Guest blogger Katharine Beals, PhD is the author of “Raising a Left-Brain Child in a Right-Brain World: Strategies for Helping Bright, Quirky, Socially Awkward Children to Thrive at Home and at School.” She teaches at the University of Pennsylvania Graduate School of Education and at the Drexel University School of Education, specializing in the education of children on the autistic spectrum. She blogs about education at Kitchen Table Math and on her own blog, Out in Left Field.
By Katharine Beals
Why underestimate what children understand?
Recent anecdotes from parents and recommendations from educators suggest that the underestimation of American children is alive and well in the world of K-12 education. In particular, more and more teachers and education experts seem convinced that kids don’t really understand the words they read or the numbers they manipulate nearly as well as their parents claim they do. Thus, one mother learns from her daughter’s 2nd grade teacher that her child doesn’t understand the chapter books she’s been reading for pleasure since kindergarten. She should be reading picture books instead. Another mother learns that the multi-digit arithmetic that her 3rd grade son has been doing since preschool is mere calculation, devoid of conceptual understanding. He should be doing simpler calculations using manipulatives and repeated addition.
How, and why, have so many educators become so skeptical about children’s understanding?
How to become skeptical is child’s play. Simply ask the child a question that ostensibly probes comprehension, but is either vague enough, open-ended enough, or verbally challenging enough that the child is unlikely to give the “correct” answer: What is that? What is it about? Why did you do that? If further probing seems necessary, ask equally difficult follow-up questions.
Ground-breaking math education theorist Constance Kamii has shown how this works with place value in particular:
1. Show the child a number like this: 27
2. Place your finger on the left-most digit and ask the child what number it is.
3. When the child answers “two” rather than “twenty,” immediately conclude that he or she doesn’t understand place value.
4. Banish from your mind any suspicion that a child who can read “27″as “twenty seven” might simultaneously (a) know that the “2″ in “27″ is what contributes to twenty seven the value of twenty and (b) be assuming that you were asking about “2″ as a number rather than about “2″ as a digit.
How might you convince yourself that a 3rd grader doesn’t understand multi-digit arithmetic? Why not tap into her immature verbal skills? Ask her to elaborate how she subtracted 562 from 831. When she stumbles, ignore any suspicion that articulating why one borrowed from the 8 in the hundreds place and reduced the 8 to a 7 is beyond the verbal skills of your typical 8 or 9-year-old.
How might you convince yourself that a 2nd grader doesn’t understand his above-grade level chapter book? Here, a sufficiently open-ended question may do the trick. Ask him what the book is about, or what will happen next, or how the text relates to himself. Then interpret any hesitation, stumbling, vagueness, or reluctance to respond as an unequivocal sign of deficient comprehension. Dismiss any suspicion that this line of reasoning implies that a teenager who answers “What did you do today?” with “I don’t know” doesn’t comprehend his day.
Perhaps less obvious is why some educators seem determined to underestimate understanding. Here are a couple of possibilities. First, doing so may level the range of apparent abilities in a class of twenty-something children. Parents might think their children are ahead academically, but if they don’t really understand what they are doing, there’s less pressure to provide them with an accelerated curriculum. There’s also less of an apparent achievement gap to be troubled by.
Underestimating comprehension may also serve to avoid or postpone teaching harder material that, frankly, can be a pain in the neck to teach. Believing that children don’t understand place value, for example, gives you an excuse not to teach those pesky standard algorithms of arithmetic. Why? Because if children don’t understand place value, then they can’t understand borrowing and carrying (regrouping), let alone column multiplication and long division. And unless they understand how these procedures work from the get-go, educators claim (though mathematicians disagree), using them will permanently harm their mathematical development.
What’s particularly striking about this underestimation is how much it seems to have permeated the establishment’s take even on those children it itself identifies as “gifted.” For example, at the recent New England Conference on the Gifted and Talented, most of the math talks either expressed concerns about children’s comprehension of place value, and/or advocated the use of manipulatives in place of abstract math. The mathematically gifted kids I know, however, grasp place value and other aspects of arithmetic with only minimal exposure to manipulatives, and quickly advance to higher levels of abstraction by the time they hit first or second grade.
So, indeed, do children in other developed countries around the world (see examples on my blog, Out in Left Field here, here and here)–whether or not we’d consider them “mathematically gifted.”
To stop holding our students back relative to their international peers, we need to stop asking them the wrong questions. Sometimes, indeed, no questions are necessary. If a child enjoys reading a particular book, then even if she fails to tell you what it’s about, she probably has a reasonable understanding of its content. If his multi-digit calculations are error-free, then even if he can’t clearly explain his steps in words, he probably has a reasonable understanding of his calculations. Comprehension may not be perfect—when is it ever so? — but the fact that it may need refinement is reason to encourage a child forward, not to stand in his or her way.
