on the 5th grade students she assessed:
I assessed a whole [5th grade] cohort in math (only one was a special education student, and he was no worse than the rest), with similar results.
They scored well in understanding "concepts." Whoop-de-doo. But none could reliably do computation with regrouping, do mental computation of any sort, measure accurately, use a number line, name fractions, or do any operations with fractions or decimals. Surprisingly they could not even count money correctly!! They could not figure out elapsed time, nor read non-digital clocks.
What good is all this great "conceptual understanding" if you can't count coins under $3.00, measure the length of a board, find the perimeter of a triangle or determine whether to add or subtract to compare two numbers? The one thing most were good at was reading graphs -- pictographs, bar graphs and simple tables.
Some of these students were quite bright and had good reasoning ability but what they ALL lacked was knowledge of number facts, facility with algorithms, precise vocabulary (perpendicular, acute angle, numerator, range), an organized approach (if guessing didn't work they were stuck), in short MASTERY at any level. Like [instructivist's] students, all have had nothing but fuzzy blah-blah since entering school. Few to none can afford Kumon and most don't have computers at home or access to them, so even that kind of practice is not available to them.
A recent study of teacher competence in our district found that most teachers in 5-8 grade mathematics did not themselves show mastery of the subject at that level. I think we may have come full circle. The system is now being run by people who are its products, and many are quasi-literate and numerate, however bright and caring they may be. Also, the lack of any kind of intellectual rigor or scientific and statistical training in their preparation has left them vulnerable to every fad that comes down the pipeline.
conceptual understanding w/o procedural knowledge:
The best way to illuminate my point would be to contrast my findings with what might have occurred in "the old days." Time was -- the 50's? 60's? when you might often find students who could proficiently, or at least adequately, perform basic operations -- including, in many cases, operations with fractions and decimals -- but would have been at a loss to explain what they were doing, or why they were (for instance) regrouping in subtraction, or inverting fractions to divide them. It was not considered necessary for students to have a "deep" understanding of the number system per se. Most of us (I remember this myself) "got it" in the process of learning the algorithms and how to apply them, and did (eventually) understand why you had to regroup, what you were doing when inverting fractions (I would have wanted to show how it works, even now it would be hard to explain succinctly in words, but it's easy enough to demonstrate).
The children I was assessing -- with a detailed, widely-used norm-referenced diagnostic math test and also with a locally developed "performance assessment" -- showed the opposite pattern. They understood about place value (haven't played with Base 10 blocks for years for nothing), knew that multiplication is repeated addition, that fractions can name parts of an object, or members of a group, and so on. They could show you (with the ever-present manipulatives), or draw a diagram and explain. They could tell you why you have to rename the ones as tens, why you have to keep the decimal points lined up, what the value of various coins etc. is, what the "big hand" and the "little hand" on a clock indicate, and so forth.
What they could NOT do was reliably apply a procedure to come up with an answer. Given a problem like 24X86, they knew this means you make 24 groups of 86 (or 86 groups of 24), but would get lost trying to build them with blocks or count out the tally marks. If they tried to use the algorithm, typically they got directionality, order of steps, etc. all mixed up, and they didn't know the number facts. The brighter ones would figure out the answer had to be something around 2000, but many did not even get that far. When counting coins, they lacked a strategy such as, counting the quarters first, then the dimes, then the nickels, etc. They randomly counted each one separately and continually lost count. They didn't know how to "count up" to find a difference (or an interval between numbers or clock times).
This is so helpful. It makes sense to me that a student could have some degree of conceptual understanding without procedural knowledge, and yet when I try to think of how that would work I come up with a blank. It's clear to me, for instance, that my "conceptual understanding" of unfamiliar subjects -- economics, say -- is thin at best. God is in the details.
hyperspecificity for conceptual knowledge -- ?
One thing I think I see happening with palisdesk's 5th graders is "hyperspecificity" for conceptual understanding. I'm used to seeing hyperspecificity for concrete knowledge. I hadn't really thought about hyperspecificity for concepts such as the meaning of multiplication. It makes sense, though. I'm pretty sure it happens to me all the time, teaching myself algebra 2. I'll have a basic conceptual understanding of a concept -- logarithms, say -- that doesn't immediately transfer to a problem type I haven't done before. I'll try to come up with an example to post.
As usual, I'm hamstrung by a lack of terminology. Our sturdy workhorse words -- procedural, conceptual -- are failing to give me the distinctions I need within the category of "conceptual understanding."
