Monday, June 30, 2008

Parents and math homework

An MSNBC article about a recent study showing that 50% of U.S. parents rate their schools poorly asks why parents are saying that schools should spend more time teaching math. The answer they propose:  parents don't want to teach math at home.

In a subsection headed with this very (bold-faced) subtitle, the author quotes a certain education professor as saying that:

Parents may want more math in school because they feel unprepared to help at home.

and that:

"Math is the subject that parents are often intimidated by. We've allowed a lot of kids to just say, 'I'm not good at math,' .... and those kids become parents."

Said professor, one of whose interests is homework "as a boundary object that links home and school practices," overlooks the following:

1. Raising a generation of kids who think they're good at math but are actually poorly prepared is far worse for future generations than raising a generation of kids who esteem themselves more critically but are better prepared.

2. Reform Math programs prepare kids much more poorly than traditional math programs do, as I've argued here, here, and here.

3. Parents at Reform Math schools, including the "model school" my kids attend whose curriculum said professor has hand-picked, are assuming more, not less, responsibility for teaching their kids at home--precisely because they're so dismayed by the math curriculum.

4. In fact, it was only a few days ago that yet another parent asked me how to get her hands on Singapore Math.

5. The only complaints I've heard from parents about doing math at home concerns the Reform Math homework (the so-called "liaison object"), which is often so poorly explained and so apparently unrelated to actual math that many parents have no idea what they and their kids are supposed to be doing.

6. My parents never had trouble understanding my math homework, nor were they were expected to be anywhere near as involved in helping their kids as I am (all those math games, "applied math" arts and crafts activities, and "math projects.").

No, today's parents aren't shying away from teaching their kids at home; in fact, we are doing more of it--perhaps more than any previous generation. At least those of us who have the time, resources, and education to do so (there are some disturbing class issues here).

Furthermore, much of what we do is above and beyond what schools are asking us to do. Indeed, many of us view what we do as unsanctioned, on the sly, possibly subversive. "I taught my daughter how to borrow and carry last night; I hope that was OK," the parent of a mathematically under-challenged 7-year-old recently confessed to me.

My concern is about the next generation of parents, too many of them Reform Math graduates, not all of them enriched by their parents, who won't be able to do for their kids what these intrepid parents are doing.

And about the widening class rifts that all this so-called "progressive education" is causing.

Friday, June 27, 2008

Minimizing grammar in autism therapy: in whose best interests?

My collaborator just reported to me her experience at this year's IDC (Interaction and Design) Conference at Northwestern University.  It seems our paper was controversial. Apparently there were a number of staunch ABA supporters in the audience. And our paper apparently offended them by making favorable mention of Noam Chomsky and his critique of behaviorist approaches to language.

Chomsky's work has highlighted the complexities of grammar and argued that this complexity cannot be acquired through external stimulus alone.  So compelling have Chomsky's arguments against Skinner and his behaviorist approach been that, for decades now, no serious linguist takes Skinner seriously.

Chomsky is bad news for the two most popular therapies that purport to teach language to autistic children:  Floor Time (DIR) and ABA (Lovaas/Discrete Trials).  Neither approach acknowledges the complexities of grammar, and most practitioners lack the linguistics training necessary to appreciate it.

Instead of taking Chomsky seriously, devotees of both approaches, traditionally rivals, have made him their common enemy.  

Greenspan has co-authored a book, The First Idea, in which he tries to argue that Chomsky is wrong about innate grammar acquisition modules, and that language is acquired entirely through nurturing and socio-emotional reasoning. 

And ABA supporters, at conferences like IDC, dismiss papers that:

1. take seriously people like Chomsky who criticize behaviorism;
2. suggest that there's such a thing as complex grammar; and 
3. propose that that there are, just possibly, more principled ways of teaching grammar to children with autism than through discrete trials of stimulus-response.

As with too many education experts, so too with too many autism therapists:  in whose best interests are they acting?

Thursday, June 26, 2008

Math problems of the week: Grade 4 Trailblazers vs. Singapore Math

1. From the beginning of the Trailblazers Grade 4 Student Guide (p. 63):


Dear Family Member: 

Your child is learning the multiplication facts for 5s and 10s. He or she might find skip counting (5, 10, 15, 2, etc.) helpful in solving the problems below. Remind your child to bring home the flash cards for the 5s and 10s. Help him or her study these handy facts. Thanks for your cooperation.

