Tuesday, September 29, 2009

Out today: "Raising a Left-Brain Child in a Right-Brain World"

While this book is most obviously for parents of left-brainers, it originated as more of a general education critique. That is, I've always used lay terms "left-brain" and "right-brain" (with the concepts they casually denote as organizing principles), but initially spent more time arguing that "right-brain" trends--the emphasis on sociability and uninstructed visual "creativity" over analytical essays and mathematically challenging math problems for example-- are bad for all students, left- and right-brained.

But focusing, specifically, on the very real needs of left-brain students gives me a more personal, specific-child-focus that should bypass some of the political polarization of the education debate. And I do strongly believe that left-brainers, in particular, are being shortchanged by The System--in all the ways I've discussed in this blog as well as in the book.

On the other hand, I continue to believe that the "right-brain" trends I talk about are bad news for all students.

On the third hand, if you know anyone with a bright, quirky, and/or social awkward child who is frustrated and/or under-performing at school, I certainly won't discourage you from mentioning this book to them.

Saturday, September 26, 2009

Reading "all about me" replaces analytical reading

...codified as early as second grade in this Text-to-Self Connections T-Chart, duly completed by my daughter:

Making Text-to-Self Connections T-Chart

The author said: Tom and Lucy got lost in a dark cave.
That reminds me of: I got lost at the beach because I couldn't find my grandma.
-----------------------------------------------------------------------------------------------------------
The author said: The Littles went on a hike. It was so far.
That reminds me of: I went on a hike that was faraway too.
-----------------------------------------------------------------------------------------------------------
The author said: The Littles got stuck in a fridge.
That reminds me of: I got trapped in my room before but not in a refrigerator.
-----------------------------------------------------------------------------------------------------------
Figuratively trapped in her room by the many assignments like this one, not to mention uninspired and resentful ("Why do they want to spy on us?!"), my daughter nonetheless earned a ☺on this T-Chart.

Thursday, September 24, 2009

Autism Diaries XIV: The Wikipedia Entry Thought Experiment

Last night, J informed me that he'd managed to hack through the firewall blocking our household computers' IP-addresses from Wikipedia--a ban that resulted from J's repeated "Wiki-vandalism" of their pages over the last school year. (No sooner had I shown him Wikipedia as a great source on black holes, time travel, and the Grandfather Paradox than he figured out he could edit it, thereby entering a whole new arena for mischief--and earning us a 6-month ban).

Once again able to edit Wikipedia articles--at least temporarily--he cautiously added a line or two to the ceiling fan entry about ceiling fan chains (if you pull them too hard, they might break), to the beach house entry (some beach houses have ceiling fans), and to the restaurant entry (some restaurants have ceiling fans).

I'd be surprised if any of J's edits are still there--we've seen how alacritous Wikipedia's established editors are about damage-control. But what is surprising is that J would be surprised as well.

I know this because of the Wikipedia Thought Experiment I conducted on him during the long hikes we took on our summer vacation. One of the things he'd carry on about was his Wiki-vandalism, and after dozens of conversations about this it finally occurred to me to ask him about which of his proposed Wikipedia edits would survive Wikipedia's administrators. For edits like "I am going to kill you," he already knew the answer; but right away he also realized that obvious entries ("some fans are on fast"), trivial entries ("some fans have five blades"), or entries that aren't of general interest ("Ari's house has 10 ceiling fans") also wouldn't endure.

So here's yet another Theory of Mind/perspective-taking exercise for children on the autistic spectrum: along with the Sally-Ann and Smarties Tests, the Wikipedia Entry Thought Experiment.

Tuesday, September 22, 2009

Math problems of the week: 3rd grade Investigations (TERC) vs. Singapore Math

I. From the second 3rd grade Investigations homework assignment (Unit 1, Session 2.1, "Trading Stickers, Combining Coins", p. 38):

Which addition combinations are you practicing?

