kitchen table math, the sequel: 10/31/10 - 11/7/10

Friday, November 5, 2010

back to Illinois

Robert H. Johnson

off-topic - social media

Do the people you know on blogs affect your odds?

Algebra II Before Geometry

This is happening in our area (not at my son's HS), but I'm not sure why. Is it being driven by the CCSSO standards? Do high schools want to get most students through the material (pseudo-Algebra II) before it's too late or while the algebra "iron" is hot? If this is true, then the top math track in high school is under attack. It's one thing to channel students off to integrated math, like Core Plus, but with the current emphasis on the traditional sequence of math classes, there seems to be only one direction for the content to go. Traditional math won, but ends up losing. Is that what will happen in lower schools if they decide to use Singapore Math? Perhaps I'm reading too much into this. I don't have any strong feelings about whether geometry or algebra II should come first, but I do have strong feelings about rigor. What is the real driving force behind the switch? Do they use the same textbooks? One comment I heard once was that this would allow the geometry classes to delve more deeply into proofs because the students would be more mature and would have more math background. I don't buy it. What direction would algebra II go, even for the honors version?

Wednesday, November 3, 2010

Knowledge is good

Just heard from Barry - new article from Sweller, Clark, & Kirschner is out --
Problem solving is central to mathematics. Yet problem-solving skill is not what it seems. Indeed, the field of problem solving has recently undergone a surge in research interest and insight, but many of the results of this research are both counterintuitive and contrary to many widely held views. For example, many educators assume that general problem-solving strategies are not only learnable and teachable but are a critical adjunct to mathematical knowledge. The best known exposition of this view was provided by Pólya (1957)....[I]n over a half century, no systematic body of evidence demonstrating the effectiveness of any general problem-solving strategies has emerged. It is possible to teach learners to use general strategies such as those suggested by Pólya (Schoenfeld, 1985), but that is insufficient.


The alternative route to acquiring problem-solving skill in mathematics derives from the work of a Dutch psychologist, De Groot (1946–1965), investigating the source of skill in chess. Researching
why chess masters always defeated weekend players, De Groot managed to find only one difference. He showed masters and weekend players a board configuration from a real game, removed it after five seconds, and asked them to reproduce the board. Masters could do so with an accuracy rate of about 70% compared with 30% for weekend players. Chase and Simon (1973) replicated these results and additionally demonstrated that when the experiment was repeated with random configurations rather than real-game configurations, masters and weekend players had equal accuracy (±30%). Masters were superior only for configurations taken from real games.


The superiority of chess masters comes not from having acquired clever, sophisticated, general problem-solving strategies but rather from having stored innumerable configurations and the best moves associated with each in long-term memory.


[L]ong-term memory, a critical component of human cognitive architecture, is not used to store random, isolated facts but rather to store huge complexes of closely integrated information that results in problem-solving skill. That skill is knowledge domain-specific, not domain-general. An experienced problem solver in any domain has constructed and stored huge numbers of schemas in long-term memory that allow problems in that domain to be categorized according to their solution moves.


[D]omain-specific mathematical problem-solving skills can be taught. How? One simple answer is by emphasizing worked examples of problem-solution strategies. There is now a large body of evidence showing that studying worked examples is a more effective and efficient way of learning to solve problems than simply practicing problem solving without reference to worked examples (Paas & van Gog, 2006).


For novice mathematics learners, the evidence is overwhelming that studying worked examples rather than solving the equivalent problems facilitates learning. Studying worked examples is a form of direct, explicit instruction that is vital in all curriculum areas, especially areas that many students find difficult and that are critical to modern societies. Mathematics is such a discipline. Minimal instructional guidance in mathematics leads to minimal learning (Kirschner, Sweller, & Clark, 2006).

Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics (pdf file)
by John Sweller, Richard Clark, and Paul Kirschner

the Arizona case

School Choice on Trial

Tuesday, November 2, 2010

punctuating by breath

from Sentence Diagramming: A Step-by-Step Approach to Learning Grammar Through Diagramming:
Before the 1960s, grammar and punctuation were taught as foundation blocks for writing instruction. In the 1960s, some research questioned the value of teaching grammar, and new ways of teaching grammar cast doubt on the traditional methods. In the midst of all this change, the baby was thrown out with the bath water where grammar was concerned, and when the 1970s rolled around, a new generation of teachers had not been trained to teach grammar and punctuation.

I am a member of that new generation of teachers, and a product of a writing education with little structured or sustained lessons in grammar. Thankfully, one of my teachers believed in teaching grammar and punctuation through sentence diagramming. Before this instruction, I lacked confidence in my writing because I didn't know for sure if my sentences were really sentences.

I can still remember the great "aha!" feeling I had when I realized that I could analyze a sentence without the teacher's assistance--I could mentally diagram the sentence to determine if it was grammatically correct. What a sense of power that gave me!

Marye Hefty
I find this remarkable.

Mary Hefty earned a Masters degree in English and then worked as an editor in a research laboratory, but as a child (or teen?) she could not tell whether she had or had not written a sentence.


She goes on:
When I left the research laboratory to become a college professor, teaching English composition and technical writing, I noticed during the first term that many of my students' papers were riddled with grammar and punctuation errors. I didn't know how to add the necessary instruction in grammar and punctuation skills to our limited class time without letting it take over the class like a weed. Typically, I tried a band-aid approach to teaching grammar and punctuation. When I saw sentence fragments in the students' papers, I talked about sentence fragments. When I saw comma splices, I talked about those. It didn't take long to realize that many of my students didn't recognize a sentence, so they couldn't solve the sentence problems. My students were just like I had been--needing structure and an organized way to learn grammar and punctuation without having the approach overwhelm them or make it difficult for them to learn the writing process in class.

In several classes, I decided to discard the band-aid approach and devote 10 percent of the class time to teaching the students grammar and punctuation, starting with the basics--What is a simple sentence? How do you diagram it? And guess what? It worked.

I've been teaching sentence diagramming in some of my courses for eight years now, and the students who begin my classes not being able to identify or define a simple sentence leave the class with the vocabulary and knowledge to identify simple, compound, and complex sentences; fragments; run-ons; and comma splices. Most importantly, the students have a foundation that enables them to learn more--without my help--after they leave the class.

Surprisingly, my university students don't mind having to learn grammar and punctuation through sentence diagramming because this approach quickly gives them the skills and confidence to fix the problems in their papers on their own. I have heard enough anecdotal evidence from my students to know that sentence diagramming works. For example, one of my former students told me that she was asked to edit letters for her boss. She said that before taking my class, she just put in the commas in where she thought she heard a pause and just guessed that the sentences were correct. "Now I know for sure, and I can really help," she said.
A lot of my students -- these are college freshmen -- "punctuate by breath."

Punctuating by breath is OK as far as it goes, I think. Now that I'm learning the formal rules of punctuation, I realize I've been breaking some of those rules for decades. I've been breaking the rules because a) I didn't know them, so b) I've been punctuating by breath.

Having thought it over, I've decided to carry on punctuating by breath when the occasion calls for it. If I want or don't want a comma somewhere,  then a comma there will or will not be. I'm the decider.

Still and all, the reason this works for me is that I have never, ever, in my entire adult life, failed to recognize a complete sentence. Nor have I failed to write a complete sentence if that's what I wanted to write.

I have come to the realization that a course about writing is a course about sentences.

Sentence Diagramming: A Step-by-Step Approach to Learning Grammar Through Diagramming

Sunday, October 31, 2010


My dad died this week.

Didn't see it coming.

My mom reminded me that she and my dad stayed at our house when I was pregnant with the twins. Or maybe it was right after I had the twins, I don't remember. Every morning, Jimmy would get in bed with them and wake my dad, whom we all called Bob, because that was his name. We called my mom Pat & my dad Bob.

After a few days, Bob left to go back home and my mom stayed on.

Every morning after that, Jimmy would get in bed with my mom and chant, "No more Bob, no more Bob."

Back to Illinois on the weekend for the funeral.