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I was kibbutzing with a compatriot in the high school parking lot last night. He's a math person who is an administrator with a major college. Very knowledgeable.
He told me he wants kids to be taught math, not arithmetic; "arithmetic isn't math."
I've heard that before but still don't know what it means.
Speaking of arithmetic, Hung Hsi Wu's article "What's Sophisticated about Elementary Mathematics" (pdf file) is out!
And remember Ron Aharoni: What I Learned in Elementary School.
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Catherine, next time you see him, ask him what he means. Does he mean, he wants the kids to learn why the arithmetic algorithms work? or does he want them to learn something not based on arithmetic? this makes no sense.
Arithmetic isn't something that mathematicians do as mathematicians. But that's because it's a solved problem, not because it isn't math.
There is a tongue-in-cheek aphorism that "If it has numbers, it isn't math", but that's more about a snobbish attitude that applied math is (in some sense) not real math. If it has a current practical application, it's engineering, or physics, or economics, or something; real math is purely intellectual.
After reading Wu, one realizes why the reform math people would think of him as bonkers. His math reasons are solid, but his pedagogy is impossible. I use better methods to teach fractions everyday, but I greatly respect and support what Wu and associates are trying to accomplish.
As far as I can tell, the hard core reformers have gone too far into trying to get everyone into math to the detriment of learning the math that Wu correctly suggests is the goal. This is a politically charged and frankly incoherent, yet dominant area of debate.
What most teachers try to do is to use reform math approaches (such as Algebra blocks) to teach traditional that is new math: Dolciani with style. The Mind Research Institute's Algebra Readiness program would be a great model of this. Certainly, the "failure" of approaches such as COMAP's either to show strong success or adoption must be considered very sad and an area that needs greater research. The binge of project-based mathematics is probably a very dead end.
Singapore Math (at least the old version used in California) tries to model doing math in a way that supports Wu's goals. For fractions, I've found that using the identity property and making conversion factors (ratios or rates) has great efficiency in re-teaching.
If we are talking about the so-called traditional math, then no, hundreds of random 4+3 or 6X7 problems is not math. But a program that shows that 4+3=7 and 3+4=7 as examples of the associative property of addition and that 7-4 = 3 and 7-3 = 4 because subtraction is the inverse of addition, then *that* is math.
Yes, and I want my son to learn writing rather than spelling or grammar.
My take is that he thinks that math is some sort of deep thinking process that solves problems. But what skills are necessary to do that? The implication is that mastery of basic arithmetic skills is not necessary; that there are other "mathy" things that are more important. You know, critical thinking and stuff like that.
However, I'm sure he wants his kids to master the times table. It's a loaded, but ultimately meaningless, comment.
More drilling on facts, naturally. But with facts should come some reflection (and like, actual reflection, not "omg let's write a reflective essay in math class") about the facts.
Take the simplest axioms of (Euclidean) geometry. I remember as a child being fascinated by the idea that all the angles of a triangle always added up to 180 degrees. I'd try to make triangles of more or less than 180 degrees but noticed the sides would fail to properly connect. And then I'd attempt squares with more or less than 360 degrees, and so on.
There are some very profound concepts in there, that of equilibrium, and also "degrees of freedom". (An increase in one of the angles must result in a decrease in at least one of the other angles to maintain connectivity.) The same reason why the angles of a polygon sum up to a constant is responsible for phenomena observed in that infinitely-sided polygon known to all children, the circle. Equilibrium there is classic: the enigmatic sin(x) and the differential equation y'' = -ky are intricately linked to that little axiom you learnt in elementary school.
(An interesting to show before pulling out the proofs of circular trigonometric functions or why tan x = sin x / cos x and so on is to start with finitely-sided polygons. Heck, start with a point, then a line, then a triangle, then a square, and show what happens to the graph of height (y) as a function of cyclical distance as you traverse around and around the polygon (x) as you add more sides. With a point you get a constant horizontal line for a function. With a vertical line you get triangular waves. With an equilateral triangle you start to get some curvature, though there are still sharp/indifferentiable points. With a square (vertices centred on the axes), you get a graph that would pass off for a hand-sketched sine wave. As you put more sides on your polygon, you get closer and closer to a sine wave. (With more sides, it also matters less how you orient your shape -- with a line , depending on how you oriented it with respect to the y-axis, the the graph of the corresponding function could be a y=c sort of graph, or a triangular wave.)