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Pingback by Confirmation Bias: When Educators Underestimate Children « The Core Knowledge Blog « Parents 4 democratic Schools — November 10, 2010 @ 12:13 pm
I’d say it’s a desire to level; that seems to be the underlying motive for most of what is happening in schools. The establishment is devoted to the pretense that all kids are equal in ability and effort and have a particular dislike of acknowledging that some kids are capable of covering more material, in more depth, in less time.
Comment by anonymous — November 10, 2010 @ 12:18 pm
I may have a different slant on this given that I am a high school teacher. But… I see the opposite of what this article describes.
I see teachers who WANT to teach more complex and challenging content, but are pressured by parents and administrators to dumb things down because complex and challenging content leads to a high student failure rate.
I see students who have been cheated of a truly rigorous experience who THINK they can write well, can comprehend what they read, etc., but then prove otherwise. I concede, though, that this might be because they haven’t been challenged enough in the early and middle grades.
I see students who think they know how to study, but then when they tell me how they study they prove otherwise.
I see students who are generally apathetic, make no connection between their performance in school and their quality of life as an adult, and generally prefer a less rigorous experience.
Having said that, I agree that many teachers assess in ways that do a pretty poor job of measuring student achievement. But, I think that means we should improve our assessment practices, not press blindly forward with more challenging concepts because we ASSUME students can do more.
That seems like an awfully big assumption.
Comment by Anthony Guzzaldo — November 10, 2010 @ 1:21 pm
I agree with Anthony (above) that, in my experience as a former high school teacher, I was often in the position of over-estimating the amount my students knew or were capable of doing. Granted, I started off teaching in a school for lower income students with learning disabilities, but I remember, for instance, asking my class on the first day of school to name some of their favorite books — only to be told they didn’t have any because they’ve never read a book for fun (in ninth grade!).
Comment by Attorney DC — November 10, 2010 @ 2:18 pm
I think Ms. Beals makes an important point –just because a students can’t demonstrate comprehension in a prescribed mode, does not mean that no comprehension has occurred. It seems to me that understanding is like painting a wall: often several “coats” of paint are required to finish the job. The first few coats are riddled with gaps, but they represent progress nonetheless.
Comment by Ben F — November 10, 2010 @ 3:10 pm
Another possible culprit is Bloom’s Taxonomy, or at least current interpretations of Bloom’s. Because administrators and “experts” are forever pressuring teachers to aim for the vaunted higher-levels (nevermind that a taxonomy, by definition, is not a heirarchy), the result is often lessons that attempt to manufacture higher-level outcomes, i.e. those that appear to require critical thinking and evaluation. Lost and devalued are the skills of remembering and applying vital concepts.
I would personally like to see this educational ouija board–and relic of the 1950s–go the way of bomb shelters, Hula-Hoops, and Formica.
Comment by James O'Keeffe — November 10, 2010 @ 3:51 pm
Ben F,
Interesting point. But, if a student can’t demonstrate comprehension, how can we tell it has occurred?
Comment by Anthony Guzzaldo — November 10, 2010 @ 4:25 pm
I think Ben’s point was not that the students can’t demonstrate understanding, but something more subtle. I think as teachers — hell, as human beings — we have a strong tendency to look for evidence that supports our point of view. Show a constructivist a kid who can compute flawlessly, for example, and Katharine’s place value example is “proof” that the learning is merely rote. Ask a different question (What is the value of the “2″ in 27?) and you might get a different answer. We tend to stop asking questions one we get the answer we want.
Comment by Robert Pondiscio — November 10, 2010 @ 5:00 pm
I buy into the idea that teachers are susceptible to confirmation bias and I can see how that might lead teachers to underestimate their students’ achievement and abilities.
I also agree that our susceptibility to this kind of bias should cause us to exercise extreme care when we assess and question students. It is generally a poor practice anyway to use a small number of measures to assess a student’s achievement and abilities.
However, I think this highlights, more than anything, the need for valid and reliable assessements.
If a student performs poorly on an assessment, the article above would argue, then I should either conclude that the student does not comprehend OR the student does comprehend but can’t demonstrate that because of the assessment method I have prescribed. Given that, I should press forward with more complex material?
Wouldn’t it make more sense for me to ensure my assessments are valid and reliable so I can tell, without having to assume, whether my students are or aren’t comprehending?
Or…. is that kind of your point?
Comment by Anthony Guzzaldo — November 10, 2010 @ 5:39 pm
When we’re talking about a child who can compute flawlessly, why do we need proof that s/he understands what s/he’s doing? It’s the understanding of those children who *can’t* compute flawlessly that warrants further investigation.
Comment by Katharine Beals — November 10, 2010 @ 5:41 pm
Katharine, You are being too kind towards elementary school teachers. You state that judgment of students regarding their students’ math ability is likely due to a confirmation bias, when it is entirely possible that those teachers have no understanding of the underlying mathematical concepts themselves and therefore are not confirming their bias but displaying their lack of understanding/knowledge. They can’t teach what they do not know.
