Making Math Make Sense
By Doug Buehl,
Madison East High School teacher
Member, Wisconsin State Reading Association
October 1998
Tactics help kids understand math language
“An angle is the union of two rays that have the same endpoint.
The sides of angles are the two rays; the vertex is the common endpoint
of the rays. Angles may be formed by segments, as in polygons, but the
sides of the angle are still considered to be rays.”
Bookmark
Keys to Reading Math - Read carefully and make sure each sentence makes sense.
- Try to summarize what you read, in your own words.
- When you encounter a tough word, try thinking of easier words
that mean the same thing and substitute.
- Talk over what you read with a partner:
- To make sure you got it right.
- To clear up anything you don’t understand.
- Be on the lookout for:
- Things the author thinks you already know.
- Things you have learned in math before.
- Read with a pencil - Work the examples as you read them.
- Make your own definitions for key terms and keep them in a section
of a notebook.
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Um . . . let’s see here. You get an angle when two rays (straight
lines) come together and touch. The parts of the angle are the sides (the
rays) and the vertex (point where they touch). Figures like polygons (a
square for example) have angles because lines (segments) touch here too.
I know that segments and rays are both straight lines, but why does the
author say that segments (lines with beginnings and ends) are the same
as rays (lines which keep on going)?
Increasingly, mathematics textbooks and assessments are requiring students
to use reading as a means to learn and demonstrate knowledge. But as illustrated
in the geometry example above, prose in math textbooks presents special
challenges for students. Math language is very precise and compacted –
each sentence conveys a heavy conceptual load of information. In addition,
textbook authors assume that readers are already versed in some of the
content being presented. Students must therefore take a different approach
when reading math compared with social studies, science, or literature.
The Strategy
Many students have a mindset that math is only the manipulation of numbers.
They glide over the reading in an attempt to jump right into solving problems,
hoping to rely on the teacher to clear up any misunderstandings. Strategies
that help students key into the unique features of math text will help
them learn more effectively from their reading:
Step 1: Start by having the students establish the identity of
the textbook author(s). The authors presumably know a lot about math,
but how connected are they to students? Emphasize that sometimes university
professors or math experts use unfamiliar vocabulary or expect that students
know more than they actually do. Explanations that seem clear to mathematicians
may indeed be confusing to students. Students need to be prepared to confront
math text that requires careful deliberation.
Step 2: Next, model how to read through a challenging section
of text. Reproduce the pages on overhead transparency film and have students
follow in their textbooks as you think aloud. Especially tune into knowledge
that the author assumes of readers, and math concepts that were previously
learned.
For example, a passage on “decimal notation” in a pre-algebra
text states: “The decimal system of writing numbers is based on the
number 10. The digits we use in the decimal system are 0, 1, 2, 3, 4,
5, 6, 7, 8, 9. Numbers written in the decimal system are said to be in
decimal notation. In our system, the smallest 10 whole numbers are written
with only a single digit.”
Your think-aloud on this passage might unfold as follows:
“Decimal system – I know about decimals. Decimal points are
used for a part of a number, like .4, .59, or .823. But the author doesn’t
talk about decimal points here. The author must think I know what whole
numbers and digits are, because he doesn’t define them. He gives
examples (0, 1, 2, etc.) for digits, and I remember that from before.
I’m not clear about the statement: based on the number 10. Does that
mean like four tenths, or five tenths? This part on decimal notation is
not clear. I better go over that again, or ask for clarification.”
And so on.
Step 3: Provide each student with a “Keys to Reading Math”
bookmark and point out how your think-aloud followed these steps. As you
elaborate on these keys, use an analogy, like reading the operating manual
for a piece of equipment or instructions for assembling an item. Often,
documents such as these are frustrating to read, and it is tempting to
discard them and try to figure out what to do without them. But you then
run the risk of making an important error that could be costly. Instead,
you may need to read the material several times, consult with another
person, and eventually translate the confusing information into something
that you can understand.
Step 4: Finally, encourage students to compile their own definitions
of key terms in a section of their notebook or on index cards. For example,
the book definition of “decimal notation” – a notation
in which the 10 digits are used to write numbers, with each place in the
number standing for a power of 10 – can be rewritten in a more student-friendly
way.
Advantages
Math reading keys provide students with strategies that can aid them
in understanding conceptually dense text. In particular,
- Students are encouraged to consider how effective the author has communicated
with them and to problem-solve when things aren’t clear.
- Students are reminded to translate what they are learning into more
personal and understandable language, and to make connections with what
they have learned before.
Further Resources:
Beck, I., McKeown, M., Hamilton, R., & Kucan, L. (1997) Questioning
The Author: An Approach For Enhancing Student Engagement With Text. International
Reading Association, Newark, DE.
Posted October 2, 1998