Beliefs About “Understanding” in Math, Dept.

Here are some of many beliefs about “understanding” in math.  It was hard to choose from so many candidates, but feel free to add some of your own.  

We shouldn’t be teaching kids algorithms before they have the conceptual understanding.

The belief is that standard algorithms for mathematical operations (like adding/subtracting multidigit numbers, multiplying and dividing multidigit numbers, multiplying/dividing fractions, etc) eclipse the conceptual underpinning.  That is, why the algorithm works.

The standard way used to be taught first, and alternate ways later, after mastery of the standard algorithm. Now it’s other way around in the belief that std algorithms eclipse “understanding”. Side dishes now become the main course and students grow confused—sometimes profoundly so. 

Problems are to be solved in more than one way, in the belief that doing so imparts and gives evidence of “understanding”.  You have students being required to solve simple problems in multiple ways supposedly to enhance discovery and impart understanding.   You have students drawing pictures for much longer than necessary, serving as both a means to simultaneously understand and explain an otherwise simple procedure.

Many of these are the same methods for operations that are taught in the traditional manner. But these alternative methods were taught after students had achieved mastery with the standard algorithms—and not for weeks on end. And many students discovered these methods themselves.  Adding 56 +68 and being made to say 60 +60 instead of 6 +6 when carrying and adding the tens column for example defeats the purpose of the algorithm which is to free up working memory.

Robert Craigen, a math professor at University of Manitoba describes this approach as “the arithmetic equivalent of forcing a reader to keep his finger on the page sounding out every word, with no progression of reading skill. It amounts to little more than a “rote understanding” for procedures that unfortunately students probably cannot perform for problems they cannot solve.”  

A blogger recently summarized it this way:

Next year’s teachers that are used to students using an algorithm for multiplication are aghast when students use unsophisticated strategies like counting by ones by drawing pictures or partial product by drawing boxes, or when the students seem to not have any idea what to do. “What do you mean, just multiply!” But to “just multiply” by mimicking an algorithm isn’t part of what students had been doing. These teachers shrug in frustration and teach “the only right way”. Students are left feeling either shafted by the previous teacher or, most likely, that they must just not be “good at math”. “

“Students who fail to understand a concept are unable to know how to use it or build upon it. They will end up with misconceptions that can go undetected for months or years.”

How is “understanding” defined?  And what do they mean by failing to use it or build upon it?  Yes, if a student only knows that 3 x 2 = 6 but does not know what the multiplication represents or what types of problems can be solved, then that student will not be able to use multiplication. 

But does lack of conceptual understanding always have this result?  While progressivists all want to teach for “understanding”, they do so without fully defining what understanding is. A definition of “conceptual understanding” does exist; it appeared in the National Research Council’s 2001 report “Adding it Up”:

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)

But this has been interpreted to mean that “procedural understanding” is rote memorization and does not entail connections to mathematical ideas.  An example of a student having procedural fluency but lacking conceptual understanding was given in a popular blog about math:

[The student} can accurately compute the area of a triangle, but doesn’t recognize how its formula was derived or how it can be extended to other shapes. (eg. trapezoids, parallelograms, etc.)

and elsewhere, someone referred to calculation as being only procedural:

“Calculation is the price we used to have to pay to do math. It’s no longer the case. What we need to learn is the mathematical understanding.”

Does this mean that a student cannot solve problems that involve areas of various shapes because he or she does not know how the formulas are derived? Is a student who does not know the derivation of the invert and multiply rule for fractional division unable to solve problems involving such operations?  

There is a difference between a novice and an expert. A student who knows a procedure but not necessarily the conceptual underpinning may later gain more understanding as they work with such procedures in solving problems.  And some students may never understand it.  What level of understanding are we talking about here?  Do we expect students to acquire expertise in a top-down fashion, with understanding first and procedure and application later?  Is it wrong to let them solve problems using the standard procedures. Or must we always sacrifice proficiency on the altar of the often undefined but cherished “conceptual understanding” ?

And last but not least, this old chestnut which has appeared in just about every math textbook written from every era you can imagine:

“In the past, math classes were about teaching facts, skills and procedures with no understanding,and mechanized drills.”

Despite this claim, it’s interesting that many adults who were educated in the eras caricatured as “failing thousands of students”, are much more capable at solving the arithmetic problems that today’s students struggle with, even those entering high school.  They are even capable of solving the open-ended, “rich”, depth of knowledge questions that are generally ill-posed one off problems that do not generalize and are assumed to lead students to “deep understanding”.

One example of such a problem is “What are the dimensions of a rectangle with a perimeter of 24 units?” I’ve seen adults who claim they are not good at math get further with such a problem–as ill-posed as it is–than the students judged to lack understanding because they cannot solve them. Apparently, knowing how to calculate the perimeter of a rectangle given its length and width is viewed as “mere memorization”. Subsequent scaffolding of such problems, like “What is the length of a rectangle with a perimeter of 24 and a width of 8?” are viewed as just more memorization. It evidently helped the adults able to come up with answers to the “rich” problem.

Again, we are dealing with levels of understanding along the spectrum of novice vs expert–a spectrum that is conveniently ignored as students are forced to endure a top-down approach to understanding with–it goes without saying but I’ll say it–disastrous results.


13 thoughts on “Beliefs About “Understanding” in Math, Dept.

