In the first post of our Lower School math curriculum series, we explored how Singapore Math guides students through concrete, pictorial, and abstract reasoning. The second post examined how Singapore’s spiral design allows students to revisit concepts with deepening understanding across several years. Today’s post looks at various benchmarks for student comprehension and provides helpful guidance for developing student confidence and a growth mindset in math.
From Fluency to Automaticity: Math Facts at Clapham
At Clapham, it’s common to hear teachers discuss “math facts” or “math fact practice” as an important way for parents to support their students at home. Math facts are the sums and problems students aim to answer quickly and correctly. As students ascend the spiral of Singapore’s curriculum, math fact mastery is an essential component to student success.
True math fact mastery comes in stages: fluency, reliability, and automaticity.
In the fluency stage, children apprehend answers through counting and using concrete manipulatives. When solving a problem, a child may begin with what she knows and use strategies to arrive at a solution. A child who is fully fluent is flexible (begins with what is known), efficient (understands which strategy is useful for solving a math fact), strategic (uses appropriate methods to solve a math fact), and accurate (arrives at the right answer). Clapham faculty point out that speed is not synonymous with fluency. In this stage, students may take more time to be accurate. A student counting on her fingers is generally in the fluency stage of math facts.
In the reliability stage, a child quickly uses the fluency strategies to arrive at the correct answer.
In the automaticity stage, a child “knows” the math fact, retrieving the answer swiftly and without pausing more than a moment to calculate the answer.
In Class One (for example, as it is illustrated above), students will ideally answer 40 math fact questions in two minutes to demonstrate automaticity by mid-year.
Scaffolding Math Facts
As parents walk with their students through math facts, a good place to start with younger students is simply to add one, then two to single digits. From there, move to adding three (“three more is two more plus one”), then four (“four more is two more plus two”). Then six (“six more is three more plus three”), seven (“seven is five more plus two”), eights from a combination of twos or fours, and then nine and ten (“nine is ten more minus one”). As students practice working with increasingly large numbers, their fluency, reliability, and automaticity grow. This scaffolding is crucial for a student’s understanding the why behind larger math facts and provides the strategic patterns for problem solving throughout their math education.
Class One
Class One students will begin practicing math fact flashcards at home in September, once scaffolding strategies have already been introduced and practiced at school, with the aim of mastering addition to ten by mid-year (or 40 problems in 2 minutes). Subtraction is introduced in the middle of the year, with the aim of mastering subtraction within 0-10 by May (again, 40 problems in 2 minutes).
Class Two
Class Two students practice addition and subtraction to ten within the first two weeks of school to reinforce automaticity. They then begin to practice addition to twenty, with the aim of automaticity by November (40 problems in 2 minutes), and subtraction within twenty by December (40 problems in 2 minutes). Multiplication of twos, threes, and fours are introduced in January, with fives and tens introduced in February, aiming for mastery by May.
Clapham aims to have students with computational reliability for multiplication and division up to ten by ten by Class Four, and multi-digit computational fluency by Class Five.
So, for Class Four, here’s an example: Rather than first teaching 7×8=56, Singapore breaks the larger numbers down into more easily retrieved data: 5×8+2×8=40+16=56. Students using these strategies are computationally reliable. For Class Five, students are ideally recalling 7×8=56 without significant time spent in computation.
What if my student is not demonstrating fact mastery by the ideal timeframes?
Students struggling to answer math facts with automaticity are almost always simply in need of practice. They can practice using strategies at first, with the aim of eventually “skipping” over the strategy step by remembering the solution with quicker recall. Flashcards can help with this.
Math Fact Practice
As parents consider how to organize or design flashcards, Class Two homeroom teacher Andree Escobar emphasizes the value of students memorizing number bonds, or fact families, to assist in math facts recall (we briefly discuss number bonds in the first Singapore Math post).
Here’s her example:
5 + 7 = 12
7 + 5 = 12
12 – 5 = 7
12 – 7 = 5
Handing a student a stack of one hundred flashcards will most likely lead to discouragement. A better strategy? Focus on four to eight facts at a time, slowly adding more as the student memorizes their facts. Spending just five to ten minutes a day will eventually lead to long-term recall as they use smaller facts to work out larger equations throughout their math education.
Here’s a suggested routine:
- Present four to eight facts
- Have the child cover or hide the answer, looking only at the problem
- Have the child say or write the answer from memory
- Check the answer immediately. If the answer is incorrect, briefly explain or rebuild the fact, then try again after a short pause
- Return to the fact later in the same session and again on later days
Tips!
- Do not make every practice a pile of unknown facts. Chronic struggle or failure may burn a child out. Instead, mix in a “high success” ratio of known facts, with just one or two unknown facts in the mix. As the student masters the unknown fact, continue to add new and less familiar facts to the mix as the student’s memory expands to retrieve more and more facts.
- Make correction non-shaming and brief: “You said 6×7=36. That’s 6×6. Let’s add one more group of six. Now tell me 7×6.”
Mindset Matters
In her book Mindset, Carol Dweck addressed the notions of a “fixed” vs. “growth” mindset. Later, Angela Duckworth engaged these ideas with an emphasis on persistence in her book Grit. Both works explored how influential mindset is in a person’s success and personal happiness. A fixed mindset assumes limits to personal capability, often quite early on. For example, a student might say to herself, “I’m just not very good at math.”
A growth mindset, on the other hand, takes a significantly different approach: “I haven’t mastered this yet.” Grit adds the hope of success with the commitment to persevere: “I haven’t mastered this yet, but if I work hard and practice, I will almost certainly succeed.”
Parents and teachers can support a child’s growth mindset in math in several different ways. One way is to point out where the child is clearly learning:
“You used a number bond to check your answer. That was solid problem solving.”
“You recognized your own error and corrected it yourself.”
“You practiced these facts for five minutes a day over the past three days and you’re already remembering most of them.”
“You haven’t mastered this yet, but you have all the strategies you need to arrive at the answer. What’s one place you could start?”
Ultimately, mindset is about reminding the child that they can learn math using models, strategies, feedback, and time, and that math ability is not a fixed trait, like eye color or height. As Charlotte Mason emphasized, children can learn. With the right approach, math assessments ideally become a celebration of what your child can accomplish, rather than a dreadful ordeal to be endured.
