There are six IQ blocks shown on the chart. From left to right:
Block One: IQ below 71
Block Two: IQ between 71 and 90
Block Three: IQ between 91 and 100
Block Four: IQ between 101 and 110
Block Five: IQ between 111 and 130
Block Six: IQ above 130
For each IQ block the mean standard score has been graphed at the end of grades 1, 2, and 3.
There are arrows (<, <<, <<<) along the Y axis (mean standard score) that show the national median for each grade. I've (helpfully) drawn a blue line at the third grade national mean, as you can see, only the kids in blocks with IQs above 100 are performing above about the national median for math and only those above 110 for reading. (The blue line only has meaning with respect to the third grade scores (the top point). You could draw horizontal lines from the double arrow (second grade) and compare it to the middle point and from the single arrow and compare it to the bottom point.)
Click on each chart to enlarge.
This chart is for total reading for the Metropolitan Achievement Test.
This chart is for total math for the MAT.
Here is Becker's interpretation of the charts:
The data showed almost no contribution to "learning rate" (pretest to posttest gains) for IQ. If IQ were correlated with gains, lower-IQ children would make smaller gains and higher IQ children would make larger gains. This does not happen for Reading on the Wide Range Achievement Test (decoding) [Ed: Not shown.] or comprehension on the MAT, there is no IQ effect gains from the end of grade one to the end of grade two (most of the gains are about equal), but there is an effect for the gain from the end of grade two to the end of grade three. I believe this effect is due to the fact that the end of third grade test for Reading Comprehension on the Metropolitan uses an uncontrolled, adult-level vocabulary (as found in fourth grade texts). Since vocabulary instruction in school does not progress gradually to the adult level (but jumps from a carefully controlled vocabulary to an adult vocabulary after third grade), the test at this level is now measuring something not taught in school. Thus, students who score higher on a test of verbal skills (IQ) do better on a test of verbal skills (Reading Comprehension) when the content was not systematically taught in school. (A caution: The data may have imposed a ceiling effect on the brighter students; the program stressed preventing failures and thus teachers may have given more effort to teaching lower performers. Even if this is the case, however, the data are noteworthy in showing what can be done "gainwise" for lower-IQ children.)
Here is my observation. I understand Becker's comparable gains argument, but look at the mean percentile ranks for each IQ block:
Math End of Third Grade
Block One (IQ below 71): 24th
Block Two (IQ between 71 and 90): 39th
Block Three (IQ between 91 and 100): 47th
Block Four (IQ between 101 and 110): 61st
Block Five (IQ between 111 and 130): 69th
Block Six (IQ above 130): 88th
Reading End of Third Grade
Block One (IQ below 71): 11th
Block Two (IQ between 71 and 90): 29th
Block Three (IQ between 91 and 100): 34th
Block Four (IQ between 101 and 110): 44th
Block Five (IQ between 111 and 130): 58th
Block Six (IQ above 130): 81st
Also notice the gradual slippage from first to third grades in Reading even for the smartest kids. There is no slippage in math. Interesting.
I don't see how the lower IQ kids are going to be able to learn in a regular classroom given these percentiles. That would seem to foreclose a college education for these students and probably an academic high school education. Am I wrong?
And for the Broader, Bolder crowd, given that many low-SES students have lower IQs and that SES inerventions have not been able to to show a significant effect on IQ past about third grade, how exactly are your proposed SES interventions going to get around this IQ conundrum. Look the high-IQ, low-SES kids are performing well. The low-IQ ones aren't. I'd like to hear a rational argument that makes sense of this.
37 comments:
. I'd like to hear a rational argument that makes sense of this.
That's simple Ken, and if you reflect for a moment you'll most likely recall that you've run into the explanation countless times - IQ is just a number and it doesn't really exist.
So in a sense your analysis is using a fictional factor and is thus meaningless.
Now let's talk about what we can all do to boost children's self-esteem.
It was but twelve days ago that our good friend Stephen Downes made that preceise argument.
tangoman,
were you serious about your last comment about self-esteem?
why not attack the problem directly and boost their academic achievement, which will boost their self-esteem?
it's hard to feel good about yourself when you are not proficient.
and I've seen full DI schools where practically everyone achieves at or above grade level in reading by 8th grade, without cherry-picking, red-shirting, or other such enhancements
Anon, Tangoman was kidding about self-esteem.
But more seriously, could you provide a litle more info about your school, which doesn't necessarily involve disclosing its name. We don't have much data on DI schools up to the eighth grade level.
Here's a start:
Is it a public school with open admission or a private/charter school with selective admission (selective admission would include merely having to obtain nd fill out an admission form)?
What's your measure of grade level.
What do you do after RM6 and CMC-F? How do you get them from a grade five reading level to a grade eight reading level? What do you use for algebra? (I assume that any student completing CMC-F is prepared for algebra).
What is the transition, assuming there is one, from DI to a more traditional pedagogy? HOw do the lower-performers handle it?
Anon, Tangoman was kidding about self-esteem.
Are you sure I was kidding? After all we know that marking with purple markers, instead of red markers, is a popular theory being implemented in order to improve student self-esteem.
Hey, if it's popular then it can't be wrong, right?
I was giving you the benefit of the doubt.
If I reconsider I'll make sure I make that correction using a purple typeface, so it doesn't come off as overly-judgmental.
Ken,
A couple questions.
You say: “I don't see how the lower IQ kids are going to be able to learn in a regular classroom given these percentiles. That would seem to foreclose a college education for these students and probably an academic high school education.”
First, which group do you mean when you say “the lower IQ kids”? Are you referring just to the Block One kids (i.e., IQs below 71), or to additional blocks?
Additionally, I’m not understanding the logical step from “these students scored at X level” to “these results foreclose a college education and probably an academic high school education”. What do you see as a baseline level of reading and math achievement necessary to take (and pass) high school academic courses and to go on to college (and would you define a baseline in terms of percentile rankings, or in terms of some absolute standard, such as a scale score passing level)? What evidence exists to support these baseline levels?
Parry
Parry,
I was referring to the bottom three groups. The 91-100 group might be able to struggle by in math, but those reading comprehension scores are fairly low and will likely be a high water mark if current trends are any guide.
And bear in mind that I'm sure some of the students in these groups scored well, but those scores are hidden by the averages. IQ is not destiny, there are other important environmental factors. By the same token, some of the students in the higher groups most likely scored poorly.