Okay, we were talking about arithmetic? Well there are a lot of taken-for-granted ideas that come back later too.
Like take 5x5. Why is that a higher figure than 6x4? Or 7x3? Or 8x2? Or 9x1? Or 10x0? As a child, this fact disturbed me a lot, right up until I discovered calculus and the derivative of a quadratic function.
Here is an interesting Excel experiment I remember doing as a kid.
My curiosity basically arose out of my laziness in trying to memorise times tables. I noticed firstly, the redundancies. The 5s times tables were always easy, and best of all, it came up in every times table, especially that awful 9s times table, or that evil 12s times table. But because you knew the 5s times table, this gave you two facts right away (9x5, 12x5). This in turn, made knowing the facts near the 5s a little easier (9x6, 9x4, 12x6, 12x4, etc.).
So you really didn't have to master 144 multiplication facts if your parents wanted you to know everything up to 12x12, because redundancy and symmetry did some of the work for you. (We take this redundancy for granted ... but really think about it. Why should 6x4 necessarily equal 4x6?)
So if you alter the factors on both sides in equal and opposite directions, 6x4 = 4x6... but, what about 5x5? Why wouldn't 5x5 give me the same thing as 6x4 and 4x6?
It makes sense that the larger your factors, the larger your products. But why was it that you could get different products when the sum of the factors was the same? After all, 7 and 3 put together is just as big as 5 and 5, but 5x5 > 7x3.
The second interesting thing is that the highest product tended to be when the two factors were equal. So say your sum constant was 11 -- 5.5 * 5.5 > 6*5.
Very interesting for me at that time. Part of my fascination was that "it just is, duh" (the philosophy of memorising times tables). But I wanted to know a deeper reason why this was so.
So one day (after a few years of pondering) I was fed up and played around with Excel in study hall.
I chose a constant (I chose 10), and I had integers running from 0 to 10 in column A, and integers running from 10 to 0 in column B, and column C was basically column A * column B.
But how to map this? So I assigned each pair a sort of x-value, while the product was the y-value. So the row where A was 0 and B was 10 was assigned an x-value of 1, the row where A was 1 and B was 9 was assigned an x-value of 2, etc.
I plotted it and got an interesting trend, but nothing satisfying.
So I thought -- well, why stick to whole numbers? So I redid it, but instead now I had 1000 rows (yay for Excel), such that each row increased A by 0.01 and decreased B by 0.01.
I hit graph.
A parabola??!! That thing we're doing in algebra? What?! I didn't get it. I was surprised at how my childhood problem was related to parabolas. Especially since I couldn't see how y=ax^2 could have arisen in anything I plotted.
Then I wondered what would happen if I didn't stop at say, 10x0. Why not 15x-5, and -5x15?
Rather than getting some interesting kinks or anomalies ... the parabola continues being a parabola .... to negative infinity ... whoaaa.
Then I decided to do this one thing. I would map the differences between each of the products to the corresponding x-values. So, the difference between 6x4 and 5x5 was 1, the difference between 6x4 and 7x3 (and 3x7 and 4x6) was 3, the difference between 7x3 and 8x2 was 5....
And the graph of these differences to the corresponding x-value was ... a downward-sloping straight line(?!!)
I was intrigued but massively confused. What do straight lines have to do with times tables?! But then the following year I took calculus AB.
In his article, Hung Hsi Wu says: "The fact that many elementary teachers lack the knowledge to teach mathematics with coherence, precision, and reasoning is a systemic problem with grave consequences. Let us note that this is not the fault of our elementary teachers. Indeed, it is altogether unrealistic to expect our generalist elementary teachers to possess this kind of mathematical knowledge - especially considering all the advanced knowledge of how to teach reading that such teachers must acquire.” (Emphasis is mine).
How is it then, according to Liping Ma, average elementary teachers in China with much less formal training than their counterparts in the US not only have a solid grasp of elementary mathematics, but also possess the skills necessary to teach it? That makes me wonder what is exactly taught to our elementary education majors in our colleges and universities.
beta said:
"That makes me wonder what is exactly taught to our elementary education majors in our colleges and universities."
Exactly . . . they aren't taught how to teach. They're taught how to "construct" knowledge in children's brains (whatever that means).
The reason why PRC (Chinese) teachers perform better in teaching math is that they were taught correctly in elementary school in the first place!!!!! They know what to teach.
