Osterhaus, S.A. (2004). Susan's math technology corner: Secondary Mathematics Education: The Years of Growth and Challenge.* Division on Visual Impairments Quarterly*, 50(1), 17-20.

How I wish that someone would ask me the question: How do you teach secondary mathematics to **sighted** students? Then I could reply: The same way I teach **blind** students! I do not mean that each individual student should be treated exactly the same. Each student is unique, but all students need access each year they are in school to a coherent, challenging mathematics curriculum that is taught by competent and well-supported mathematics teachers. (NCTM, 2000) I strive to appeal to as many senses as possible, so I encourage all of my students to read, speak, listen to and look at, touch and feel, sing, and sometimes even smell and eat mathematics - basically completely immerse themselves in the problem at hand. In my experience the more in-roads math concepts have to access the brain, the more likely your student will be able to out-put a correct solution to a problem and transfer that knowledge when learning a new concept. Too many students: especially students who are poor, not native speakers of English, disabled, female, or members of minority groups - are victims of low expectations in mathematics. (NCTM, 2000) Unfortunately this has often been the case with the majority of blind and visually impaired students, who fall under this umbrella - frequently in several categories. Classroom mathematics teachers must provide high expectations for **all** their students, and they should be strongly supported by staff trained in the special needs of students with visual impairments.

## A Few Secondary Math Education Links to Get You Started

Sources for quality manipulatives and other math materials:

- http://www.delta-education.com
- http://www.didax.com
- http://www.etacuisenaire.com
- K-12 math and science teacher center. http://www.enc.org/
- Math Forum - Online resource for K-12+ math teachers: Ask Dr. Math, discussion groups, Internetnewsletter, Teacher2Teacher, search engine, and more. http://mathforum.org
- TSBVI's math forum.
- http://www.nctm.org/
- Science Access Project at Oregon State University - Latest research on making math accessible to people with print disabilities. http://dots.physics.orst.edu
- Teaching strategies, resources (more vendors, people, and places links, math packet, videotapes, publications, Nemeth reference sheets, and Nemeth tutorial downloads), suggested adaptive tools and technology, and more. Math Resources.

## What major challenges are encountered when teaching math concepts to blind and visually impaired students?

One of the most difficult challenges for me has been teaching concepts involving three-dimensional objects. When I first did my student teaching (over 35 years ago), I taught geometry in a regular education classroom. My nickname was The Tinker Toy Lady because I was always coming to class with some kind of physical 3-D model to illustrate the day's lesson. 3-D problems are found in all levels of mathematics. They are often difficult for students with vision to understand, especially when trying to create 3-D objects in a two-dimensional drawing. Hey! They are difficult for me!! Such a drawing, even when tactually raised, makes little sense without sighted perspective. Yet, the textbooks continue to draw these 3-D raised line drawings that seem to contradict what the math teacher has just taught the student. For example, a teacher may have just explained to a student that a cylinder has two bases, which consist of two congruent circles and their interiors, and let them examine several real cylinders. Then, when the homework is assigned or the test is administered, they are given a two-dimensional drawing that would seem to indicate that a cylinder only has one base, which consists of an ellipse and its interior. Sometimes my students would be better off without the picture. Whereas, it may help the sighted student, it often causes confusion for the blind student. In addition, the blind student has to learn what the 3-D object really feels like, and then what it feels like as a sighted person would see it. Talk about extra work!

The next most immediate challenge is keeping up with the advancement in math technology tools for the sighted. The scientific graphing calculator is now becoming a requirement for coursework and even standardized tests. There is not yet an accessible equivalent for the very popular TI-83 for example. The blind student can work the majority of these problems without a scientific graphing calculator, but the point is that they are at a disadvantage if they must do everything manually. Nevertheless, the Accessible Graphing Calculator combined with the ORION TI-34 talking scientific calculator allow these students to at least approach a level playing field. (Osterhaus, 2003; 2002; 2001) My latest dilemma is finding an accessible equivalent to the Geometer's Sketchpad.

