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In this article they use hands on learning to explore probability. Students use a spinner to determine if their number is odd or even and have to decide if there is a higher probability of spinning an odd or even number. There are multiple avenues for students to explore with this concept and allows students to play around with the numbers and make sense of their solutions.

Backer and Chick (2007) say, "The activity suits group work, and fosters problem-solving skills."

Baker, M., & Chick, H. (2007). Making the Most of Chance. Australian Primary Mathematics Classroom. 12(1), 8-13.

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Multiplication, division and fractions

This article explains the benefits of students exploring open-ended tasks with manipulatives. Dean, a student in this article, explored the relationship between multiplication, division, and fractions by using an array, exploration and guided questions supported from the teacher.

Vale and Davies (2007) writes: "Crucial for Dean's discovery was the openness of the task, the classroom teacher's suggestion that he try it with different numbers and the university teacher's persuasion to record his findings systematically."

"This case illustrates the importance of discussion with students while they are working on open-ended tasks."

Vale, C., & Davies, A. (2007). Dean's Great Discovery: Multiplication, Division and Fractions. Australian Primary Mathematics Classroom. 12(3), 18-22.

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How does the skill of memorizing multiplication facts increase fluency?

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How does implementing Learning Stations improve student comprehension of multiplication?

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How does number sense affect students ability

to memorize multiplication facts?

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Teaching Student-Centered Mathematics Grades 3-5

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Teaching Student-Centered Mathematics Grades 3-5

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Van de Walle describes the importance for students understanding numbers before being introduces to the rules. This section focused on fraction computation and the importance of students developing number sense before being taught the rules dealing with fractions and the different operations. Van de Walle suggests allowing students to explore fractions and have the opportunity to relate the fractions with whole numbers. Students need to develop an understanding of fractions before being taught the rules for the different operations. If students do not develop an understanding of fractions and have the opportunity to explore the different operations the rules will not have any validity and will be easily forgotten or misused.

Van de Walle (2006) says, "Armed only with rules, students have no means of assessing their results to see if they make sense" (p.161).

Van de Walle (2006) says, "To put it simply, children construct their own knowledge" (p.1).

Van de Walle, J.A. (2006) Teaching Student-Centered Mathematics Grades 3-5 (Volume 2). Boston, MA: Pearson.

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Marilyn Burns explains the value of number sense and describes it's importance for children in grades K-8. Children who have not developed number sense have difficulty applying their learning. They see one way to answer questions and are often troubled with questions that require them to apply their knowledge. Burns suggests having students solve problems in their head and building on their estimation computation and sharing their solutions with the class. Students need to be provided with rigorous activities; so they can build on their comprehension which stems from building on their number sense.

Burns (2007) says, "Students with good number sense can think and reason flexibly with numbers, use numbers to solve problems, spot unreasonable answers, understand how numbers can be taken apart and put back together in different ways, see connections among the operations, figure mentally, and make reasonable estimates" (p.24).

Burns (2007) explains, "When children think that there is one right way to compute, they focus on learning and applying it, rather than on thinking about what makes sense for the numbers at hand" (p.24).

Burns, M.(2007) About teaching mathematics: A k-8 resource (3rd ed). Sausalito, CA: Math Solutions

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Developing Automaticity in Multiplication Facts: Integrating Strategy Instruction With Timed Practice Drills

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Developing Automaticity in Multiplication Facts: Integrating Strategy Instruction With Timed Practice Drills

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This article explains the importance of students building automaticity with their multiplication facts and explores building automaticity through different approaches. These approaches include solely working on timed practiced tests, building automaticity through procedures and incorporating an integrated learning approach where the focus pertained to fact fluency and building procedures. Fluency is important because it allows students to solve complicated problems because they don't have to spend time focusing on finding the correct answer to the multiplication fact, rather they can concentrate on the multiple step process. Similarly, it is also important for students to have number sense, if they only have fact fluency, they will not have a complete understanding of how to solve the problem. Therefore, both approaches are important and an integral method incorporates both methods.

Woodward (2006) says:

"Without the ability to retrieve facts directly or automatically, students are likely to experience a high cognitive load as they perform a range of complex tasks" (p.269).