_________________ _________________
_________________ _________________
_________________ _________________

2. Write two addition combinations that are hard for you, and explain what helps you remember them.
Addition combination: _________________
What helps me:
________________________________________________________________
________________________________________________________________

Addition combination: _________________
What helps me:
________________________________________________________________
________________________________________________________________

3. How did you practice your addition combinations? Who helped you?
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________

II. From the second 3rd grade Singapore Math exercise (Primary Mathematics 3A Standards Edition, Unit 1, "Numbers 10-10,000", p. 11):

Write the missing numbers:

(a) 999 = 900 + ____ + 9
(b) 2658 = 2000 + 600 + 50 + ____
(c) 4955 = 4000 + 900 + ____ + 5
(d) 1773 = 1000 + ____ + 70 + 3
(e) 3332 = ____ + 300 + 30 + 2
(f) 5690 = 5000 + ____ + 90
(g) 6206 = 6000 + ____+ 6

III. Extra Credit:
Consider a third grader child who isn't practicing any particular addition combinations because he or she has been after-schooled using the Singapore Math curriculum, and thus has mastered all addition combinations worth remembering, as well as the standard algorithm for adding arbitrary numbers. How is he or she to complete the Investigations problem set above?

Sunday, September 20, 2009

How to be Cool in Third Grade

This year's 3rd grade summer reading assignment, How to Be Cool in Third Grade, seems not to be having its intended effect on the third graders I know.

The book's message is that being cool isn't about abandoning your superhero underwear and avoiding your mother, but about being nice to your classmates.

But what one boy I know learned from this book was that he should give up his superhero underwear. It hadn't occurred to him, before reading the book, that superhero underwear wasn't cool.

By the time most kids are in third grade, even the less socially savvy ones, the idea that coolness comes from kindness sounds as phony as "why can't people just get along?" does to adults. Indeed, as another boy observed when explaining why he's not popular: "the popular kids are the ones with lots of friends who are mean to everyone."

Friday, September 18, 2009

7th Grade Literacy Assignment: more uninstructed art

Read Shipwreck-The Island Trilogy #1. Write a one-page summary of the events of the book. Create a 3-D image of one scene from the book. This can be on a poster board or in a shoebox, but it must have color. Make sure the details are from the book, and that anyone could easily identify the scene you pick.
Yet another example, as Dawn and Beth write below, of how the "creative" projects, in Beth's words, "are an insult to real creativity and art."

Wednesday, September 16, 2009

Math problems of the week: 7th grade 1920's math vs. Connected Math

TWO WAYS TO THINK ABOUT MATHEMATICS

I. From the beginning of Hamilton's Essentials of Arithmetic: Higher Grades (7th & 8th grades), p. 25:

How to Solve Problems

I. Before you try to solve a problem you must find out exactly what it means. That is, you must consider:
a. What facts are stated or implied in the problem.
b. What kind of answer the question asked for.
c. By what steps the required answer can be found from the given facts.

II. The most important habit to acquire is accuracy. A wrong answer is worthless. Always test your work. Also make a mental estimate of the answer.

III. The second essential is rapidity. To secure rapidity, always choose the shortest method of work where several methods are possible. It is sometimes well to indicate the necessary operations before performing any of them. Then the work may often be shortened by cancellation.

1. Find the cost of 3 3/4 lb. of lamg at $.39 a pound.
Facts stated:
Amount of lamb bought; price per pound.
Question: What is the cost of the total amount bought?
Mental estimate: 3 3/4 lb @ $.39 cost about 3 3/4 × $.40, or $1.50.
Written work: 3 3/4 = 15/4; 15/4 × $.39 = $1.47.
Test: 3 × $.39 = $1.17; 3/4 of $.39 = $.30; $1.17 + $.30 = $1.47.

