmerman, B. (2000). Attaining selfregulation: A social cognitive perspective. In M. Boekaerts, P. R. Pintrich, & M. Zeidner (Eds.), Handbook of selfregulation (pp.1339). San Diego, CA: Academic Press.
 ↑ Zimmerman, B. (2000). Attaining selfregulation: A social cognitive perspective. In M. Boekaerts, P. R. Pintrich, & M. Zeidner (Eds.), Handbook of selfregulation (pp.1339). San Diego, CA: Academic Press.
 ↑ Zimmerman, B. J., & MartinezPons, M. (1990). Student differences in selfregulated learning: Relating grade, sex, and giftedness to selfefficacy and strategy use. Journal of Educational Psychology, 82, 5159.
 ↑ Zimmerman, B. (2000). Attaining selfregulation: A social cognitive perspective. In M. Boekaerts, P. R. Pintrich, & M. Zeidner (Eds.), Handbook of selfregulation (pp.1339). San Diego, CA: Academic Press.
 ↑ Camahalan, F. G. (2006). Effects of SelfRegulated Learning on Mathematics Achievement of Selected Southeast Asian Children. Journal Of Instructional Psychology, 33(3), 194205.
 ↑ Zimmerman, B. (2000). Attaining selfregulation: A social cognitive perspective. In M. Boekaerts, P. R. Pintrich, & M. Zeidner (Eds.), Handbook of selfregulation (pp.1339). San Diego, CA: Academic Press.
 ↑ Camahalan, F. G. (2006). Effects of SelfRegulated Learning on Mathematics Achievement of Selected Southeast Asian Children. Journal Of Instructional Psychology, 33(3), 194205.
 ↑ Camahalan, F. G. (2006). Effects of SelfRegulated Learning on Mathematics Achievement of Selected Southeast Asian Children. Journal Of Instructional Psychology, 33(3), 194205.
 ↑ Camahalan, F. G. (2006). Effects of SelfRegulated Learning on Mathematics Achievement of Selected Southeast Asian Children. Journal Of Instructional Psychology, 33(3), 194205.
 ↑ Renkl, A. (1999). Learning Mathematics from workedout examples: Analyzing and Fostering selfexplanations. European Journal of Psychology of Education, 14(4), 477488. doi: http://dx.doi.org.proxy.lib.sfu.ca/10.1007/BF03172974
 ↑ Tu, CT. (2011). An Instructional experiment: Using workedout examples in mathematics problemsolving of elementary school students. Bulletin of Educational Psychology, 43(1), 2550.
 ↑ Tajika, H., Nakatsu, N., Nozaki H., Neumann, E., & Maruno S. (2007). Effects of SelfExplanation as a Metacognitive Strategy for Solving Mathematical Word Problems. Japanese Psychological Research, 49(3), 222233. doi: 10.1111./j.14685884.2007.00349.x
 ↑ Tajika, H., Nakatsu, N., Nozaki H., Neumann, E., & Maruno S. (2007). Effects of SelfExplanation as a Metacognitive Strategy for Solving Mathematical Word Problems. Japanese Psychological Research, 49(3), 222233. doi: 10.1111./j.14685884.2007.00349.x
 ↑ Mevarech, Z. R., & Kramarski, B. (2003). The Effects of Metacognitive Training versus Workedout Examples on Students’ mathematical reasoning. British Journal of Educational Psychology, 73(4), 449471. doi: http://dx.doi.org.proxy.lib.sfu.ca/10.1348/000709903322591181
 ↑ Mevarech, Z. R., & Kramarski, B. (2003). The Effects of Metacognitive Training versus Workedout Examples on Students’ mathematical reasoning. British Journal of Educational Psychology, 73(4), 449471. doi: http://dx.doi.org.proxy.lib.sfu.ca/10.1348/000709903322591181
 ↑ Scheiter, K., Gerjets, P., & Schuh, J. (2010). The Acquisition of ProblemSolving Skills in Mathematics: How Animations Can Aid Understanding of Structural Problem Features and Solutions Procedures. Instructional Science, 38(5), 487502. doi: http://dx.doi.org.proxy.lib.sfu.ca/10.1007/s1125100991149
 ↑ Scheiter, K., Gerjets, P., & Schuh, J. (2010). The Acquisition of ProblemSolving Skills in Mathematics: How Animations Can Aid Understanding of Structural Problem Features and Solutions Procedures. Instructional Science, 38(5), 487502. doi: http://dx.doi.org.proxy.lib.sfu.ca/10.1007/s1125100991149
 ↑ Scheiter, K., Gerjets, P., & Schuh, J. (2010). The Acquisition of ProblemSolving Skills in Mathematics: How Animations Can Aid Understanding of Structural Problem Features and Solutions Procedures. Instructional Science, 38(5), 487502. doi: http://dx.doi.org.proxy.lib.sfu.ca/10.1007/s1125100991149
 ↑ van Garderen, D., Scheuermann, A., & Poch, A. (2014). Challenges students identified with a learning disability and as highachieving experience when using diagrams as a visualization tool to solve mathematics word problems. ZDM Mathematics Education, 46, 135–149.
 ↑ Larkin, J., & Simon, H. (n.d.). Why a Diagram is (Sometimes) Worth Ten Thousand Words. Cognitive Science, 65100.