"Methods vary from the use of visual displys such as ten frames, number lines (Thompson & Vande Walle 1984; Van de Wall, 2003) to more general techniques such as classroom discussions where students share fact strategies with their peers (Steinberg, 1985; Thorton, 1990; Thorton & Smith, 1998)" (p.270).

"Math educators argue that emphasis on strategies helps students organize facts into a coherent knowledge network (Isaac & Carroll, 1999; Rathmel, 1978)" (p.271).

"Frequent timed practice is essential. However, strategies help increase a student's flexible use of numbers, and for that reason, Cumming and Elkins advocate the use of strategy instruction for all students through the end of elementary school" (p.271).

"However, research cited above (Geary, 1993; Goldman et al., 1998) indicates that students with LD do not develop sophisticated fact strategies naturally" (p.271).

"Results from this study indicate that an integrated approach and timed practice drills are comparable in their effectiveness at helping students move toward automaticity in basic facts" (p.287).

"While Automaticity in facts is still relevant to proficiency in traditional algorithms, automaticity has become important to estimation, mental calculations and approximation skills" (p.287).

Woodward, J. (2006). Developing Automaticity in Multiplication Facts: Integrating Strategy Instruction with Timed Practice Drills. Learning Disability Quarterly. 29(4), 269-289.

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When Multiplication Facts Won't Stick: Could a Language/ Story Approach Work?

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When Multiplication Facts Won't Stick: Could a Language/ Story Approach Work?

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This article explains the effects of using a Language/Story based approach when teaching students multiplication facts. There are students who have a difficult time memorizing multiplication facts and Mahler (2011) explains, "when weak mental representations are strengthened by a story, mnemonic, visualization, or activity, the chances of effective storage and retrieval can be greatly increased because they serve to anchor and connect that information" (p.7). A study was conducted and students who participated in this approach showed gains, especially students who had lower multiplication memorization skills at the beginning of the study.

Mahler (2011) says:
"We could drill a particular fact many times one day, and by the next day, that fact had vanished" (p.5).
"We were solely concerned with the act of memorization" (p.5).
"Most research speculates that delayed automaticity in facts is due primarily to a memory-based deficient (Levine, 2001)" (p.5).
"While episodic memory collects our experiences in a way that allows us to replay them, semantic memory involves knowledge of facts, rules, symbols, meanings, and ideas that are not necessarily connected to specific incidents or events" (p.5).
"When there are problems with retrieval from memory, mastery of the multiplication math facts can be difficult" (p.6).
"Children with math disabilities tend to do less mental rehearsal, a function of the central executive component, than children without math disabilities; it is that subvocal rehearsal that reduces memory decay (Swanson et al., 2008) and strengthens memory traces" (p.6).
"To be successful at one step, a child must have properly mastered each prior step (VanDerHeyden & Burns, 2009).
"Students with insufficient working memory already have a tendency to lose track of what they are doing, forgetting one part of a task while working on another; thus it is even more important that these particular students attain automaticity with multiplication facts" (p.6).
"When weak mental representations (e.g., 3 x 4) are strengthened by a story, mnemonic, visualization, or activity, the chances of effective storage and retrieval can be greatly increased because they serve to anchor and connect that information" (p.7).
"Needing to translate a story made his retrieval slower than if he had memorized the numerical answers, but compared with not being able to retrieve the fact at all, this route was acceptable" (p.

Mahler, J.D. MEd.(2011). When multiplication factos won't stick: Could a language/story approach work? Retrieved May 6, 2013, from http://www.eric.ed.gov

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This article explores using math games a whole lesson approach. Students in the article explored multiplying and dividing by decimals to develop a better understanding of a complex concept. Through exploration, students were able to experience the results when multiplying and dividing a number by a decimal and the results which contradict previous multiplication and division understandings.

Bragg (2006) explains how a students, "initial cognitive conflict was overcome through the use of trail and error, which seems to have helped him develop an understanding of the effect of multiplying decimals and thus he achieved cognitive equilibrium" (p.6).

Bragg (2006) explains how after the game, "Children were able to articulate the effect of multiplication by numbers larger and smaller than one" (p.6).

