The human brain is a knot of billions and billions of neurons and cells for storing memories. Research tells us 90 percent of information transmitted to the brain is visual, and we process visual information faster than text (Jensen, 2008). Yet too often, we present math as numbers, equations, worksheets, and facts to memorize. Learners need to envision numbers, patterns, sequences, equations, and relationships between numbers to accomplish tasks. For example, the Fibonacci sequence is not just a formula; we often explain it by posing the question, "How many pairs of rabbits will there be in one year?" Visualizing the image (as in Figure 1) of the baby and adult rabbits helps the learner visualize the formula, x<SUBSCRPT>n+1</SUBSCRPT> = x<SUBSCRPT>n</SUBSCRPT> + x<SUBSCRPT>n−1</SUBSCRPT>
Figure 1. Fibonacci Rabbits
So, how does your brain see numbers? Do you envision math as a numbers, formulas, and answers, or do you see groupings and images?
Some learners remember math facts and formulas easily. That's because these learners had good experiences with math and made connections to prior knowledge. By habit, these learners take numbers and regroup, round up, round down, work with base ten, and use estimation as an error check. For example, learners like these see 31 × 13 as simply three steps, all done through visualization.
Figure 2. Visualizing the Problem
Other learners experienced math as facts to memorize through drill-and-kill exercises, practice, and lots and lots of worksheets. Each formula is unique, often with few connections or relationships with other formulas. In the example 31 × 13, learners who don't know who to visualize math need paper to write out the five steps, and need to remember place value, placeholder, and how to carry a number.
Figure 3. Solving a Problem Without Visualization
Math comes easy for learners able to visualize the problem, because at an early age, they envisioned math as chunks of images, which speeds up information processing and gives the brain time to make new connections based on previous knowledge. Other learners must rely on sequential steps to solve a math problem. They follow instructions and struggle to memorize the steps and the rules, but often do not understand or remember the foundation beneath the concept. These learners focus on the answer to the problem, not learning the solution.
So how do we help this second kind of learner? First, listen to him or her to find out how they see numbers. Do they see patterns, sets, parts, number lines, and shapes? The key is to get the learner to envision numbers and manipulate them in different ways so that they make sense. Developing these visualization techniques requires exposing the learner to multiple ways of working with numbers, using manipulatives and items other than digits, so that numbers become fluid and fewer sequences need to be memorized.
Memorization is still necessary, but creating memorable images is crucial. Seeing math as images improves mental capacity, because the brain processes images in long-term memory and anchors them permanently. With text, information is processed in short-term memory, and we only retain small bits of information at a time (Burmark, 2004). To test this concept, look at figure 4. Which did your brain process the fastest?
Figure 4. Processing Images vs. Text
Math Activities
At one time in our lives, we counted on our fingers. This was the start of envisioning math. Now, the five-finger image is implanted in our brains. And according to research done by Jo Boaler, professor of mathematics at Stanford University's Graduate School of Education, this early visualization strategy results in higher math achievement. How might other visualizations help learners see numbers differently? Here are four additional activities to help students envision math.
TouchMath
TouchMath is a good multisensory teaching method for math skills such as number sense and operation skills. The organization'swebsite contains items for purchase and free downloads. TouchMath combines vision, movement, hearing, and touch senses for working with numbers. This method uses numbers that feature touch points. This multisensory approach helps students ground abstract mathematical concepts in concrete experience. TouchMath is not just for early addition and subtraction; it has value in higher math, as well. For example, students learn how to use the pencil-tapping method, visual cues, and skip count to work with concepts like place value, multiplication, division, time, money, fractions, story problems, and pre-algebra.
Figure 5. Example of numbers with reference points
Paper Folding for Whole and Part
Paper circle folding helps students learn about circumference, ratio, area, and fractions. Start with a paper plate, which equals the whole. Fold the circle diameter-axis by touching any two imagined points on the circumference together and crease with a hard-edged tool. This folds the circle in a ratio of two parts of the whole. Fold again opposite diameter-axis resulting in four parts, and so on. Focusing on the folding process that forms the creased circle combines kinesthetic and visual learning to embed these concepts in a learner's memory.
Figure 6. Paper Folding
Number Line
A number line helps students visualize number sequences, counting, comparing, adding, subtracting, multiplying, and dividing. Values on the number line can be whole numbers, negative numbers, or decimals. Ask the student to touch numbers on the number line and at different intervals. For example, ask students to count by 2s, or intervals of 2, along the number line. As students engage in these activities, they build a visual image of the number line and each number's place within the sequence.
Figure 7. Number Line
Playing Cards
The Positive Engagement Project created Acing Math, where students play cards to learn math. The site provides multiple ideas for math games requiring only one deck of playing cards. For example, there are games for skills such as basic operations, fractions, percents, decimals, patterns, and positive and negative integers. These games take advantage of the images and patterns built into normal playing cards.
By helping students envision math, educators tap into the first and best way children learn, while also building a foundation of long-term math knowledge that will help students skillfully persevere as math concepts become more challenging and more abstract.