 ↑ Banerjee, B. (2011). The effects of using diagramming as a representational technique on high school students' achievement in solving math word problems. Dissertation Abstracts International Section A, 71, 394
 ↑ Uesaka, Y., Manalo, E., & Ichikawa, S. (2007). What kinds of perceptions and daily learning behaviors promote students' use of diagrams in mathematics problem solving?. Learning And Instruction, 17(3), 322335.
 ↑ Uesaka, Y., Manalo, E., & Ichikawa, S. (2007). What kinds of perceptions and daily learning behaviors promote students' use of diagrams in mathematics problem solving?. Learning And Instruction, 17(3), 322335.
 ↑ Uesaka, Y., & Manalo, E. (2012). Taskrelated factors that influence the spontaneous use of diagrams in math word problems. Applied Cognitive Psychology, 26, 251–260.
 ↑ van Garderen, D., & Scheuermann, A. (2015). Diagramming word problems: A strategic approach for instruction. Intervention in School and Clinic
 ↑ Walker, D. W., & Poteet, J. A. (198990). A comparison of two methods of teaching mathematics story problemsolving with learning disabled students. National Forum of Special Education Journal, 1, 4451.
 ↑ Cobb, P. (1991). Reconstructing Elementary School Mathematics. Focus on Learning Problems in Mathematics, 13(2), 3–32.
 ↑ Brown, J.S., Burton, R.B. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2, 155192
 ↑ Nagel, N., & Swingen, C. C. (1998). Students’ explanations of place value in addition and subtraction. Teaching Children Mathematics, 5(3), 164–170.
 ↑ Nelson, P. M., Burns, M. K., Kanive, R., & Ysseldyke, J. E. (2013). Comparison of a math fact rehearsal and a mnemonic strategy approach for improving math fact fluency. Journal Of School Psychology, 51(6), 659667. doi:10.1016/j.jsp.2013.08.003
 ↑ Jeon, K. (2012). Reflecting on PEMDAS. Teaching Children Mathematics, 18(6), 370377.
 ↑ Jan, S. and Rodrigues, S., (2012). Students' difficulties in comprehending mathematical word problems in English language learning contexts. International Researcher, Vol. 1, Issue 3, Accessed Dec 4, 2015 fromhttp://www.academia.edu/2078056/STUDENTS_DIFFICULTIES_IN_COMPREHENDING_MATHEMATICAL_WORD_PROBLEMS_IN_ENGLISH_LANGUAGE_LEARNING_CONTEXTS
 ↑ Jan, S. and Rodrigues, S., (2012). Students' difficulties in comprehending mathematical word problems in English language learning contexts. International Researcher, Vol. 1, Issue 3, P. 156, Accessed Dec 4, 2015 fromhttp://www.academia.edu/2078056/STUDENTS_DIFFICULTIES_IN_COMPREHENDING_MATHEMATICAL_WORD_PROBLEMS_IN_ENGLISH_LANGUAGE_LEARNING_CONTEXTS
 ↑ http://www.councilforlearningdisabilities.org/mathematicsdisabilities
 ↑ ^{a} ^{b} ^{c} ^{d} Anderson, J. R., Corbett, A. T., Koedinger, K. R., & Pelletier, R. (1995). Cognitive Tutors: Lessons learned. Journal of the Learning Sciences, 4(2), 167.
 ↑ ^{a} ^{b} Koedinger, K. R., & Corbett, A. (2006). Cognitive tutors. The Cambridge handbook of the learning sciences, 6177.
 ↑ Koedinger, K. R. & Anderson, J. R. (1993). Effective use of intelligent software in high school math classrooms. In Proceedings of the World Conference on Artificial Intelligence in Education, (pp. 241248). Charlottesv
 ↑ Koedinger, K., Anderson, J., Hadley, W., & Mark, M. (1997). Intelligent tutoring goes to school in the big city. HumanComputer Interaction Institute.
 ↑ Corbett, A. T., & Anderson, J. R. (2001). Locus of feedback control in computerbased tutoring: Impact on learning rate, achievement and attitudes. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (pp. 245–252). New York, NY, USA: ACM.
 ↑ What Works Clearinghouse (ED). (2016). Carnegie Learning Curricula and Cognitive Tutor[R]. What Works Clearinghouse Intervention Report. What Works Clearinghouse.
 ↑ Morgan, P., & Ritter, S. (2002). An experimental study of the effects of Cognitive Tutor® Algebra I on student knowledge and attitude. Pittsburgh, PA: Carnegie Learning, Inc.
 ↑ Cabalo, J. V., Jaciw, A., & Vu, M. (2007). Comparative effectiveness of Carnegie Learning’s Cognitive Tutor® Algebra I curriculum: A report of a randomized experiment in the Maui School District. Palo Alto, CA: Empirical Education, Inc.
 ↑ Campuzano, L., Dynarski, M., Agodini, R., & Rall, K. (2009). The effectiveness of reading and mathematics software products: Findings from two student cohorts. Washington, DC: U.S. Department of Education, Institute of Education Sciences.
 ↑ Pane, J. F., Griffin, B. A., McCaffrey, D. F., Karam, R., Daugherty, L., & Phillips, A. (2013). Does an algebra course with tutoring software improve student learning?.