Mental math isn’t explicitly part of most curricula, but students who can’t answer relatively-simple equations in their heads with speed or automaticity will likely struggle with harder content.
But before answering the question, “How can I improve my mental math?”, it’s helpful to know the definition(s) of mental math.
What is mental math?
The Manitoba Association of Mathematics Teachers defines mental math as:
A combination of cognitive strategies that enhances flexible thinking and number sense. It is calculating mentally without the use of external memory aids. It improves computational fluency by developing efficiency, accuracy, and flexibility.
Or, from students’ perspectives, it’s:
- Math done in your head
- Math that is done in the mind, quickly and efficiently
- Warming up your head with math
- To do math instantly, without the effort put into operations and processes
- Math that you understand so well that you don’t need to write anything down to do calculations/find the answer
Prince Edward Island’s Department of Education believes “mental mathematics must be a consistent part of instruction in computation from primary through the elementary and middle grades.”
Wayne Watts, educator and author of numerous math textbooks, once said: “Number sense cannot be taught. It can only be developed.”
The science behind it is convincing, too.
Research-backed benefits of mental math
For example, an oft-cited study of a 1st grade class found that students who quickly recall addition facts had more cognitive resources to learn other skills and concepts.
In the journal Cogent Education, researchers ran another study with 118 students in 5th grade exploring how mental computation and mathematical reasoning affect each other.
The evidence was fascinating:
[There] is a significant positive correlation between mental computation and mathematical reasoning. It is noteworthy that rather than exposing students to familiar classical problems, students need to be enabled to deal with exceptional/non-routine problems, and especially young children should be encouraged to do mental computing in order for developing both skills.
Duke researchers published a study in Clinical Psychological Science about mental math — from a health perspective.
After brain-scanning 186 undergraduates, results suggested engaging the brain’s prefrontal cortex during mental math exercises is linked to better emotional health.
Thankfully, you’re already helping students build core mental math skills when you teach rounding, estimating and fact fluency — developing number sense, as well as how they remember and reproduce steps and solutions.
It’s mental math practice time!
To improve how your students build and practice these mental math skills, try the 12 strategies below. Use the ones that best work for you, and keep the downloadable list at your desk for quick reference.
1. Introduce mnemonic devices
Students who struggle with basic fact fluency can improve by using mnemonic devices — cues such as rhymes and acronyms to help recall information.
In her master’s thesis, Teaching Through Mnemonics in Elementary School Classrooms, Arianne Waite-McGough found teachers understand the positive impact this device can have on students within and “beyond the classroom walls.”
Current research shows that singing, moving and overall enjoyment of a subject enhances the learning process and long term recall of material. All of these requirements are present when using mnemonics in the classroom. My research proved similar findings. All of the teachers that I surveyed noted higher levels of learning, engagement and fun while singing songs based on the core content material.
Take this mnemonic device for a multiplication fact as an example: I need to be 16 years old to drive a 4×4 pickup truck.
Because they must be easy to remember, it helps if the cues involve:
- Tangible objects or scenarios
- Quick stories that distill larger chunks of information
Although you can think of mnemonic devices yourself and share them with students, it’s beneficial if you run an activity that gets them to make their own.
They’ll likely find it easier to remember mnemonic devices they create.
2. Read math books
There are many math books that effectively contextualize the processes behind solving equations, helping students commit them to memory.
Depending on student age, consider:
- Each Orange Had 8 Slices — This book focuses on counting and addition, presenting problems in easy-to-process sentences. It sets a new scene, complete with questions, with each turn of the page.
- The Grapes of Math — Containing basic multiplication problems, this book is a series of illustrated riddles. Each riddle offers clues and secrets to solving a specific equation, helping students improve reading comprehension along with math skills.
- Sir Cumference — Set in medieval times, this book series focuses on measurement and geometry. With occasional help from his son and wife, Radius and Lady Di of Amater, the knight Sir Cumference must solve math-related challenges that pose threats to his family and kingdom.
- Secrets of Mental Math — As opposed to a children’s book, this guide promises to “have you thinking like a math genius in no time” with the help of “mathemagician” Arthur Benjamin. Since it’s 200+ pages long, you might find more success in selecting key excerpts and reading — and applying — the mental math tricks with your students. There’s also a foreword by Bill Nye the Science Guy!
As you read books out loud, your students can practice their mental math. Alternatively, you may use books as a way to leverage the benefits of peer teaching.
Just pause after identifying an equation, giving them time to work through the problems in their heads. After they share their responses, read on to discover the answer.
3. Provide relevant word problems
Many students will be more receptive to math drills and practice if the material is engaging.
David Kember, a professor in curriculum methods and pedagogy, and his team published an article in Active Learning in Higher Education about the motivators of student learning.