## What advice would you give to a general education teacher who has a student with a visual impairment?

These are my collaborative/inclusive strategies:

- Adapted educational aids are a necessary component of any mathematics class. They are especially needed to supplement textbooks that have omitted tactile graphics or contain poor quality ones. However, they are also needed to help in interpreting mathematical concepts - just as their sighted peers benefit from various manipulatives. It is very beneficial to the entire class when the braille student's aid is a fun and useful tool for the sighted students and teacher as well.
- Math teachers need to verbalize everything they write on an overhead, blackboard, or whiteboard and be precise with their language. If the braille learner still has difficulty keeping up, the math teacher should be encouraged to give the student/vi teacher a copy of their overhead transparencies prior to class if pre-prepared or immediately after. Another alternative might be for a classmate to make a copy of their notes to share. The use of whiteboard technology, which allows transmission of the board contents to a low vision student's laptop works very well.
- Math teachers need to give worksheets, tests, etc. to VI teachers to transcribe into Nemeth far enough in advance, so that the braille student can participate with their fellow students in class - not later alone. Print copies should be legible as well. One way to insure this is for the math teacher to prepare their print materials using Scientific Notebook; then all students can receive high quality materials in a timely fashion in regular and large print and Nemeth code.
- Relate various mathematical applications to student activities enjoyed by blind students as well as the sighted students -

- Put various mathematical concepts to song or at least teach similar to an athletic cheer.
- The FOIL method for multiplying binomials F - O - I - L: First, Outside, Inside, Last!!!!
- Quadratic formula sung to the tune of Pop Goes the Weasel
- Be sure to include athletic experiences that a blind student can relate to; include the parabolic curve of a diver, as well as the football quarterback's pass.
- Math teachers need to realize that it is their job to teach the mathematical concepts to their students. This is not the job of the VI teacher. The VI teacher can be very helpful by insuring that all materials are in proper Nemeth code and all graphics are of good quality if the math teacher is able to supply these in print in a timely manner. However, any math teacher will tell you that there is always that teachable moment that you cannot anticipate. This is when it is imperative that the math teacher has some tools at his/her disposal. It is the responsibility of the VI teacher to expose the math teacher to the various tools and aids available to him/her. Math teachers can be quite creative, as many VI teachers have discovered. See Assistive math tools for a list of suggested tools and technology.
- Blind students should not be excused from learning a math concept because they are blind: "Blind students can't graph." "Blind students can't do geometric constructions." Not only can they graph and draw geometric constructions, with the right tools, they can often do so better than their sighted peers. Consideration should be taken into account however with regard to number of problems assigned. It is permissible to shorten the assignment, as long as the student can demonstrate competence in the content area.
- It is very important for all students (and especially for the VI student) to use as many senses as possible when learning a new math concept. They need to read a new math problem, write it, listen to it, tactually explore it through manipulatives, and when possible move their body and/or manipulative through space. If it's a fractional problem involving food for example, they can even taste and eat the problem.
- There is an ongoing need for four-way communication among the math teacher, the VI teacher, the family, and the student. Braille textbooks, materials, and aids need to be ordered early. The source of a problem needs to be discerned as quickly as possible - is it the math concept, the braille, or the quality of the tactile graphic? Vocabulary in itself can be a problem. Fractions have numerators and denominators in print and braille; however, they have "tops" and "bottoms" in print and "lefts" and "rights" in braille.
- For classroom test taking, the student should be given the test in their reading medium (with an option for partial oral administration; for example, in the case of students with learning disabilities who need word problems read) and supplied with appropriate enlarged/tactile graphics, aids, abacus, and/or a talking/large display scientific/graphing calculator. Blind students should be given at least twice the time to complete tests. At times, it may be desirable for the blind student to take the test separate from the group due to the needed extra time, use of aids (especially those involving speech), and/or partial oral administration.

## References

NCTM (2000). *Principles and Standards for School Mathematics* [On-line] Available: http://www.nctm.org/standards/

Osterhaus, S.A. (2003). Susan's math technology corner: Back-2-School: What's new and what's improved. *Division on Visual Impairments Quarterly*, 49(1), 5-8.

Osterhaus, S.A. (2002). Susan's math technology corner: The Accessible Graphing Calculator (AGC) from ViewPlus Software. *Division on Visual Impairments Quarterly,* 47(2), 55-58.

Osterhaus, S.A. (2001). Susan's math technology corner: The ORION TI-34 talking scientific calculator from Orbit Research. *Division on Visual Impairments Quarterly,* 46(3), 37-41.

Susan A. Osterhaus, M.Ed.

Secondary Mathematics Teacher

1100 West 45^{th} Street

Austin, TX 78756

Phone: (512) 206-9305

Fax: (512) 206-9453

E-mail: