Imagine a second-grade teacher pulling a small group of students during her math block and reinforcing concepts from a two-digit addition lesson. She notices several students counting on their fingers to solve single-digit sums as they work on their math problems. She realizes that the work is taking them a bit longer than they should, and by the time they reach the step where they have to regroup in their two-digit addition problems, the students make careless addition mistakes. As a result, the work that students need to complete takes longer than it should.
She quickly notices that her students lack automaticity (the ability to quickly recall their math facts). They are also not demonstrating fluency (the ability to solve problems accurately and within a reasonable time frame). She recognizes what many math teachers immediately recognize as well- that her students require more practice. So she begins to plan math drills every day for her students until automaticity and fluency improve.
Why fluency and automaticity are important
Procedural fluency is described as one of the five strands of mathematical proficiency [https://www.nationalacademies.org/read/9822/chapter/6], and the skill is defined as the “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately”. Within procedural fluency, automatic recall, or automaticity, of basic facts plays an important role in students being considered fluent in mathematical skills. Math fluency progresses through three phases:
Phase 1 is the counting phase, where students have to actually count, for example, to add numbers.
Phase 2 is the reasoning strategy phase, where students will reason their way through a problem, for example, knowing that 9 plus 1 is 10, therefore 9 plus 2 is just one additional, or 11.
Phase 3 is mastery, where students can accurately and automatically produce answers without counting or reasoning.
What Kinds of Math Practice are Important for Fluency?
Students cannot jump from counting to mastery; they still need to master the reasoning phase to become fully fluent in math facts. What this means is that students need to develop number sense and an understanding of how to add, subtract, multiply, or divide numbers; after that, recall and practice of math facts can be incorporated to help students move towards mastery.
After the explicit instruction where students are practicing concepts and reasoning through problems, they can then move into retrieval practice, the strategy of actively recalling information again and again, as well as spaced repetition, the strategy of completing short, frequent practice sessions over a long period of time (https://www.edutopia.org/article/incorporate-spaced-learning-lesson-plans/ ).
Finally, students need to receive immediate and specific feedback to ensure learning sticks and that their practice is accurate and moving them toward automaticity and fluency. [https://growbeyondgrades.org/blog/immediate-feedback]. Taken together, strong math practice looks like:
Explicit instruction focused on the conceptual understanding of math facts
Retrieval practice and spaced repetition to strengthen math fact mastery
Immediate feedback to correct mistakes and misconceptions, and reinforce correct understanding
Explicit Instruction
Explicit instruction by a classroom teacher is considered an incredibly effective way for students to learn new content [https://www.readingrockets.org/topics/curriculum-and-instruction/articles/16-elements-explicit-instruction]. Explicit instruction requires teacher modeling of how to complete a procedure or problem, grouped work, where all students, along with their teacher, work together to complete the procedure, and then independent practice specific to each student. By combining instruction to promote conceptual understanding of math facts with additional tools and resources to support retrieval practice and math fact mastery, students have a clear path to true procedural math fluency. This is where math practice apps come in!
The Model Teaching Math Facts App
The Model Teaching Math Facts App was designed around the research principles mentioned above. Designed to be used after explicit instruction and after conceptual understanding practice, it allows a simple and easy-to-use method of practicing math facts applications covering addition, subtraction, multiplication, and division. It does nothing else, so it is targeted to improve math fluency in students. It is also highly customizable, allowing students to practice exactly the concepts and even the exact numbers that they need to work on. It contains several modes, all able to be customized based on the operation and numbers you wish for students to practice:
Quiz Mode:
One problem at a time, with immediate feedback. This supports retrieval practice and immediate feedback to students so that they can see the correct answer immediately before moving on.
Flashcards:
Flashcards are one of the most studied retrieval practice formats and support long-term retention of math skills.
Workbooks:
Fully customizable, printable worksheets with answer keys provide a low-tech option for practice, which is important for students to limit their interactions with digital products, especially at a young age.
