Verification

I said in my last post that I would talk about role of mental representation in comprehension.  Before I get into that topic, however, I need to discuss a somewhat related idea I have been thinking about quite bit as I have worked with some of my students, namely, the importance of structuring lessons to allow students to verify their answers and their thinking.

The following story about my son provides a nice example of the importance of what I am calling verification.  One day when my wife went to pick my son up from kindergarten (nine years ago)  he wanted to read to her a story that the class had read together that day.  The reading of the story was done not by showing the students how to “sound out” the words, but through repetition and predictability of the text.  My son apparently read the whole story correctly, except that he read some small function word incorrectly, perhaps “in” for “on”  Some of the other students who heard him read the story corrected him immediately.  However, because he had simply memorized the text pattern rather than learned how to translate the letters into sounds, he became upset.  Moreover, there was no really good way to repair the error so that he understood the mistake:  I suspect in his mind, it was not an error of translation, but an error of memory.

This sort of thing happens countless times with our students, and those of us who work especially with struggling students see that such instructions ends up teaching students to guess constantly and to approach their work un-strategically, confident that it will never entirely be fully understandable to them.  The problem of such mis-understanding in education is quite large:  Howard Gardner has an interesting book called The Unschooled Mind that discusses a significant body of research that demonstrates that even students with high achievement in typical measures of academic performance have persistent and fundamental misunderstandings about what they seem to have learned.

As I’ve worked with students, the curricula that I find the most effective as well as enjoyable to use are those that are founded upon sense-making and require students to prove their answers in terms and with explanations that make sense to them.  This can be as simple as having a student verify a number fact by counting on his fingers, and demonstrating that his subtraction calculation was correct by using base-ten blocks or some similar physical representation.

Much of this can be built into a curriculum systematically.  But one of the great arts of teaching, it seems to me, is to figure out how to help a student assimilate new information and new experiences within the terms of his own understanding.  

 

Math and Reading Workshop

Last week I had the opportunity to help lead a professional development workshop on the topic of Math and Reading for middle school and secondary level teachers in the Ithaca City School District. I was invited by my friend and neighbor Dani Novak, a math professor at Ithaca College who is dedicated to bringing the joy of math to everyone, as well as working to eliminate the fear and anxiety that riddle so many math classrooms (www.ithaca.edu/dani/).

There were eight teachers who participated in the workshop. All were extremely interesting. Five were from DeWitt Middle School, and most of these did not know that the others had registered. Propitiously, I suppose, the final report of the National Mathematics Advisory Panel was released the day before the workshop (www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf). I haven’t had time to read the document carefully, but at first glance it seems very sensible, pointing to the critical importance of algebra instruction, and looking at how to focus elementary and middle school instruction to better prepare students for algebra. (A key lack now seems to be instruction in fractions.)

In preparing for the workshop I read a book called The Number Sense by Stanislas Dehaene. The book was written in 1999, but there was just a New Yorker profile about Dehaene this month, and the parent of one of my students alerted me to it. Dehaene argues that humans have a hard-wired number sense, observable in the behavior of human babies as well as in other mammals, and that a great deal of the problems children have with math instruction is because the instruction usually fails to build sensibly upon these sets of inborn mathematical intuitions.

There is much to ponder here. One of the questions I raised with the group was the difference between understanding a nonsensical and a “sensical” sentence: “The morphius is under the zinderfloss,” v. “The cat is under the table.” At first the teachers wanted to say that they didn’t understand the first sentence at all, and yet, they could all correctly answer the question “Where is the morphius?” As we discussed the matter further, the key difference appeared to be that they could make a mental image of the second, but not the first, even though they understood some purely formal, grammatical relationships expressed in both sentences. The interesting question that thus emerges with regard to comprehension of mathematics (and, indeed, of comprehension generally) is the role of mental representation. I’ll share some more of this discussion in my next post.

The teacher’s themselves came with some excellent questions, including:

• what are good ways to teach mathematical vocabulary
• how can we best help students who don’t read thoroughly and completely
• what are the underlying brain activities that are related to math
• how can we get students motivated about math
• is it wrong to give middle school students a printed multiplication table when doing their work
• what are cultural differences that influence student performance

I’ll try to discuss all of these topics as well in future posts.

Thoughts on “sight words”

Although the term “sight words” is used frequently in discussing beginning reading instruction, it is the source of considerable confusion.  The reason for the confusion is that people use the term to refer to three related but nonetheless distinct concepts.