Upon interviewing 36 undergraduate students, Kember concluded:
Teaching abstract theory alone was demotivating. Relevance could be established through: showing how theory can be applied in practice, establishing relevance to local cases, relating material to everyday applications, or finding applications in current newsworthy issues.
In other words, if students don’t find your math lesson relevant, their motivation to learn will greatly diminish.
A straightforward, yet effective, way of enlivening content is by creating math word problems. This is because you can tailor questions to students.
For example, you can:
- Reference Student Interests — By framing your word problems with student interests, you should grab attention. If most of your class loves baseball, a measurement problem could involve the throwing distance of a famous outfielder. Using cross-cultural and cross-curricular connections with help strengthen students’ neural loops.
- Make Questions Topical — Word problems based on current events or issues can engage students by providing clear, tangible ways to apply knowledge. Not only will students find your lessons more interesting, they’ll believe it’s worth knowing.
- Include Student Names — Naming a question’s characters after your students is an easy way make it relatable, motivating your class to tackle the problem.
By capturing interest, student motivation should increase when practicing skills important for mental math.
Note: If they struggle with world problems, teach the mnemonic, “STAR”:
Search the word problem
Translate the words into an equation
Answer the problem
Review the solution
4. Play estimation games in class
Estimation games are fun math activities that encourage students to develop skills and techniques they can use to simplify equations in their heads.
Easy to run but challenging to play, a popular estimation game in many classrooms involves only two dice and a sheet of paper that’s divided into two columns. One column lists the values on each dice face, whereas the other contains numbers of your choosing.
To play, pair students together. Taking turns rolling the dice, they must add the corresponding numbers together in their heads. For example, if a student rolls five and six, the equation is 878 + 777. Without pencil, paper or calculator, the student must solve the equation. If he or she is within a range of five numbers — verifying the solution with a calculator — the answer is considered correct.
The first student to answer five questions right wins.
For more advanced classes, you can simplify the numbers but require multiplication instead of addition.
5. Play fact fluency games in class
A fun alternative to flashcards, fact fluency games allow students to build recall and reproduction skills important for mental math.
Engaging options for 1st to 8th grade classes include:
- Math Facts Bingo — Create bingo cards that contain answers to different equations. Then, hand them out to students. Instead of calling numbers, state equations such as 8 x 7. After determining the product is 56, they can check off the number if it’s on their cards.
- Stand Up, Sit Down — Pick a number and share it with students. Then, read equations out loud. Sitting in a circle, students must stand if the answer matches the number you picked. If they incorrectly stand or remain seated, eliminate them until one student remains.
- 101 and Out — As the name implies, the goal is to score as close to 101 points as possible without going over. Start by dividing your class into groups, giving each a die along with paper and a pencil. Groups take turns rolling the die, deciding if it’s best to count the number at face value or multiply it by 10. After each roll, the number is added to the group’s total. The game ends when a group hits 101 points or goes over — whichever comes first.
As skill-building as they are engaging, your students’ improvement in fact fluency should be clear after playing a few rounds of these math games.
6. Encourage the use of math apps and websites
An alternative or supplement to drills and worksheets, consider using a digital program that features a range of problems aligned with different skills.
Such math apps and websites prompt students to continuously answer questions in an often-engaging environment, building a range of skills important for mental math.
Popular options include:
- Prodigy Game — Free and aligned with curricula across the English-speaking world, Prodigy automatically differentiates content and gives adaptive feedback tailored to each student. Teachers can also make in-game assignments to deliver custom content, making it a favorite of more than a million educators.
- NRICH — An ongoing project by the University of Cambridge, this website features math games, articles and problems. It divides resources by United Kingdom key stages and United States grade levels, allowing your students to easily access the right content.
- Math Is Fun — This website contains content suitable for younger students, using concise sentences and cartoon characters. On top of exercises that cover essential math skills, there are games and puzzles.
Because all students need is a computer or mobile device to use these programs, it’s likely some will voluntarily practice at home.
7. Round up when multiplying by 9
There are simple ways to alter difficult equations, making them easier to solve with mental math.
Students can use existing rounding and fact fluency skills when multiplying by 9, 99, 999 and any number that follows this pattern.
First, tell students to round up the 9 to 10. Second, after solving the new equation, teach them to subtract the number they just multiplied by 10 from the answer.
For example, 67 x 9 will lead to the same answer as 67 x 10 – 67. Following the order of operations will give a result of 603. Similarly, 67 x 99 is the same as 67 x 100 – 67.
Despite more steps, altering the equation this way is usually faster and allows students complete it in their heads.
8. Double and halve
When mastering multiplication beyond basics, students can quickly use mental math skills to multiply two integers when one is an even number.