My logic is that standardized tests correlated highly, so I'm thinking that a student that is performing at the 34th percentile in reading comprehension at the end of third grade, and is learning at a slower rate thatn his peers, is not going to score any higher than the 34th percentile on the SAT or ACT and therefore will not be ready for college. AT best they'll be placed in remedial classes and the likelihood of passing those are exceedingly slim.
Hmmm, I took away the opposite reaction, especially in math. You've got low SES kids with below average IQs (especially the 71-100) group, and you're getting them pretty darned close to scoring at the national average, with a good slope to that line. For Group 2, it also looks like they've closed the distance a little between the end of 1st grade and the end of 3rd grade.
What is 34th percentile for the SAT? What evidence exists to say that a kid scoring at the 34th percentile on the SAT isn’t ready for higher learning?
If anything, these graphs look like the best argument I’ve seen for DI. My take-away is, you can have a kid from a disadvantaged background with an 85 IQ, and he’ll still make steady academic progress from year-to-year, learning at least at the same rate as his higher SES peers in math (which is often the gateway subject for academic scheduling in middle and high schools). And he’ll even outperform a third of his peers in math. I’ve seen plenty of kids who score at the 34th percentile on our state standardized test (NC) who are on grade level by a criterion-referenced standard.
I guess part of the question is (and I don’t know the answer), where’s the score cut-off on the MAT that roughly corresponds to grade-level proficiency?
Parry
"What is 34th percentile for the SAT? What evidence exists to say that a kid scoring at the 34th percentile on the SAT isn’t ready for higher learning?"
The 34th percentile for the kids that *take* the SAT would be a low-900s score (math plus verbal). But I'm going to guess that most high school dropouts don't take the SAT. In fact, not all high school students take the SAT ...
If I had to guess, I'd guess that 34th percentile here would translate into an 800-850 SAT score.
-Mark Roulo
"If I had to guess, I'd guess that 34th percentile here would translate into an 800-850 SAT score."
A followup ...
From a VDare article (I didn't fact check it ... but the claim sounds plausible): "For example, at Cal State Northridge in the San Fernando Valley, the average SAT score is 930 and only 25 percent graduate within six years..."
http://www.vdare.com/Sailer/070128_scores.htm
-Mark Roulo
Why am I doing this?
Okay ... more data on SAT and graduation rates from a University of Hawai'i Report:
University of California:
Average SAT: 1223
6-year graduation rate: 75.9%
Cal State University:
Average SAT: 988
6-year graduation rate: 37.7%
UH Manoa:
Average SAT: 1095
6-year graduation rate: 54.0%
http://www.hawaii.edu/iro/adhoc/grssatavg.pdf
I'll suggest that an 800-850 SAT score means *maybe* a 10-15% chance of graduating in 6 years. I also suggest that the degree obtained won't be one requiring math (so no engineering, science, math, etc) and won't be pre-law or pre-med.
-Mark Roulo
Mark,
Thanks for the report on SAT scores vs graduation rates.
I once had a school administrative official tell me that SAT scores don't really matter because they aren't correlated with grades in college.
This helps explain where they DO matter.
"I once had a school administrative official tell me that SAT scores don't really matter because they aren't correlated with grades in college."
I'm not sure that I do believe this, but to the extent that it is true, it can only be true because of compression-of-range.
Consider this thought experiment: Next year MIT (or CalTech) takes 100 students who score 350 on the Math SAT. Does the administrator *really* think that these kids won't be at the bottom of the class (assuming they don't flunk out)?
Given that most colleges have a restricted range on students (the school is looking for test scores above some value ... different value for different schools. The students that get in to both Stanford and Rural State University go to Stanford, so this puts an effective *upper* range on the SAT scores at a given college), I'm willing to believe that *within* a college, SAT scores aren't completely correlated with GPA. But ... I also note that the "smarter" students seem to go into the harder majors, too (e.g. Engineering instead of Ethnic Studies).
So ... given that a college will have most students within a few hundred SAT points of each other (the Hawai'i study reports that UC students in the 25th to 75th percentile of SAT score ranged from 1083 to 1311 ... but UCBerkeley has a different student body from UCIrvine) *and* that students self select into majors (if you are flunking engineering, you might go into economics and do fine ... I doubt that very many flunking econ majors move to engineering), there might be only mild correlation between SAT scores and GPA at a given campus.
But the implication in the claim above is that SAT scores don't tell very much at all about college academic success. This is clearly nonsense.
-Mark Roulo
I'm no data analyst, but it looks to me like the kids with lower IQs continue to get closer and closer to the national mean each year. While the kids with IQs over 100 attained this benchmark in 3 years or less (the highest IQs having already reached that benchmark before receiving any instruction), the lower IQ groups seem to me to be marching steadily towards that goal -- thus "overcoming" the plight of low SES statistical fate.
Therefore, I would think that the IQ condundrum may not apply after all. If those groups continue to make the progress they are showing in these early grades (Ken, you do the math because I suck at it), it would seem to me they will not only reach national medians, but also exceed them by the time they finish school.
This is what I am thinking, but maybe I am wrong and I know you'll explain to me why I am! :)
If those groups continue to make the progress they are showing in these early grades (Ken, you do the math because I suck at it), it would seem to me they will not only reach national medians, but also exceed them by the time they finish school.
Not gonna happen. The heritability of IQ increases as we age through childhood. Think of the issue this way: results are mediated by nature and nurture. When we are very young and in school, the teachers can control our environment to a incredible degree. For the teachers on this blog, how many can control the environment of their 12th graders to the same degree as kindergarteners? As the child grows they begin to act out on their inclinations to a greater extent and those inclinations often stem from the nature side of the equation.
I would argue that it is also a dangerous trap to define post-secondary options largely in terms of four-year colleges (not that Ken was necessarily doing that). Community colleges are enrolling increasing numbers of students.
Does a student scoring at the 34th percentile have the requisite reading comprehension, math, and writing skills to successfully complete an associate's degree?
Parry
hey KD does this contradict you?
http://www.edletter.org/current/ferguson.shtml
How much progress has been made in closing black-white achievement gaps?
Huge progress since 1970, not much progress since 1990. Sixty-two percent of the overall black-white reading-score gap for 17-year-olds disappeared between 1971 and 1988. About one-third of the math-score gap disappeared during the same period. Over the last several years the gap has narrowed significantly for both 9- and 13-year-olds, but there’s been a bit of backsliding for the older teens.