Our teachers (YOU AND ME) were taught poorly. Now we need to teach ourselves how to teach correctly and then the next generation of teachers may be competent. Ed schools don't have the time or talent to get us up to speed. We have to try harder and think better. For example, UNLEARN mastering times tables, learn to teach 9+6 = 10+5 = 15.
Well, maybe there is too much emphasis on HOW to do things in a classroom, so that not much attention is given to WHAT the elementary education students know, or think they know. After all, if the elementary education students never understood, let's say division of fractions, when they were in elementary school themeselves, they are not going to magically figure it all out when they become a teacher without any deliberate effort. It seems to me that the only way to break the cycle is for teacher education programs to go back and teach the content to the teachers. If they can't do that (not enough time or resources), then I really don't see any reason for them to exist at all.
Why would one want to UNLEARN mastering times tables?
Mastering times tables is a trained seal activity -> speed means very little from a math perspective, but learning that 7+5 = 10+2 allows students to understand math issues while learning basic skills. Learning what 3*5 means, not what 3*5 is, matters more. Take some Singapore Math training and "duh." I started where you are, you can be better, but you gotta start.
BTW Americans do better than Singapore Math students on speed tests on time tables. It's a big joke.
"The fact that many elementary teachers lack the knowledge to teach mathematics with coherence, precision, and reasoning is a systemic problem with grave consequences."
That's why the selected curriculum is so important. Look at the sample NAEP test questions and results. The problem has more to do with basic competence. Wu is running the risk of overstating the problem.
Teachers who know more might be nice, but it really isn't necessary for schools to do a whole lot better. How much training does it take to follow a good curriculum and to ensure timely mastery of the basics? Unfortunately, you have the big hurdle of convincing schools that mastery is important. They talk about balance and understanding, but they won't accept the responsibility of mastery. Many schools use Everyday Math that tells teachers to "trust the spiral". Big mistake. You need to look at actual test questions and results to get a better feel for the problem.
If you master the times table like a trained seal then that frees your mind up to solve more complex problems. Long division and algebra problems are now easier because you don't have to think about the basics and deal with the new concept at the same time.
ari-freedom
This is a hazy area to discuss, but doing problems, isn't doing math with rigor (just difficulty). Many of my students can do two-step problems - they have no idea why from many levels. They are just more accomplished seals. Of course, the turing test applies in a funny way.
Yes, that's a simpler solution that works pretty well - "an engineering" solution. Perhaps, Singapore Math with Professional Development for K-5 would be a great solution. This is one reason so many people support Singapore Math. It's the easiest way to get us on track with sustainability.
"For example, UNLEARN mastering times tables...."
Ummm, no. In fact, hell no. (Further iIntemperate language deleted for civility's sake.)
Facility (automaticity) with basic arithmetic operations is critical to the ability to do any higher math. When you are trying to solve a partial differential equation, you don't have the time to be thinking about the arithmetic you need to crank through. When you're working with fractions with disparate denominators, you don't have time to worry about how to find the LCM.
I disagree vehemently and categorically with your thesis. Ignoring the basics in the hope of reaching the "interesting problems" is one of the most pernicious ideas in math education today.
Regarding automaticity:
It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.
Alfred North Whitehead : British mathematician & philosopher (1861 - 1947); Source: An Introduction to Mathematics.
Orange Math -- let's think for a minute about strategy of learning to add numbers such as 7 and 5. Singapore and other curricula teach this strategy of "making 10":
7+5 = (7+3) + (5-3) = 10 + 2 = 12
Singapore regularly introduces strategies like this. It is great because it does two things. First it teaches the student a way to figure out the sum. Second it introduces the student to the over-arching idea of strategizing to solve problems. (The problem to solve is, for example, what's the sum 7+5; the strategy, make 10).
In the meantime we should also be teaching the student to automatically recall such facts as that 7+5 is 12. Hence flash cards and timed worksheets and so on. That way they can put such simple things behind them on go on to harder stuff. That harder stuff will also have TEMPORARY strategies that help students to solve the problems, grok the concept of strategizing, while also practicing facts to mastery -- and then moving on.
"speed means very little from a math perspective"
I beg to differ.
My son will have no trouble following a lecture in advanced mathematics since he can perform the basic math calculations in his head - automatically. But those students who failed to memorize their math facts will be getting lost when the instructor blasts through the basic math calculations of any particular equation.
In fact my 6th grade son's speed and level of memorization has far exceeded what I was taught and I am now the slow learner. He often has to explain the short cuts he has taken to his rapidly aging and slightly addled mother.
Failing to memorize math facts is the number one reason why students begin to struggle with math from grade 2 and on.
OK, I've been politely beaten up enough. Thank you. I was a bit strong, and maybe wrong, in my comments. Still,...
Many of you give arguments in knowing math facts and moving on and how that the key is not having to think with delays to move on - a very Gladwellian argument. Would Singapore Math students do better if memorization was pushed? I think not.
Building heuristics takes content mastery, not automaticity - just smoothness. Continuity is more important than velocity. In other words, automaticity is necessary, but not sufficient for understanding. Continuity of concepts to form a foundation is necessary also. This is the point of a common core: a consistent minimalization of what is necessary. Simply memorizing math facts inhibits the opportunity of learning mathematics in the early years.
Singapore math does push memorization. It just is expected that the drill is done separately. Ask on the forums at Singaporemath.com. My children memorized the addition facts before starting Primary Math 1A, and the multiplication tables before (I believe) 3A.
Fascinating discussion. A few points:
--
Chinese elementary math teachers are specialists: they teach elementary math, they don't teach everything.
As someone who tries to teach elementary math to future elementary teachers, I can tell that content as well as teaching techniques is in the curriculum, but teacher knowledge of elementary math is a big topic, and while we are training generalists, we're not going to be able to to justice to all of it.
For instance, OrangeMath has been talking a lot about learning make 10 strategies for figuring out addition facts. That's important teacher knowledge.
that doesn't mean don't memorize the times table--but the strategies for mental calculation should come before the memorization. I recommend to anyone who is teaching basic addition-multiplication table stuff Edward Rathmell's explanation of how to use the strategies to get to student competence with math facts:
http://thinkingwithnumbers.com/QuestionsAnswers/index.html
Personally I'd like to see more memorization of multiplication facts rather than less, but I don't mean that I want students to get faster at answering 100 facts, what I want is for children to practice multiplication facts backwards instead of forwards: being able to look at 36 and say that's 6x6 and 4x9 is a more important skill later on than being able to figure out 6x6 or 4x9, and that's a skill that only comes after you have that seems to come after the facts are memorized.
humbug--I need to proofread myself better. I think the only type, though, that changes what I meant to say in an important way is that I think children should practice multiplication facts backwards _as well as_ forwards (not instead of forwards).
My son mastered his math facts BEFORE we started on mental math exercises presented in Saxon 5/4 homeschool version.
"strategies for mental calculation should come before the memorization."
There didn't seem to be any downside to putting the cart before the horse.
Necessity is the mother of invention.
If you drill hard enough, the incentive to learn strategies will quickly take hold.
There are also other fun ways to put both brain and automatic reflex into math facts, like coming up with custom operators that are composite of more basic operators, and have them do a large set of drill problems,
"Simply memorizing math facts inhibits the opportunity of learning mathematics in the early years."
It does nothing of the kind -- and when you present an actual argument I might bestir myself to rebut it. Again.
OrangeMath has convinced me. Knowing stuff makes learning harder!
Lsquared, thank you for the link.
>>what I want is for children to practice multiplication facts backwards instead of forwards: being able to look at 36 and say that's 6x6 and 4x9 is a more important skill later on than being able to figure out 6x6 or 4x9, and that's a skill that only comes after you have that seems to come after the facts are memorized.
Doesn't this acquisition of the understanding of the Properties depend on both the student and the way the material was presented? It seems that some teachers/districts totally omit the part of the course that presents and uses the communative, associative and distributive properties. In our case, it was because the section that our child was placed in was deemed a 'concrete learner' section and that material was omitted in favor of finger tricks to 'aid' memorization of the facts. Much less understanding gained from that approach than his sibling's non=sped class with the same teacher where use of properties and abstract thought was expected.
Neither one of my children ever drilled on math facts. Singapore Math textbooks & workbooks were enough for the child stuck in the 'concrete learner' class to master the material over the summer. The other mastered the tables in class and amused himself by mentally working out how to get the 11s and 12s as well as the squares to 20.
In my experience as a volunteer at ele. school, a child that solely knows their facts without understanding can't use the properties until much later when they are formally presented in honors Alg. I..if the child makes the cut to get into the class. They simply get stuck on thinking that they know what the teacher is presenting, being only 8 or so in third grade, and totally skip the idea that they need to understand visually what multiplication is and how the properties work. This comes to bite them later in fractions and pre-algebra as they don't grok the true meaning of multiplication or the properties. But at 6,7, or 8, with the child having the tables perfectly memorized, what teacher is going to argue with the parents...much easier to give in...
"There didn't seem to be any downside to putting the cart before the horse."
The problem is that many don't think the cart is necessary or that the cart will magically appear if you ride the horse.
There are many levels of understanding, but mastery is not some rote add-on. There is linkage. Trained seals reflect bad teaching, not bad pedagogy.
Some students require more justification or motivation before learning new skills, but full understanding is not possible without the skills. The trained seal conclusion leads to a devaluation of the real importance of mastery, but trained seals have no mastery.
Is this a matter of definition? Perhaps, but it leads some to come to the wrong conclusion.
"The trained seal conclusion leads to a devaluation of the real importance of mastery..."
It's heartbreaking that the followers of constructivism fail to recognize that deeper understanding is precisely what students gain from mastering math facts.
"It seems that some teachers/districts totally omit the part of the course that presents and uses the commutative, associative and distributive properties."
Much instruction for mastery of the standard algorithms is also omitted in K-5 classrooms these days. Games and finger tricks, as lgm points out, supplant knowledgeable instruction.
There is a tongue-in-cheek aphorism that "If it has numbers, it isn't math", but that's more about a snobbish attitude that applied math is (in some sense) not real math. If it has a current practical application, it's engineering, or physics, or economics, or something; real math is purely intellectual.
Right!
Yes, I've come across that.
My friend believes that you don't need to teach the times tables because kids pick them up quickly and easily at home.
I probably argued him out of that one, seeing as how I could cite more than one adult who never learned his math facts -- and didn't 'pick them up' even though his parents very much wanted him to.
The problem is that many don't think the cart is necessary or that the cart will magically appear if you ride the horse.
That is a great way of putting it.
I see this all the time.
True of reading instruction, too.
"Expose" children to authentic literature and the cart will magically appear.
However, I'm sure he wants his kids to master the times table.
His own child did master the times table quickly, with very little teaching or practice, it seems.
He told me something fascinating.
For a time, the elementary school here had what I gather is a 'reform' curriculum: CSMP. I think this is it.
My friend told me that SIX of the kids in his child's 2nd grade class, which wasn't tracked, ended up taking BC calculus in high school.
--"Simply memorizing math facts inhibits the opportunity of learning mathematics in the early years."
This is still ludicrous. What's more, it's so wrong because arithmetic is a perfect example of why constructivism NEEDS math facts to mastery!
My son is learning addition. He knows how you do addition: you count all of the things you add together! He can tell you so explicitly. In this sense, he knows how to add! He knows the general principle by which addition works, but it is a useless way to know how to add. It does not derive for him the answer to 2 + 5. He still has to count it to get 7.
He knows some addition facts. For instance, he knows that 3 and 3 is 6,that 4 and 4 is 8, that 5 and 5 is 10. He knows that 2 + 3 is 5. He also knows that 3 + 2 is 5. He does not yet comprehend that 2 + 3 is the same as 3 + 2 is the same as 5, but he is close. He is going to learn about commutativity for that sum soon, and that will be because he has learned his math fact, not because he was taught commutativity. It will be constructivist, and happening in his own head, and it will be a direct result of repeatedly knowing that 3 and 2 is 5 and 2 and 3 is 5.
And more, there's no inductive step in his head yet. He isn't going to suddenly grok that 3+4 is the same as 4+3 when he does figure out 3+2 is the same as 2+3. That means he needs a LOT more math facts before he can make that inductive leap.
In fact, he's going to need to count on his finger several thousand times before he figures that out, or he's going to need to start memorizing some sums.
He does not yet know that 2 and 5 is 7, though he can add on his fingers. If you then ask him to add 2 and 5, he can't even start with "5" on one hand, and just add 2. HE ALWAYS counts from 1. The sheer number of math facts that we adults KNOW are the same is huge! We know that adding two numbers, we don't have to start counting from 1 again; we know that 0+5 is the same as 5. Kids don't know this. Even if you believe in constructivism, especially if you believe in it, why wouldn't you want to give kids the math facts so that they can make the correct inferences?
If you believe in constructivism, then you need to admit that real guiding from the side requires ultra-specific-tailored perfect knowledge of each child and their current mental state, their current mental knowledge. There's no way that can be done in a modern classroom. Parents are significantly more capable of it. But even parents know that it's easier to teach memorization so you KNOW WHAT YOUR CHILD KNOWS, so you can teach them to infer from there.