2. Tell by mental estimates which is greater: 99 × $5 or 100 × $4; 7/8 or 5 3/4 or 6;
1 7/8 × 6 or 7.

3. How much cheaper is it to buy 3 dozen plums at 15 cents a dozen than at 3 for 5 cents?

4. How much is saved by buying 3 1/2 lb. of sugar for 35 cents instead of at 10 1/2 cents a pound?

II. From the beginning of Connected Mathematics 7th grade booklet, Accentuate the Negative, p. 17:

Mathematical Reflections

In this investigation, you worked with positive and negative numbers. You analyzed sequences of events in the MathMania game, looked at temperature, and extended the number line to represent numbers less than 0. You also learned how to decide whether one number is less than or greater than another number. These questions will help you summarize what you learned:

1. Describe what positive numbers, negative numbers, and 0 mean in terms of
a. keeping score in MathMania.
b. temperature readings.

2. Describe how you can compare the following types of numbers to decide which is greater. Use examples to illustrate your thinking.
a. two positive numbers
b. two negative numbers
c. a positive number and a negative number

3. Describe how to locate numbers on a number line. Use examples to illustrate your thinking. Be sure to include positive and negative numbers as well as fractions and decimals in your examples.

Think about your answers to these questions, discuss your ideas with other students and your teacher, and then write a summary of your findings in your journal.

III. Extra Credit: 

Consider accuracy and rapidity in problem solving vs. mathematical reflections. What are your mathematical priorities? Can we have it both ways? Might one problem set do a better job than the other one does of encouraging accuracy, rapidity, and mathematical reflection?

Monday, September 14, 2009

Must secondary school biology be boring?

I wondered this last night as I helped my 7th grader study for his first biology quiz last night.

Over and over again, we went through the organelles and their functions. Lysosomes "digest food particles and foreign invaders;" ribosomes "make proteins"; mitochondria "break down nutrients and makes ATP." Most forbidding-sounding of all, the Golgi Complex "packages proteins and sends them out of the cell."

Why was all this so unsatisfying?

It's not just the memorization, but the meaninglessness of it all. These facts raise more questions than they answer: how/why do lysosomes digest food, ribosomes make proteins, etc.? At this microscopic level, inaccessible to daily experience, what does it mean to break things down and make energy? What *is* ATP?

In other grade school subjects, the how's and the why's are more transparent. When we multiply together two polynomials, we have some sense of why we use Distributive Law; when we learn about the Pilgrims landing at Plymouth Rock, we learn not just the names and the dates involved, but why and how they came; when we discuss the digestive or the circulatory systems, we have real-life observations at our fingertips.

But with 7th grade cell biology, we get little more than a bunch of labels and associated sentences. Because the why's and how's are so nontransparent, because we can reach them only at a great depth of analysis, I'm wondering whether cell biology has any place in middle school, or even high school biology. Why not wait until after chemistry and molecular biology, and even physics? Only after mastering these subjects, I suspect, are students ready to get something more than a list of spelling words and dictionary definitions of the Golgi Complex et al.

After all, with a very few exceptions, things in life only get really interesting after some serious left-brained analysis.

Saturday, September 12, 2009

Myths about left-brain schooling

A recent Harper's Magazine article perpetuates the myth that our schools focus narrowly on academics at the expense of everything else. In particular, writes Mark Slouka, a national obsession with creating jobs and keeping big businesses happy is causing math and science (or what he calls "mathandscience") to rule the school at the expense of everything else.

While Slouka claims that mathandscience has crowded out the humanities, others, faulting No Child Left Behind, have claimed that reading, writing, and arithmetic have crowded out art and music.

Of course, if one were to get out of one's armchair and visit an actual classroom, one would observe that:

1. "math" and "science" classes involve less actual math and science than ever before.

2. while art instruction may no longer exist, neither does penmanship instruction, phonics instruction, or the direct instruction of mathematics.

3. art production is alive and well in the myriad posters, cartoons, dioramas, 3-D models, props, and costumes that students are required to produce for their language arts, social studies, "math," and "science" classes.

Today's students aren't being trained to do much at all, let alone to be cogs in the capitalist machinery. Rather, they are guinea pigs in the Constructivist experiment that dominates the least capitalist sector of our society.

