Fact Set Sequencing BrainLift

• Owner
  • Serban Petrescu
• Purpose
  • To establish a definitive, research-backed knowledge base on the optimal sequencing and grouping of arithmetic facts for building mathematical fluency.
  • This document will explain the cognitive science and pedagogical rationale behind why facts are introduced in a specific order, moving from foundational principles to an actionable, opinionated strategy for curriculum design.
  • It will serve as the governing "constitution" for how we structure learning paths in our products.

Experts

Aligned

• Jennifer Bay-Williams
  • Why: A leading voice in modern math education who focuses on "basic fact fluency that is foundational, not tedious." Her work provides models for moving from conceptual understanding, through strategy use, to automaticity, directly informing the pedagogical basis for strategy-driven fact grouping.
  • Locations: University of Louisville, X/Twitter
• Arthur Baroody
  • Why: A prominent researcher in the development of children's mathematical thinking. His work emphasizes a "phases of learning" model for basic facts, which outlines a progression from counting strategies, to reasoning strategies, to mastery (automatic recall). This provides a cognitive science framework for why sequencing matters.
  • Locations: University of Illinois, Google Scholar
• Daniel Ansari
  • Why: A cognitive neuroscientist who studies the brain basis of numerical and mathematical skills. His research helps explain the cognitive mechanisms underlying how children develop number sense and automaticity, providing a neurological basis for curriculum design.
  • Locations: Western University, Numerical Cognition Lab
• Nicole M. McNeil
  • Why: The lead author of a comprehensive 2025 paper, "What the Science of Learning Teaches Us About Arithmetic Fluency." Her work synthesizes the current state of research on sequencing, retrieval practice, and explicit instruction, confirming that deliberately organized, strategy-driven practice is superior to random practice.
  • Locations: University of Notre Dame, Google Scholar

Counter

• Jo Boaler
  • Why: A professor of mathematics education at Stanford who argues that an over-emphasis on timed, rote memorization can cause math anxiety. Her work provides the strongest research-based counter-argument, emphasizing deep, flexible number sense over speed, which forces us to justify our focus on automaticity.
  • Locations: youcubed.org, X/Twitter

DOK4: Spiky Points of View

• To ensure the most effective path to automaticity, we deliberately restrict user control over the intra-skill curriculum sequence.

  • Why it's controversial: The mainstream approach in many educational and consumer apps is to maximize user agency, allowing students or teachers to select specific facts or sets to practice (e.g., "just the x7 table"). This is believed to increase engagement and honor learner choice.
  • Our Bet: While we allow users to select a skill domain (e.g., Multiplication), we are betting that the massive efficiency gains from a perfectly sequenced, machine-controlled curriculum within that skill will create more long-term motivation through genuine progress than the fleeting engagement of intra-skill choice. Our facts show that derived strategies depend on automated anchors; allowing a user to practice x7's before x2's and x5's are automatic encourages brittle rote memorization, which is pedagogically unsound and undermines the entire strategy-driven model.

• A fact is automated once via its simplest path, then maintained via spaced repetition; we do not re-teach known facts with new strategies.

  • Why it's controversial: A common pedagogical view is to revisit facts with multiple strategies to build flexible thinking (e.g., re-teaching 2+8 as a "making ten" problem after it was already learned via "counting on"). This is thought to deepen conceptual understanding.
  • Our Bet: We are betting that for the specific goal of automaticity, this approach is inefficient. A fact should be automated once via the earliest, simplest applicable strategy. After that, it enters a pool of mastered facts to be reviewed through spaced repetition to ensure long-term retention. We do not move it back to an earlier strategic phase to re-practice it with a new strategy. This maximizes efficiency and accelerates the path to 100% automaticity across all facts. While this may trade some strategic flexibility for speed, that trade-off is acceptable for the primary goal of achieving automaticity.

• To maintain student motivation, we will enforce a maximum size on fact sets; pedagogically-pure 'catch-all' sets that become too large must be broken down into smaller, digestible chunks.

  • Why it's controversial: A pure instructional designer might argue that all remaining "Think-Addition" facts belong to the same strategic category. Splitting them is an artificial distinction based on surface-level characteristics rather than the underlying cognitive strategy.
  • Our Bet: We are betting that the motivational benefit of completing several small, manageable "levels" (e.g., 10-15 facts) far outweighs the cost of this pedagogical impurity. A student who sees they have 40+ facts left in one giant "All Else" set may feel overwhelmed, whereas a student who sees a few smaller, achievable sets feels a constant sense of progress. The faster feedback loop of completion is the stronger motivator.