These 5th graders have for arithmetic what I have for logarithms: some kind of start-up understanding of the concept that won't take them very far when confronted with an actual logarithm problem in the flesh.
thank you palisdesk, pissed-off teacher, instructivist, redkudu, dy/dan, nyc educator, exo, smartest tractor (I'm sure I'm leaving others off ....)
For parents and the broader public schools are a black box. Mike Schmoker says that's by design; the official term for management in schools is "loose-coupling," which means, I gather, that the goal of management is to protect the core functions of the organization from outside scrutiny. (more later...much, much later)
Teachers who share their experiences with outsiders are functioning as the education reporters our country needs but does not have.
Robert Slavin on transfer of knowledge
hyperspecificity posts
loose coupling & instrutional leadership
Brownell and other reformers from the 20's through the 50's were after teaching math with meaning, so that students understood why the procedures worked. I had a fairly good understanding of place value and how it related to addition, subtraction and multiplication. For fractional division, many books simply established a pattern, showing students that 1/3 of 6 for example is the same as 1/3 x 6 which is 6 divided by 3. Later, the student is reminded that 6/3 = 6 x 1/3 just by virtue of the procedure he/she has mastered previously. Similarly, problems like 1/2 divided by 3 are presented so that the student understands 1/2 is divided into three parts, hence it is 1/3 OF 1/2 or 1/3 x 1/2. The student is again reminded that 1/2 divided by 3 is 1/2 x 1/3. With the pattern of invert and multiply shown, the book I had then made the leap to apply such concept to all fractional division problems.
ReplyDeleteThis of course didn't fully explain why one inverted and multiplied but we had a sense that it followed the pattern we had been seeing and that sufficed for the time being. I notice that Singapore Math provides an illustration of 12 divided by 3/4, showing why invert and multiply is the solution, but then makes the leap from whole number divided by fraction to fraction divided by fraction.
The point is that the procedural with some explanation gives a sense of how things work. The deep understanding is not really necessary to know how to solve problems, and as students progress in later grades, more information as to the "why" of such procedures is given, particularly when they reach algebra.
Ironically, students in today's reform math who are supposedly taught the deep understand can neither do the procedure nor understand why anything works.
Skills without understanding can be fixed. Understanding without skills cannot.
ReplyDeletePalisdeck, I was wondering if you could tell us the name of the widely-used norm referenced diagnostic math test you referred to? I need something like this. Thanks.
ReplyDeleteMost likely, it is the Stanford Diagnostic. A terrific test.
ReplyDeletehttp://harcourtassessment.com/haiweb/cultures/en-us/productdetail.htm?pid=015-8893-45X
Love the Stanford Diagnostic. But we don't have it available.
ReplyDeleteThe test I used was Key Math (Revised) from American Guidance Service. It has flexible uses; for example individual student assessment (often used to develop IEPs for exceptional students), but it is closely keyed to curricular expectations and can be of assistance in identifying instructional areas to focus on when used with a cohort of students (that's why I was using it); because it has both a "Form A" and "Form B"it can be used pretest/posttest style to evaluate the effectiveness of an instructional intervention or curriculum module.
Here's a blurb about it:
The Key Math-R test provides both content- and norm-referenced assessment of a student's quantitative strengths and weaknesses. This revised version has been updated to reflect the NCTM standards so that additional emphasis has been placed on such skills as estimating, prediction, reasoning, interpretation of data, and solving situational problems.
The content is organised into three areas: Basic Concepts, Operations, and Applications. Each area has three or five strands which are further subdivided into three or four domains. Each domain is assessed by six individually administered items, so that each content strand is a subtest of 18 to 24 items; the total test has 13 subtests (516 items) to provide measurement of a student's mathematical abilities. Each item is also referenced to an objective.
I surfed for some links that would spell out more details and found these:
http://ericae.net/eac/eac0125.htm
http://www.tki.org.nz/r/assessment/two/pdf/reviews.pdf
This latter document on pp. 51-55 has an extensive description of the test. Some of the caveats don't apply unless you're from New Zealand;-)
A major limitation is that it is individually administered and so rather time-intensive. If you have a lot of experience with it (I guess I can claim this), you can halve the time usually required by accurately identifying the starting level. Following the guidelines suggested for age and grade can be very time-consuming as you go backwards and forwards to get to the required basal and ceiling items on each subtest.
Within a half dozen students I noticed a distinct pattern and could be almost 100% thereafter in correctly picking the starting item that would give me three correct responses in a row. I find the test provides an opportunity to observe the student's thinking and approaches to problems in a way a paper-and-pencil test rarely does. Certain patterns and instructional needs were common across students, regardless of perceived ability or achievement.