How many elementary school teachers really understand and can teach place value? How many can use place value to explain the standard algorithms. If my experience is characteristic of the system, then that number would be embarassingly low.
However, there is a world of difference between a child who can compute flawlessly and one that can explain why that computation works the way it does. I would hardly expect young children to have the verbal acuity to express those concepts in words, but older children (5th-6th grade) should be able to flawlessly execute the standard compuational algorithms and explain why they work.
Comment by Erin Johnson — November 11, 2010 @ 2:39 am
“older children (5th-6th grade) should be able to flawlessly execute the standard compuational algorithms and explain why they work.”
Not if they have an autistic spectrum disorder, or some other expressive language disorder. Again, why bother probing the comprehension of students who can flawlessly execute the standard algorithms? The teacher’s precious time, rather, should be focused on those who can’t, while those who can should be allowed to move ahead.
Comment by Katharine Beals — November 11, 2010 @ 9:57 am
Katharine, it’s great to see you blogging here.
Another take: I think that teachers (at the elementary level, at least) have a thought-style that leads to the things you describe. They tend to think that words==thoughts. Their thinking is very “surface-y.” They also tend to be more emotional and relational (as opposed to intellectual). In short, there is a set of traits that’s selected for in elementary teachers that makes them ill-equipped to deal with students who differ from them.
I don’t know how true this is. I am just speculating here. Put I wanted to put the idea out there.
Comment by Hainish — November 11, 2010 @ 9:59 am
In addition, it’s certainly true that becoming proficient at an arithmetical operation, and doing it many times, can lead to understanding of why it works. It’s kind of the same principle as muscle memory in athletics or musical performance. Familiarity makes the underlying structure or relationship become more apparent.
Comment by pinetree — November 11, 2010 @ 12:28 pm
Clearly, if a child has a language disorder that level of verbal discription may be beyond them. That doesn’t ameliorate the value of being able to explain the standard algorithms verbally in children without those disabilities.
The benefit of being able to verbally explain how the standard algorithms work: This process illuminates what concepts and understanding that the child has developed regarding place value, the distributive property, the concepts of equal groups, etc… It is this profound understanding (and not just memorization of the definitions) that enables students to transition to manipulating symbols in abstract forms (e.g. algebra).
It is possible for students to intuit those foundational math concepts from mastery of the standard algorithms alone, but too many students can become computationally proficient without developing that intuitive understanding. And when those students move into Algebra, the abstract nature becomes overwhelming.
But in all fairness to our students, our teachers generally lack that foundational understanding of arithmetic. Asking students to describe why the computational algorithms work, when their teachers are generally clueless themselves, seems to be an exercise in futility.
Comment by Erin Johnson — November 11, 2010 @ 12:46 pm
My preference would be to focusing on *teaching* how the distributive property works, and how place value works (the brighter students might explore examples in base 2, base 8, etc), rather than on trying to assess through the messy prism of language whether students understand it. (As Hainish insightfully points out, thought and words are not equivalent.)
A combination of explicit teaching of the concepts in question and repeated practice (as pinetree points out) would make it highly unlikely that a child who consistently performs the algorithms correctly doesn’t understand them.
Indeed, I’m curious whether anyone has encountered a child who:
1. has been explicitly shown by a competent teacher how place value and the standard algorithms work
2. has had repeated opportunities to practice the standard algorithms
3. consistently performs the standard algorithms correctly
and yet:
4. doesn’t understand place value and the standard algorithms?
Of course,
Comment by Katharine Beals — November 11, 2010 @ 2:10 pm
This discussion brings to mind an anecdote from my childhood. One year for Christmas, I received two wonderful gifts: The original Baseball Encyclopedia, a massive volume of statistics of every player to have ever appeared in a major league game, and a calculator. (It was the early 70s; both of these were a big deal). I remember playing with the calculator and stumbling by accident on the formula for computing batting averages.
I understood the difference between a good and bad batting average. I understood that it was a function of the number of hits divided by the number of at bats. I probably understood — but could not articulate — that batting averages were a way to compare two players who did not have the same number of at bats. I might have even understood (I don’t recall) that if you got three hits for every ten at bats, you were a .300 hitter.
I definitely did not know the concept of ratios or percents. But when I learned percents in school, I remember clearly thinking. “Hey! That’s exactly how you figure out batting averages!”
Comment by Robert Pondiscio — November 11, 2010 @ 2:21 pm
“..make it highly unlikely that a child who consistently performs the algorithms correctly doesn’t understand them.” I wasn’t speaking to the specific algorithm but to the understanding of the foundational concepts used to create the algorithms.
So no, it is not obvious that most children will intuit the distributive property (or any of the other foundational concepts) from the standard algorithm for multiplication etc. Nor will they necessarily intuit the equivalence of a fraction to division of whole numbers from learning to add, subtract, muliply and divide fractions. But that equivalence is essential in algebra.
The foundational concepts are so embedded in the standard algorithms that it is difficult to tease out whether kids understand those concepts or not. Procedural competence is necessary but not sufficient. But to move to abstract manipulations of symbols, that conceptual knowledge is absolutely required (particularly with fractions).
There are at least two ways (that I am aware of) of assessing whether the students have made the connections between the foundational concepts and the procedures that they learned: 1) get them to verbalize their knowledge or 2) give them a non-obvious problem to solve that requires them to use that foundational conceptual understanding.
Either method requires a fairly competent teacher who understands what those foundational concepts are; not something that the vast majority of our elementary school teachers can claim to have.
Comment by Erin Johnson — November 11, 2010 @ 3:55 pm
Erin, I appreciate your insistence on making sure that students understand why algorithms work. But I’m also the product of a K-7 education during which we were never taught why they worked, and certainly not expected to be able to verbalize why. And yet, most of us progressed smoothly into Algebra. Why do you think that might have been?
Comment by pinetree — November 11, 2010 @ 4:54 pm
Giving them a “non obvious problem” to solve may or may not be an adequate means to judge understanding. As Dan Willingham has pointed out in various articles of his, the “inflexible knowledge” that you are characterizing as “non understanding” eventually matures with much practice and experience so that they foundational concepts can be applied to new problems. And it isn’t as if students are not given challenging problems. They are. Good teachers (and good textbooks) provide challenging problems that take them beyond what the repetition aspect of a set of problems, but taking students too far beyond what can be applied at certain ages is what I see as a common flaw. “Oh they lack understanding, otherwise they could solve it.” No, I don’t think so. Providing a context for what is happening when you do a particular algorithm is important. Some kids will use that information to gain a deeper understanding. Other kids will use only what they need to know (the procedure) to do the math asked of them.
When students get to algebra, some of them find they can articulate what is happening with arithmetical procedures, which they couldn’t articulate when they were younger. The difficulty in articulation is not limited only to those students with autistic spectrum disorders. It is common to a lot of kids. As they progress through math, they acquire new tools with which to express their understanding. Too much is made of this; that if kids don’t understand why the “invert and multiply” rule of fractional division works, they aren’t capable of math reasoning. Not true. And even Singapore’s math series doesn’t provide an explanation of that rule in the 6th grade text. They present an explanation in 7th grade when the student has had experience with algebraic symbol.
Comment by Barry Garelick — November 11, 2010 @ 5:04 pm
Pinetree, Some kids intuit enough foundational concepts from practice and apply this to algebra but by no means is this a majority of children. There were some kids who learned to read using Whole Language. It just wasn’t the best approach for the vast majority of school children. Likewise, asking children to pick up the foundational concepts from just repeated practice of the algorithms is slow and leaves way to many children behind.
Barry, That “inflexible knowledge” *may* mature. By no means is that a certainty. Certainly there are many kids who intuit the foundational arithmetic concepts from repeated practice of the standard algorithms. But is this the optimal use of their time?
One of the advantages of Signapore math is that their students are able to rapidly progress though arithmetic (and begin algebra in 6th grade) by using conceptual knowledge to enable students to learn faster and better. By understanding the “why”, students learn faster and better retain what they learn. And learning the “why” better prepares them for the symbolic manipulations in algebra.
As for the “non-obvious” problem, this is a tool that a teacher can use to see (in a non-verbal way) if a student got the fundamental concept or not. Most US elementary teachers do not even know what those concepts are, let alone how to see them in problem-solving and so using the non-obvious problem is probably worthless in the vast majority of elementary school classrooms.
To Katharine’s point above where she states “My preference would be to focusing on *teaching* how the distributive property works…” I agree. But teaching and learning are two entirely different animals. A teacher could think that he/she is illuminating the distributive property, but how is he/she to know whether each and every child got it? And frankly, the standard algorithms are awesome tools to show children how these foundational principals/concepts are applied in mathematics.
Conceptual understanding is a rather ill-defined proposition. Even Liping Ma had to use examples to show conceptual understanding in practice. There is no singular list of concepts that have to be mastered and I would suspect that there could be flexiblity regarding which concept and the level of mastery that is needed to be successful in algebra. But there are advantages of using conceptual knowledge to facilitate the learning of arithmetic: faster learning of arithmetic, better problem solving and easier transition to symbolic mathematics.
I realize that the “describe what you know” has been highjacked by the Fuzzy math guys and frankly, their version of “description” is misguided and painful.
My version is probably more akin to Liping Ma’s description of Chinese teachers:
“…Chinese teachers often cited an old saying to introduce further discussion of an algorithm: “Know how, and also know why.” In adopting this saying, which encourages people to discover a reason behind an action, the teachers gave it a new and specific meaning -to know how to carry out an algorithm and to know why it makes sense mathematically- … Verbal explanation of a mathematical reason underlying an algorithm, however, seemed to be necessary but not sufficient for the Chinese teachers. …after giving an explanation the Chinese teachers tended to justify it with a symbolic derivation. …they tended to point out that the distributive law is the rationale underlying the algorithm. … they showed how it [multiplication algorithm] could be derived from the distributive law in order to illustrate how the distributive law works in this situation and why it makes sense.”