Bragg (2006) says, "the students were able to grasp the complexity of the mathematics involved in the games" (p.7).

Bragg, L.A. (2006). "Hey I'm learning this." Retrieved May 8, 2013, from: http://www.eric.ed.gov/PDFS/EJ793935.pdf

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Efficient Class-Wide Remediation: Using Technology to Identify Idiosyncratic Math Facts for Additional Automaticity Drills.

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Efficient Class-Wide Remediation: Using Technology to Identify Idiosyncratic Math Facts for Additional Automaticity Drills.

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This article focused on the importance of students mastering their multiplication facts. The problem occurs when students can't complete complex problems because they are becoming stuck on their facts and wasting their time figuring out the answer to the multiplication problems rather than focusing on what the problem is asking. The article places an importance on students practicing facts they have not memorized. In this study teachers used DRP (Direct, Practice, Repair) which allows students to the opportunity to practice facts they haven't mastered yet by using the CCC (Cover, Copy, Compare) method.

Parkhurst, J., Skinner, C. H., Yaw, J., Poncy, B., Adcock, W., & Luna, E. (2010).

The article Efficient Class-Wide Remediation explains how, "Researchers have drawn an important distinction between fluency and automaticity (Skinner & Daly, in press). The term fluency reflects the ability to respond to a group of stimuli both quickly and accurately. However, when discussing mathematics researchers have used the term automaticity to describe a students' ability to respond to a specific fact rapidly, accurately and with minimal effort or cognitive resources (HasselbringPoncy, Skinner, & Jasper, 2007 et al., 1987 1988; Poncy, Skinner, & Jasper, 2007; Poncy, Skinner, & O'Mara, 2006)" (p.111,112 Efficient Class-Wide Remediation: Using Technology to Identify Idiosyncratic Math Facts for Additional Automaticity Drills. International Journal of Behavioral Consultation and Therapy. 6(2), 111-123.

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Comprehension and fluency go hand-in-hand. In this article Janette Bobis explores the strategies students experience when solving multiple step problems and solving problems with multiple strategies. When implementing a new process or performing a multiple step process, it is very important for students to have a foundation of number sense in order to carry out multiple step problems. Students who do not have a basic number sense understanding often give up because it's too much work and they are not provided with appropriate experiences based on their knowledge.

Bobis(2007) says:
"In fact, computational fluency, whether employing mental or written methods, and number sense are intertwined and should be developed together" (p.23).
"Research has shown that those who are good at mental computations possess a well -developed sense of number(McIntosh & Dole, 2000)" (p.23)
"The worry with an early emphasis on standard algorithms is that students will shift their focus to executing convenient procedures rather than on understanding the mathematics" (p.23).
"The challenge for teaches is to encourage the development of, and consistent use of, more efficient and appropriate strategies for solving mathematical problems without it being "too hard" for children" (p.24).
"However, it is clear that the promotion of number sense is critical to a basic understanding of mathematics and to a child's ability to compute easily" (p.27).

Bobis, J. (2007). From Here to There: The Path to Computational Fluency with Multi-Digit Multiplication. Australian Primary Mathematics Classroom. 12(4), 22-27.

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In this video teachers in Palmer Primary School explain the importance of number sense and building depth in children's understanding of times tables. At an early age they explore the 4 times tables by looking at patterns and applying these patterns to solve more challenging problems. The Teaching Channel(2013) says, "Children need to make connections with their math so they can apply their learning."
Teaching Channel (Producer). (2013). Progression in primary math [video]. Available from http://www.teachingchannel.org/videos/teaching-multiplication

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Enthusiastic ways to teach multiplication and math to students in grades 3, 4 and 5. Using patterns and other learning tip can help elementary school students to learn multiplication faster and allow them to apply it to other problems.

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Topic: I plan to study how building number-sense influences the memorization of multiplication because I want to find out how to increase automaticity to show the relationship between comprehension and fluency through learning centers so that I may better understand the best way for students to memorize their multiplication facts.

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Topic: I plan to study how building number-sense influences the memorization of multiplication because I want to find out how to increase automaticity to show the relationship between comprehension and fluency through learning centers so that I may better understand the best way for students to memorize their multiplication facts.