By Diane Brauner on December 12, 2019

Tactile number lines can be very beneficial to building strong number sense for students who are visually impaired. In the first number line post, we discussed teaching *basic* number line concepts and activities. The second number line post focused on using the number line to teach addition and subtraction. The third number line post introduced an accessible digital number line app, Cosmic Numbers. This post will focus on how students can *apply* their number line skills and how number lines can be used to reinforce additional skills.

## Importance of Number Line Concepts

Teachers of the Visually Impaired struggle with prioritizing student goals and balancing pull out time and time in class. In today's classroom environment, emphasis is given to tasks that will be on the high-stakes exams. Sometimes, the end goal is considered and not the concepts that are taught before the end goal. Example: The end goal might be to master math facts; however, the number sense taught through number lines might not receive priority time or can even be overlooked. When these number sense concepts are not fully taught and comprehended, students are apt to have gaps in their math skills, which may not show up until more complex, higher ed math. For students who are visually impaired, these gaps can become a serious problem in high school level math classes. It is critical that students with visual impairments have opportunities to experience and embrace foundational math concepts along with the ability to independently think through math problems and knowing which math tools to pull from their math toolbox.

Sometimes educators might assume that a number line is "too visual" for students who are visually impaired. "Visualizing" does not necessarily mean that the student has to "see" with his eyes - students with visual impairments need to be given tactile materials under their fingers that help build the same concepts. Students like Logan demonstrate that touching is simply a different way of "visualizing"; visually impaired students use a different sensory channel - they substitute touch for vision. Basically, "visualizing" is *remembering* what was seen or touched.

## When to Use a Number Line

When students are first introduced to the number line, they are learning about the physical layout of the number line, how to find a point or place a point on the number line, and other basic concepts. Once the student understands these basic things, he is now ready to use the number line to help solve math equations. Students learn to use and practice math tools during set activities in the classroom; however, students also need opportunities to figure out **when **to use the tool! Having a number line allows students to figure out calculations beyond memorized math facts. Math facts, although important, don't represent the size of numbers or help a student make sense of a word problem. When students understand number line concepts, they have a reliable tool that they can use to check their strategies.

In this video, Logan (first grade) is sitting a table with a brailled math worksheet; there is an APH tactile number line adhered to the desk edge closest to Logan. The current word problem starts with 9 birds and 3 birds flew away. While wiggling in his seat, Logan reads the problem and first tries to mentally figure out the answer; he tentatively whispers 6. He turns sideways in his seat and with his index finger in the air, he 'air counts' (bounces his finger three times moving from the right to the left) and then says, "Now there . . . were . . .." Although he has a mental image of a number line and knows to move to the left, he seems lost for a second. He realizes he needs to look at the number line and he turns and loudly drops his fingers on the APH number line. Logan says, "I'm going to start on 9 and subtract 3." However, he actually started on the number five (Note: the braille numbers 5 and 9 are reversals) and counted three tick marks to the left. Logan used both hands to count the tick marks, then leaving his right index finger on the last tick mark, he dropped his left index finger to find the braille number. Jessica allowed him to count over three, then prompted, "Did you start on 9? Double check your 9." Logan confidently used both hands to quickly read the braille numbers, stopping on 9. He moved both hands up to the tick mark, then the left hand led/guided while the right index finger counting the tick marks; Logan used three fingers of each hand to track across the tick marks. To confirm, Logan quickly and efficiently scooted his fingers back to check the number 9 and recounted the tick marks.

Logan initially did not know the answer to 9 minus 3 (or was not confident in his answer); however, he chose to use the number line to figure out/confirm the answer. Jessica shared that this was a transition point for Logan as he was really learning that there is just one correct answer. The number line, a concrete kind of imagery, helped him realize that there could only be one exact answer. In the video above, Logan used the air gesture of subtracting on a number line, demonstrating that he has built a mental "visualization" of a number line!

## Long Number Line

This activity uses numbers from 0 - 500 (counting by tens) to create one long strip of numbers. The numbers are in braille with the print number transcribed above. Cut the braille paper in narrow strips - about 2" high - and put the strips together to form one long strip. Adhere the strip of numbers to the edge of a long counter top in the classroom. The student literally walks to the right as he reads the number line - reinforcing the concept of a "long" number line.

In the video below, Logan is standing and walking as his hands read the brailled numbers. Logan did start at the beginning of the number line; the video begins at 450. Logan uses both hands on the braille as he reads aloud. Saying the numbers aloud while reading them and taking small sideways steps is a great way to build number sense!

Note: The students are taught to call out the numbers without "and". Example: "four hundred fifty" not "four hundred and fifty".

Long Number Line Video (Counting by Tens to 500)

Logan's math teachers have commented that the entire class has benefited from many of the tactile resources and activities (such as the physical long number line above). In an effort to help Logan, these wonderful teachers took a deep dive into the real concepts being taught and their classroom teaching methods have expanded these concepts - which in turn has helped all their students have a deeper understanding of these concepts!

## Reversals

Jessica, TVI extraordinaire, shared her observation about braille reversals. It is developmentally expected that all children have reversals which they usually work out. Braille students have to develop spatial orientation before fully understanding and identifying braille reversals. Braille has opportunities for many reversals including the letters, "d", "f", "h", and "j", the numbers "0", "4", "6" and "8" and the punctuation symbols "period", "exclamation mark", and "quotes". These braille symbols all have three dots in the same formation, just turned in a different vertical or horizontal plane.

Rather than drilling these braille reversals in isolation, use activities that provide context clues to help the student determine the braille symbol. Do not focus on identifying letters in isolation but rather use reading passages and CVC words that include the braille letter. Students like Logan become frustrated when unable to identify isolated braille letters; this frustration often led to avoidance behaviors. As educators, we want to set our students up for success not failure! By providing context clues, the student is more successful in figuring out the letter - these activities provide "errorless learning" opportunities. The number line can also provide context clues for braille number reversals. When given the - to 10 number line, Logan instantly determined that he was on the far right of the number line, so the number must be a "9" and not a "5". In this case, he used the spatial clue/location to help identify the number.

## Resources

Logan's class is using Bridges Math from the Math Learning Center. Want to dive deeper into number lines? In this Math Learning resource, *The Learning to Think Mathematically with the Number Line a Resource for Teachers, a Tool for Young Children,* Jefrrey Frykholm, PhD., spells out the "reasons for developing the number line as a foundational tool".