Learning Math In addition to the information provided in Accessible Instructional Materials (AIM) and Designing Instruction, this section provides an introduction to some additional considerations when designing instruction that includes students who are using already using AIM in order to read for math learning. Even in math, printed text is the primary method for gaining instructional information in many classrooms.
The questions stated in the introduction (and stated as headings below) can help guide the design of instruction when incorporating AIM into lessons.
What is the instructional goal?
When the instructional goal is to learn math, accessible instructional materials (AIM) can provide necessary access to information that is being provided to other students in printed text. However, while difficulties reading print do not necessarily result in math challenges, the special circumstances involved with learning math can be impacted by the use of AIM. In addition, some of the same things that present challenges in reading also can present challenges for students trying to learn math. Conversely, some of the compensatory strategies used by students who use AIM can be advantageous when instruction takes this into account. This section will highlight some additional considerations when designing instruction that includes AIM.
Accessible materials are intended to be used by all students with print disabilities. However, because the way students take in information (and learn) is highly variable, the use of AIM is not likely to benefit all students with print disabilities in the same way.
Therefore, consider how a student’s—
- challenges can present barriers to math learning,
- strengths and compensatory strategies can be used as an advantage in math learning
What tasks do students need to do in order to achieve the instructional goal? Will requiring these tasks present barriers for some students?
- use tactile manipulatives with hands or computer mouse
- read a math textbook to gain information and understanding
- use the information gleaned from the text to learn math
- estimate, count, and perform number operations
- learn, recall, and use rote math facts
- retain and manipulate symbolic information (numerals)
- use and understand special math language
The role of language in math learning is often underestimated. Language is the tool that is used to bridge concrete math concepts to higher-level abstract math concepts. Math, as any other subject, has its own subject-specific vocabulary; but it also makes use of several other special sentence structures such as conditional statements (if/then), comparatives, negatives, inferentials, words with different meanings inside and outside of math (e.g., interest, square), varied forms of words with related but distinct meanings (such as hundred, hundreds, or hundredth), abbreviations and symbols, etc. Consideration should be given to developing consistency with the language that is used year to year. In addition, students need to be exposed (even passively) to this language early to be able use it for math learning.
In using a specialized format, how is the student gaining access to the content? In other words, what sensory modalities are being used (e.g., vision, auditory, touch)? Will the use of the modality conflict with any other tasks required to reach the instructional goal?
If the student is using a specialized format, consider what sensory modalities are being used to access the content (e.g., vision, auditory, touch) and whether they will conflict with other tasks that may be required. The use of one modality (visual, audio, or tactile) prevents the student from using the same modality for another purpose. For example, if a student is using braille, he or she will not be able to do other things with his or her hands at that time. In addition, the use of braille may prevent a student from counting on his or her fingers.
Although this may seem obvious, it is an important consideration for instructional design so that planning is done in advance to reduce barriers that prevent the student from performing the same tasks or reaching the same instructional goals as other students. Indeed, at the elementary level, the literature suggests that teachers found they needed to include physical manipulatives to demonstrate concepts before braille or audio information was useful.
The literature suggests that children who are blind often do not count on their fingers. The reason for this is that their fingers serve as a vehicle for gaining information rather than as a way to perceive numbers. The absence of counting on fingers affects these students’ ability to subitize.1 To be able to subitize, a student needs to determine, without counting, the total number of a set. Young children are able to subitize 1, 2, or 3 items. For a child who is blind, this would require him or her to use hands to feel items and to determine how many items are in a set. However, as soon as children who are blind begin to use hearing to count, they can subitize a set of spoken numbers (Ahlberg & Csocsan, 1999).
What materials are students using to achieve the instructional goal?
For example, students in most classrooms use math textbooks, workbooks, worksheets, calculators, and manipulatives.
Math textbooks, workbooks, and worksheets require the student to—
- Handle a textbook and turn its pages
- See text, formulas, and other visual graphics
- Decode text
- Comprehend the information being presented (have sufficient understanding of the language, vocabulary, formulas, visual graphics being used, etc.)
- Remember what is being read in order to make meaning from it
Calculators require the student to be able to—
- Handle and press buttons
- Know which buttons correspond to each number or operation
- Understand how to use and obtain information from the calculator
Manipulatives require the student to be able to—
- Handle, move, stack, etc. the manipulatives
- See or feel them
- Make sense of what they mean
- Remember and apply the information
How do the challenges that prevent the student from gaining information from printed text present barriers in math learning?
Some students, including those who use AIM, exhibit math-learning challenges, despite lack of underlying math learning disabilities (number sense or understanding). For some students with print disabilities, the same differences in the phonological loop (common in students with dyslexia) that make decoding text challenging also make memorization and recall of rote math facts challenging. While some students are able to use verbal strategies to recall math facts, students with challenges in the phonological loop likely use strategies such as repeated addition to figure out their multiplication facts—a much slower processing and computation.
Research suggests that the use of physical manipulatives can assist students to develop recall of rote math facts. Additionally, the use of a calculator to compensate for poor math fact recall can improve computational speed. In fact, research suggests that when these students with dyslexia use a calculator, they do not exhibit an overall math learning difference (Simmons & Singleton, 2009).
How do the challenges that prevent the student from gaining information from printed text present strengths or compensatory strategies for math learning?
Audio and e-text may provide a distinct advantage over braille materials for math learning in students who have visual challenges. The literature suggests that these students may be at a distinct advantage when subitizing spoken numbers because they use auditory information to compensate for a lack of visual information. For example, while children who use vision can typically subitize as many as four objects at once, children with visual challenges can subitize as many as seven or eight objects auditorily (Ahlberg & Csocsan, 1999).
It is believed that these students’ use of hearing not only compensates for their lack of sight in learning math, it could help them to develop their phonological loop capacity. This is not only helpful for subitizing, but also for efficient retrieval of math facts. In addition, this may have implications for working memory. Although not mentioned in the literature, due to above-average development of auditory memory capacity, it is possible that these students may be able to hold more digits in working memory.
How will using AIM impact cognitive load?
Although there is no specific research on the impact of AIM on working memory, research into cognitive load theory has investigated the efficiency of learning with multimedia presentation of materials, including audio, video, and visual presentations. Despite research describing conflicting findings, one overall picture emerges: When describing a concept, simultaneous spoken and written words enhanced learning, but with the addition of a diagram, learning is depressed (Mayer, Heiser, & Lonn, 2001; Moreno & Mayer, 2002).
Research suggests that when audio and written materials are describing a diagram, learning is reduced. This indicates that requiring a student to split attention between a visual diagram and written material may create an increased load on visual working memory. Conversely, when audio and visual materials (either written words or diagrams) are presented together, learning is enhanced. This suggests that splitting the cognitive load between the visual and phonological working memory is an effective teaching technique. This has implications for math learning, since diagrams are frequently used to demonstrate mathematical concepts.
Cognitive load theory maintains that working memory capacity is limited, so that if a task creates too much working memory load, learning will be hindered (de Jong, 2010). However, in the case of students with math disabilities, if cognitive load can be minimized, AIM may be helpful to students for whom reading a textbook would hinder math learning.
1. Subitize: to perceive at a glance the [total] number of items presented (Dictionary.com, Unabridged; based on the Random House Dictionary, &c; Random House, Inc. 2010.)