The readability of a mathematical text is difficult to ascertain because of the complex nature of mathematical text. Robert Kane defined "mathematical english" as:

"Mathematical english (ME) is a hybrid language. It is composed of ordinary english (OE) commingled with various brands of highly stylized formal symbol systems" (Kane, 1967, p. 296).

More recently this definition has been expanded to include graphics:

"Mathematics textbooks and other teaching materials are almost always multi-modal, containing text, symbolic notation, and graphics" (Hammill, 2010, p. 1).

And, as observed by Mary Schleppergrell:

"Students need to be able to work simultaneously with all aspects of this multi-semiotic system - natural language, the language of mathematics symbolism, and the visual semiotic constructed in the graphs, charts, and diagrams that are integral to mathematical reasoning." ((Schleppergrell, 2010, p. 74).

Difficult as it may be, estimating the readability of mathematical texts is critically important. If students are to attain any measure of independence as mathematical learners, they must have access to mathematical texts pitched at the level of their mathematical reading ability. That is, the readability of the text must match the reading ability of the student, which is also difficult to determine.

What is meant by "readability?" Some dictionary definitions include:

- Easy or interesting to read. (Random House College Dictionary, Revised Edition, 1980, p. 1099)
- The ease with which a text may be scanned or read; the quality in a book, etc., of being easy to understand and enjoyable to read. (Oxford English Dictionary, http://www.oed.com/view/Entry/158852?redirectedFrom=readability#e)
- The quality of written language that makes it easy to read and understand. The FreeDictionary,(http://www.thefreedictionary.com/readability)
- Readability is the ease in which text can be read and understood. (Wikipedia, http://en.wikipedia.org/wiki/Readability)

For mathematics, readability has been described as:

- "In the broadest sense, readability is the sum total (including interactions) of all those elements within a given piece of printed material that affects the success which a group of readers have with it. The success is the extent to which they understand it, read it at optimum speed and find it interesting." (Dale and Chall, 1948, quoted in Shuard & Rothery, 1984)
- "A readability formula is a linear equation which predicts reading difficulty." (Kane, Byrne & Hater, 1974, p. 24)
- "By readability we here shall mean the matching of reader with material. To do this it is necessary to take into account the reader's comprehension, fluency and interest and the way in which these interact. all influence the degree of attainment of both cognitive and affective goals. Not only the choice of words, sentence length, etc. will be significant factors, but also the other features such as content style, format, organization, illustrations, humor, .... " (Austin & Howson, 1979, p. 171)
- "For the purpose of this study the concepts of readability or reading will be taken to mean the matching of the reader with the material, which by implication means that it is necessary to take into account the reader's comprehension, fluency and interest in the text." (Noonan, 1990, p. 62)
- "Readability includes all factors related to reading and comprehending written text." (Wiest, 2003, p. 1)

Then readability of a mathematical text then involves characteristics of a text that facilitate or inhibit the ability of a reader to comprehend the text. It is analogous to the notions that different automobiles are easier or harder to drive, or that different aircraft are easier or harder to fly.

Readability is often estimated by formulas. The large majority of these formulas are inapplicable to mathematical text because they have no ability to evaluate symbolic notation, much less graphical information. A good overview of readability formulas as applied in mathematics is found in Chapter 7 of *Children Reading Mathematics*, edited by Shuard and Rothery (1984). They give an overview of readability formulas generally, and describe the difficulties of applying formulas to mathematics. They also describe a readability formula develop by Kane, Burne & Hater specifically for mathematical text, which is described in full in *Helping Children Read Mathematics* (Kane, Burne & Hater, 1974):

Predicted Readability = -0.15A + 0.10B -0.42C - 0.17D + 35.52 where:

A = Words not on the Dale list of 3000 common (non-mathematical) words and also not on the list "Mathematics Words Known to 80 percent of Children."

B = Number of changes from a word token to a mathematics token and vice versa in the 400-token passage selected.

C = Number of different mathematics terms not on the list "Mathematics Words Known to 80 percent of Children," plus the number of different mathematics symbols not on the list "Symbols Known to 90 percent of Children."

D = Number of question marks in the 400-token passage.

Another approach to estimating mathematical readability was developed by Talisyon (1983). It is a "feadback-based" readability formula for mathematical text based on asking samples of students to mark the elements of a text they find "unclear" be they words, sentences, paragraphs, notation, or diagrams.

"The measure of readability presented in this paper is the reader's perceived clarity of a reading material. This is expressed as the communication index. C. l. of the material given by

Communication Index (C. 1.) = (X-to-Reader Communication Failure) / ((No. of X) (No. of Readers))

where X, an element of the reading material, can be a word, sentence, paragraph, figure, table, graph or equation;

and where "X to Reader Communication Failure" is the number of readers indicating unclear X summed up for all unclear X's." (Talisyon, 1983, p. 7)

A third approach to evaluating the readability of mathematical texts is to use Cloze Tests. Noonan (1990), Talisyon (1983), Green and Tomlinson (1983), and Hater and Kane (1975) all discuss how to use the cloze procedure to evaluate the readability of mathematical texts. Kane, Byrne and Hater (1974) inf fact used regression analysis of cloze data to identify the parameters of their readability formula described above.Cloze tests have the advantage of directly matching a give text with a given student.

This brief overview of mathematical readability suggests the following:

- If students are to be successful at reading and learning from mathematical texts, they must have access to texts that are appropriate to their mathematical reading ability.
- It is therefore, important to gage the readability of mathematical texts, not to mention the mathematical reading ability of students.
- The multi-semiotic nature of mathematical text makes the use of standard procedures and formulas for estimating mathematical readability unusable.
- Special mathematical readability procedures and formulas have been created, but they are difficult and awkward to use and likely of little practical value. The Cloze procedure might be an exception.

---Mark Horney. Last updated 27 March 2012

**Cited References:**

Austin, J. L., & Howson, A. G. (1979). Language and mathematical education. Educational Studies in Mathematics, 10(2), 161–197.Dale, E., Chall, J.S. (1948).

Formula for predicting readability. Education Research Bulletin, 27

Green, D. R., & Tomlinson, M. (1983). The Cloze Procedure Applied to a Probability Concepts Test. Journal of Research in Reading, 6(2), 103–118.

Hammill, L. (2010). The Interplay of Text, Symbols, and Graphics in Mathematics Education. Transformative Dialogues: Teaching & Learning Journal, 3(3). Journal for Research in Mathematics Education, 121–127.

Kane, R. (1967). The Readability of Mathematical English. Journal of Research in Science Teaching, 5, 296-298.

Kane, R. B., Byrne, M. A., & Hater, M. A. (1974). Helping children read mathematics. New York, NY: American Book Company.

Noonan, J. (1990). Readability problems presented by mathematics text. Early Child Development and Care, 54, 57–81.

Schleppegrell, M. J. (2010). Language in mathematics teaching and learning: A research review. In J. Moschkovich (Ed.), Language and Mathematics Education: Multiple Perspectives and Directions for Research (pp. 73-112). Charlotte, NC: Information Age Publishing, Inc.

Shuard, H., & Rothery, A. (Eds.). (1984). Readability formulae and their limitations. Children Reading Mathematics (pp. 76-88). Longdon: John Murray Publishers Ltd.

Talisyon, V. M. (1983). A feedback-based readability formula for science and mathematics curriculum materials. Journal of Science and Mathematics Education in Southeast Asia, 6(2), 5–10.

Wiest, L. (2003). Comprehension of mathematical text. Philosophy of mathematics education journal, 17.

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