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\r\nWhen math feels slow, frustrating, and full of one step forward, two steps back, the issue may not be ability so much as fluency. Math is like learning to read. A traditional math curriculum relies on rote memorization, similar to some older reading programs, which teach students to memorize sight words rather than decode letter sounds. Memorization can provide a quick win because students learn to match memorized answers to math facts or to read by recognizing sight words.<\/p>\r\n\r\n
In the long run, this hinders their ability to read at a higher level because it conditions their brains to rely primarily on the region responsible for memorization. They haven\u2019t had much practice using all their reasoning capabilities to decode words, so the mental muscles are flabby.<\/p>\r\n
Math works the same way. A child may memorize a multiplication fact like 5 x 6 = 30, but have no real understanding that 30 represents 5 groups of 6 objects or 3 groups of 10. For example, without math fluency, they may not be able to solve a word problem that asks them to find the total number of marbles when 5 children each have 6 marbles.<\/p>\r\n\r\n\r\n
Developing math fluency takes time, consistency, and mastery of concepts. All About<\/em>®<\/sup> Math<\/em> incorporates a research-based method to teach each math concept:<\/p>\r\n The National Research Council defines fluency with math as the ability to carry out mathematical procedures accurately, efficiently, and flexibly, emphasizing that true fluency involves more than speed or memorization.1<\/sup> As the NRC says, fluency also includes flexibility with numbers; the ability to understand a concept so well that you can apply it to new situations.<\/p>\r\n Math drills and worksheets build speed and memorization. This gives your student a shallow grasp of math operations with no real understanding of how numbers work.<\/p>\r\n Memorizing math facts<\/strong> is knowing that 5 x 6 = 30<\/p>\r\n Math fluency<\/strong> is the understanding that 30 is 5 sets of 6, or 6 sets of 5, and is part of a larger number pattern. Your child is fluent when they intuitively grasp that 5 x 7 is one more group of 5 than 5 x 6.<\/p>\r\n When your child is fluent in math, often called \u201cproficient,\u201d he can explain why adding, subtracting, multiplying, or dividing numbers works to find the correct answer and when to use each operation. He demonstrates an understanding of the concepts behind the math, rather than relying on rote memorization.<\/p>\r\n\r\n The importance of mathematics fluency lies in its role as a bridge between basic skills and higher-level thinking.<\/p>\r\n Fluency:<\/p>\r\n Going back to our reading comparison, when your student is fluent at decoding words, he can read much more smoothly and with less effort. This allows him to focus on comprehension. Fluency leads to confidence and a willingness to tackle harder material.<\/p>\r\n A math fluency example in real life: When you are baking cookies, and the recipe calls for \u00be C of flour, what do you do if you don\u2019t have a 3\/4C measuring cup? Because you are fluent in math, you aren\u2019t stymied by the problem; you know that you have other options. You could use a 1\/4C measured 3 times. Or you could use 1\/2C and 1\/4C to equal 3\/4C (or you could use my method and just wing it!).<\/p>\r\n Finding a solution doesn\u2019t take much mental effort when you are fluent in math. You understand the concept of fractions and can easily devise a different way of achieving the outcome.<\/p>\r\n A child with strong math fluency can play with the concept, develop a higher-level model of it, and apply the concepts to new situations, even creating unique solutions to problems.<\/p>\r\n\r\n The Concrete-Representational-Abstract approach gives your child a powerful way to build the how of mathematical understanding and fluency. The National Research Council\u2019s five strands of math proficiency tell us what develops as fluency grows.<\/p>\r\n The National Research Council describes mathematical proficiency as five abilities that develop together: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Rather than treating these as separate skills, think of them as different tools for becoming comfortable and capable with math. All About Math's hands-on instruction, visual representations, practice, problem solving, and discussion help develop these tools.