Comment by Erin Johnson — November 11, 2010 @ 8:49 pm
Erin,
Are you under the impression that Chinese teachers ask their students to explain their answers?
I stand by my original question (with my “and”‘s spelled out, and a couple of other clarifications, in case they weren’t clear the first time):
Has anyone anyone has encountered a child who:
1. has been explicitly shown by a competent teacher how place value and the standard algorithms [plural] work [including the distributive law that underlies them]
*and*
2. has had repeated opportunities to practice the standard algorithms
*and*
3. consistently performs the standard algorithms correctly
and yet:
4. doesn’t understand place value and the standard algorithms?
Comment by Katharine Beals — November 11, 2010 @ 9:26 pm
That “inflexible knowledge” *may* mature. By no means is that a certainty.
With experience and practice, it does. And not just in mathematics.
I agree with Katharine’s observations. And Singapore’s success is that the conceptual is a context for the procedural. Again: they don’t teach why the invert and multiply algorithm works until 7th grade when the student has the algebraic tools with which to see how it works.
When I was in school, if I didn’t understand the explanation of why something worked, I got it by working the procedures. It’s a combination. And the books I used were not confined to just repeated practice.
As for the non-obvious problem good teachers make sure their students have the prior knowledge and use such problems as a means of exploration. Some students may get it after some work; some may not, but they learn from the discussion. Those that don’t get the problem are not necessarily lacking in the conceptual underpinning. I stand by what I said and what Katharine says.
Comment by Barry Garelick — November 11, 2010 @ 10:09 pm
I have a child who can provide long, extremely detailed narrations of the books she reads but who scored just okay on the reading comprehension portion of the standardized test she took. Why? Because she is still very literal and has difficulty “reading between the lines” to answer inference questions. She’s young so I’m not particularly concerned at this point. But I can see where certain educators would look to that and claim that she doesn’t really understand what she’s reading.
Comment by Crimson Wife — November 11, 2010 @ 10:29 pm
From Liping Ma’s translation of how a Chinese teacher described her class teaching of subtraction with regrouping:
“We start with the problems of two-digit number minus a one-digit number, such as 34-6. I put the problem on the board and ask students to solve the problem on their own, either with bundles of sticks or other leaning aids, or even with nothing, just thinking. After a few minutes, they finish. I have them report to the class what they did. They might report a variety of ways. One student might say “34-6, 4 is not enough to subtract 6, But I can take off 4 first, get 30. Then I still need to take 2 off. Because 6=4+2. I subtract 2 from 30 and get 28. So my way is 34-6 = 34-4-2=30-2=28.” Another student wo worked with sticks might say, “When I saw that I did not have enough separate sticks, I broke 1 bundle. I got 10 sticks and I put 6 of them away. There were 4 left. I put the 4 sticks with th original 4 sticks together and got 8. I still have another 2 bundles of 10s, putting the sticks left all together I had 28.” Some students, usually fewer than the first two kinds, might report … I put all the ways students reported on the board and label them with numbers, the first way, the second way, etc. Then I invite students to compare: Which way do you think is the easiest? Which way do you think is most reasonable? Sometimes they don’t agree with each other. Sometimes they don’t agree that the standard way I am to teach is the easiest way. Especially for those who are not proficient and comfortable with problems of subtraction within 20, such as 13-7, 15-8 etc., they tend to think that the standard way is more difficult.”
My read on this transcript is yes: Chinese teachers do ask their students to explain their reasoning.
Comment by Erin Johnson — November 11, 2010 @ 11:13 pm
27? Is this octal?
There are many levels of understanding and many times you have to stick with a basic understanding to start with. But also, there is linkage between understanding and mastery. Doing a homework set is not just about speed. People rarely understand something before practice. The big bogeyman is supposedly rote application of mathematical algorithms. This is an indication to me of not enough practice. Look at any proper math textbook and study the homework sets. Most are carefully constructed to test the student to make sure they really understand the concepts. I taught my son Algebra and Geometry the past two years and he always claimed that he understood it when I introduced the material. This, of course, was followed by many questions and “Aha!” discovery moments while working the problem sets. (Imagine, discovery while working alone on problem sets.) When I see a student who tries to solve a problem using some sort of rote, plug and chug technique, I know that the issue is not a poor introduction to the concepts. It is a lack of effort with the homework.
Comment by SteveH — November 11, 2010 @ 11:34 pm
“Parents might think their children are ahead academically, but if they don’t really understand what they are doing, there’s less pressure to provide them with an accelerated curriculum. There’s also less of an apparent achievement gap to be troubled by.”