Thursday, September 10, 2009

Math problem of the week: 7th grade Connected Math vs. 6th grade Singapore Math

I. From the first problem set in the 7th grade Connected Mathematics Accentuate The Negative booklet ("Investigation 1: Extending the Number Line"), p. 9:

1. Copy each pair of numbers below, inserting > or < to make a true statement.

a. 53 35
b. -50 0
c. -30 15
d. -70 -90

2. Order the numbers below from least to greatest.

25, 2, 5, -3, 15, -7, -25, 12, 1, -4, 0

II. From the first problem set of the 6th grade Singapore Math curriculum, Primary Mathematics 6A, ("Algebra"), p. 5:

1. A watermelon weighs m kg and a pineapple weighs 2 kg.
(a) Express the total weight of the fruits in terms of m.

(b) If m = 4, find the total weight of the fruits.

(c) If m = 6, find the total weight of the fruits.


2. Sumin bought a pen and a book for $x. The pen cost $5.
(a) Express the cost o the book in terms of x.

(b) If x = 11, find the cost of the book.

(c) If x = 15, find the cost of the book.

III. Extra Credit:

Order the two problem sets in order of:

(a) grade level

(b) difficulty

(c) degree of higher-level thinking

Tuesday, September 8, 2009

Hands-on activities: "Let's just say we did it!"

My daughter is still debriefing from last school year.

She recently described a situation in which her math group was required to work out an arithmetic problem using blocks. Not only did she know the answer without the help of the blocks: the blocks were actually a hindrance because the towers kept falling over. She apparently advised her group that they could cease the frustrating block play, write down the answer, and simply pretend they'd found it with the help of the blocks.

Among other things, today's laborious hands-on "solutions" are promoting deceit.

...And perhaps giving those hands-on methods that actually are effective a bad name.

Saturday, September 5, 2009

Postmodern Math: Does 2 + 2 always = 4?

Postmodern views of math constitute another front on traditional, left-brain-friendly mathematics.

Consider, for example, the following conversation at education.change.org, which I felt compelled to join:

Ira Socol:

One of the reasons teachers have picked up on my "real world math" lesson ideas is that they teach students that math is a series of functions based on agreed upon rules. Two apples only equal two other apples if we agree that many things either matter or do not matter. Or would you trade two dollar bills for two 100 dollar bills? Or do we, for your sake, average all students and treat them as if they are the same. (why isn't a plate appearance an at bat? why isn't a person without a job considered unemployed? why do we think of certain kinds of numbers as different from others?)

Katharine Beals:

Have you asked a mathematician whether they think their field is culturally constructed? I'm willing to bet that you'd find few mathematicians, any where in the world, who would say that algebra, geometry, calculus, differential equations, topology, or applied math (as in physics or engineering) is culturally constructed--except in the trivial sense that math is the product of culture. Math education is another matter entirely; if math education were less culturally constructed, and more mathematical, it would be a better thing for students of *all* cultures.

Ira Socol:

Actually Katharine, the mathematicians I know all understand the cultural construct and the sense of "rules." Once you get past 'high school math' this all becomes obvious. No mathematical system works without a shared understanding of its rules.

Whether you read Kline from 1953 or Lave from 1988 or anyone before or since, you'll get the picture.

Let's take a very simple notion - the "prime number." What makes it "prime"? Yes, you've got the answer, but that requires you to believe that, say, the number "1.3" is somehow fundamentally different than 1.0, 3.0, 13.0, or 130.0. Which is not "a fact" but a cultural choice.

This is why any math major knows that there are different geometries, etc, depending on the choice of accepted rules. We are not really stuck with Newtonian Physics, you know.

Katharine Beals:

If prime numbers are socially constructed, so is the speed of light. Your argument applies equally to this concept. What makes it "the speed of light"? Yes, you've got the answer, but that requires you to believe that say, the speed of 299,792,458 is somehow fundamentally different from, say, 200,000,000 meters per second, or 400,000,000 meters per second. Which is not a "fact" but a cultural choice. (For a much more sophisticated argument on the social constructedness of the speed of light and other physical concepts, see Sokal's Transgressing the Boundaries).


