I do these more for myself than anyone else, but here, I am quoting the most useful parts of this ASCD book (click links to read online for free). Basically, I'm editing out the boring. You're welcome.
I'm also just doing a couple of chapters at a time because it's kind of dry and I never know how much time I will have to read. So consider it a series if you like.
Joan M. Kenney
*An ESL student thought of 'whole' numbers as 'hole' numbers, as in how many holes a number add. He thought 6 and 10 were odd numbers because they each only have one hole. He didn't know if 3 was even or odd because it could be considered as having two holes or two half holes which would make one whole hole. He knew the definition of even or odd but misunderstood 'whole'.
Younger students can be quit mystified by the fact that changing the orientation of a symbol- for example, an equal sign (=) from horizontal to vertical- can completely change its meaning.
Vocabulary can be confusing because the words mean different things in mathematics and nonmathematics contexts, because two diferent words sound the same, or because more than one word is used to describe the same concept.
Symbols may be confusing either because they look alike (e.g., the division ad square root symbols) or because different representations may be used to describe the same process.
Graphic representations may be confusing because of formatting variations or because the graphics are not consistently read in the same decision.
One strategy we arrived at is for teachers to model their thinking out loud as they read and figure out what a problem is asking them to do. Other strategies include dialoguing with students about any difficulties they may have in understanding a problem and asking different students to share their understanding.
James Bullock (1994) defines mathematics as a form of language invented by humans to discuss abstract concepts of numbers and space.
The meaning that readers draw will depend largely on their prior knowledge of the information and on the kinds of thinking they do after they read the text (Draper, 2002): Can they synthesize the information? Can they decide what information is important? Can they draw inferences from what they've read?
In English there are many small words, such as pronouns, prepositions, and conjunctions, that make a big difference in student understanding of mathematics problems. For example:
- The words of and off cause a lot of confusion in solving percentage problems, as the percent of something is quite distinct from the percent off something.
- The word a can mean “any” in mathematics. When asking students to “show that a number divisible by 6 is even,” we aren't asking for a specific example, but for the students to show that all numbers divisible by 6 have to be even.
- When we take the area “of” a triangle, we mean what the students think of as “inside” the triangle.
- The square (second power) “of” the hypotenuse gives the same numerical value as the area of the square that can be constructed “on” the hypotenuse.
- What is the major concept?
- How can I help students connect this concept to their lives?
- Are there key concepts or specialized vocabulary that needs to be introduced because students could not get meaning from the context?
- How could we use the pictures, charts, and graphs to predict or anticipate content?
- What supplemental materials do I need to provide to support reading?
As the reading progresses, the teacher should ask process questions that she wants the students to ask themselves in the future. They may be asked to predict what the reading will be about simply by reading the title of the piece (if there is one, such as a graph or story problem). Next the students should make two columns on a piece of paper, one headed “What I Predict” and the other headed “What I Know.” Once the students have silently read each section of the piece, they should fill out each column accordingly. At this point, the teacher should ask students questions such as the following:
- What would you be doing in that situation?
- Does this make sense?
- What does the picture/graph/chart tell you?
- How does the title connect to what we're reading?
- Why are these words in capital letters?
- Why is there extra white space here?
- What does that word mean in this context?
Figure 2.4. Guided Reading Example
Solving Systems Using Substitution
1. What does the title tell you?
From a car wash, a service club made $109 that was divided between the Girl Scouts and the Boy Scouts. There were twice as many girls as boys, so the decision was made to give the girls twice as much money. How much did each group receive?
2. Before you read further, how would you translate this story problem into equations?
Translate each condition into an equation.
Suppose the Boy Scouts receive B dollars and the Girl Scouts receive G dollars. We number the equations in the system for reference.
3. What do they mean here by “condition”?
The sum of the amounts is $109.
(1) B + G = 109
Girls get twice as much as boys.
(2) G = 2B
4. Did you come up with two equations in answer to question 2 above? Are the equations here the same as yours? If not, how are they different? Can you see a way to substitute?
Since G = 2B in equation (2), you can substitute 2B for G in equation (1).
B + 2B = 109
3B = 109
B = 36 1/3
5. How did they arrive at this equation?
6. Do you see how it follows?
7. Does it make sense? How did they get this?
To find G, substitute 36 1/3 for B in either equation. We use equation (2).
8. Do this, then we'll read the next part.
G = 2B
= 2 × 36 1/3
= 72 2/3
So the solution is (B, G) = (36 1/3, 72 2/3).
The Boy Scouts will receive $36.33, and the Girl Scouts will get $72.67.
9. Did you get the same result?
Are both conditions satisfied?
10. What conditions do they mean here?
Will the groups receive a total of $109?
Yes, $36.33 + $72.67 = $109. Will the boys get twice as much as the girls? Yes, it is as close as possible.
11. How would you show this?
Where did they get this equation?
Note: Text in the left column above is adapted from University of Chicago School Mathematics Project: Algebra (p. 536), by J. McConnell et al., 1990, Glenview, IL: Scott Foresman.
Students are helped not by having their reading and interpreting done for them, but rather by being asked questions when they don't understand the text. The goal is for students to internalize these questions and use them on their own.