There’s been enough progress to establish firmly that these gaps are not written in stone. Even IQ gaps are narrowing. Measurements of the intelligence of kids less than one year old show virtually no racial or social-class differences, yet racial and social class achievement gaps are firmly established by the time students start kindergarten. Something happens before kindergarten that produces differences in proficiency.
Achievement gaps are not facts of nature. They are mostly because of differences in life experience. We’ve got to figure out how to get all kids the kinds of experiences that really maximize access to middle-class skills. That’s the challenge.
I'd like to know how a school in the Phila School District managed to improve on the state tests 54% in 5th grade reading with balanced literacy, Harcourt/Brace Trophies, leveled BL texts, PSSA prep, a reading specialist, site based management( school selects all its own teachers), and a dedicated staff. As far as I know there was no DI in this school, nor any explicit instruction of the type posted on this site.
There are plenty of low IQ kids in the school with half the kids transferring each year. Stories like this make is almost impossible to get schools to listen to stories that DI or explicit instruction is the only way to save the kids from reading mal-instruction. Seems this school, located in a poor area of the city, has managed huge gains with balanced literacy.
kathy
Phila Inquirer:
http://www.philly.com/inquirer/local/
20080816_One_school_s_steps_to_PSSA_success.html
How much progress has been made in closing black-white achievement gaps?
Not much.
Huge progress since 1970, not much progress since 1990.
Which tells us that we've probably squeezed out all of the improvement in the system, after all, closing the gap became mission #1 during that era and it's likely that black children were underperforming as a group compared to the level that they should have been performing at if their environment were improved.
Once those gains were realized it brought us to natural limitations, kind of like being able to only squeeze so much juice out of a lemon. Taking a neglected lemon, all shriveled in the sun and trying to get juice out of it will yield subpar levels compared to taking a lemon fresh off the tree. In both cases one lemon will never yield a pitcher full of juice though the fresh lemon will yield closest to its natural juice content.
Over the last several years the gap has narrowed significantly for both 9- and 13-year-olds, but there’s been a bit of backsliding for the older teens.
This is no surprise. Slower performing children are getting more attention, ceterus paribus to the detriment of others, and this additional time on task and attention will yield improved performance, however the gains are only temporary as we show here by breaking down the Flynn-Dickens study by age-cohort.
Measurements of the intelligence of kids less than one year old show virtually no racial or social-class differences, yet racial and social class achievement gaps are firmly established by the time students start kindergarten.
Actually, they're firmly established by the time the child is 3 years old. So tell us, how much does a 1-3 year old child get influenced by media, peer relationships (not acting white) and the other environmental bugaboos that are hypothesized to be the cause of the gap?
Achievement gaps are not facts of nature. They are mostly because of differences in life experience.
Before making such definitive statements it would do you well to know your subject matter. Clearly you don't. Please tell us the biological mechanism that provides every reproductive group, be it family, tribe, clan or race, with equal intelligence?
Tangoman,
"Please tell us the biological mechanism that provides every reproductive group, be it family, tribe, clan or race, with equal intelligence?"
Are you suggesting that some races are genetically superior to others in terms of intelligence?
Parry
Are you suggesting that some races are genetically superior to others in terms of intelligence?
Define "genetically superior."
Look, if you believe in evolution then there is no way around this issue. Races developed through partial inbreeding and through partial reproductive isolation.
Here are two examples. The first is the plight of Pakistani underachievement in the UK while their phenotypic neighbors, Indians, are doing well. The key fact that distinguishes the two groups is the long-standing practice of cousin marriage:
. . new research showed Pakistani families produced an alarming 30% of the UK's genetically diseased children. . . .found that at least 55% of the community was married to a first cousin. . . .British Pakistani family is at least 13 times more likely than the general population to have children with recessive genetic disorders. . . .while British Pakistanis accounted for just 3.4% of all births, they had 30% of all British children with recessive disorders and a higher rate of infant mortality.
The ugly new health vs culture focus on British Pakistanis comes just days after separate new research described them as languishing at the bottle of the social mobility league table.
To put that news into perspective, consider this:
The 50 inbred children were products of marriages between first cousins; their mean age was 7.7 years (range, 6-11 years). A significant (p 0.001) negative association was found between inbreeding and score on the Weschler Intelligence Scale for Children (WISC-C). In addition, the weighted mean IQ of inbred children )88.4 + or - 1.37) differed significant (p 0.001) from that recorded among 50 noninbred controls of similar age and socioeconomic status (99.6 + or - 2.0).
Are you prepared to argue that a millenium or more of institutionalized cousin marriage (this isn't just an occasional cousin marriage occurring once every few generations) has no impact on population characteristics? Are you going to argue that the plight of Pakistani underachievement is discrimination when the phenotypically similar Indians are doing quite well in the UK?
The second case focuses on Jewish Achievement.
At least 176 Jews and persons of half-Jewish ancestry have been awarded the Nobel Prize,1 accounting for 23% of all individual recipients worldwide between 1901 and 2007, and constituting 37% of all US recipients2 during the same period. In the scientific research fields of Chemistry, Economics, Medicine, and Physics, the corresponding world and US percentages are 27% and 40%, respectively. (Jews currently make up approximately 0.25% of the world's population and 2% of the US population.)
Harpending, Cochran and Hardy lay out their evolution-based model for Jewish neurological development.
So Parry, just so we all know where we're coming from, let me ask you a question - do you believe in evolution or do you view the world through creationist eyes?
TangoMan asks:"Please tell us the biological mechanism that provides every reproductive group, be it family, tribe, clan or race, with equal intelligence?"
Parry dodges: Are you suggesting that some races are genetically superior to others in terms of intelligence?
In light of the following:
"These observations suggest that some genetic variants that influence g will vary between populations rather than within populations. For instance, certain Asian populations have a frequency of 0.60 in COMT Met158 allele, which predicts lower COMT-enzyme activity and thereby better cognitive performance, while Caucasians have a frequency of 0.42 for the same allele.
TangoMan asks Parry the same question which he dodged earlier, that is, to inform us all of the biological or mystical mechanism which stops evolutionary pressures from acting above the neck.
TangoMan,
I must admit, I find your language choice to be interesting. I’m not sure how I can “dodge” a question that wasn’t asked of me (from my reading, your question about biological mechanisms and intelligence was directed at an anonymous poster, and not at me). But now you have asked me directly, so please let me provide an answer.