Incredibly simple and focused, the Model Teaching Math Facts App does one thing well: allow students to move toward math fluency mastery in an easy-to-use and fully customizable app. Custom number ranges and operation selection allow teachers and parents to target specific tasks, and will allow complete differentiation for students.
How the Model Teaching Math Facts App Compares to Other Practice Tools
There are many math apps available to both teachers and students, and they offer a variety of functions! Some math apps are comprehensive, focusing on content across the entire student’s grade level, while other math apps offer a narrow and thorough dive into a particular concept. Math facts apps generally fall into two categories. The first is the gamified app, which incorporates math practice with characters, stories, rewards, or animations. The second is apps focused more on school accounts, which require teachers to create accounts, roster students, assign PINs, or track student usage through a teacher dashboard. Both types of apps have their place in the classroom, depending on the teacher, parent, or students’ needs! But these kinds of apps aren’t always the best choice for all students. Gamified apps could potentially slow down practice and redirect students’ attention to the visual stimuli of the game, or shift students’ attention to the features of the game and not necessarily the math itself. Account-based apps could require additional setup time or include additional cumbersome features that a teacher may not always need. The Model teaching Math Facts App is designed for a specific purpose: to allow students to practice addition, subtraction, multiplication, and division math facts in varied ways as they work toward math fluency and mastery. It removes any additional “fluff” or steps, focusing solely on math skills for mastery. If a student or teacher is looking for a specific solution to a classroom problem, our app can provide it. The Model Teaching Math Facts app requires no account and stores no student data. A teacher, parent, or student can open the app and get started immediately. It is fully customizable for the exact operations and numbers a teacher may wish for students to practice. And, most interestingly, it pairs digital practice with customizable and printable worksheets, which is uncommon among free math facts apps.
Considerations and Model Teaching Recommendations
The Model Teaching Math Facts App, like all Model Teaching Apps, is designed as a supplement, not a substitute, to instruction. You can teach operations conceptually before students begin retrieval practice, so that students understand what each operation represents before they begin mastering facts. Explicit instruction in strategies by the classroom teacher is important when students are introduced to new concepts. When a teacher feels students are ready for practice, introduce the Model teaching Math Facts App. But keep sessions short! Research, especially for elementary students, supports brief, focused practice over longer drill sessions. In addition, mindful use of digital technology is an important component of effective teaching and can enhance learning, but not become a focus of how students interact with math concepts. Five to ten minutes of daily practice on the app is sufficient. Finally, pair any digital practice with pencil-and-paper instruction using the worksheet builder feature. Then, pair that independent practice work with additional classroom discussion or other group math activities to allow for varied interaction of math facts within your classroom. You can even recommend the app to your students’ parents, with guidance on how they can practice their math facts digitally or by completing worksheets at home.
Getting Started
The Model Teaching Math Facts App is available at apps.modelteaching.com/mathfacts. No account is required to begin using it. You can use it in three ways:
Worksheets
Practice mode
Flashcards
Simply choose your operation, set your number ranges, and create your practice. It's ready to use within 30 seconds, so you have a ready-to-go tool your students can use immediately.
About Our Apps
Model Teaching's apps are focused digital tools designed to reinforce the great instruction that is already happening in your classroom. We at Model Teaching are pro-instruction and know that the best learning will come from a child's teacher and their parents. Our apps exist to reinforce that instruction with focused practice or targeted instructional support, not to substitute for it. Our apps are built to support teachers in their planning and students in their practice, designed around what research tells us actually works in the classroom. Our position on AI and digital tools is straightforward: they should improve thinking, not replace it. Every Model Teaching app is built with that standard in mind. To explore the full suite, visit apps.modelteaching.com.
References:
Baroody, Arthur J. “Why children have difficulties mastering the basic number combinations and how to help them.” Teaching children mathematics 13.1 (2006): 22-31.
Bay-Williams, Jennifer, and Gina Kling. Math fact fluency: 60+ games and assessment tools to support learning and retention. Ascd, 2019.