Sidney Andrews:

As a computer science graduate I can tell you with certainty that at an upper-level math elective, you are taught right away that the foundation of grade school math is based on assumptions. I remember the first time we did a proof dealing with polar math that proved everything we learned about geometry in grade school is simply based in baloney.

The basis of all geometry is the rule that for any two points, you can create two parallel lines. When you get to a high-level mathematics curriculum, You are challenged to prove this and it is simply not possible to prove. It is just an agreed upon assumption so that we can teach geometry.

I may not necessarily see it as a cultural difference, but mathematics is taught in a context that is vastly different dependent upon who teaches it or where you are in the world. Values hold constant, but numbers do not.

Katharine Beals:

What you're calling "assumptions" mathematicians would call "axioms". Euclidean geometry uses one set of axioms; Lovbachevskian geometry (where there are no parallel lines) uses another. Mathematicians use both systems; there's no contradiction or baloney. I learned both systems in 9th grade geometry, and was intrigued by their conceptual coexistence. It's actually rather beautiful. Ask a mathematician!

Sylvia Martinez:

Math as an abstract idea may be culture-free, but we don't teach it or test it that way. We test math with the expectation that you have memorized and internalized definitions and rules that we have deemed "age-appropriate".

If you see the question "what is 2+2" on a test, you are expected to not think about anything real, but to answer based on the abstraction of a linear counting system. You are not supposed to imagine about what you are counting (that might be apples and oranges), or if perhaps you are counting non-linear events (like 2 laps plus 2 laps around the track get you exactly nowhere, which could be represented as zero.) But because we teach 2+2=4, we expect students to give us back that very simplistic answer on the test.

Students with any imagination often give "wrong" answers for interesting and potentially correct reasons. We are testing the compliance of the student to accept the teaching, not math.

If you asked mathematicians what 2+2 is, you would get a range of answers, questions, and demands for more clarification. It's hardly cut and dry. I can absolutely guarantee that NO mathematician would answer "4" without qualifying the answer with additional information.

People think that because math is logical, that implies that there is always a "correct" answer for every situation. Which if you think about it, is a very Western way to approach things.

Katharine Beals:

Re 2+2, please provide the names of the mathematicians who would not simply answer "4". I know many mathematicians, and I'm curious whether the ones you say would give a "range of answers, questions, and demands for clarification" are people that I know. I can absolutely guarantee that MANY of the mathematicians I know would simply answer 4; but I have not surveyed all of them.

Ira Socol:

Katharine, you keep asking. Each time you've asked I've not just reached for my bookshelf but put "culture and mathematics" into Google Scholar, pulling up tens of thousands of articles on this issue - most by, yes mathematicians.

It is nothing new. I have a friend who wrote his 1962 Masters Thesis in Math on this subject, and I know it is spoken of constantly in our math education program (which is tied to a 'fairly reputable' mathematics department).

I'd again encourage you to read Morris Kline's work - going back to the early 50s - or to look at the "NYU School" exploring the "fictions of mathematics." I think you will enjoy the conversations.

Katharine Beals:

Ah, but have you searched "culture and mathematics" & "2+2=4"? That's specifically what I was asking about...

Also, there may be a bit of a selection bias in the theses of the articles that turn up under your search... I know many mathematicians (algebraists, analysts, number theorists), but I'm guessing that not a single one who would say that mathematics is, in any way that is deeply significant to math itself, a cultural construct.

Ira Socol:

PJ Davis in Mathematics (1988) provides for you: http://www.people.ex.ac.uk/PErnest/pome22/Davis%20%20Applied%20Mathematics%20as%20Social%20.doc:

Take any statement of mathematics such as ‘two plus two equals four', or any more advanced statement. The common view is that such a statement is perfect in its precision and in its truth, is absolute in its objectivity, is universally interpretable, is eternally valid and expresses something that must be true in this world and in all possible worlds. What is mathematical is certain. This view, as it relates, for example, to the history of art and the utilization of mathematical perspective has been expressed by Sir Kenneth Clark ("Landscape into Art"): "The Florentines demanded more than an empirical or intuitive rendering of space. They demanded that art should be concerned with certezza, not with opinioni. Certezza can be established by mathematics.