DOK3: Insights

• Strategy-driven sequencing is the only effective path to automaticity. Random practice is inefficient. The most effective path to automaticity is to sequence practice in a deliberate order that mirrors the logic of derived strategies. By practicing "doubles" before "near-doubles," the brain first strengthens and automates the anchor facts, making the derived strategy easier and faster to apply. Repeated, timed execution of the derived strategy is what eventually "chunks" the process into a single, automatic retrieval.

• Inverse operations must be automated as an extension, not a separate topic. Subtraction and division automaticity are not developed in isolation. They are built almost exclusively by leveraging the inverse relationship with addition and multiplication ("Think-Addition/Multiplication"). Therefore, to be effective, practice for an inverse operation fact set (e.g., ÷2's) must be tightly coupled with and sequenced immediately after the mastery of the corresponding primary operation fact set (e.g., x2's).

• Procedurally simple 'rule-based' facts should be automated first to build momentum. For the specific goal of automaticity, a rule's procedural simplicity is more important than its conceptual complexity. Facts governed by simple, consistent rules (e.g., n+0, nx1, nx0) should be front-loaded in the curriculum. While the underlying concepts (like zero) can be abstract, the procedures are trivial to execute. Automating this large block of facts quickly provides students with early success, builds momentum, and significantly reduces the total cognitive load of un-mastered facts.

• 'Doubles' are the foundational anchor facts and must be mastered before their derivatives. The "Doubles" facts in addition (7+7) and their multiplication equivalent (x2) are the most critical strategic pillar in the curriculum. They are memorable, pattern-based, and serve as the cognitive anchor for a large family of more difficult derived facts (near-doubles, x4, x8). A curriculum must therefore ensure that these anchor facts are practiced to a high degree of automaticity before introducing practice for the facts that are derived from them.

• Subtraction practice must be sequenced to automate two distinct strategies: "Counting Back" for small subtrahends and "Think-Addition" for larger ones. Because subtraction strategies are asymmetrical, a one-size-fits-all approach is inefficient. Practice must be structured to first automate the simple, procedural "Counting Back" strategy (n-1, n-2). For all other facts, practice must be designed to explicitly automate the more powerful "Think-Addition" strategy, leveraging the existing network of automated addition facts.

• Fact sets must be grouped by a single, independent strategy and sequenced by dependency. To minimize cognitive load, each fact set should represent a single, discrete cognitive strategy (e.g., 'doubles', 'making ten'). These sets should then be sequenced based on their strategic dependencies. Anchor fact sets (like x2 and x5) must be automated before the derived sets that rely on them (like x4 and x6). Once a critical mass of anchor facts and strategies is automated, the final remaining facts are grouped to encourage the rapid selection between already-mastered strategies.

• Cognitive interference between similar-looking facts can be minimized through deliberate set design. In addition to grouping by strategy, research suggests avoiding the inclusion of facts with high surface-level similarity in the same practice set (e.g., practicing 6x7=42 and 6x8=48 together). This reduces the risk of "interference," where the memory of one fact disrupts the retrieval of a similar one, thereby accelerating the path to automation for both.

• Timing is a tool for consolidation, not a performance evaluation; it must be introduced carefully to mitigate anxiety. While timed practice is the necessary catalyst to force the leap from strategy to retrieval, its primary risk is affective (creating math anxiety), not cognitive. To mitigate this, timing must be introduced only after a student demonstrates high accuracy with a strategy. All feedback related to timing must focus on progress and mastery, not on speed as a measure of worth.

• The Commutative Property must be explicitly leveraged to maximize practice efficiency. A curriculum that treats 3x7 and 7x3 as two distinct facts to be learned is inefficient. An effective system must be designed to explicitly reinforce the Commutative Property, ensuring that practice on one form of the fact (e.g., 3x7) contributes directly to the automaticity of the other (7x3). This effectively halves the required practice time for non-square facts.