1. How much is 5 dimes?

2. How much is 8 nickels?

3. How much is 4 dimes and 4 nickels?

5. Jerome placed 5 plates on the table for dinner. Each plate received 4 pieces of silverware (fork, spoon, and knife). How many pieces of silverware were placed on the table.

6. Write a story to show 5 + 9. Draw a picture to do with your story. Write a number sentence on your picture. 

7. Skip count from 0 to 100 by 5s.  Write down the numbers as you count.

8. Skip count from 0 to 20 by 10s.  Write down the numbers as you count.

2. From the beginning of the Singapore Math Grade 4 workbook (Primary Mathematics 4a) (p. 24):

Write the missing factors.  Use only prime numbers.

(a) 64 = 2 X 8 X ____ X ____
(b) 84 = 6 X ____ X ____
(c) 45 = ____ X ____ X 5
(d) 72 = ____ X 4 X  ____ X ____

Write the missing factors represented by n.
(a) 24 + 3 X 2 X n
(b) 18 = 3 X n X 2
(c) 25 X 4 = 5 X n X 4
(d) 21 X 20 = 21 X  2 X n
(e) 3 X 32 = 6 X n
(f) 16 X 2 = n X 4

Tuesday, June 24, 2008

Any summer projects to share?

Has your school assigned your child an excessively complex, open-ended, off-topic, and/or demanding of time and/or "creativity"? If so, please share it here as a comment. Having excerpted my own children's summer project assignments here and here, I'm hoping to post an anthology of additional summer project assignments in a later entry.

Monday, June 23, 2008

Summer math projects: grade 6

Then there's the 6th grade summer math project that my autistic son will have to do.

I initially misread this as "6th Grade Summer Project." 

Nope:  it's "6th Grade Summer Math Project"


A Festive Meal

For this project, you will help to plan and prepare a festive meal with family and friends.  It can be for a special occasion, like a holiday barbecue or a birthday party, or just a nice dinner at home.  You must:

-Assemble your project as a booklet or on a poster.
-Write a brief introduction that describes the event--What was the occasion? When did it take place? Where was it held? Make a list of everyone who came to the meal.
-Show the menu for the meal.
-Make an itemized list of everything you bought for the meal showing how much each item cost and find the total, or you can include the receipts from stores.
-Write down a recipe for a main dish, dessert or salad that was served at the meal.
-Save the label from a purchased item and copy everything included in the "Nutrition Facts."
-Describe something from the meal that involves numbers.  Be creative. For example, how many candles were on the cake? How many miles did your cousin from out-of-sate have to drive? How many coals were in the grill? How many cookies did you eat? How many trash bags did you need when you cleaned up? Etc.
-Include photos if you have them or draw an illustration.


Of special interest:

1. The exhortation, ubiquitous with this sort of open-ended project, to "be creative."

2. The ratio of effort to learning.  Particularly if we factor in the efforts of parents and factor out that which has nothing to do with math.

Saturday, June 21, 2008

Summer math projects: grade 2

Accompanying my daughter's report card yesterday was the following assignment for a "summer math project":

*Select a community worker/helper. Write 3-5 sentences about he or she uses math at work. Provide an illustration or model. Make it colorful and be creative. :-)

*Design a phone directory. Include the fire department, police department, hospital, library, school, ____, ____, (The blank lines represent two additional choices that the student should provide for the directory [e.g., church, neighborhood store, community center]).


My daughter is rather shy, and I'm having trouble thinking of an appropriate "community worker/helper" for her to interview.

Then it hits me: her grandfather!  A retired--but active--math professor, he might have some interesting things to say about how he uses math at work.

...As well as whether "math projects" such as this one make students better mathematicians.

Friday, June 20, 2008

My daughter's grades are in: music and shyness don't mix

Good grades in reading, writing, and even in math and science (where she's learned to play by the language-arts rules).  Then there's music and social studies.

It's perhaps no surprise that a private, unsocial child earns mediocre grades in the kind of social studies class that increasingly asks students to make "personal connections."

Less obvious is how someone who's been picking out tunes on the piano since she was three, sings complicated songs with perfect pitch, and, after 9 months of piano lessons, plays minuets with precision and feeling, consistently gets mediocre grades in music class.

I don't think that elementary school music teachers gave out grades back when I was in school. Now that they do, the easiest thing for them to base them on is participation:  singing along with your classmates.

And there is perhaps nothing that embarrasses my shy, musically sensitive girl more than public singing.

Wednesday, June 18, 2008

Math problems of the week: Grade 1 subtraction in Investigations vs. Singapore Math

My daughter just finished up a year of 1st grade Investigations, and (at home) a year of 1st grade Singapore Math.  In honor of this, let's compare how far each curriculum gets with subtraction:

I. From the final subtraction problems in 1st grade Investigations (Today's Math, p. 142; p. 153):

There are 4 butterflies on a leaf.   
Three of the butterflies fly away.
How many butterflies are on the leaf?

1. Are there more or fewer butterflies on the leaf at the end of the story?

2. How many butterflies are on the leaf at the beginning of the story?

3. How many butterflies fly away?

4. How many butterflies are left on the leaf?


There were 30 grapes in the bowl.  
Chester ate 12 of them.
How many grapes are left?


II. From the final subtraction problems of 1st grade Singapore Math (Primary Mathematics 1B, p. 180; p. 184)


(a) 39 - 28

(b) 55 - 11

(c) 49 - 28

(d) 67 - 43

(e) 75 - 30

(f) 96 - 83


Juan has 54 marbles.
He gives away 6 of them.
How many marbles does he have left?


Extra Credit (parallel to last week's assignment):

Enumerate the math skills involved in each problem.

Explain which problem set involves less hand-holding by authority figures and more student-centered discovery.

Discuss how the two problem sets reflect the cultural and political differences between American and Singaporean societies.

Tuesday, June 17, 2008

Middle school: the science that might have been and the pseudo-science that beckons

I'm just back from a meeting with the middle school teachers who will be teaching my mainstreamed autistic son next year.  They're bright, energetic, and well-intentioned.  And they're caught up in all the latest educational trends that have so shortchanged children on the autistic spectrum.

Massive, interdisciplinary projects.

Group work.

Extensive oral and written "communication" in all disciplines, including math and science.

As the science teacher explained to me, "It's not like we go in and do an experiment, and then the next day we go in and do another experiment." Subtext:  doing experiments is somehow suboptimal. Too narrow and rigid? Too coldly analytical? Too authoritarian a notion of truth? Too outdated a notion of science education?

Rather, he said, virtually quoting chapter and verse from the National Science Education Standards, the important thing is communicating about science, presenting science to others, working in groups, and integrating science with daily life and with the other middle school subjects--especially the subject now known as "literacy."

"And will literacy class cover topics from science?" I was tempted to ask.  But I already knew the answer.

And I was distracted by the dread of navigating my son through all this language intensive pseudo-science. And by the longing for all that science class might have been for my budding natural scientist, who loves doing experiments and learning about how things work, but doesn't work well in groups and takes forever to read a chapter, write an essay, and design an acceptable poster.

So suddenly we're discussing incentives, rewards, and punishments.  We talk about Hershey's kisses and after-school detentions. 

"Aren't there some intrinsic rewards that would work for you son?" someone finally asks.

"Yes," I reply.  "Pure science, and pure math."

Too bad such rewards are no longer intrinsic to school.

Sunday, June 15, 2008

Where right is left and left is right

Both George Lakoff's new book, The Political Mind, and Leonard Boasberg's review, which appears in this weekend's Philadelphia Inquirer, make claims that further perpetuate anti-left-brain stereotypes.

The first is to claim that political campaigns based on analyzing facts, figures, and policies don't work, because most people, quoting Boasberg on Lakoff, think in "frames, metaphors, and symbols." 

From here, it's a slippery slope to giving up on rational debate altogether.

The second is to equate, again quoting Boasberg on Lakoff, "the essence of progressive morality" with empathy, or nurturing parents, and "conservative thought," which Lakoff considers "profoundly antidemocratic," with obedience to an authority figure, or "decider," assumed to know "right from wrong."

From here, it's a slippery slope to saying that any commitment to objective analysis and correct answers, particularly if it involves criticizing instead of nurturing, is inherently reactionary and antidemocratic. Particularly since so many people in the worlds of education and the humanities have been promulgating such notions for years.

Of course, that's why Lakoff's political metaphors serve him so well.  

On the other hand, they ignore:

1. The free-market and libertarian wings of the political right, which are more anti-authoritarian than most progressives.

2. The transcendent nature of the cult of personality, which, historically, has arisen as much on the political left as it has on the political right.

But would a book equating left-brain thinking with leftist politics and right-brain thinking with rightist politics sell as well as Lakoff's?

Saturday, June 14, 2008

Fact, facts, facts

Moments ago, my daughter finished her 39th Magic Tree House book (and will have to wait until September for number 40).  Besides meeting her at her zone of proximal development, with plot-driven stories and minimal psychological complexity, they've filled her with just what she needs to cope--in school and in Life.

Namely, facts.

Facts bout dinosaurs, dingoes, and octopuses; about pirates, knights, and cowboys; about the rain forest, the outback, the arctic, the ocean, the prairie; about the Greek olympics, Vesuvius, the civil and revolutionary wars; about Thomas Edison, Leonardo DaVinci, and William Shakespeare.

All that cultural currency that she's too often too spaced out to pick up from unstructured environments like school (and Life), but does absorb through focused reading.

All that cultural currency that schools no longer teach explicitly, but that surfaces nonetheless in a fleeting, haphazard kind of way--in class discussions, in social studies, and in reading assignments. (Not to mention Life).

The more cultural currency my daughter has in her pocket, the greater the number of familiar references that can attract her fleeting attention in unstructured learning environments.  

Stuffing her with facts--far from squelching her creativity and higher-level thinking--thus precipitates the following chain reaction:

The more attentive she becomes (drawn in by all those familiar references), the more she listens. 

The more she listens, the more she realizes the virtues of listening.

And vice versa...

And the more she listens to others, the more she ventures out of her own world--however creative and fantastical it is--into those that her imagination has not yet dreamt of.

Any suggestions on what to read next?

Thursday, June 12, 2008

Autism pride

A recent ABC news story showcases the Autism Self Advocacy Network:

...[A] non-profit group aimed at advancing autism culture and advocating for "neurodiverse" individuals.

Their message, according to ABC:

Stop the search for a cure and begin celebrating autistic people for their differences.

In the words of founder Ari Ne'eman, a 20-year-old man with Asperger's Syndrome:

We believe that the autism spectrum and those on it, are important and necessary parts of the wide diversity present in human genetics.

The Autism Self Advocacy Network increasingly includes parents, like Kristina Chew, mother of a severely autistic boy named Charlie.  Quoting ABC:

  Chew now believes that autism treatments and so-called cures are a waste of time. She said she'd rather see Charlie, now 11, benefit from better support services and education.

In Chew's own words:

My son is who he is. He's not going to change; he's always going to be Charlie. And at the same time, I loved him just for what he was.

But isn't autism a severe disability?  To this, Ne'eman retorts:

Where does disability come from? It comes, in many respects, from a society that doesn't provide for an education system that meets our needs. From people who often discriminate or bully or even injure us, and from a society that is largely intolerant.

OILF's assessment:

1. Autism is an extremely heterogeneous disorder.  Those most able to speak out and self-advocate represent only a tiny portion of the spectrum.

2. In particular, such people cannot represent the interests of those with more severe autism, or the concerns of their parents.

3. Many parents do spend too much time pursuing fruitless "miracle cures."

4. Some are too quick to give up on more promising treatments like education.

5. Except that, as Ne'eman notes, and I've argued here, here, here, here, and here, we have an education system that decreasingly meets the needs of those on the autistic spectrum.

6. Which, in turn, exemplifies how society discriminates, in particular, against those at the milder end of the spectrum, via bullying by peers, excessive labeling and medication by professionals, and marginalization of their analytical and calculation skills in math and science by so-called education experts.

The Autism Self Advocacy Group should steer clear of those for whom they cannot speak, and focus its efforts on points 5 and 6.

Wednesday, June 11, 2008

Math problems of the week: grade 5 Investigations vs. Singapore Math

1. From grade 5 Investigations, Investigation 7, "Name that Portion:"

a. For the first game of the session, 100 people came, and 1/4 of them walked.  How many people walked?

b. For the second game of the season, 200 people came, and 1/4 of them walked.  How many people walked?

c. What fraction of the 100 people did not walk?  What fraction of the 200 people did not walk?

d. How many of the 100 people did not walk?  How many of the 200 people did not walk?

2. From grade 5 Singapore Math (workbook 5a) exercise 13 of "Fractions:"

After spending $30 on a dress, Mary had 3/8 of her money left.  How much money did she have at first?

4/7 of a group of children are boys.  If there are 18 more boys than girls, how many children are there altogether?


Extra Credit:

Enumerate the math skills involved in each problem.

Explain which problem set involves less spoon-feeding by authority figures and more student-centered discovery.

Discuss how the two problem sets reflect the cultural and political differences between American and Singaporean societies.

Monday, June 9, 2008

Science as broadcast journalism

In your journal, you are a local newscaster reporting the news. During your broadcast, you receive information that a major earthquake has struck a city in another part of the country. You must interrupt the regular news to present an emergency news bulletin. Write that bulletin, following the five W’s. [What, Where, When, Who, and Why]

From the Science Explorer series, published by Prentice Hall and intended for students in middle school.

Would Brian Greene approve?

Sunday, June 8, 2008

Innumeracy at the New York Times

If a woman has two children and one is a girl, the chance that the other child is also female has to be 50-50, right? But it’s not... The possibilities are girl-girl, girl-boy and boy-girl. So the chance that both children are girls is 33 percent. Once we are told that one child is female, this extra information constrains the odds. (Even weirder, and I’m still not sure I believe this, the author demonstrates that the odds change again if we’re told that one of the girls is named Florida.)

From George Johnson's review of Leonard Mlodinow's The Drunkard’s Walk: How Randomness Rules Our Lives, in this Sunday's New York Times Book Review.
The fact that the Book Review editors chose the first sentence of the above excerpt as a call-out sentence (in the paper edition) is especially alarming.

Friday, June 6, 2008

Right-brained epiphanies, Part III: selling science to the masses

So strongly right-brained is modern-day American pop culture that even scientists--perhaps on the insistence of their publishers and publicists--are marketing their field as a predominantly right-brained enterprise.

This past week, we have Brian Greene, in his New York Times Op Ed, marginalizing the view of science as a logical, linear progression of technical problems like solving equations, balancing reactions, and "grasping the discrete parts of a cell." 

"[S]cience is so much more than its technical details," he argues. Indeed, such details are dispensable: "cutting-edge insights and discoveries can be clearly and faithfully communicated to students independent of [them]."

Greene recasts science as artistic, literary, emotional, and holistic, focusing on its "breathtaking vistas," its distinction as "the greatest of all adventure stories," its equivalence to a "language of hope and inspiration," and its capacity to "instill a sense of connection to our lives and our world" and transport us "out beyond the stars."

Then, in today's New York Times, we have a review of Sherwin Nuland's latest book on medicine, suggestively entitled The Uncertain Art, in which we see both author and reviewer zealously dismiss the reliance on technology and the scientific method, and recast medicine as an intuitive, humanistic "art" that extends beyond the logic of "Western" science.

Reviewer Barry Gewen touts Nuland as striving "to undermine smug certainties about modern science," with its "unreflective reliance on technology," and "restore the doctor-patient relationship, the touchy-feely human connection, to the center of medical practice:"

Doctors, he insists, have to be more than technicians. They should be, first of all, humanists, intuitionists, appreciative of each patient’s individuality and particular situation, practitioners of a quirky, unpredictable, uncertain art. True healers understand this. “To become comfortable with uncertainty,” Dr. Nuland writes, “is one of the primary goals in the training of a physician.”

Gewen is most taken with Nuland's discussion of miracles that "can't be explained by current scientific research and perhaps never will be:" acupuncture, electroshock therapy, and the placebo effect.  Here, muses Nuland:

Perhaps philosophies may be required beyond those that have been so successful since the scientific method became a major current of Western thought.

I doubt that any sane scientist, in the privacy of his or her lab, would abandon the scientific method just because there are things he or she doesn't yet understand, or dismiss technical details just because there are larger phenomena that emerge from them, or insist on dichotomies between details and wonder, technology and humanism, and uncertainty and the dogged search for answers. 

It's just that those who wish to sell science--or themselves--to the right-brained public seem to think it best to pretend otherwise.

Thursday, June 5, 2008

Please visit an actual classroom before you make recommendations

One of the biggest obstacles to rescuing our classrooms from the impoverished reforms that education experts have enacted is lack of public awareness.  
One culprit:  the many public intellectuals who speak out as if completely unaware of what's changed, perpetuating long-outdated stereotypes of public school classrooms.  

Worse, echoing the education experts, some of these wise men don't hesitate to make recommendations based on such stereotypes.

Here, for example, is what best-selling cosmologist Brian Greene has to say, in this past Sunday's New York Times Op Ed., about science education:

A great many studies have focused on this problem [of children losing their interest in science], identifying important opportunities for improving science education. Recommendations have ranged from increasing the level of training for science teachers to curriculum reforms.

But most of these studies (and their suggestions) avoid an overarching systemic issue: in teaching our students, we continually fail to activate rich opportunities for revealing the breathtaking vistas opened up by science, and instead focus on the need to gain competency with science’s underlying technical details.

Greene claims that:

...our educational system fails to teach science in a way that allows students to integrate it into their lives.

His conclusion:

We must embark on a cultural shift that places science in its rightful place alongside music, art and literature as an indispensable part of what makes life worth living.

If Greene were to enter any number of today's grade school science classes, he wouldn't see a focus "on the need to gain competency with science’s underlying technical details.” Rather, he'd see a botched attempt to implement precisely what he suggests.

As practiced in actual classrooms, this means things like having 5th graders read Dr. Seuss's "The Lorax" (with its "breathtaking vistas" of Truffula trees), choose something they care about as much as the Lorax does his trees (thus integrating "science" into their lives), and give presentations in which they speak for that something, complete with colorful props and costumes (thus placing science alongside art and literature).

Perhaps if schools did focus more on science competency, students would be more interested in science, more filled with scientific wonder, and more able to take college and graduate level classes which tackle the big questions.

Wednesday, June 4, 2008

Math problems of the week: grade 3 Investigations vs. Singapore Math

1. From the end of Today's Math 3, the Investigations workbook for grade 3:

How can 6 people share 8 hot dogs?  You may draw a picture to help you.  Then explain your answer.

How can 6 people share 8 pencils?  You may draw a picture to help you.  Then explain your answer.

2. From the end of Primary Mathematics 3B, the Singapore Math workbook for grade 3:

Give your answer in its simplest form.
1/8 + 5/8
1 - 7/12
9/10 - 5/10
2/5 + 1/5 + 2/5

Write each amount of money as a fraction of a dollar.

Perhaps there's a pencil shortage in Singapore, where the schools don't give children opportunities to investigate, and reflect on, how breaking pencils into pieces isn't the best way to share them. 

But perhaps the Singaporeans' superior quantitative skills training makes them better than we at manufacturing, distributing, selling, and buying new pencils.

Tuesday, June 3, 2008

Math projects and children with autism:

I just finished talking with a fellow "autism mom," whose son, like mine, attends our local public school. She told me how stymied he is by the middle school math projects.  

The organizational and the language arts requirements, the lack of structure and clear expectations, the general exhortation to "be creative"--all these pose problems for her son.

His most recent math projects included designing a playground, and designing a city.  He was stumped, and she ended up doing most of the work.

My son fares no better.  His most recent project:  

Design a board game using everything you know about math.  Make it colorful.  Be creative.

Stymieing another autistic spectrum boy:

Choose a number and design a wanted poster about it.

He eventually chosen the number 7, "wanted for being odd."

Perhaps these kids would feel less odd about math if it tapped their strengths (math) rather than their weaknesses (organization, language arts, graphic design).

Monday, June 2, 2008

How to earn high grades in Reform Math

I just got back from a parent-teacher conference with my daughter's 1st grade teacher in which, following my earlier bafflement, I finally learned the answer to this question.

-It's not enough to get the right answer.  (Trivial, since Reform Math math is so easy).

-It's not enough to be mathematically advanced.  (Teachers don't assess skills that exceed the low, state-mandated standards).

What matters is whether/how you explain your answers.

We're talking about problems like 7 + 8.

You can say things like:  

I subtracted 1 from the 8 to make it 7, and then added 7 and 7 to get 14, because I know my doubles, and then added 1 to get 15.


I added 1 to the 7 to make it 8, and then added 8 and 8 to get 16, because I know my doubles, and then subtracted 1 to get 15.

But you can't say:

I just know that 7 + 8 is 15 because I've done this problem so many times that I've memorized the answer.

Apparently, my daughter is now providing acceptable explanations--if not about how she actually solved the problem, then about how she would have solved the problem had she not been stricken with that unfortunate side-effect of repeated exposure to addition problems.

Namely, rote memorization of addition facts.