The view that mathematics represents a timeless ideal of absolute truth and objectivity and is even of nearly divine origin is often called Platonist. It conflicts with the obvious fact that we humans have invented or discovered mathematics, that we have installed mathematics in a variety of places both in the arrangements of our daily lives and in our attempts to understand the physical world. In most cases, we can point to the individuals who did the inventing or made the discovery or the installation, citing names and dates. Platonism conflicts with the fact that mathematical applications are often conventional in the sense that mathematizations other than the ones installed are quite feasible (e.g., the decimal system). The applications are of ten gratuitous, in the sense that humans can and have lived out their lives without them (e.g., insurance or gambling schemes). They are provisional in the sense that alternative schemes are often installed which are claimed to do a better job. (Examples range all the way from tax legislation to Newtonian mechanics.) Opposed to the Platonic view is the view that a mathematical experience combines the external world with our interpretation of it, via the particular structure of our brains and senses, and through our interaction with one another as communicating, reasoning beings organized into social groups.

The perception of mathematics as quasi-divine prevents us from seeing that we are surrounded by mathematics because we have extracted it out of unintellectualized space, quantity, pattern, arrangement, sequential order, change, and that as a consequence, mathematics has become a major modality by which we express our ideas about these matters. The conflicting views, as to whether mathematics exists independently of humans or whether it is a human phenomenon, and the emphasis that tradition has placed on the former view, leads us to shy away from studying the processes of mathematization, to shy away from asking embarrassing questions about this process: how do we install the mathematizations, why do we install them, what are they doing for us or to us, do we need them, do we want them, on what basis do we justify them. But the discussion of such questions is becoming increasingly important as the mathematical vision transforms our world, often in unforeseen ways, as it both sustains and binds us in its steady and unconscious operation. Mathematics creates a reality that characterize our age.
Other "open" readings on the same issues

http://www.economics.pomona.edu/widner/courses/econ58/ps/whatmath.pdf

http://markelikalderon.com/wp-content/uploads/2006/12/EpistemicRelativism.pdf

http://www.members.tripod.com/~jan_dejnozka/peano_russell_quine_number.pdf

And I'd urge you to go the the library, and if you are resisting Kline, read Reuben Hersh's "What is Mathematics, Really?" (1997)

Katharine Beals:

But he doesn't actually deny that 2+2=4. Find me a math professor (a mathematician, not a philosopher!) who says, specifically, that 2+2 doesn't equal 4 in all possible universes, and I'll eat my hat.

I'm a big fan of foundational problems (computation theory; model theory; incompleteness the unprovability of certain mathematical statements--have you taken courses in any of these?) And I'm all for openness, as long as its meaningful!

Ira Socol:

Kline and Hersh both say this. Read, it'll give you something to chew on.

and
Believe it or not, sometimes 2 + 2 does not equal 4. It depends on what type of measurement scale you are using. There are four types ofmeasurement scales - nominal, ordinal, interval, and ratio. Only in the last two categories does 2 + 2 = 4.

Thus, even on number lines, 2+2=4 is only sometimes true. You need a formal set of, yes, culturally applied, rules to make 2+2=4.

I understand the desire to have something absolute and "always true" in the world. And I am sure it is stunningly frustrating for a pure rationalist to argue with a post-modernist like me, but I challenge you to dig deeper into this.

As Bain's book demonstrates, the shattering of the knowledge system created by K-12 math and science teachers is the first task of professors at top universities.
Katharine Beals:

Apples and oranges! "Nominal scales" use a different definition of number from that used in the statement "2 + 2 = 4"

From the paragraph that follows the one you cite from:
Each number merely represents a category or individual. For example, numbers on baseball or football uniforms are only nominal. Having the number "1" on your uniform does not necessarily mean you are"numero uno" (the best) in your sport. Social security numbers are also nominal. All they do is name or classify the individual.
In making statements like 2+2=4, people are not referring to numbers on uniforms. If they were, then the statement would be no more meaningful than "too plus too equals for"!