My school has recently tanked in our standardized tests in math (down to about 20% proficient), so a district "improvement team" has been sent in to help us boost our results. Get ready for this: the improvement team is spending a lot of time, money and energy helping staff develop skills in teaching...... reading comprehension strategies!!! Our reading results are not bad (about 60% proficient or advanced), but ......I guess our reading results will be even better this year. Meanwhile our math scores continue to plummet. Something about this bizarre scenario seems to me to epitomize the state of public education as we know it.
Anyway, I convinced the admin team that we needed data on what students actually knew and what specifics needed to be emphasized instructionally if we are going to improve our outcomes.
Unfortunately what I have found out is that they need to be taught pretty well everything.
Not sure how we will do that. We're so busy doing all this extra stuff on reading comprehension......
oh wow!
ReplyDeleteKey Math is supposed to be great!
I spent a lot of time cessing it out two years ago when I was seriously panicked about C's math knowledge (not that I'm not seriously panicked now...)
We desperately need a REAL diagnostic assessment of where he is.
Key Math is far too expensive for an individual family to use. My school apparently uses it for SPED kids, not for regular kids.
The good news is that the new assistant super is bringing in some kind of formal formative assessment in math -- but I wonder why they're not using Key Math for everyone?
I'm remembering she said Scott Foresman?
Does that ring a bell?
(Must find my notes.)
My school has recently tanked in our standardized tests in math (down to about 20% proficient), so a district "improvement team" has been sent in to help us boost our results. Get ready for this: the improvement team is spending a lot of time, money and energy helping staff develop skills in teaching...... reading comprehension strategies!!! Our reading results are not bad (about 60% proficient or advanced), but ......I guess our reading results will be even better this year. Meanwhile our math scores continue to plummet. Something about this bizarre scenario seems to me to epitomize the state of public education as we know it.
ReplyDeleteSame here. My district is obsessed with "literacy." ALL time and energy are being poured into literacy -- WAC, writing workshop, etc. -- while the math department is in absolute crisis from a parent's point of view. (The middle school math teachers are giving writing assignments, too.)
In our case the administration isn't exactly "wrong" in its focus, however. The superintendent was hired at a point where there was enormous parent demand for decent writing instruction, so that was her mandate.
There was no mandate for improved math instruction. There was widespread rejection of TRAILBLAZERS, but of course widespread rejection of a program -- as opposed to widespread interest in the implementation of a brand-new program -- never translates to a mandate.
One of the things I see happening -- I'm sure I'm right about this -- is that the steady exit of Tier 1 & Tier 2 teachers is radically changing the quality of the school but no one is picking up on it. People don't realize that these aren't the same schools their older kids attended.
The introduction of Trailblazers combined with the loss of the middle school math teacher who chaired the department for so long -- combined with the refusal to interview or hire middle-aged, experienced math teachers to replace her -- mean that the math program has taken catastrophic hits.
Unless people can tutor their kids up to the level the school would have gotten them in previous years, we'll see our scores decline.
Of course, parents here are on top of things. They can be counted upon to step in and remediate the school.
Ed and I would love to have data on how much parent remediation is going on here. It's huge, but I'd like to know what it consists of specifically.
ReplyDeleteAlso, we're curious about differences amongst parents.
Some parents remediate; some don't -- are there consistent differences between these two groups?
I wonder whether kids who are getting As and Bs in the "regular" track are most vulnerable to the loss of experienced teachers. Parents get activated by bad grades.
Barry - which book is that lesson in?
ReplyDeleteThe school administered the Key Math test to my daughter (who has an IEP) last year after I requested a beginning-of-year assessment. Coincidentally, I was at a SPED workshop yesterday, and the speaker downplayed the value of tests such as the Key Math that generate a grade-equivalent score. The problem is that the overall score is an average, and may not necessarily reveal a student’s weak areas. They’ve probably had parents who misunderstand this and may be satisfied thinking their child is performing at a particular grade level when in fact there are significant deficits that should be addressed.
ReplyDeleteIn my daughter’s case, she scored above grade level partly because her computing skills are so high due to Kumon. Some of the other test areas were average or below.
I’ll have to check to see if they will administer this again, and plan to do it annually. It seems to me that this is a good test to measure “present levels of performance” as required in an IEP.
"Skills without understanding can be fixed. Understanding without skills cannot."
ReplyDeleteHuh?!?!
Since the kids start with neither, this suggests that if they understand the concept, but need more practice to build up skill, the cause is hopeless.