Carpenter, Shana K., Steven C. Pan, and Andrew C. Butler. “The science of effective learning with spacing and retrieval practice.” Nature Reviews Psychology 1.9 (2022): 496-511.
Murray, Ewan, Aidan J. Horner, and Silke M. Göbel. “A meta-analytic review of the effectiveness of spacing and retrieval practice for mathematics learning.” Educational psychology review 37.3 (2025): 75.
Ophuis‐Cox, Fieke HA, Leen Catrysse, and Gino Camp. “The effect of retrieval practice on fluently retrieving multiplication facts in an authentic elementary school setting.” Applied Cognitive Psychology 37.6 (2023): 1463-1469.
Powell, Sarah R., et al. “The NCTM/CEC position statement on teaching mathematics to students with disabilities: What’s in it and what’s not.” (2025).
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Math Facts Practice App
This Math Facts app provides guided practice as well as printable worksheets for teachers and parents. Students can complete quizzes, flashcards, and build math fluency and automaticity. Teachers can use the app to design custom worksheets for their students.
Math fact fluency is the ability to solve problems accurately and within a reasonable time frame. Procedural fluency is one of the five strands of mathematical proficiency, defined as the skill of carrying out procedures flexibly, accurately, efficiently, and appropriately. Within fluency, automatic recall of basic facts (automaticity) plays an important role.
Automaticity is the ability to quickly recall math facts from memory. Fluency is the broader ability to solve problems accurately and within a reasonable time frame. A student who lacks automaticity may count on their fingers for single-digit sums, slowing their work on larger problems and producing careless mistakes during regrouping steps in two-digit addition.
Math fluency progresses through three phases. Phase 1 is counting, where students physically count to add numbers. Phase 2 is reasoning, where students reason through a problem (for example, knowing that 9 plus 1 equals 10, so 9 plus 2 equals 11). Phase 3 is mastery, where students accurately and automatically produce answers without counting or reasoning.
Strong math practice looks like this: explicit instruction focused on conceptual understanding of math facts, retrieval practice and spaced repetition to strengthen mastery, and immediate, specific feedback to correct mistakes and reinforce correct understanding. Students cannot jump from counting to mastery; they need the reasoning phase before recall-based practice will move them toward fluency.
The Model Teaching Math Facts App is a focused digital tool that lets students practice addition, subtraction, multiplication, and division math facts after they have received explicit instruction. It is highly customizable, allowing teachers and parents to target the exact operations and number ranges each student needs. It does one thing well: move students toward math fluency mastery.
The app offers three customizable modes. Quiz Mode shows one problem at a time with immediate feedback, supporting retrieval practice. Flashcards is one of the most studied retrieval practice formats and supports long-term retention. Workbooks are fully customizable, printable worksheets with answer keys, providing a low-tech option useful for limiting young students' screen time.
Gamified math apps can slow practice by redirecting attention to characters, stories, rewards, or animations rather than the math itself. Account-based apps add setup time and features many teachers do not need. The Model Teaching Math Facts App removes that fluff, requires no account, stores no student data, and pairs digital practice with printable worksheets, which is uncommon among free math facts apps.
Research, especially for elementary students, supports brief, focused practice over longer drill sessions. Five to ten minutes of daily practice on the app is sufficient. Pair digital practice with pencil-and-paper work using the worksheet builder, and add classroom discussion or group math activities to vary how students interact with the facts.
No. The Model Teaching Math Facts App requires no account and stores no student data. A teacher, parent, or student can open it and start practicing immediately, with no rostering, PIN assignment, or teacher dashboard tracking required.
The Model Teaching Math Facts App is available at apps.modelteaching.com/mathfacts. Choose your operation, set your number ranges, and create your practice within 30 seconds. Use the app after explicit instruction and conceptual understanding practice, and pair it with classroom discussion or group math activities for varied interaction with math facts.
Suggested Citation: Pond, Adam. Building Math Fact Fluency: A Model Teaching App For K-12 Teachers. Model Teaching, 2026.
https://www.modelteaching.com/education-articles/math-instruction/building-math-fact-fluency