Allison,
Please consider "simply teaching math facts" as equivalent to using flash cards to memorize. There is a continuum, but this is on the far side away from abstraction, even experience.
The problem with the flash card approach is that it misses an opportunity to teach math along with the arithmetic.
This conversation isn't about constructivism. It's about correlation research (which has problems of its own). In Singapore, the use of flash cards (allegedly) is rare, Singapore does well on international tests; better than nations that use flash cards (eg USA). Hmmm. Maybe there is something to investigate. At the very least, the use of memorization may not be the best way to learn math even for facts.
In my world of students in high school. Next to none understand place value. They cannot read outloud a decimal fraction. They cannot round numbers. You can teach tricks like "five or more means add one," but you should see their faces when shown a number line. Duh. Visualization and handson helps greatly, but simple memorization and tricks like "cross-multiply" serve little purpose.
"Next to none understand place value. They cannot read outloud a decimal fraction. They cannot round numbers. You can teach tricks like "five or more means add one," but you should see their faces when shown a number line."
I'll say it again. This isn't pedagogy. It's incompetence. (I'm not talking about the students.) Consider that place value is taught over and over and over in all sorts of ways. It's a staple of reform math. Are you claiming that these students have never see a number line?
There is nothing mysterious about what's going on here in terms of how the brain works. You are seeing a problem and coming to the wrong conclusion. "Simple memorization" is a strawman that is used to justify your own bias.
>>And more, there's no inductive step in his head yet. He isn't going to suddenly grok that 3+4 is the same as 4+3 when he does figure out 3+2 is the same as 2+3.
Why do you feel this way? In my experience as an elementary parent volunter, there are preK, K, and gr. 1 students that do reason this out for themselves. They are the students that understand number bonds, or as R. Aharoni puts it in the linked article, the 'forming of a unit'. Jumping to memorization will defeat the attempt to lead the student to understand this extremely important concept.
OrangeMath,
You complained about "simply memorizing math facts". Now you equate memorization with flash cards.
I have never used flash cards in my life for anything, and I can memorize anything. They two are not the same. So your straw man that flash cards may or may not be used to teach something is irrelevant.
I can teach commutativity much more effectively to kids who already see that it's true. What part of that concept do you deny?
If they don't know that 2+4 if 6 and 4+2 is 6, then you aren't going to teach commutativity effectively.
Then you bring up other things that are different straw men. Your kids can't do place value? What does that have to DO with math facts? Either they were taught place value well or they weren't, but if you don't know single digit addition automatically, then you won't EVER comprehend the value of place value.
If your students can't comprehend a number line, they weren't taught it. This is about bad teaching, plain and simple. Are you claiming that they were taught using flash cards INSTEAD of being taught the number line?
That's just silly. They weren't taught at all.
-->>And more, there's no inductive step in his head yet. He isn't going to suddenly grok that 3+4 is the same as 4+3 when he does figure out 3+2 is the same as 2+3.
--Why do you feel this way? In my experience as an elementary parent volunter, there are preK, K, and gr. 1 students that do reason this out for themselves.
What I said didn't say he won't eventually reason it out for himself. Please read what I said. The inductive step isn't there yet. He isn't going to *suddenly* grok it after seeing one more example. What I expect he will do is *master* a few dozen such pairs and then *start* to grok it.
I witnessed this with his counting. He learned his numbers. Then he started to "count", but he didn't comprehend the one to one correspondence between the objects and the integers. He slowly did, over time. He would count incorrectly, and we'd stop him and have him recount. Repeatedly. Then, after several weeks or months, he stopped making errors when he counted objects to ten. He still makes errors counting on his fingers when he doesn't wiggle the correct one, as his motor skills are not quite able to do what he wants. Again, this is improving over time.
The idea that we see examples and then do an AHA never to again fall backward is an adult idea and probably wrong at that. It is not how children learn.
He has AHAs, but then he forgets the AHA and needs to relearn it. He again makes mistakes. Over time, the error rate drops.
Memorization does not defeat the attempt to lead the student. It supports it. Because learning is not linear. Teaching my son that 4 plus 4 is 8 has improved his counting skills--because he KNOWS he's supposed to get to 8. The inferences keep reinforcing in all the directions. Bootstrapping from something KNOWN is the important part.
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