We ran into this starting in Kindergarten. It really surprised my wife and I. I call them preemptive strikes. In Kindergarten, they tested our son for reading but they didn’t want to tell us the results. It didn’t matter, since we taught him phonics and he started reading when he was 4. When we did ask about the test, the teacher launched into a lecture on how some kids can read encyclopedias, but they don’t understand a word of it. We thought: “OK, where did this come from?” We decided that they really wanted to prevent us from daring to ask for something more or different. Of course, they never tested him for understading.
In first grade, it didn’t matter that we wanted the teacher to assign him chapter books and to write book reports. (they are supposed to differentiate) No. He had to do what the other kids were doing and she immediately launched into a lecture on how kids need to find their “voice”. When we once told her that our son loved geography and could find any country in the world, she said: “Yes, he has a lot of superficial knowledge.” Ouch! We were quite naive back then. Ironically, later that year, our son had to show the student teacher where Kuwait was when they were doing a thematic unit on sands from around the world.
We can have a nice discussion here about exactly how much understanding a student should have for each math topic and grade, but the real problem is not understanding. It’s something else. Understanding is a cover for low expectations and trying to treat all kids as equals.
Comment by SteveH — November 12, 2010 @ 12:00 am
On the one hand, nearly 50 years ago my 1st grade teacher, at a well-regarded private school, insisted to my parents that I couldn’t *really* know how to tell time (she was pretty snide to me about it too…).
On the other hand, when my daughter was in kindergarten I remember being terribly cowed by the number of parents who talked about how their children were reading Harry Potter, and in the end I concluded that their definition of “reading” was pretty flexible. I’m not surprised that teachers tend to be skeptical of parents’ assessments of their child’s abilities.
Comment by Rachel — November 12, 2010 @ 12:33 am
Barry, I am not sure why you are rejecting the need for conceptual understanding and allowing the fuzzy’s such a firm claim on one of the most critical elements that could actually dramatically improve US arithmetic instruction. Liping Ma’s treatise, the Teaching Gap and the Singapore math materials are all fairly consistent: conceptual understanding should underpin building procedural fluency.
This is not to say that conceptual understanding needs to be first, procedures second (Another canard, mistakenly promoted by the fuzzy math guys). To the contrary, proper math instruction is somewhat iterative and that conceptual understanding can/does build over time and experience.
Even more importantly, using conceptual understanding makes learning the procedures *much* faster as the students have a framework for catching the common errors seen in procedures alone. And those core ideas, which are the foundation of all math, are essential for symbollic manipulations seen in algebra and higher math.
We have no tradition in our country of teaching using conceptual understanding (and what the fuzzy’s are doing using TERC et al. does *not* qualify!). The way that you and I learned math, while perhaps thorough, did not teach conceptually the way that Liping Ma describes teaching in China, nor how the Teaching Gap describes teaching in Japan.
As for the non-obvious problem, I don’t prefer that approach. In my experience having children explain their thinking verbally was/is infinitly more informative. But given Katharine’s concerns about children with verbal difficulties, that approach is an alternative that can be used to elicit how the child is thinking.
SteveH, Elementary school teachers who do have a profound understanding of arithmetic (the way the Liping Ma describes) are few and far between. The term “conceptual understanding” has been mistakenly used to promote horrific (non-)math programs (e.g. TERC, Elementary Math, etc.). The fact that incompetent teachers are using that term to cover for their inability to teach (or low expectations in Katharine’s terms) does not negate the benefit that students get from using real math concepts in developing their procedural fluency.
Comment by Erin Johnson — November 12, 2010 @ 1:05 am
Erin,
“Elementary school teachers who do have a profound understanding of arithmetic (the way the Liping Ma describes) are few and far between.” Of course you know all about this charade but I often thought, if the public only knew.
It clearly explains why so many of these “teachers” reluctantly teach math at all and many find any excuse under the sun to CANCEL math for the day (We had an assembly. We simply didn’t have enough time today. We had to have extra time to do reading. It was a half day today due to teacher professional development. We had to have Tommy’s birthday party this afternoon. The fire drill took too long. Etc., etc., etc.). It was almost comical after awhile until you thought about what those kids missed.
SteveH,
“Understanding is a cover for low expectations and trying to treat all kids as equals.” Where to begin? Treating all kids as equals is code for whole group instruction – at all costs. That, of course, translates into convenience for the teacher at the expense of meeting the individual needs of students. And trust me Steve. This is one tough nut to crack in the public school monopoly.
“…(they are supposed to differentiate) No. He had to do what the other kids were doing and she immediately launched into a lecture on how kids need to find their “voice”.” Again, it’s all code for operating at the convenience of the teacher and the differences in the kids be damned. I often thought the same here: IF THE PUBLIC ONLY KNEW. A basic query: What’s easier, teaching one math lesson to a class of twenty-two third graders (all over the map in terms of ability, levels, degrees of motivation and readiness, etc.) or attempting to address the aforementioned differences, especially the different paces at which kids learn? It’s somewhat rhetorical, isn’t it Steve?