Your first question was: “Please tell us the biological mechanism that provides every reproductive group, be it family, tribe, clan or race, with equal intelligence?” Your second question was: “Do you believe in evolution or do you view the world through creationist eyes?”
Let me start with the second question. I believe that evolutionary processes continue to impact human development, and I believe it is difficult to argue (and I would not) that these processes do not impact what is commonly called “intelligence”, however defined. Cognitive ability (however defined) clearly has a powerful genetic component. I find the Ashkenazi Jews to be a particularly compelling example of how both social and genetic factors can influence the development of a sub-population over time, in terms of overt cognitive ability.
I find your second question to be much more difficult to answer. First of all, your question seems to lump together multiple reproductive groups, which I find to be problematic. From my understanding, the degree of impact of evolutionary biological mechanisms would seem to be inversely proportional to the size of the group in question. The Ashkenazi Jews are an example of a tribe or clan, as are (I believe) the specific Pakistani families that you mention. However, I would absolutely agree (as I said earlier) that evolutionary factors influence the development of sub-populations over time, assuming that there are high levels of within-group breeding, and very low levels of out-of-group reproduction.
But this doesn’t seem to address my question, which was “Are you arguing that some races are genetically superior to others in terms of intelligence?” Or, clarified, are there substantial genetic differences in terms of intelligences between races? The Ashkenazi Jews or the Pakistani families you mention are not races, but sub-populations within races. In Cochran, Hardy, and Harpending’s paper “Natural History of Ashkenazi Intelligence”, they make the point that “It is noteworthy that non-Ashkenazi Jews do not have high average IQ test scores (Ortar,1967), nor are they overrepresented in cognitively demanding fields.”
When you originally posed your question to the anonymous poster about biological mechanisms and intelligence, the intimation that I read into your question was that you believe that achievement gaps between various races in the United States (African Americans, Caucasian Americans, Hispanic Americans, Asian Americans, etc.) are a result of genetically-explainable differences in intelligence between the races, i.e., some races are more intelligent than others. So I guess a more specific question to you would be: Do you believe that Caucasian Americans are genetically more intelligent than African Americans and Hispanic Americans, and that the achievement gap is largely a result of genetic differences in intelligence? I do not believe either of those things.
Parry
So I guess a more specific question to you would be: Do you believe that Caucasian Americans are genetically more intelligent than African Americans and Hispanic Americans, and that the achievement gap is largely a result of genetic differences in intelligence?
The preponderance of evidence indicates this to be the case. Keep in mind that we're not speaking of absolutes here, but of distributions, and while there is significant overlap in distribution, the overlap doesn't imply identicalness.
I'm also curious why you framed the question such that you left out Asians (105) and Ashkenazi Jews (115) whose group mean IQ exceeds that of Caucasians? Are you angling for some white supremacy smear?
I do not believe either of those things.
Believing something in the face of evidence to the contrary implies that the basis for one's beliefs rely more on faith than evidence. That's the exact charge usually leveled at Creationists.
I find the Ashkenazi Jews to be a particularly compelling example of how both social and genetic factors can influence the development of a sub-population over time, in terms of overt cognitive ability.
The terms population and sub-population are applied arbitrarily . Using the same arbitrary model we can argue that African-Americans are a sub-population of Africans. Or that WASPs are a sub-population of Europeans, or that Han Chinese are a sub-population of Asian. The main point though, is that if you concede that "sub-populations" can diverge from the metrics that characterize their host population by way of evolutionary mechanisms, then it's incumbent on you to explain why the host population is immune to the same evolutionary dynamics.
Secondly, it seems that you concede that some populations, or if you prefer, sub-populations, have a greater measure of cognitive ability than other populations, races, or sub-populations, so perhaps you can explain the mathematics of how every group can have equal intelligence when one group has more than equal intelligence. If it's possible for one group to exceed the level of equal intelligence then why is it impossible for another group(s) to fall short?
To put the question more concretely, if the mean level of intelligence is determined by the following formula:
(x + 100 + 100 + 100 + 115) /5 = 100 IQ
and 100 is determined to be the mean level of intelligence and 115 is the mean level of Ashkenazi Jewish intelligence, then how can the value of x also be 100?
First of all, your question seems to lump together multiple reproductive groups, which I find to be problematic. From my understanding, the degree of impact of evolutionary biological mechanisms would seem to be inversely proportional to the size of the group in question.
Yes, but don't neglect the fact that the groups are hierarchically nested. A clan consists of multiple tribes and those tribes consist of multiple families. It's not the case that a clan consists of 10 tribes, with each tribe from some different part of the world. Therefore there are boundaries on diversity. Further, group size isn't the sole determinant of evolutionary effect. Lastly, the partial inbreeding occurs within the network, repeated continuously over many millennia. This is why, even though Europe and Africa consisted of many clans, tribes and families, there exist today phenotypic and genotypic distinctions between race that are greater than between clan or tribe. The key point here is that there existed more inter-clan and inter-tribe gene flow than inter-race gene flow, even with the fuzzy boundaries involved.
the specific Pakistani families that you mention.
You mischaracterize the dynamic by labeling it as something that is specific to select families. Take a look at the rate of cousin marriage in Pakistan as a whole - numerous surveys put the figure between 40% and 60% of all marriages.
And it's not just Pakistan, but a wide swath of the Middle East and North Africa.
Here is a study on Consanguinity among the Saudi Arabian population:
This study was conducted on 3212 Saudi families to investigate the prevalence of consanguineous marriages. The families were interviewed and the information on the relationship between the husband and wife was obtained. The overall rate of consanguinity shows that 57.7% of the families screened were consanguineous.
So, if I understand your position correctly, you're entirely willing to concede that the genetics of intelligence comes into play so long as the group under question is arbitrarily small so that evolutionary dynamics can have effect, but when those same evolutionary dynamics are at work on a larger scale are you going to say that you're just a little bit pregnant? In other words, if Pakistani's in the UK are languishing at the bottom of the social mobility ladder due to lower group measure of intelligence, then what's the reluctance in making the same argument for the population of Pakistan, or for that matter, most of the Middle East? And for that matter, if you're willing to make that argument then why label it out of bounds for other groups, when so much evidence indicates genetic factors being at work?
Lastly, keep in mind that the institutionalization of cousin marriage is a long standing social practice, so if you chart the family tree of most families from the region, you find that, generation after generation, the woman who should be the aunt is actually the mother. This means that even if an individual doesn't engage in the practice,they have a strong likelihood of being born to a lineage that did engage in the practice, in other words, the are part of a population, not an island onto themselves. This where we see regression to the mean come into play. To put this concept into the American mileau, this is why we see the off-spring of very intelligent African Americans experience a greater gap in IQ between parent and child than we see in the off-spring of very intelligent Jews, Caucasians, and Asians.
the woman who should be the aunt is actually the mother.
Typing faster than I should - this should read "the woman who should be the great-aunt is actually the grandmother."
I think Parry @ 4:48 has been misled by Ken's helpful blue line -- "and you're getting them pretty darned close to scoring at the national average, with a good slope to that line."
They do get closer to the third-grade average from the first to the second to the third grade -- one would certainly hope so -- but they are less close to the second grade average at the end of the second year than they were to the first grade average at the end of the first year, and less close to the third grade average (the blue line) at the end of the third year than they were to the corresponding averages the previous year.
That is, it's a BAD slope.
(And these were DI classes. The controls were probably worse.)
Linda,
I was referring to math—sorry if that wasn’t clear. From what I can tell, groups 2 and 3 shave a couple points off their separation from the mean from 1st grade to 3rd grade. Group 2 goes from about a six-point difference (42-35.8) to about a 3-point difference (71-67.7), and Group 3 goes from about a two-point difference (42-40) to about a one-point difference (71-69.7). If nothing else, low-SES kids are keeping pace with a national cohort over the three-grade span, which seems positive to me.
TangoMan,
We’re starting to range pretty far outside of my area of expertise. But I will do my best to muddle through a coherent response.
You say that it is incumbent on me “to explain why the host population is immune to the same evolutionary dynamics.” I disagree. You appear to be arguing that there are significant genetic differences in intelligence between various races in the US, and that these genetic differences explain the achievement gap. This would be a pretty bold statement, and one that, as far as I can tell, is in opposition to mainstream scientific beliefs. Now, there’s nothing inherently true about mainstream scientific beliefs. But I would say, given the breadth of your apparent argument, that it is incumbent on you to explain the specific evolutionary dynamics that would lead Asian Americans, Caucasian Americans, African Americans, and Hispanic Americans to have significantly different race-based levels of intelligence.
The reason I believe the Ashkenazi Jew example is powerful is because there is a coherent line of reasoning to explain the phenomenon. There existed a relatively small group of people who maintained social and reproductive boundaries to outsiders over an extended period of time, with specific environmental pressures that could have selected for certain cognitive traits that correspond well to the traits measured on an IQ test.
What coherent line of reasoning exists to explain genetic differences in intelligence between the various races in the US?
And you know what—if you don’t want to answer the question, just ignore it. This rabbit hole of discussion is not particularly productive, in my opinion. There is a tremendous amount of evidence that suggests that the achievement gap is the result of a host of environmental factors, and I find that evidence to be persuasive. You apparently find a different set of evidence to be more persuasive. But as a K-12 educator, my job is to make sure that students receive a high quality education, no matter their race, SES, or native intelligence (whatever that may be). I choose to believe (because I am persuaded by a preponderance of evidence) that large-scale achievement gaps are not “facts of nature” (the phrase that started us on this discussion), but rather facts of circumstance, and that it is within our power as a society to address these gaps. If you believe otherwise, then more power to you, but don’t expect my vote for School Board.
Parry
But I would say, given the breadth of your apparent argument, that it is incumbent on you to explain the specific evolutionary dynamics that would lead Asian Americans, Caucasian Americans, African Americans, and Hispanic Americans to have significantly different race-based levels of intelligence.
I've already shown two scenarios which lead to genetic consequences for level of cognition (Ashkenazi Jews and people born into cultures that are engage in systemic consanguineous mating) which clearly establishes that environmental factors can lead to genetic consequences.
Now when a child is having difficulty in a school environment and is sent for diagnostic testing to determine whether the child is learning disabled, it's unlikely that you or any other teacher or administrator requires that the diagnostician provide you with a full and complete explanation for HOW the learning disability developed biologically before you accept the information and develop an action plan on how best to aid the child. You see that the child is having difficulty, the diagnostic testing reveals a biological factor at play, and that's good enough for you.
So, that being the case, why do you pose such an exacting level of proof, which you don't even apply to your favored environmental models, when the nature argument for cognition can already meet the standards found at the diagnostic level?
Here's what we know:
-The Black-White Gap appears by age 3 when children are matched for gender, birth order and level of maternal education.
-IQ is correlated to a subject's reaction time at a level of 0.67. This is a physical test that isn't at all dependent on a child's vocabulary, how many books are in the child's household, or other fuzzy metrics. We see a similarly strong correlation to reverse digit span tests, which are also immune to cultural and familiarity factors (after all, there aren't many White, Jewish and Asian families who make a habit of regularly repeating strings of numbers in reverse order.)
-Brain volume, as measured by MRI, is correlated to IQ at a level of 0.4. Brain volumes (corrected for body size) differ by race. In fact, the Chinese have set up a Brain Bank to study Chinese brains because much of the existing brain research has been conducted on Caucasian brains.
-Research on transracial adoption shows that the children's home environment has less influence on cognitive development than the genetic inheritance they received. Such studies have looked at children born of two black parents, one black parent, two white parents, and Korean children. The Korean study found that the children performed at a level comparable to other Koreans in America irrespective of the SES of their adopted families and that their adoptive siblings performed at levels significantly correlated to levels of parental education.
- As I've already indicated elsewhere in these comments, we're now finding alleles associated with cognition that have differing distributions by race. COMT Val158Met is only one of many.
These lines of evidence rely on physical measurement, genetic screening, and there are no plausible environmental factors that explain outcomes as succinctly as the hereditarian hypothesis. The standards of evidence are similar to what goes into diagnosing biologically caused learning disabilities.
You say that it is incumbent on me “to explain why the host population is immune to the same evolutionary dynamics.” I disagree.
You accept that some small groups can develop cognitive abilities that differ from an accepted norm, but you reject the case that larger groups are affected by such evolutionary developments.
If I proposed that falling off of a 3 story building would harm a person but falling off a 30 story building would cause no harm, you'd place the burden of proof onto me to show why the same principle doesn't apply to the larger case as it does to the smaller case. Similarly, if I argued that sending a young child to school for a year will improve their mastery of the content taught but sending the child to school for 12 years will have no effect on their level of content mastery, again you'd place the burden of proof onto me to demonstrate why the principle of a relationship between learning time and content mastery completely breaks down after the first year of education.
You accept the causal model for Jews and Pakistanis and I assume that you accept the methods of measurement (IQ tests, school performance, socioeconomic status, behavioral metrics, etc) so why the difficulty in accepting the fact that the identical methods of measurement when applied to American Blacks and Hispanics, which yield results consistent with the causal model applied to the above groups, could also have causes similar in nature?
There is a tremendous amount of evidence that suggests that the achievement gap is the result of a host of environmental factors, and I find that evidence to be persuasive.
Don't you feel uncomfortable taking such an extreme position (environment responsible for 100% of outcome) rather than the more moderate position (environment responsible for 50% and biology responsible for 50%.)
When you a priori discount 50% of the independent variables, and concentrate all remedies on the other 50% and treating that 50% as though it is encompassing 100% of the independent variable, then your model will NEVER yield the results that you predict for you've grossly mismodeled the problem. In fact, that's likely why, after nearly a half century of concentrated and expensive effort aimed at reducing the racial achievement gap, that every approach taken has been deemed a failure.
But as a K-12 educator, my job is to make sure that students receive a high quality education, no matter their race, SES, or native intelligence (whatever that may be).
No one, and certainly not I, is telling you to treat the children under your charge differently because of race. The goal of providing a high quality education is NOT DEPENDENT on whether one believes in evolution or holds to creationist views. In other words, the naturalistic fallacy or Hume's Is-Ought problem, doesn't hold, meaning that simply because we accept the consequences of evolution doesn't imply that social policy must bend a particular way. So, the goal of providing a high quality education remains unchanged. What changes is a.) the quest for the impossible is not futilely mounted, b.) the more efficient use of resources to tactics and strategies that are known to achieve results, and c.) the reduction of unwarranted blame directed at innocent parties.
I choose to believe (because I am persuaded by a preponderance of evidence) that large-scale achievement gaps are not “facts of nature” (the phrase that started us on this discussion), but rather facts of circumstance, and that it is within our power as a society to address these gaps.
If you hold that remedying the gap is within the power of society then you should have little trouble in providing the evidence that supports that position, evidence of society actually closing the gap. Holding a belief in the absence of evidence is better described as faith and if faith becomes the basis for policy formulation or the development of a world view, then schools have no grounds to argue against the implementation of religious instruction or the introduction of Creationist science lessons.
I'd rather pull rational folks over to the side of evidence-based reasoning.
But I would say, given the breadth of your apparent argument, that it is incumbent on you to explain the specific evolutionary dynamics that would lead Asian Americans, Caucasian Americans, African Americans, and Hispanic Americans to have significantly different race-based levels of intelligence.
Similarly, because we have no way of explaining the evolutionary aspect to these findings:
Race had a significant and differential effect on the bones in the axial and appendicular skeletons. In the axial skeleton, black children had greater cancellous bone density, but similar cross-sectional area of the vertebral bodies. In contrast, in the appendicular skeleton, black children had greater femoral cross-sectional area, but similar cortical bone area and cortical bone density. Compared to white children, vertebral bone density and femoral cross-sectional area at sexual maturity were, on the average, 10.75% and 5.7% higher, respectively, in black children.
This study indicates that skeletal muscle mass as a fraction of body weight is smaller in Asian compared with African-American and Caucasian children.
We must therefore presume that discrimination is the cause and that increased efforts of social intervention will lead to equalization of bone mass and skeletal muscle. To assume that there is a biological cause without understanding the evolutionary process that would bring about this disparity is an unwarranted and dangerous position especially when we know that things like diet and exercise can affect body characterization.
TangoMan,
Here is where I believe we agree:
The set of cognitive skills measured by IQ tests has a strong genetic component. Furthermore, all things being equal, a child with a significantly higher IQ will likely perform better in school than a child with a significantly lower IQ, given that many of the academic tasks emphasized in K-12 education are particularly amenable to the set of cognitive skills measured by IQ tests.
Finally, there is at least one example of a group (or population or sub-population, whatever term you wish to use) of specific people—i.e., the Ashkenazi Jews—with specific characteristics who, as a result of specific environmental factors that appear to have selected for the types of cognitive skills measured by IQ tests, developed atypically high mean group IQ scores over an extended period of time. I know you would likely argue that the Pakistani example you cited also belongs in this category, but I don’t know enough about that example to agree or disagree with you on it.
Here is where I believe we disagree:
From my reading of your argument, you appear to be suggesting that, because there is at least one example of a group of people achieving higher-than-expected mean IQ scores as the result of selective evolutionary pressure, one can therefore extrapolate this finding to considerably larger groups that share neither the particular characteristics, nor experience the particular circumstances of the much smaller group.
I believe this argument is an attempt to jump from the particular to the general without much evidence to support the jump.
You mention the evolution/creationism debate multiple times, and I believe that debate is analogous to our discussion, but not in the way you describe it. I find evolutionary theory to be so persuasive because it presents a comprehensive, detailed narrative that explains not only where we are now, but how we got to where we are now. This is, in my opinion, one of the biggest stumbling blocks for creationists who wish to challenge evolutionary theory using scientific arguments. A creationist may argue that a zebra is simply too complex to have resulted from evolutionary forces, but the creationist has no counter-narrative to provide a more compelling explanation. If zebras didn’t evolve, then how did they appear? Did they simply spring out of thin air one day on the African Savannah? If so, how did God cause that to happen? When it comes down to it, creationists (from my reading of their argument) can’t come close to the level of explanatory detail provided by evolutionary theory. They are left with the argument, “Well, God just made it happen”.
This is why I continue to find the Ashkenazi Jew example so persuasive. Cochran, Hardy, and Harpending don’t just say, “Well, this group has higher IQs, so their must be some genetic and evolutionary forces at play.” They provide a detailed, persuasive narrative that doesn’t just document the phenomenon, but explains how the phenomenon could have occurred.
This is where I find your argument to be unpersuasive. There is a comprehensive, detailed narrative built around explaining how environmental factors could have led to the specific achievement gaps that exist between specific groups in our country at this time. Are there holes in that narrative? Yes. Are there data that appear to contradict pieces of that narrative? Absolutely. In fact, you refer to multiple pieces of data that poke holes in the narrative. But poking holes is not the same thing as constructing a compelling counter-narrative.
Which is why I return to my central point. If you want to argue that the achievement gap in this country is entirely or largely (or even pretty significantly) a result of genetic differences in intelligence between various races, I believe it is incumbent on you to provide, not just disparate data that challenge the environmental factors narrative, but rather a counter-narrative of at least the same detail as Cochran, Hardy, and Harpending’s Ashkenazi Jew example. In other words, identify the specific evolutionary selective factors that led to differences in intelligence that manifest in the specific achievement gaps between specific races that we see in this country at this time. When I have asked you to do so in our discussion up to this point, you have not, saying instead “I've already shown two scenarios… which clearly establishes that environmental factors can lead to genetic consequences.” To me, that kinda sounds a lot like “Well, evolution just made it happen.”
I will end with these final thoughts, because you mention that the goal of providing a high quality education is not dependent on one’s opinions related to our discussion (and I do mean end—I think we have gotten to the point in our discussion where, much like the non-French-speaking American tourist in a Parisian department store, we keep repeating the same things, only more slowly and more loudly).
From a professional/practical perspective, this discussion doesn’t have much value to me. My job is to ensure that the students in my building receive a high quality education, no matter their IQ, race, SES, etc.
From an intellectual perspective, this discussion is of value to me. We are talking about the human condition, and I believe that just about any good-faith discussion of the human condition is valuable.
From a moral perspective, this discussion is incredibly important to me. For moral reasons, I like to believe that a certain sense of equality exists across racial lines. Now, might my moral perspective influence my intellectual perspective? Who’s to say. I am a pretty analytic individual, but I think it would be hard to argue that our personal morals and values don’t color our judgments.
From a policy perspective (and this is the one that may matter the most), I believe this discussion is tremendously important. If policy makers were to believe that there were significant genetic differences in intelligence between various races in the US, this could have huge policy implications. Much of NCLB is predicated on the notion that all children can learn and that we can close achievement gaps (two beliefs to which I subscribe)—but a belief in significant genetic differences in intelligence might suggest that we abandon those notions, accepting the achievement gap as simply a fact of nature. That, in my opinion, would have disastrous consequences.
Anyway, thank you for taking the time to participate in this discussion with me. It challenged and forced me to refine my own thinking, which is something I always appreciate.
Cheers!
Parry
Much of NCLB is predicated on the notion that all children can learn and that we can close achievement gaps (two beliefs to which I subscribe)—but a belief in significant genetic differences in intelligence might suggest that we abandon those notions, accepting the achievement gap as simply a fact of nature. That, in my opinion, would have disastrous consequences.
Bear in mind that the way NCLB is structured, the achievement gaps can appear to close without them actually ever changing at all. If teaching is improved or tests made easier (or a combination of the two), pass rates will increase to the point where pass rates are close to identical. Nonetheless, there will remain considerable variability above the pass score.
So even if genetic factors predominate, it remains a worthy goal to improve instruction, and thereby improving student outcomes.
I believe it is incumbent on you to provide, not just disparate data that challenge the environmental factors narrative, but rather a counter-narrative of at least the same detail as Cochran, Hardy, and Harpending’s Ashkenazi Jew example
These narratives exist, but because they're not testable, they're no more than Just So stories. Even the HCH paper relies on a Just So narrative. They certainly provide testable hypotheses, in that they predict that people with certain alleles will have higher IQs but there is no way to actually test their narrative.
The point is that their work doesn't hinge on the narrative, rather it is built on the testable hypotheses. We've already identified scores of alleles associated with cognition so we can make predictions on levels of mean IQ based on the distribution of the alleles with populations. Alternatively, we can do admixture studies relating IQ to proportion of European, Asian, Jewish, Black admixture.
From a policy perspective (and this is the one that may matter the most), I believe this discussion is tremendously important.
Indeed, this is important for policy reasons. For instance, even back to the 1970s, early 80s, we know that the racial wage gap, with IQ being controlled, was closed and has been since. However, even today there is concentrated effort to close the wage gap across race and no attention at the policy level to the IQ question.
So, what happens is that society squanders resources on "remedying injustice" which isn't really injustice and secondly, blame and ill-will is fostered, in that many are taught to blame others for their station in life when the blame is undeserved in our current times. The playing field is mostly level, in that a white man, with an IQ of 115, earns the same as a Black man with an IQ of 115.
Much of NCLB is predicated on the notion that all children can learn and that we can close achievement gaps (two beliefs to which I subscribe)—but a belief in significant genetic differences in intelligence might suggest that we abandon those notions, accepting the achievement gap as simply a fact of nature. That, in my opinion, would have disastrous consequences.
Policy as it pertains to education right now squanders resources on futile efforts that are lysenkoist in nature. Right now teachers are blamed for student shortcomings, and they will be forever so long as we operate under a false notion of the human condition.
Throwing more resources at problems results in diminishing marginal returns. Society does not have infinite resources to deploy to education. The revised goal of education should be to insure that the pathways for progress are cleared for students so that no students are held back from reaching their potential due to race, SES, or other factors beyond their control.
Right now the mission of the educational establishment is to close the achievement gap and other priorities are given lower priority. Setting off on an unachievable quest predicated upon a false understanding of reality will result in guaranteed failure and harm countless of millions of students who could have made better marginal use of resources if they were more fairly distributed and discourage many thousands of teachers who will feel like failures for not being able to achieve their goal of closing the gap.
In closing, the nightmare policy universe you imagine won't come about because folks like you will still be in the catbird seat. Your passion and dedication are clearly evident, so I'm quite certain that you could respond to a rational worldview with policies and pedagogies that balanced opportunity, fairness and efficient and equitable deployment of scarce resources.
the Ashkenazi Jews—with specific characteristics who, as a result of specific environmental factors that appear to have selected for the types of cognitive skills measured by IQ tests,
You make it sound like IQ is only good for test taking, when in fact, the cognitive skills are instrumental to success in our modern society.
I believe this argument is an attempt to jump from the particular to the general without much evidence to support the jump.
The evidence is identical with respect to the particular case and the general case, that is, the evidence is found in school achievement rates, SES, learning disabilities, crime rates, IQ tests, etc. I suspect that when you write "evidence" what you really mean is an engaging narrative. We can measure Jewish success in many venues and that success is often dependent on cognitive skills. All of the metrics that are used in the Jewish case, and that correlate to high IQ, are also available for use with other groups and those results correlate significantly with IQ.
There is a comprehensive, detailed narrative built around explaining how environmental factors could have led to the specific achievement gaps that exist between specific groups in our country at this time.
