“Young children reinvent arithmetic. Spontaneously or by imitating their peers, they imagine new strategies for calculation. They also learn to select the best strategy for each problem. The majority of their strategies are based on counting, with or without words, with or without fingers. Children often discover them by themselves, even before they are taught to calculate.”
― Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics
Quotes to Ponder
01
Quotes from YouCubed.org
“Math facts are a very small part of mathematics but unfortunately students who don’t memorize math facts well often come to believe that they can never be successful with math and turn away from the subject.”
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"There is a common and damaging misconception in mathematics – the idea that strong math students are fast math students."
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"In order to learn to be a good English student, to read and understand novels, or poetry, students need to have memorized the meanings of many words. But no English student would say or think that learning about English is about the fast memorization and fast recall of words. This is because we learn words by using them in many different situations – talking, reading, and writing."
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"As educators we all share the goal of encouraging powerful mathematics learners who think carefully about mathematics as well as use numbers with fluency. "
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"...Number sense, something that is much more important for students to learn, and that includes learning math facts along with deep understanding of numbers and the ways they relate to each other."
02
Quotes from "Eight Unproductive Practices in Developing Fact Fluency" by Kling & Bay-Williams (2021)
"Use visuals and contexts in ways that lead students to be the ones discovering and explaining the strategies (how they work and why they work)."
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"Too often, teachers at a particular grade level feel pressured to ensure their students have mastered their basic facts by the end of the school year. And yet year after year, facts memorized through rote techniques do not tend to stick."
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"It is better for students' future learning and their mathematical identities to focus on learning strategies and not feel pressured to memorize."
"Be the tortoise in the fable of the tortoise and the hare. It is better for students’ future learning and their mathematical identities to focus on learning strategies and not feel pressured to memorize."
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"Playing facts games with classmates or family members is one of the most beneficial ways to encourage children to master their basic facts and can greatly increase their taste for mathematics, ..."
03
Quotes from the Heinemann blog "What is Mathematical Fluency and How Does it Support Learning?"
"We want fluency and understanding and if we want that we also need to be purposeful about how we develop that."
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"I think we've been hoping for too long that fluency will come out of just doing a lot of good problems with kids. And I don't think that's enough."
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"This feature, we call mental mathematics... it's not intended to be fast recall, it's intended to [kinda] build on your prior understanding."
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"Fluency is built on the conceptual understanding. It comes from understanding and strategies."
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"Fluency comes from understanding and those strategies [learned during number sense development]."
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04
Quotes from "Number Sense on the Number Line" by Woods, Geller, & Basaraba (2017)
"Learning these concepts and skills by working with number lines may help students develop a mental representation of the order and magnitude of numbers that can then be used to make comparisons, understand place value, and model mathematical operations (Diezmann & Lowrie, 2006)."
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"For those students with LP or MD or who are at risk, the number line may provide an important link between concrete representations of early mathematics concepts and more abstract mathematical notation. Some students may need explicit instruction on how to interpret the number line or may need to work with a number path prior to using the number line."
05
Quotes from "Building Computation Systems through Place Value" in Building Number Sense through the Common Core by Witzel, Riccomini, & Herlong (2012)
"If mathematics procedures of whole numbers are based on concepts of place value, then students can scaffold those procedures further to concepts such as fractions and polynomials. However, this requires a focus on long-term growth rather than single-grade expectations." (p. 72)
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“Understanding place value may be difficult for several students. Focus on concrete objects and verbal practice before applications. It may be that the child cannot make the connection between what is more commonly spoken and the place value language.”
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"...other nations use place value consistently within their language, in the United States the use of place value in mathematics computation is restricted to mature learners. We need to teach place value understanding to students earlier."
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"Matching the concrete materials to the abstract equation helps when teaching students to reason through the algorithm."
Quotes about Dyscalculia and Numeracy Difficulties
"Teaching numeracy requires introducing new questions and ideas and guiding pupils to develop numerical concepts and knowledge through using equipment and developing vocabulary. Teachers do need to teach the conventional symbols, signs, and calculation procedures. However, this is best done by exploring the concepts first and then using the mathematical notation to express the ideas." (p. 115)
"Repeated failure leads to maths anxiety; in extreme cases it may lead to maths phobia. The pupil is unable to take in or process information and may appear to 'freeze'." (p.55 Ch. 4).
Numeracy needs to be taught in a way that is structured, cumulative and multi-sensory (p.116).
Pupils with a fixed mindset are afraid of making mistakes in case it shows them to be less able than they are perceived to be.