Speaking of postmodermism and math, have you read Sokol's Fashionable Nonsense?

Ira Socol:

Wait! A student would have to know your definition of "number" in order to answer that question? I thought this was "absolute" and "culture free"?

But no, I don't read intentionally fraudulent academic writing. I have other things to do with my time.

Katharine Beals:

I've taught math for a number of years, to a number of different students from different cultural backgrounsd, and have *never* encountered a student who needed to know my definition of number (i.e., that I wasn't referring to numbers on uniforms) in order to answer 2 + 2 = 4.

Have you?

Greg Cruey:

Ira says that 2+2 doesn't always have to equal four: it's a qualified truth.

Katherine wants Ira to find a math professor who will stipulate to the idea that 2+2 does not equal 4 (which doesn't strike me as quite what Ira said). And Katherine says that if Ira finds such a math professor, it proves only that the math professor is a closet philosopher. If I were Ira, I wouldn't spend much time looking...

Without investing the time to read Ira's citations (I have a life), I thought he was pretty convincing. And I thought Katherine was circular: all mathematicians agree with her; if they don't, they're not pure mathematicians.

I think we've all agreed that education (its value in a society, its pedagogy, etc.) is culturally defined. The math discussion has been entertaining; I just can't decide if it has a point in the context of Jon's article.

I will say this: the ability of pure mathematicians to articulate great truths abstractly (which I take to mean hypothetically, in the absence of any real context) is something that I see as a cultural exercise in itself. Most great truths can be articulated concretely or abstractly. You can talk about materialism and the nature of reality or you can talk about Plato's Cave. You can talk about Grace or about the Prodigal Son. Some cultures prefer the concrete presentation. Most Western European cultures prefer abstraction. And that preference is in itself cultural. Even for mathematicians, I think...

Katharine Beals:

To clarify, what Ira said was:
If you asked mathematicians what 2+2 is, you would get a range of answers, questions, and demands for more clarification. It's hardly cut and dry. I can absolutely guarantee that NO mathematician would answer "4" without qualifying the answer with additional information.
I then said:
Find me a math professor (a mathemtician, not a philosopher!) who says, specifically, that 2+2 doesn't equal 4 in all possible universes, and I'll eat my hat.
Note that Ira is making a claim about what "NO mathematician would do;" I'm asking him to find ANY mathematician who believes that 2+2 doesn't equal 4 in ANY possible universe.

You're also reading too much into what I said about philosophers. There are some really good mathematician/philosophers out there.

Greg Cruey:

Fair enough.

So you'd take an answer from a mathametician-philosopher?

Katharine Beals:

Yes. The question, again, is whether the statement "2+2=4", when it is a mathematically well-defined rather than a mathematically nonsensical statement (which was implicit in my initial question but which I now feel the need to state explicitely!), is true in all possible universes.

----------

I'm still waiting for an answer from Ira, or Greg, or Sidney.

But, since, like Greg, I have a life, I've moved on...

Thursday, September 3, 2009

Math problems of the week: 2nd grade Investigations vs. Singapore Math

1. From the 2nd grade Investigations (TERC) Stickers, "Number Strings and Story Problems" unit, (assigned in late January):

Assessment: Number Strings

Use combinations that you know to solve these problems.
Show your work.

1.
6 + 3 + 4 + 6 =


2.
7 + 5 + 9 + 3 + 5 =


3.
9 + 6 + 7 + 1 =


4.
6 + 8 + 6 + 7 =


2. From the beginning of the 2nd grade Singapore Math workbook, volume 2 (of 2), Primary Mathematics 2B (Standards Edition), p. 17:

1. Add.

(a) 183 + 99 =
(b) 246 + 98 =
(c) 199 + 99 =
(d) 206 + 98 =
(e) 99 + 556 =
(f) 98 + 235 =
(g) 99 + 408 =
(h) 98 + 399 =

3. Extra Credit

Which problem set gives students more freedom and promotes more discovery learning?