<\/p>\r\n Conceptual understanding:<\/strong> Knowing why the math works. When your child builds 6 groups of 4 with manipulatives, draws those groups, and eventually writes 6 × 4 = 24, he is learning more than a multiplication fact. He is developing an understanding of what multiplication means and how 6 × 4 connects to ideas such as repeated addition and 24 ÷ 6. All About Math\u2019s use of the CRA framework, starting with concrete objects, drawing a representation, followed by writing numbers and symbols, acts as scaffolding that explicitly builds neural connections by systematically moving through and engaging different functional regions of the brain. This develops mastery in a way that simple worksheets cannot.<\/p>\r\n Procedural fluency:<\/strong> Knowing how to do the math efficiently and accurately. Understanding a concept doesn't eliminate the need for practice. As skills develop, your child should become increasingly accurate and efficient at performing familiar mathematical procedures. But procedural fluency also includes flexibility. A child who doesn't immediately recall 6 × 7 might use a known fact such as 5 × 7 = 35 and add another 7. All About Math combines conceptual understanding with purposeful practice to develop mastery of each concept, then uses regular cumulative review to build accuracy and procedural fluency with learned skills.<\/p>\r\n Strategic competence:<\/strong> Figuring out how to approach a problem. Real problems don't always make it clear which procedure to use. As your child develops strategic competence, he learns to ask, \"What do I know? What am I trying to find? What could I try?\" A story problem, for example, might require him to determine which information matters, choose an operation, and decide how to represent the problem. All About Math gives him opportunities to reason through problems and use what he already knows to find the solution.<\/p>\r\n Building a habit of seeking new strategies rather than giving up when you don\u2019t know the answer on the first try is also a critical step in building executive functioning. This skill equips him to adapt his thinking when faced with any of life's problems.<\/p>\r\n Adaptive reasoning:<\/strong> Thinking about whether an approach makes sense. Your child needs to consider whether their reasoning is sound, explain why an answer makes sense, and try another approach when necessary. Simple questions such as \"How did you know?\" or \"Could you solve it another way?\" encourage him to reflect on mathematical relationships instead of depending on a single memorized solution path. All About Math builds this kind of reasoning through oral discussion questions that encourage him to explain his thinking.<\/p>\r\n Productive disposition:<\/strong> Learning that math makes sense and \"I can figure this out.\" The National Research Council's final strand may be the one parents notice most. Your child develops a productive disposition when he begins to see math as something understandable and useful and himself as capable of learning it. When he can build a problem, see how the quantities relate, explain his reasoning, and find another strategy when he gets stuck, math becomes less a collection of mysterious rules and more something he can make sense of.<\/p>\r\n These five strands work together. Concrete objects and visual representations help build understanding. Practice helps procedures become efficient. Problems provide opportunities to choose strategies. Discussion encourages reasoning. And every experience of making sense of a mathematical idea helps your child build confidence in his ability to tackle the next one.<\/p>\r\n That's the fuller goal of mathematics fluency: not simply knowing answers, but developing proficiency with numbers and their relationships so your child can understand, use, explain, and adapt.<\/p>\r\n\r\n Memorization is useful for automaticity after the concepts are mastered, but speed with fact cards doesn\u2019t demonstrate conceptual understanding. Facts are often quickly forgotten after the lesson ends and the test is over. Here\u2019s a quick way to prove this: what is 11 x 12? If you don\u2019t use multiplication daily, you likely had to pause and figure out the answer; it didn\u2019t jump to mind instantly. My favorite trick is to find a known fact near the problem; in this case, 10 x 12 is an easy 120, plus one more group of 12 for 132.<\/p>\r\n If you are like me, you forgot the answer you likely memorized at 12, but you had the tools to figure it out another way. You didn\u2019t have to rely on memorization; you were using your understanding of multiplication to find the answer.