Comment by Paul Hoss — November 12, 2010 @ 8:09 am
Everyone seems to be assuming that this is a misunderstanding, probably pushed by schools of education and professional development. That’s true but if you track back to the underlying reports themselves it’s hard not to see a deliberate manipulation of the mission of K-12 schools.
And before someone asserts “conspiracy”, let’s remember that people who want to push an agenda that an electoral majority would not support have misrepresented what they are actually up to throughout history.
I have been reading the Common Core implementation documents especially those that have been prepared in anticipation of the new science standards. They all assume a US that is a planned economy where the federal government determines how resources are allocated and will find meaningful employment for citizens. They are full of false statements, plausible fallacies, and weak research. Classic examples of what one writes when you are trying to justify an already chosen path.
If you desire a future that is not free market, a republic, or based on the rights of individuals, you would not want schools that foster students who are independent readers capable of understanding whatever they choose to read. You would not want students trained through math and science to analyze for themselves.
All these “learning to learn”, “child centered” theories once implemented have the effect of creating citizens who are less capable of functioning independent of the state.
Somewhere Dewey must be celebrating what is possible with patience, misleading rhetoric, and a sufficient number of teachers, administrators, and professional interest groups who earn their livelihood from the blog not functioning. We are in the process of finally adopting his view of what the school should be.
We should probably remember what type of polity he thought his students would be living in.
Simply too many coincidences for there not to be coordination. Why?
Comment by Student of History — November 12, 2010 @ 8:28 am
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This discussion reminds me of a meeting between middle school teachers and my former superintendent. He said, “Of course you teachers know what your kids already know and what they still need to know.” He looked shocked when I admitted that I didn’t. Who they heck knows –really knows –what’s in each head? Johnny may be ignorant of almost everything, but because he has an uncle who lived in Patagonia he has detailed knowledge of vaquero gear. Each child has his own unique mental “fingerprint” –but it’s way more complex than a fingerprint. Until we develop some type of knowledge/understanding MRI machine, this “fingerprint” will never be visible.
If we had such a MRI machine, I reckon we’d see subtle gradations of understanding. The word “utopia” may suggest just “a kind of place” in one kid’s mind; “an impossible place” in another kid’s mind; “a good place” in another kids’ mind. These variations dependent on the various types and frequencies of exposure to the word. The MRI images might show differently shaped neuron constellations.
When I think about what’s happening in kids’ minds when I give a lecture or show a video, I know that neurons are firing and connecting in directions that we’ll never know. This is not to say we shouldn’t probe, but that probes don’t come close to measuring the ramifications of a rich learning event in the neuronal ecosystem.
Core Knowledge helps us fathom what’s in the brain’s “black box” because it lets us know for certain a good chunk of what went into the black box.
Comment by Ben F — November 12, 2010 @ 12:34 pm
Katharine, To answer your question, I have yet to see many US teachers really understand those foundational math concepts (distributive propery, place value, etc.) let alone use them to teach the standards algorithms. If you have then surely you have been quite fortunate.
Barry, Teaching the invert and multiply is rather simple to explain (algebra is not necessary) so I am not sure why you think that Singaporean teachers do not do so.
One possible explanation:
Problem: (3/4) / (1/2)
Solution:
1. Transform the problem into a simplier problem
2. Multiply the number (3/4) by 1
(3/4) / (1/2) = (3/4) x 1 / (1/2)
3. Multiplication of recipricals is equivalent to 1
= (3/4) x (2/1) x (1/2) / (1/2)
4. Identity property of division (any number divided by itself is equal to one
= (3/4) x (2/1) x [ (1/2) / (1/2)]
= (3/4) x (2/1)
By transforming the problem into a simplier problem it becomes easier to solve (one of the foundational concepts that students should be learning in arithmetic).
Comment by Erin Johnson — November 12, 2010 @ 1:00 pm
Good post and good discussion. It seems to confirm my general impression that it’s seldom easy to assess understanding. Failure of a student to succinctly verbalize an idea is usually not conclusive evidence of a lack of understanding. And the ability of a student to succinctly verbalize an idea is usually not conclusive evidence of the presence of understanding. Indeed, I’m not so sure I understand just what is meant, or ought to be meant, by understanding. Since fractions enters into this conversation may I mention an article of mine about teaching and learning fractions? It’s at http://www.brianrude.com/fracteach.htm.
Comment by Brian Rude — November 12, 2010 @ 4:34 pm
Reading this blog has convinced me to confess that I have been guilty of underestimating what my math students know. I taught high school juniors last year, and I am currently teaching 7th graders in the same district. I thought that the 4-year difference between my students would mean I would have to cover much more basic mathematical concepts. The various forms of assessment that I have used so far this year actually suggest otherwise. My seventh graders (three levels worth of students) are much stronger mathematically than I had expected. In fact, I have found myself using some of the same lessons and projects that I used with my high school students.