• Timed practice is the mechanism that forces the leap from strategy to retrieval. To prevent "strategic entrenchment"—where students simply get faster at calculation instead of automating retrieval—practice must be strictly timed. The time limit must be short enough to make multi-step derived strategies non-viable. This controlled time pressure is the essential catalyst that forces the brain to abandon the slower, effortful Phase 2 strategy and build the fast, direct neural pathway required for Phase 3 automatic retrieval.

DOK1-2: Facts

The Cognitive Path to Automaticity

The primary goal of fluency practice is not to teach concepts, but to convert slow, effortful, and working-memory-intensive strategies into fast, effortless, and automatic retrieval from long-term memory. Cognitive science shows this happens in a predictable, three-phase developmental sequence. Our product's role is to provide structured practice that accelerates students from Phase 2 to Phase 3.

• Phase 1: Procedural Strategies (Counting & Modeling). This is the starting point for a child's understanding of an operation. They solve problems by physically or mentally modeling the action, for example by counting on their fingers or drawing groups. This phase is accurate but extremely slow and cognitively demanding. We assume students have largely passed this phase before using our product.

• Phase 2: Derived Fact Strategies (Reasoning & Relating). In this critical intermediate phase, students use a toolkit of reasoning strategies to solve problems. They leverage known facts to solve unknown ones (e.g., solving 6+7 by reasoning from the known double 6+6=12). This is faster than counting but still requires conscious effort and consumes working memory. Our curriculum should be designed to take students who are reliant on these strategies and guide them to the final phase.

• Phase 3: Automatic Retrieval (Automaticity). This is the end goal. The fact is no longer "solved" or "derived"; the answer is retrieved directly from a well-consolidated, long-term memory network, typically in under a second. This process is fast, effortless, and consumes virtually no working memory, freeing the student's cognitive resources to focus on higher-order problem-solving.

• The Commutative Property Halves the Learning Load. A foundational principle of arithmetic is the Commutative Property (a+b = b+a and axb = bxa). By having students understand that 3x8 is the same as 8x3, the number of unique multiplication and addition facts a student must automate is nearly cut in half.

• The Risk of "Strategic Entrenchment". A primary risk in fluency practice is that students can become extremely fast and efficient at executing derived fact strategies (Phase 2) without ever making the cognitive leap to automatic retrieval (Phase 3). For example, a student might get very fast at solving 6+7 by thinking (6+6)+1, but they are still performing a multi-step calculation. This "strategic entrenchment" occurs when practice lacks sufficient time pressure to make the slower, procedural strategy non-viable.

• Subtraction Involves Two Distinct and Asymmetrical Strategies. Unlike addition and multiplication, subtraction is not commutative. Research shows that students naturally develop two different strategies based on the relationship between the minuend and subtrahend:

  • "Counting Back" (or "Take Away"): Used when subtracting a small number (e.g., 10-2). The student starts at the whole and counts backward. This is a procedural, counting-based strategy.
  • "Counting Up" (or "Think-Addition"): Used when the numbers are close together (e.g., 10-8). The student starts at the part and counts up to the whole, reframing the problem as a missing addend problem (8+?=10). This is a more advanced, relational strategy.

Optimal Sequencing for Automating Strategies

• Addition: From Counting to Retrieving

  • Automating +0 (The Identity Rule): Practice begins by automating the Additive Identity Property. While the concept of zero is abstract, the rule ("the number stays the same") is procedurally the simplest in arithmetic. Its consistency allows for rapid automation, even if deep conceptual understanding is still developing.
  • Automating +1 and +2 (The "Counting On" Strategy): Practice next focuses on making the foundational strategy of "Counting On" so fast it becomes automatic. This leverages a child's most basic, concrete understanding of the number line.
  • Automating the "Doubles" as Anchors: The Doubles facts (e.g., 7+7) are practiced next. Because they are memorable and pattern-based, they are the easiest set of non-counting facts to commit to long-term memory. They become the primary "anchor facts" from which other facts are derived.
  • Automating "Near-Doubles" via the Doubles Anchor: Practice then moves to Near-Doubles (e.g., 7+8). The practice is explicitly designed to leverage the now-automatized "Doubles" facts. By repeatedly and quickly executing the procedure (7+7)+1, the brain eventually "chunks" this two-step process into a single, automatic retrieval: 7+8=15.
  • Automating "Making Ten": Practice for this strategy is sequenced later as it's more cognitively demanding. It requires repeatedly practicing the decomposition of a fact (e.g., 9+7 -> 9+1+6 -> 10+6) until the top-level association (9+7=16) is stored and retrieved as a single unit.
  • Automating +9 and +10 (The Place Value Strategies): The +10 facts are practiced to automaticity based on their simple place-value pattern. Then, the +9 facts are practiced using the efficient derived strategy of "add 10, subtract 1."