<\/p>\r\n All About Math<\/em> helps students master math by:<\/p>\r\n Explore a sample of the All About Math<\/em> Level 1 student activity book<\/a><\/p>\r\n These lessons engage multiple regions of the brain to construct mental models, allowing mathematical understanding to become a set of flexible tools that transfer naturally into other areas of learning, such as:<\/p>\r\n All About Math<\/em> builds conceptual understanding, and students see why math works. Review the pages of the level 3 teacher's manual sample to see this in action.<\/a><\/p>\r\n\r\n Memorization has an important place in learning math. But mathematical fluency becomes visible when your child can do more than remember an answer or reproduce a procedure. A fluent learner understands enough about math to use what she knows flexibly, explain her thinking, and make adjustments when a familiar approach doesn't quite fit.<\/p>\r\n One sign to watch for is whether your child can explain how she knows. Suppose your child quickly tells you that 7 + 8 = 15. Instead of asking for another fact, try asking, \u201cHow did you know?\u201d She might explain, \u201cI know 7 + 7 is 14, so one more is 15,\u201d or \u201cI took 2 from the 7 to make 10 + 5.\u201d Those explanations show that she isn't relying only on memorized recall. She understands number relationships and can use them to reason.<\/p>\r\n Another sign is flexibility when the problem changes. If your child has memorized one procedure, she may become stuck when a problem looks unfamiliar. As she is developing fluency, she is increasingly able to ask, \u201cWhat do I already know that could help me?\u201d or try a different approach. For example, if she doesn't immediately know 42 ÷ 6 = 7 but knows that 6 x 6 = 36, can she figure out that one more group of 6 (or 6 x 7) = 42, so 42 ÷ 6 is 7? Being able to choose, adapt, or switch strategies is an important part of fluency.<\/p>\r\n You can also look for your child's ability to show the same idea in different ways. Can she solve a problem with objects, draw or model what is happening, explain it aloud, and eventually connect those representations to numbers and mathematical symbols? Moving between these forms provides evidence that she understands the mathematical idea underneath the procedure rather than simply remembering a sequence of steps.<\/p>\r\n And don't worry about speed drills or timed tests: In their position paper on procedural fluency, the National Council of Teachers of Mathematics states: \u201cTimed tests do not assess fluency and can negatively affect students, and thus should be avoided\u201d2<\/sup><\/p>\r\n Instead, the National Research Council recommends using assessment to uncover students' mathematical thinking and inform instruction. Questioning, observations, and tasks that ask children to explain their reasoning can reveal aspects of mathematical proficiency that test answers alone may not show. (National Research Council, 2001) At home, that can be surprisingly simple: watch your child work, ask how she found an answer, invite her to show you another way, or occasionally ask, \u201cHow do you know that makes sense?\u201d<\/p>\r\n The goal isn't to turn every math lesson into an interrogation. A few low-pressure questions can tell you much more than a page of answers completed against the clock. As your child's mathematics fluency grows, you should see not only greater accuracy and efficiency, but also increasing understanding, flexibility, and confidence in deciding what to try next.<\/p>\r\n\r\n Think of building fluency through mastery, not memorization. This requires developing the conceptual framework for why math works the way it does.<\/p>\r\n In early grades, it starts with building a concrete understanding of number sense through hands-on manipulatives. There\u2019s more about what is number sense and why it matters in this article.<\/p>\r\n Use visual representations, such as creating equal groups, to support multiplication. The child might have a problem, such as: If we have 30 guests and 5 tables, draw a picture showing how many people we need to seat at each table to model N x 5 = 30.<\/p>\r\n Make meaningful connections through story problems by encouraging the student to analyze, evaluate, and determine what data is needed to solve them.<\/p>\r\n Include verbal reflection questions that encourage your student to think through how they arrived at an answer.<\/p>\r\n Practical strategies for parents:<\/p>\r\n Fatigue and frustration when learning new math concepts are reliable predictors of future \u201cpencil allergy\u201d when it comes to doing math. If the lessons are long and cognitively tiring, your child learns to associate math with uncomfortable feelings and doubts about his ability to succeed. Keeping lessons short and, equally as important, not moving forward until he demonstrates fluency with the current topic helps avoid fatigue and negative associations with learning math. This builds a track record of wins and avoids tackling new concepts before the foundational knowledge is mastered.<\/p>\r\n All About Math<\/em> follows this practice by limiting lesson time to 20 minutes per day, for a total of an hour and 40 minutes a week spent on math fluency.<\/p>\r\n\r\n Preschool math fluency begins with number sense. At this stage, fluency is defined by familiarity with numbers rather than math operations.<\/p>\r\n Effective early activities include:<\/p>\r\n Is your child ready to start formal math? Use our free placement tests<\/a> to find the right starting level.<\/p>\r\n\r\n A key to building math fluency is more than just traditional math practice; it includes exposure to numbers in a variety of settings and applications. Well-designed math fluency activities increase hands-on involvement with numbers, helping children to see different relationships and patterns.<\/p>\r\n A game like Make Ten, using a 10 frame, offers several benefits. You\u2019ll need:<\/p>\r\n Give your student a cup with 10 two-color counters. They can shake the cup and pour out the counters. Some will land with the yellow side up; others will land with the red side up. Have the child put the yellow counters on the 10-frame, followed by red ones.<\/p>\r\n If they are still in the concrete stage of learning, she can verbalize the resulting math fact (e.g., 6 yellow plus 4 red equals 10)<\/p>\r\n Once she confidently verbalizes the math facts, you can have the child write the math problem created by the counters.<\/p>\r\n Using activities like this will:<\/p>\r\n More math fluency ideas:<\/p>\r\n\r\n Math fluency worksheets can be effective when used after explicit instruction, cumulative repetition, and demonstrated proficiency are already achieved. If you have identified a specific area that needs more practice to help your child develop automaticity, worksheets can have their place.<\/p>\r\n Research on math fact intervention using worksheets or flashcards suggests that short practice sessions, done in small doses with deliberate breaks in between, can be useful.3<\/sup><\/p>\r\n Deliberate use of worksheets for a targeted purpose, such as building automaticity with mastered facts, makes them a tool, not a teaching method.<\/p>\r\n\r\n To help build math fluency, All About Math<\/em> combines:<\/p>\r\n Students manipulate numbers physically before solving them mentally. Each step of the Concrete-Representation-Abstract approach mentioned above is assessed for mastery before moving on to the next step. For example, your student should be comfortable with addition using manipulatives before he is asked to add using pictures of objects. Once he is doing well with operations using pictures, you move on to solving written problems. Oral discussion occurs throughout the lessons to help your child develop and express their understanding. This process ensures he understands the concepts at a mastery level before he moves on to the next topic.<\/p>\r\n\r\n\r\n
What is Math Fluency?<\/h2>\r\n
Why Is Math Fluency Important?<\/h2>\r\n
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\r\nDeveloping Math Proficiency: A Fuller Picture of Math Fluency<\/h2>\r\n
Conceptual Understanding vs. Memorization<\/h2>\r\n
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How is Math Fluency Assessed?<\/h2>\r\n
Building Mathematics Fluency Step by Step<\/h2>\r\n
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How Much Time Per Week Should be Spent on Math Fluency?<\/h2>\r\n
Math Fluency in Early Learning<\/h2>\r\n
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Math Fluency Activities and Games<\/h2>\r\n
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Addition & Subtraction<\/h3>\r\n
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Multiplication & Division<\/h3>\r\n
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What about Worksheets for Teaching Math Fluency?<\/h2>\r\n
How All About Math Builds Math Fluency<\/h2>\r\n
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\r\n All About Math<\/em><\/th>\r\n Drill-Based Math Programs<\/th>\r\n<\/tr>\r\n<\/thead>\r\n \r\n\r\n Building conceptual understanding<\/td>\r\n Rote memorization<\/td>\r\n<\/tr>\r\n \r\n Develops confidence<\/td>\r\n Pressure to be faster<\/td>\r\n<\/tr>\r\n \r\n Long-term retention<\/td>\r\n Facts are quickly forgotten<\/td>\r\n<\/tr>\r\n \r\n Uses 10-frames and other tools \/ manipulatives to visualize<\/td>\r\n Flashcards<\/td>\r\n<\/tr>\r\n \r\n Pattern recognition and reasoning<\/td>\r\n Learns facts in isolation from patterns<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n