Sometimes the term “sight words” is used to refer to words that students have learned to recognize extremely rapidly, or “on sight.”  A more precise term would be a “rapidly recognized” words.  Hence, the word “mop” is a sight word for student if he can read the word quickly and it is not a sight word for that student if he cannot yet read it quickly. Recognizing a large number of words rapidly is necessary to become a good reader and it should therefore be one of the fundamental goals of all beginning instruction to help students read the several thousand most frequent words in English in this rapid, automatic way.    Researchers have operationalized the concept of “rapidly recognized” by constructing fluency tables based on the normal distribution of oral reading rates for students at different ages and grades.  (One of the most comprehensive of these tables can be found at brt.uoregon.edu/tech_reports.htm).  This sort of information can be used to identify whether students are recognizing words rapidly enough for likely reading success, and it is the basis of such prominent early reading assessments as the Test of Word Reading Efficiency (TOWRE) and the Dynamic Indicators of Basic Early Literacy Skills (DIBELS).

The term “sight words” is also often used as a synonym for high-frequency words, such as those found on lists compiled by Fry and Dolch.  It is worth noting that the use of the term in this sense refers to a characteristic of the word itself independent of the student’s skill.  Lists of high-frequency words are very useful because, as I mentioned before, one of the fundamental goals of beginning reading instruction is to help students recognize the most common words rapidly.  Using these lists of high-frequency words, educators can create lessons intelligently and assess student performance effectively.

Finally, “sight words” is also used to refer to irregularly spelled words that don’t fit common phonics generalizations, words such as “was,” “could,” and “laugh.”  Like the concept of “high-frequency” word, and unlike the concept of “rapidly-recognized” word,  the concept of “irregularly-spelled” word refers to a characteristic of the word itself and not the learner’s skill.  Most reading programs explicitly identify the irregularly-spelled words as a special category of word, labeled with terms such as “outlaw words,” “rule-breakers,” and “words that don’t play fair.”  Students are typically asked to learn these words not by repeatedly sounding them out, but as visual wholes, that is, without analyzing how the letters represent the sounds in the word.  

Although this technique for teaching irregularly-spelled words is quite common, its necessity is hardly obvious.  The reason is that irregularly-spelled words typically have only a single letter/sound correspondence that is irregular.  For instance, in the word “was,” the letters “w” and “s” represent typical sounds—it is only the “a” that is irregular.  If the words “was” is taught only as a visual whole, then the underlying relationship between its letters and sounds is hidden from the student.  This relationship is a powerful tool for helping the student learn the word accurately and quickly.  Moreover, it is rarely the case that an irregular letter/sound correspondence is unique to a particular word, but is usually shared in at least one or two other common examples.  For instance, the unusual vowel sound in “was” also appears in the word “what.”  Similarly related words include “any-many,” and “would-should-could.”  Since the human memory is associative, it would seem to be quite natural as well as helpful to make these relationships obvious to students.  Furthermore, the more extensively the letter/sound relationships in the word are analyzed explicitly, the easier it is for teachers to help students learn the difference between words with similar spellings such as “want” and “went.” 

 These comments do not mean that I think it is particularly harmful to teach irregularly-spelled words as visual wholes.  Indeed, I do believe that there is some benefit from teaching a small number of words in this way.  The benefit lies in helping a beginning student read a number of common function words without going over the sounds in those words.  One does not need to teach words with the /ew/ sound, for example, to teach a very beginning student the word “to.”  Once the beginning student knows a few of these function words, he is able to read some simple sentences, and therefore extend his reading skill.   In addition, there are a handful of very common words such as “of” in which all the letters are irregular and there are no other words with a similar spelling pattern.  However, the list of extremely useful function words like “to” and high-frequency words with unique spellings like “of” is quite small, perhaps no more than a dozen.

What is the moral of this story?  It is probably a good idea to retire the ambiguous term “sight words” and replace it as appropriate with the more precise phrases:  rapidly-recognized words, high-frequency words, and irregularly-spelled words.  Once equipped with these precise distinctions, what should the good reading teaching do with them?   She should teach all high-frequency words and words with common spelling patterns so that students can recognize them rapidly.  This work should involve prominently making the letter/sound relationships in the word explicit and only very rarely should involve presenting irregularly-spelled words as unanalyzed visual wholes.