• Multiplication: From Patterns to Properties

  • Automating the "Rules" and "Easy Patterns" (x0, x1, x10, x5): Practice starts here because these facts either follow simple, memorable rules (Zero and Identity properties) or leverage highly familiar patterns (skip-counting by 10s and 5s). Automating this large block of foundational facts first provides the student with early success and a strong base of known facts.
  • Automating the "Doubles" Family (x2, x4, x8): The x2 facts are practiced first in this group to build a bridge from the known addition "Doubles." Practice then proceeds to x4 and x8. The goal of this sequenced practice is to make the derived strategies ("double-double" for x4, "double-double-double" for x8) so rapid that the original fact (e.g., 8x7) becomes a single, retrieved memory, rather than a multi-step calculation.
  • Automating with the Distributive Property (x3, x6, x9): Practice for these final sets leans heavily on automating derived strategies that use the distributive property. x3 is practiced as "double the number plus one more group" (2n + n). x6 is practiced as "five groups plus one more group" (5n + n). x9 is practiced as "ten groups minus one group" (10n - n). Repeatedly executing these specific, efficient strategies under time pressure is what consolidates the multi-step process into a single, automatic retrieval for each fact.

• Division: From Sharing to Thinking Multiplication

  • Prerequisite: Multiplication Automaticity. Division automaticity is almost exclusively built by leveraging the inverse relationship between multiplication and division. Research and pedagogical consensus show that students achieve fluency by reframing division problems as "missing factor" multiplication problems. Therefore, the practice sequence for division must directly mirror and follow the sequence for multiplication.
  • Automating ÷1 and "Divide by Itself": Practice begins by automating two simple rules: dividing by 1 (Identity Property, e.g., 8÷1=8) and dividing a number by itself (e.g., 8÷8=1).
  • Automating via "Think-Multiplication": The core of division fluency is converting division problems into missing factor problems (e.g., 56÷8 becomes 8x?=56). The practice sequence is therefore designed to automate this conversion, leveraging the fact families that have already been practiced to automaticity during multiplication. The sequence of practice (by ÷10, ÷5, ÷2, etc.) directly follows the multiplication sequence to ensure the anchor facts are in place. For example, a student masters x2 facts, then immediately practices ÷2 facts to automate the inverse relationship.

• Subtraction: From Counting Back to Thinking Addition

  • Prerequisite: Addition Automaticity. Research strongly indicates that students learn very few subtraction facts independently. Automaticity with subtraction is achieved by leveraging the inverse relationship between addition and subtraction. Therefore, the practice sequence for subtraction must mirror and follow the sequence for addition.
  • Automating -0, -1, -2 (The "Counting Back" Strategy): Practice begins by automating the inverse of "Counting On." This is the most basic procedural strategy for subtraction and is practiced first to build initial confidence and success.
  • Automating "Minus Itself" and "Neighbors": Practice on n-n (e.g., 8-8) automates a simple rule. Practicing "Neighbors" (e.g., 9-8) automates the "one difference" rule, which is a simple extension of counting back.
  • Automating via "Think-Addition": The core of subtraction fluency is converting subtraction problems into missing addend problems (e.g., 15-7 becomes 7+?=15). The practice sequence is therefore designed to automate this conversion.
  • Automating with "Making Ten": This is a powerful derived strategy for subtraction. 14-9 is practiced by thinking "first take away 4 to get to 10, then take away 5 more." Repeatedly practicing this decomposition under time pressure automates the top-level fact 14-9=5.
  • Automating with Place Value (-10, -9, -8): Practice for -10 reinforces place value. Practice for -9 and -8 automates a derived strategy that leverages the -10 anchor (e.g., to solve 17-9, think 17-10 is 7, so 17-9 must be 8).

Addendum: Recommended Fact Set Sequences

Multiplication

Addition

Subtraction

Division