Myrtle on procedural and conceptual knowledge

Maybe I can make a convincing argument that a student who only thinks of multiplication as iterative addition and can't multiply 24X86 has neither procedural nor conceptual knowledge.

There is more to the "concept" of multiplication than iterative addition. (Try applying iterative addition to 1/8 x 2/5.) Perhaps iterative addition is appropriate for 2nd and 3rd graders learning their multiplication tables (or is it 3 and 4th graders these days?) But "the" concept of multiplication includes the fact that it distributes over addition (and that it's associative as well). The multiplication algorithm invisibly makes use of the distributive "concept," and does not employ an iterative "concept." Perhaps I'm overdoing the disdain quotes but I've been lied to too many times by people telling me that something is the "concept" of a procedure or rule and it turns out not to be.

A child with a conceptual knowledge of multiplication, and a lot of time on his hands, could successfully multiply two digits numbers without the multiplication algorithm:

24 X 86 means that
(20 + 4)(80 + 6) which means/implies that...

Etc. You see where I am going with this. One of the benefits of Singapore is that the kid does end up with a conceptual understanding of multiplication, and can apply his knowledge of concepts to come up with correct answers.

Notwithstanding operations on super hairy numbers, he is capable of doing the algorithm on paper when he needs to and can resort to "concepts" when he needs to do mental calculations.


the multiplication algorithm invisibly makes use of the distributive "concept"

I love that!

I love the whole Comment, in fact. People like me -- people who value liberal arts education in general and mathematics education in particular but who aren't expert in mathematics and probably never will be, have no way to get at these things.

I intuitively grasp the notion that there is some kind of "starter understanding" a person can have without being fluent in procedures. Seeing that 6x4 is the repeated addition of 6 4s or 4 6s as the case may be (I've spent quite a bit of time muddled over that one!) strikes me as superior to not seeing it. (I had no idea multiplication could be called repeated addition until I started reteaching myself math, and then I noticed it on my own.)

But at the same time I am gripped -- and gripped is the correct word -- by the conviction that a starter understanding is not a real understanding.

And yet because I lack a real understanding I have no way to express this and thus no means of combating the forces of reform math when they threaten to overrun my son's education.

I'm logging this post under Greatest Hits so I'll know where it is when I need it.

Climbing Mt. Parnassus

I had never heard of this book until Myrtle left this comment:

Tracy Lee Simmons "Climbing Parnassus" is even more divergent. It traces the history of classical education throughout the centuries and chronologically speaking should be read before Diane Ravitch's "A Century of Failed School Reforms" which picks up about in 1900 where Simmons leaves off.

The difference between Simmons and Wise Bauer is that Simmons gives you the "whys" of classical education and Wise-Bauer gives the "hows." I would summarize her hows as being for non-expert parents. Her area of expertise is writing and English, not physics and math and while her recommendations in those areas certainly won't send anyone into an educational death spiral, they also aren't nearly as clever and insightful as what she has to say about English, composition, history, and foreign language.


and here is an excerpt from Well-Trained Mind: left by Concerned Parent:

The Parrot Years:

Houses rest on foundations. Journalists gather all the facts before writing their stories; scientists accumulate data before forming theories; violinists and dancers and defensive tackles rely on muscle memory, stored in their bodies by hours of drill.

A classical education requires a student to collect, memorize, and categorize information. Although this process continues through all twelve grades, the first four grades are the most intensive for fact collecting.

This isn't a fashionable approach to early education. Much classroom time and energy has been spent in an effort to give children every possible opportunity to express what's inside them. There's nothing wrong with self-expression but when self-expression pushes the accumulation of knowledge offstage, something's out of balance.

Young children are described as sponges because they soak up knowledge. But there's another side to the metaphor. Squeeze a dry sponge, and nothing comes out. First the sponge has to be filled.

[snip]

So the key to the first stage of the trivium is content, content, content. In history, science, literature, and, to a lesser extent, art and music, the child should be accumulating masses of information: stories of people and wars; names of rivers, cities, mountains, and oceans; scientific names, properties of matter, classifications; plots, characters, and descriptions. The young writer should be memorizing the nuts and bolts of language-- parts of speech, parts of a sentence, vocabulary roots. The young mathematician should be preparing for higher math by mastering the basic math facts."


If you haven't noticed yet, Amazon has a dandy new rotating-books carousel feature that lets you scroll through all the other related books people who purchased the book you're looking at purchased. What a fantastic research shortcut. If I know a book is good -- or, more importantly, that it's considered good by the experts whose work I'm writing about^* --- I can instantly learn what other books are in its category.

I found Animals in Translation in the carousel for The Psychology of Learning and Behavior by Barry Schwartz, which is causing me to contemplate purchasing it just to find out what the connection is specifically.

Back to Mt. Parnassus; the carousel there has one book I own and like very much: The Laurel Wreath and Harp: Poetry and Dictation for the Classical Curriculum.

Another 3 I'm interested in:

• Composition in the Classical Tradition
  

• Fifty Famous Stories Retold
  

• Increasing Academic Achievement with the Trivium of Classical Education

^* make that trying to write about

a Yahoo with a list

Myrtle: Communal reinforcement is the process by which a claim becomes a strong belief through repeated assertion by members of a community. The process is independent of whether or not the claim has been properly researched or is supported by empirical data.

Catherine: Right, except the cool thing is it doesn't have to be "communal." One person with a Yahoo list can do it.

Myrtle: Or one yahoo with a person list.

Bar Diagrams II

Algebra without algebra:

Fifth grade word problem from Singapore: Jim and Dan have $24 altogether. If Jim gives $2 to Dan, he will have three times as much money as Dan. How much money does Jim have?

Observe the mighty bar diagram in action:

The kid thinks, "I can see that $24 will be divided into four equal groups of $6. Jim will have 3 groups of 6...that's $18.

Method #2

Let j be Jim's money and d be Dan's money: j + d = 24; j - 2 = 3*(d + 2) so j = 3(d + 2) + 2 substitute in to the first equation: 3(d+2)+2 + d = 24 simplify: 4d + 8 = 24, solve: d = $4; j = $20.

They both have their charms.


multimedia learning (Catherine)

Myrtle on teaching math with proofs

I had a six hour conversation today with my husband on proofs vs. calculations. How can I be so sure that this proof approach is the right thing to do? While I personally think it's entertaining, how do I know that the kid will do okay on the SAT or won't be counting on his fingers later on?

ooooo....

This sounds fun:

I've exhausted my interest in the topic of Fuzzy Math and I'm now interested in Junk Geometry.

Drat These Greeks

Birkhoff's Geometry, Singapore NEM (doesn't cover quadratic equation until 9th grade), Frank Allen's axiomatic algebra instruction, Apostol Calculus, and more.

Go read.

Moise and Downs TOC

Myrtle's find:

Table of Contents


also:

• SMSG Geometry at ktm-1
  

• Amazon review of Moise and Downs (at ktm-1)
  

• An A-maze-ing Approach to Math by Barry Garelick

another Myrtle find

good grief

Myrtle may have surpassed even Google Master. (I have to say....I'm wondering whether Myrtle IS Google Master!)

Check this out.

your brain on bar diagrams

Somebody really did do a study on this. MRI of brain on bar diagram vs. traditional algebra.

"In Singapore schools, algebraic word problems are taught using two methods: formal algebra and model. The latter depicts relevant quantitative relationships between unknowns in a pictorial format. In this study, we used fMRI to investigate whether the two methods are subserved by different cognitive processes. "

It would have been more interesting to see this study done on children rather than adults. I'm guessing it would have turned out more like this: