While mental math may not be a formal component of many educational curricula, students who lack the ability to swiftly and automatically solve relatively straightforward equations in their minds are likely to encounter difficulties when tackling more complex mathematical concepts.
However, before addressing the query of “How can I enhance my mental math skills?” it is valuable to understand the various definitions of mental math.
From the perspective of students, mental math can be defined as:
- Mathematics calculations performed mentally.
- Swift and efficient mental calculations.
- Preparing your mind for mathematical tasks.
- Instantaneous math without the need for written steps or processes.
- Proficiency in math to the extent that written calculations are unnecessary.
The Department of Education in Prince Edward Island emphasizes the importance of integrating mental mathematics consistently into computational instruction from primary through elementary and middle school grades.
Wayne Watts, an educator and author of numerous math textbooks, once stated, “Number sense cannot be taught. It can only be developed.”
There is substantial scientific support for the benefits of mental math. For instance, a well-cited study involving a 1st-grade class revealed that students with rapid recall of addition facts had more cognitive resources available for learning other skills and concepts.
In the journal Cogent Education, researchers conducted a study with 118 5th-grade students to explore the relationship between mental computation and mathematical reasoning. The findings were intriguing, showing a significant positive correlation between mental computation and mathematical reasoning. Additionally, students were found to benefit from tackling non-routine problems and engaging in mental computation, particularly young children.
Duke researchers published a study in Clinical Psychological Science that examined mental math from a health perspective. After brain-scanning 186 undergraduates, the results suggested that engaging the prefrontal cortex during mental math exercises is linked to improved emotional health.
Fortunately, you are already aiding students in developing fundamental mental math skills when you teach concepts like rounding, estimating, and fact fluency. These approaches contribute to the development of number sense and enhance their ability to remember and apply steps and solutions.
Now, it’s time to delve into mental math practice! To improve how your students build and hone these mental math skills, explore the following 12 strategies and select the ones that align best with your teaching approach.
1. Introduce mnemonic devices
Students who encounter challenges in acquiring basic fact fluency can make significant improvements by employing mnemonic devices. These devices involve using cues like rhymes and acronyms to facilitate the recall of information.
In her master’s thesis titled “Teaching Through Mnemonics in Elementary School Classrooms,” Arianne Waite-McGough discovered that teachers recognize the positive impact of these devices on students, both inside and “beyond the classroom walls.”
Contemporary research underscores the notion that activities such as singing, movement, and overall enjoyment of a subject enhance the learning process and contribute to the long-term retention of material. Mnemonics in the classroom fulfill all these requirements. My research yielded similar results, with all surveyed teachers noting increased levels of learning, engagement, and enjoyment when incorporating songs based on core content material.
Consider this multiplication fact mnemonic as an example: “I need to be 16 years old to drive a 4×4 pickup truck.”
For mnemonic cues to be effective, they should ideally incorporate:
- Rhymes
- Tangible objects or relatable scenarios
- Brief narratives that distill larger chunks of information
While you can certainly devise mnemonic devices yourself and share them with students, it can be highly advantageous to organize activities that encourage students to create their own mnemonic devices. They are likely to find it easier to remember these devices when they are actively involved in their creation.
2. Read math books
Numerous mathematics books are available that effectively contextualize the processes involved in solving equations, aiding students in committing these methods to memory.
Depending on the age of your students, consider the following titles:
- “Each Orange Had 8 Slices” – This book focuses on counting and addition, presenting problems in easily digestible sentences. It introduces new scenarios and questions with each page turn.
- “The Grapes of Math” – Featuring basic multiplication problems, this book presents a series of illustrated riddles. Each riddle offers hints and insights into solving specific equations, simultaneously enhancing reading comprehension and math skills.
- “Sir Cumference” – Set in medieval times, this book series delves into measurement and geometry. With occasional assistance from his son, Radius, and his wife, Lady Di of Amater, the knight Sir Cumference must tackle math-related challenges that threaten his family and kingdom.
- “Secrets of Mental Math” – While not a children’s book, this guide promises to transform readers into “math geniuses” with the guidance of “mathemagician” Arthur Benjamin. Given its length of over 200 pages, you may find it more effective to select key excerpts and read, as well as apply, mental math techniques with your students. Additionally, it features a foreword by Bill Nye the Science Guy!
During read-aloud sessions, your students can practice their mental math skills. Alternatively, you can utilize books as a means to leverage the benefits of peer teaching. Simply pause after identifying an equation, allowing them time to work through the problems mentally. After they share their responses, continue reading to unveil the correct answers.
3. Provide relevant word problems
To make math drills and practice more appealing to students, it’s crucial to engage them with the material. David Kember, a professor specializing in curriculum methods and pedagogy, and his research team, explored the factors motivating student learning, as published in the article “Active Learning in Higher Education.”
Through interviews with 36 undergraduate students, Kember’s findings highlighted that teaching abstract theory in isolation can be demotivating. Relevance in mathematics education can be established by demonstrating how theory applies in real-world scenarios, connecting it to local cases, relating it to everyday applications, or linking it to current newsworthy issues.
In essence, if students cannot perceive the relevance of a math lesson, their motivation to learn is likely to wane.
A straightforward yet effective approach to infusing vitality into math content is by creating math word problems. This method allows you to customize questions to suit your students’ interests. For example, you can:
- Reference Student Interests: Crafting word problems that align with your students’ interests can capture their attention. For instance, if your class has a passion for baseball, you could create a measurement problem involving the throwing distance of a famous outfielder. Establishing cross-cultural and cross-curricular connections can further strengthen your students’ engagement.
- Make Questions Topical: Designing word problems based on current events or contemporary issues provides tangible ways to apply mathematical knowledge. This not only makes your lessons more intriguing but also convinces students of the practical relevance of what they’re learning.
- Include Student Names: Personalizing word problems by naming the characters after your students can make the problems relatable, motivating your class to tackle them with enthusiasm.
By piquing their interest and aligning the content with their lives, you can enhance student motivation when practicing essential skills for mental math.
4. Play estimation games in class
Estimation games are enjoyable math activities designed to encourage students to develop skills and techniques that simplify equations mentally.
A popular estimation game, both easy to set up and challenging to play, only requires two dice and a sheet of paper divided into two columns. One column lists the values on each die face, while the other contains numbers of your choice. Here’s an example:
[Image: Example Image]
To play, students are paired up. They take turns rolling the dice and then mentally add the corresponding numbers together. For instance, if a student rolls a five and a six, the equation becomes 878 + 777. Without using pencil, paper, or a calculator, the student must solve the equation. If their answer falls within a range of five numbers (which can be verified with a calculator), it’s considered correct.
The first student to provide five correct answers wins the game.
For more advanced classes, you can simplify the numbers but require multiplication instead of addition for an added challenge.
5. Play fact fluency games in class
Fact fluency games offer an enjoyable alternative to traditional flashcards, allowing students to develop essential recall and reproduction skills crucial for mental math.
Here are engaging options suitable for 1st to 8th grade classes:
- Math Facts Bingo: Create bingo cards featuring answers to various equations and distribute them to students. Instead of calling out numbers, present equations such as 8 x 7. When students determine that the product is 56, they can mark off the corresponding number on their cards.
- Stand Up, Sit Down: Choose a number and share it with your students. Then, read equations aloud. Students sit in a circle and must stand up if the answer matches the number you selected. If they stand or remain seated incorrectly, eliminate them until only one student remains standing.
- 101 and Out: In this game, the objective is to score as close to 101 points as possible without exceeding that total. Divide your class into groups, providing each with a die, paper, and a pencil. Groups take turns rolling the die and decide whether to use the number at face value or multiply it by 10. After each roll, they add the number to their group’s total. The game concludes when a group reaches 101 points or exceeds it, whichever happens first.
These math games not only engage students but also enhance their fact fluency skills. You should notice a clear improvement in your students’ fact fluency after a few rounds of these enjoyable activities.
6. Encourage the use of math apps and websites
As an alternative or valuable supplement to traditional drills and worksheets, consider incorporating digital programs that offer a diverse range of problems aligned with various mathematical skills.
These math apps and websites encourage students to engage with a continuous stream of questions within an often immersive and interactive environment, fostering the development of skills essential for mental math.
Here are some popular options:
- Prodigy Math: Prodigy Math is designed to align with math curricula from English-speaking regions worldwide. It automatically adapts content and provides personalized feedback for each student. Teachers have the flexibility to create custom in-game assignments to deliver tailored content.
- NRICH: NRICH is an ongoing project by the University of Cambridge, offering math games, articles, and problems. It categorizes resources according to United Kingdom key stages and United States grade levels, making it easy for your students to access the appropriate content.
- Math Is Fun: This website is particularly suitable for younger students. It presents content using concise sentences and engaging cartoon characters. In addition to exercises covering fundamental math skills, it offers games and puzzles to make learning enjoyable.
Since these programs require nothing more than a computer or mobile device, some students may willingly practice at home, enhancing their math skills independently.
7. Round up when multiplying by 9
There are simple techniques for simplifying complex equations to make them more amenable to mental math.
Students can leverage their existing rounding and fact fluency skills when dealing with multiplication by numbers like 9, 99, 999, and others that follow this pattern.
Here’s a straightforward method:
- Instruct students to round up the 9 to 10.
- After solving the new equation, teach them to subtract the number they just multiplied by 10 from the answer.
For instance, consider the equation 67 x 9. This can be approached as 67 x 10 – 67. Following this method, the result is 603, which can often be computed more swiftly and mentally.
Similarly, for an equation like 67 x 99, students can simplify it to 67 x 100 – 67. Despite involving more steps, altering the equation in this manner typically allows students to complete it mentally with greater ease.
8. Double and halve
When advancing their multiplication skills beyond the basics, students can employ a swift mental math technique for multiplying two integers when one of them is an even number.
The method is straightforward: they halve the even number and double the other number repeatedly until the even integer can no longer be halved, or until the equation becomes more manageable.
Here’s an illustration using the example of 33 x 48:
- 33 x 48
- 66 x 24
- 132 x 12
- 264 x 6
- 528 x 3
- 1,584
This mental math approach relies solely on an understanding of the 2-times table and can greatly simplify the multiplication process.
9. Cover-copy-compare
While typically used as an intervention strategy, the Cover-Copy-Compare technique can be a valuable addition to most fact fluency lessons. This method consists of three steps:
- Creating a Math Fact Sheet: Begin by dividing a sheet into two columns. In the left column, write down approximately 10 math facts related to the same skill, including number sentences and their answers. In the right column, label it as “Responses.” Distribute copies of this sheet to your students.
- Running the Exercise: In this step, students are tasked with studying the math facts in the left column and then accurately reproducing them in the “Responses” column. Allow them some time to thoroughly review the facts. Afterward, instruct them to fold the paper to cover the left column and write, from memory, the first fact in the “Responses” column. If they get it right, they can move on to the next fact. If not, they should continue trying until they have correctly reproduced the math fact.
- Recording Mastered Skills: Once a student has completed a certain number of sheets related to a particular skill, you can recognize their achievement by awarding them a badge signifying skill mastery. This gamification strategy can add an element of engagement to the exercise.
To extend this technique beyond basic fact fluency, you can create sheets that focus on various skills such as rounding, memorizing steps for complex equations, and more.
10. Use the Taped-Problem approach
The taped-problem approach is a highly effective active learning strategy for enhancing students’ fact fluency, as highlighted by a pioneering 2004 study.
Here’s how to implement this strategy:
- Prepare the Audio Recording: Create or obtain an audio recording featuring basic math problems. Ensure that there are brief pauses between stating each problem and revealing the answer.
- Equip Students: Provide each student with a pencil and paper.
- Engage in the Activity: Play the recording for the students. Their task is to write out each equation and attempt to solve it before the answer is disclosed. If a student cannot solve the problem, they should write down the correct answer. If they initially arrive at an incorrect answer, they should cross it out and replace it with the correct response.
You have the flexibility to adjust the length of the pauses in the recording. Longer pauses can discourage students from relying on hearing the answers, while shorter pauses can promote quicker mental computation and automaticity.
11. Building blocks
If you’re looking to enhance mental math speed throughout your class, consider introducing students to foundational concepts like multiplication tables or the relationships between fractions, decimals, and percentages.
As students become increasingly acquainted with multiplication tables or equivalent representations, they will develop a swifter ability to recognize and resolve problems both within and outside the classroom.
A study published in the Journal of Neuroscience, titled “Why Mental Arithmetic Counts: Brain Activation During Single-Digit Arithmetic Predicts High School Math Scores,” examined 33 high school students to evaluate their proficiency in solving addition and subtraction equations.
12. Number talks
Ruth Parker, CEO of the Mathematics Education Collaborative, and Kathy Richardson, a prominent educator in elementary mathematics, developed this mental math exercise.
To initiate the activity, present an abstract math problem, such as 18 x 5, and challenge your students to solve it mentally.
Naturally, in a class with more than 20 students, you’ll likely observe that they arrive at correct answers in various ways.
Number talks serve as an excellent means to illustrate the creativity inherent in mathematics. They also serve as an effective way to kickstart your math lessons or encourage parents to engage in mathematical discussions with their children.
In an article titled “Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts,” Jo Boaler, a professor of mathematics education and co-founder of Stanford University’s youcubed, emphasizes:
“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.”
Hence, we trust that these activities will significantly contribute to your students’ mental math proficiency in the current school year and beyond.