When that narrative is put into testable form, it falls apart. I've played this game before. The narrative is convincing to many people because of confirmation bias. However, all of the effects associated with the factors that contribute to the narrative don't adequately explain the disparity we measure. That's the problem. If you don't want to acknowledge genetics, then you have to devise a heretofore unstated FACTOR X which explains the unaccounted variance and that FACTOR X has to be more powerful than most of the already identified factors. Pinning your hopes on an unimagined environmental factor rather than acknowledging that humans are not immune to evolutionary pressures, is taking the road that leads to epicyclic rationales.
but a belief in significant genetic differences in intelligence might suggest that we abandon those notions,
Does a belief in significant genetic differences in fast twitch muscles suggest that we steer children not of West African ancestry away from sprinting in school?
Look, here's the problem for folks like you - this data isn't going to stop. The case is only getting stronger. Leftists were successful with their anti-science protests and closed down the successor to the Human Genome Project, the Human Genome Diversity Project. However, it's hard to hold back the tide and the project was reformed, renamed, and is now completed. There are follow-on projects underway that are adding more data. Clinging to a model where evolutionary forces were inoperative above the neck is going to leave folks out in the cold, much like the Lysenkoist Soviet Union got left behind compared to the progress the rest of the scientific world made in biology.
Shift of focus:
When I look at the math graph and think of an ideal instructional implementation, I expect that the higher the IQ the higher the learning rate should be. I expect this to be particularly true for math. Further I would expect that the higher IQ group should have the highest variation in 3rd grade. Instead in the graph the rate is almost constant across IQ, and the standard deviation gets progressively lower as grades go by for the high IQ group, the opposite of the trend for all the other IQ groups.
Why?
I expect the particular implementation of DI was causing learning rate to plateau. It was designed and tuned with the lower IQ groups in focus. Having just read "The War Against the Schools: Academic Child Abuse" higher IQ kids got almost no mention or focus.
Toy model of IQ:
Higher IQ folks can make bigger jumps in logic, hold more in their memory than Lower IQ folks.
Basic DI theory:
In order to understand the ideas being conveyed despite bad instruction the learner will need to find the signal in the noise, to make leaps in logic, and put desperate ideas together, because the instruction does not do that work for the student. So under bad instruction, lower IQ students get left behind, while the higher IQ students figure things out in spite of the instruction.
A good DI program, will minimize the cognitive load needed to understand a given skill. The material will be presented in a hierarchical and careful fashion making each new level of the skill easy to learn. This allows the lower IQ students to learn the material to mastery at the rate the material is taught, if enough practice is built in. This does not immediately or necessarily raise their IQ, but it does raise their skill level for the measured task. (though if you raise enough skills you may impact IQ, did anyone ever try and replicate Engelman's data where he reports having raised preschoolers IQ's by 20 points?)
A fixed DI system is likely going to have some maximum rate of presentation, without a full re-write of the system. The focus of the particular DI curriculum used is on making sure everyone can learn at grade level rate.
One can imagine a more flexible DI system that adapts to higher IQ students by allowing bigger gaps in the material. Combined with fewer repetitions, this should allow for much faster learning. This would be very hard to do with one set of printed materials though. Ideally this would be adaptive to the learning rate of the individual student.
I need to go re-read the Diamond Age...
Robert,
There are a few factors that contribute to the effect.
1. More attention was focussed on the lower-performers. They were also given the best teachers.
2. The lower-performers sometimes received extra instructional sessions.
3. Sometimes the progress of higher-performers was sacrificed because there wasn't enough students to form a "high level" class, so they were placed with the next lower class. This is a function of the high mobility rate found in low-SES areas.
The higher performers did proceeded at a 25% faster pace on average than the middle group. The lower performers proceeded at a 25% slower rate than the middle group.
This differential does not seem to be reflected in the standardized test scores.
My understanding is that the pacing differential actuallly decreased in the later grades (2 and 3) as the lower performers' skills improved. I've also heard that in some of the more difficult material that isn't taught in the home, like logic (Reasoning and Writing), that the rate of learning for all students was very similar. I see this with my own son (age 8 doing RW level D); I cannot skip as much like I do in math.
The DI materials are adaptable for studenst at the tails. For higher performing kids, material can be skipped and practiced reduced (I've covered about half a year's worth of math with my son this summer (2 months of lessons); the rate of progress is constrained more by the need to acquire automaticity which takes practice rather than the speed and ease of which new concepts can be acquired).
For lower performers, it becomes necessary to repeat more lessons, give more review and extra practice, and circle back around for review sessions. However, counterbalancing this is that in DI the lower performers are not perfectly lost as they frequently are in the traditional curriculum, so that their rate of learning compared to higher performers is reduced.
But I will say that my experience teaching my son math is that he can proceed very quickly through the curriculum. I have little doubt that if I had an hour a day for 180 days with adapted DI materials that I could get him through the entire elementary math sequence by the end of third grade at which point he'd be ready for algebra.
"But I will say that my experience teaching my son math is that he can proceed very quickly through the curriculum. I have little doubt that if I had an hour a day for 180 days with adapted DI materials that I could get him through the entire elementary math sequence by the end of third grade at which point he'd be ready for algebra."
I'll add one more anecdotal point of evidence to what Ken is saying (at the risk of missing his point).
I'm homeschooling. The huge advantage that homeschooling has over traditional instruction is that the one-on-one instruction *can* pace the child's learning perfectly. You give back something by being an amateur (and in our case by not being focused on simply accelerating the standard sequence ... we are spending extra time on geometry, for example).
My son is on track to start algebra in 5th grade (maybe 1/2 way through 5th grade). It is most certainly possible to get at least *some* kids to algebra much earlier than we are doing in typical classrooms.
Now ... could this be done in a normal classroom environment? Probably not in a heterogeneously grouped classroom. Could this be done in a normal classroom environment with more homogeneously grouped kids? And an actually skilled teacher (rather than the amateur my kid has teaching him)? I'd be very surprised if the answer was no.
Note, however, that the homogeneous grouping would need to be by subject matter. Penmanship is a whole nuther thing for my kid ...
-Mark Roulo
We've taught my son math at home using Kumon for several years. He knew his multiplication tables before first grade.
He is now about to go into third grade, and working on the algebraic distributive property. You know, A*(B+C) = (A*B) + (A*C), in that format.
My son's quite bright, and it wasn't particularly easy, but we did do it in after-school time. And it was probably a lot harder getting through multiplication than anything else.
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