This *can't* be true. The Spanish Inquisition could arrange for the skills to be learned ...
Have I misunderstood your point?
-Mark R.
Barry - which book is that lesson in?
ReplyDeleteIt's not one lesson, but many spread out through the book. It's Arithmetic We Need, Book 6 by Buswell, Brownell and Sauble.
Skills without understanding can be fixed. Understanding without skills cannot.
ReplyDelete"This *can't* be true. The Spanish Inquisition could arrange for the skills to be learned ..."
Well, you *can* always start all over again.
But imagine students taking algebra in ninth grade after 8 years of conceptual understanding-based math. With big gaps in their skill set, they have little chance. However, a student who knows the basic skills cold need not worry much about understanding. They have more than they think.
" But imagine students taking algebra in ninth grade after 8 years of conceptual understanding-based math. With big gaps in their skill set, they have little chance. However, a student who knows the basic skills cold need not worry much about understanding. They have more than they think."
ReplyDeleteWould an accurate summary of yhour position be that "you can build on skills without conceptual understanding while repairing the conceptual understanding, but not the reverse."
??
-Mark Roulo
"Would an accurate summary of yhour position be that 'you can build on skills without conceptual understanding while repairing the conceptual understanding, but not the reverse.'"
ReplyDeleteYep, but I was trying to be more dramatic.
Reform math supporters believe that there is no linkage between understanding and skills. That's why they focus on conceptual understanding and don't worry about mastery of skills at any one point in time. I call it an excuse for low expectations.
Real understanding cannot be had without skills. If you have skills, you have some level of understanding. The next level of understanding can then be achieved.
Reform math supporters view all skills as rote; providing no additional benefits over conceptual understanding. They think that mastery adds only speed and not understanding. That's their basis for using calculators.
Years ago, I got into an argument with a well known reform math denizen of the math-learn forum. His argument was that he knew all that he needed to be a soccer referee, without ever being a soccer player. But I want my son to practice math, not appreciate it. I want my son to be on the stage playing the piano, not sitting in the audience. The difference between viewing and doing is not just rote speed.
Of course, a practicing soccer referee develops skills and experience, but that's a different goal.
This comment has been removed by the author.
ReplyDeleteI noticed that my school was trying to push “conceptual understanding without procedural knowledge” in the way they wanted to write one of my daughter’s IEP math goals. They wanted to write: "the student will correctly choose the operation(s) required to solve word problems”, with no mention of actually solving the problem correctly. I guess they just wanted to be sure that she understood how to do it. I had to push to have them change the goal to have her able to actually solve the problems.
ReplyDeleteI saw another IEP where the math goal was also written so the student would simply learn which operation to use, so I’m thinking this must be used a lot.
"They wanted to write: 'the student will correctly choose the operation(s) required to solve word problems' ..."
ReplyDeleteThis probably isn't what they have in mind, but the correct operation is 'NAND'. Seymour Cray could build an entire supercomputer using just NAND logic and RAM :-)
So ... imagine teaching your daughter to say, "I would use NAND. Many, many times," in response to all word problems :-)
-Mark R.
Maybe I can make a convincing argument that a student who only thinks of multiplication as iterative addition and can't multipy 24X86, has neither procedural nor conceptual knowledge.
ReplyDeleteThere is more to the "concept" of multiplication than iterative addition. (Try using applying iterative addition to 1/8 x 2/5)Perhaps iterative addition is appropriate for 2nd and 3rd graders learning their multiplication tables (or is it 3 and 4th graders these days?)But "the" concept of multiplication includes the fact that it distributes over addition (and that it's associative as well) The multiplication algorithm invisibly makes use of the distributive "concept", and does not employ iterative "concept." Perhaps I'm overdoing the disdain quotes but I've been lied to too many times by people telling me that something is the "concept" of a procedure or rule and it turns out not to be.
A child with a conceptual knowledge of multiplication, and a lot of time on his hands, could successfully multiply two digits numbers wihtout the multiplication algorithm:
24 X 86 means that
(20 + 4)(80 + 6)which means/implies that...
Etc. You see where I am going with this. One of the benefits of Singapore is that the kid does end up with a conceptual understanding of multiplication, and can apply his knowledge of concepts to come up with correct answers.
Notwithstanding operations on super hairy numbers they are capable of doing the algorithm on paper when they need to and can resort to "concepts" when they need to do mental calculations.
I would use NAND. Many, many times.
ReplyDeleteI'm going to have C. add that to his short response bank.