This of course begs the question of what has been covered in those four years if my high schoolers were working on some of the same concepts as my seventh graders. It’s also important for me to note that I wasn’t alone in my instructional decisions as a high school teacher. In fact, every other Algebra II teacher followed the same curriculum and sequence I did last year.
The main thing I plan to take away from this blog is to always assume my students know more than they do. I can always go back and spend more time on concepts they struggle with, but I can’t make up for valuable instructional time lost on material that they have already mastered. In fact, I have also noticed this year that the more I assume my students know, the more confidence they are learning to have in themselves.
Comment by s.harris — November 18, 2010 @ 12:21 am
This blog has been interesting to read from beginning to now. It seems to me that most people responding to this blog must not elementary teachers. In our society, there are many adults walking around that claim that they do not “do” math or they cannot solve math problems. It has become something that seems to just be accepted as normal. I am a second grade teacher and have had many parents tell me that they cannot help their child at home because they do not understand what is being taught or that this is the extent of what they can do. They will not be able to help their child in future grade levels because they lack the understanding of how to do the math. These people went through school and were taught the procedures such as the algorithm where they learned what to do, but not why and how it works. Many people that I run across have difficulty estimating when it comes to numeration.
When I hear about scores from testing situations (even though I am not a fan of standardized testing), the strand that is many times the lowest is that strand of numeration/number sense. There are many comments about people challenging the conceptual understanding of students versus procedural knowledge, but many students of today do need to have that thorough understanding before they receive the algorithm as it gives them the concrete experiences that they need developmentally before moving into the abstract. It helps them become better problem-solvers. Some children are natural connectors and if you teach them an algorithm and give them some practice, they will get it and will make the connection that the 2 in 27 means two tens or twenty in value. Most children will not do that independently and need experiences. Children learn and understand better when they are constructing their knowledge and are given opportunities to manipulate items and come up with generalizations and have class discussions about how and why things work. I agree that differentiation must occur for children, but the concepts must still be discussed and the teacher must strategically facilitate the discussion about findings and strategies. Some children need story problems to work on that have single-digit numbers while other can accomplish the same understanding by working with larger numbers. The discussion for the concept knowledge will still be the most important part of the math lesson because it is higher level.
I think that with the number of adults walking around verbalizing that they do not know math and not understanding anything about the equations and formulas that are quite important in higher levels of math, the thinking needs to change about how there are so many people that turned out this way. There will always be those who just “get it” but most are not that way from my experiences.
The complete base is in elementary school when it comes to developing a mathematical foundation which must be conceptual. Also, the standard algorithms are not the only way to solve problems. In elementary math, I teach strategies to solve problems. Students are able to explore possible strategies for figuring our their answers and of course must show their work or explain in words how they arrived at the answer, but the algorithm for adding or subtracting two or three digit numbers is not stressed at this level because there are so many ways to think about numbers and students are encouraged to find what works best for them. The grade level at my school has worked together within the last 3-4 years to be sure that we are teaching conceptually and we notice a difference in our students abilities and so do the teachers in third grade when they receive our students the next year.
Comment by Kharma Banks — November 18, 2010 @ 5:51 am
If an adult can balance their checkbook correctly 100% of the time, but does not know why or how addition and subtraction works, is this a problem?
For 99% of people I would much prefer computational accuracy over mathematical understanding. In reality mathematical operations are highly abstract and theoretical, most people never need to know these ideas, nor should they.
Comment by Matt — November 18, 2010 @ 7:30 am
I agree with Matt’s post to an extent. If someone can balance their checkbook correctly, that’s great, but are they using tools to help them or figuring it out on their own? So many people cannot even determine the tip at a restaurant without pulling out the cell phones and typing into the calculator.
I also believe that mathematical understanding leads to fluency with computations and accuracy. Perhaps most people never need to know these ideas, but, should we be the ones that determine what they need to know or should they have the option to make their own decisions based upon having the exposure and the opportunity to understand?
Comment by Kharma Banks — November 18, 2010 @ 10:40 pm
Matt, If the sole goal of schooling was to enable balancing a checkbook, then no: it wouldn’t matter. But if the goal of elementary arithmetic was to provide a solid foundation for understanding abstract math (algebra, etc.) then yes: students need to know how and know why arithmetic algorithms work. And there is no inconsistency with knowing how and being computationally accurate. The top students, internationally, know/can do both.
Comment by Erin Johnson — November 19, 2010 @ 5:51 pm
More importantly than being able to balance a checkbook or understand abstract math, I think that the goal of math teachers should be to help their students become logical thinkers and problem solvers. My goal for my students is that they will be able tackle any problem or project they are presented with by synthesizing and applying all of the knowledge they have in a logical and innovative way. After all, I think that my math major and computer science minor have shaped me into much more of a thinker than just a number-cruncher.
Comment by s.harris — November 21, 2010 @ 7:32 pm