They just need to halve the even number and double the other number. They stop this process when the even integer cannot be halved, or when the equation becomes manageable.
Using 33 x 48 as an example, here’s the process:
- 33 x 48
- 66 x 24
- 132 x 12
- 264 x 6
- 528 x 3
The only prerequisite to this mental math trick is understanding the 2-times table.
Normally used as an intervention tactic, Cover-Copy-Compare can have a place in most fact fluency lessons.
There are three steps to this mental math practice, which are:
- Creating a Math Fact Sheet — Divide a sheet into two columns, writing about 10 math facts pertaining the same skill in the left column. Include number-sentences and answers. In the right column, write “Responses.” Distribute copies of the sheet to students.
- Running the Exercise — The goal for students is to study the math facts in the left column, correctly reproducing them in the “Responses” column. To do so, give them time to study the facts. After, they fold the paper to cover the left column while writing — from memory — the first fact in the “Responses” column. If correct, the student can move onto the next fact. If incorrect, the student tries again until he or she has properly reproduced the math fact.
- Recording Mastered Skills — Once a student has completed a certain number of sheets related to a common skill, you can award him or her a badge that denotes skill mastery. This gamification strategy can make the exercise more engaging.
To go beyond basic fact fluency, you can make sheets that focus on rounding, memorizing steps to complex equations and more.
10. Use the Taped-Problem approach
First, obtain or make an audio recording of basic math problems that has short pauses between stating the problem and revealing the answer. Second, provide each student with a pencil and paper.
As you play the recording, students must write out each equation and try to solve it before the answer is revealed. If the student cannot solve the question, he or she writes down the correct answer. If the student reaches an incorrect answer, he or she crosses it out and writes the right response.
You can lengthen the pauses so students don’t depend on hearing the answers, whereas you can shorten them to encourage automaticity.
11. Building blocks
Wondering how to improve mental math speed across your class? Familiarize students with building blocks such as multiplication tables or fractions, decimal, and percent equivalents.
The more your students become familiar how multiplication tables or equivalents look, the quicker they’ll be able to recognize and solve problems in and outside the classroom.
A study in the Journal of Neuroscience titled, “Why mental arithmetic counts: Brain activation during single digit arithmetic predicts high school math scores”, tested 33 high school students on their ability to solve addition and subtraction equations.
All of them performed well, which correlated to their math PSAT scores. Interestingly, as neurobiologist Dr. Susan Barry outlined:
Those students with higher math PSAT scores engaged parts of the brain, the left supramarginal gyrus and bilateral anterior cingulate cortex, which have been associated with arithmetic fact retrieval. In contrast, those students with lower math PSAT scores engaged the right intraparietal sulcus, a region involved with processing numerical quantity. In completing the test in the scanner then, students with the higher math PSAT scores relied more on their memory of arithmetic facts.
12. Number talks
Ruth Parker, the CEO of the Mathematics Education Collaborative, and Kathy Richardson, one of the nation’s leading educators of elementary mathematics, developed this mental math practice.
To start, pose an abstract math problem. Take 18 x 5 as an example problem and ask your students to try and solve it in their heads.
Naturally, in a class of 20+ students, you’ll likely find they answered correctly — but differently.
|Five Ways to Solve 18 x 5|
|20 x 5 = 100
2 x 5 = 10
100 – 10 = 90
|10 x 5 = 50
8 x 5 = 40
50 + 40 = 90
|18 x 5 = 9 x 10
9 x 10 = 90
|18 x 2 = 36
2 x 36 = 72
18 + 72 = 90
|9 x 5 = 45
45 x 2 = 90
Number talks are a perfect way to illustrate that there’s creativity in math. They’re also a great way to begin your math lesson or encourage parents to do with their kids!
In the article “Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts”, professor of mathematics education and co-founder of Stanford University’s youcubed, Jo Boaler, writes:
Research tells us that the best mathematics classrooms are those in which students learn number facts and number sense through engaging activities that focus on mathematical understanding rather than rote memorization.
So, we trust these activities will help your students’ mental math practice this school year and beyond.
Downloadable list of the mental math practices
Click here to download and print a simplified list of the 12 mental math practices to keep at your desk.
Ready to share these mental math secrets?
Okay, they’re not really secrets. But using these mental math practices should help your students build rounding, estimating and fact fluency skills — allowing them to solve many equations with ease and automaticity, preparing them to tackle tougher content.
Armed with increased confidence, you may notice an uptick in student engagement and motivation.
These benefits, in and of themselves, make a strong case for practicing mental math.
>>Create or log in to your teacher account on Prodigy — a free platform that helps students build mental math skills by practicing in an engaging game-based learning environment. Aligned with curricula across the English-speaking world, it’s loved by more than a million teachers and 50 million students.