  1. Your goal is like shooting fish in a barrel, but will a teacher audience suddenly realize that what they were (ironically) directly taught by rote is fundamentally wrong? Then there are the educators who know exactly what they are doing – defining the learning process to match how they want a classroom to operate. It’s all about them – “the process is the product.” Beyond the very low level of the Common Core, they abdicate all responsibility for skill enforcement, and in the case of eliminating algebra in 8th grade, they ensure that many kids will never live up to their potentials.

    Most high schools fix this with proper AP and IB math course sequences, but for many kids, it’s all over and no amount of Pre-AP pushing in 9th grade and doubling up in math will fix that. Common Core let K-8 off the hook in terms of expectations and Pre-AP is the College Board’s attempt to create a magical nonlinear change to get to a STEM or proper college level by the end of high school. They KNOW what the problem is and they expect students and parents to buy this solution and accept the responsibility of failure. This is educational incompetence.

    We’ve been fighting this for over 20 years and I don’t know a solution unless it’s driven by aware parents. They are the ones who need this information to better challenge what schools are doing – or at least, fix the problem themselves. But when the in-class differentiated (self) learning groups are formed, will teachers notice that the ones in the advanced groups have all gotten help at home? Will they see how the imbalance relates to the educational background of the parents, but not figure out or care that it’s what they should be doing? With my son, some teachers thought I was a “tiger parent.” Nobody asked what I did at home.

    “In the past, math classes were about teaching facts, skills and procedures with no understanding, and mechanized drills.”

    This is fundamentally and provably wrong. Still it is used, and still they don’t define understanding – conceptual or otherwise. What logical argument can prevail? They will never accept that they have a systemic problem on a massive scale.


  2. “What are the dimensions of a rectangle with a perimeter of 24 units?”

    It is a great problem as long as it is mixed in with others with real answers and as long as the answer is the only appropriate one, some form of “The problem is ill-imposed.” Did you get it out of some printed work we could laugh at or is it just a generic example?


    • It comes from here, and there is no mention to the student that it is ill-posed.
      The author poses the question to the student after asking what is the perimeter of a rectangle with dimensions of 4 and 8 units. Then he asks the open-ended question which the student can’t answer. The open-ended question comes out of the blue and is not exactly mixed in with others as you suggest, nor is it scaffolded in any way, such as asking to find the length of a rectangle with perimeter 24 and width of 5.

      The author uses it to show “evidence” that a student who can’t answer it readily does not possess “deep understanding”, only procedural understanding of what a perimeter is.


  3. If a student can answer (by guess-and-checking etc) the length of a rectangle with perimeter 24 and width 8 but is lost finding the length of a rectangle of perimeter 10 and width \sqrt{3} then I say they lack the critical element of understanding needed for this problem. Or if you don’t like irrational numbers her, how about perimeter 1224 and width 321? I don’t care if the person solving one of these latter versions is simply pulling out and manipulating a memorized formula — they clearly have the better understanding over the guess-and-checker and even if they are the proverbial trained monkey, through repeated application of that rule over a range of problems they will gain further deepened grasp of what’s going on over time, unlike the one who can only solve the version contrived to work under ad hoc explorations.

    I think Wayne’s point about the problem being ill-posed (in reference to the other form — what are the dimensions of a rectangle of perimeter 24?) is correct, notwithstanding that Barry is suggesting that this wasn’t the point, it was about inflexibility of knowledge that is simply application of a straight-up formula and that other problems in the were not posed in this way. Granted this too, but it does not negate the ill-posed business; in fact I think it’s partly why Kaplinsky’s point is wrong.

    Note first that the jump is not from a straight-up use of a formula to a problem in which the formula is used in reverse, as Kaplinsky seems to think — the student just doesn’t grasp enough to use it in back-to-front manner — it is a leap from a well-defined problem to an ill-posed one. The meta-problem of resolving the expectations of the question poses a greater burden than the problem itself.

    A few years ago I put in time & energy tracking down what educationists are calling “rich problems”, and from what I could tell, a rich problem is almost always a classical word problem with some information taken away. In other words it is an ill-posed problem, such as “The difference of two numbers is seven. What are the numbers?” I really (really, really) dislike this. It’s dishonest, and it teaches students — and teachers — bad habits.

    Referencing the “Student 1 – Part 2” video there are many problems evident in what the teacher does here, so sorry for the long list to follow (and I’ve surely missed some stuff)

    1. The ill-posed problem. I’m complaining both about the problem and also about the use of this category leap to try to force a point about formulas not supporting understanding.

    2. “List the”. Huh? There are generally TWO dimensions for a rectangle. “List” a list of two things? This is a misdirection. Not that it’s bad — I’ll ask students for “ALL the solutions in positive numbers x, y to the equation x^2-6x+y^2-8x +25 = 0” and be happy when they provide the single unique solution x=3, y=4. But I don’t ask this of novices who are first trying to grasp what equations in two variables mean and what one means by a “solution” or who lacks the requisite algebra background to crack this one. It’s a misdirection because “All” suggests they may find more than one solution, and they won’t. A “list” of two things isn’t technically wrong, but it adds an extra complication likely to throw off the novice.

    3. Further, it’s imperative that expectations be made clear! What constitutes “a list”? This is an unfamiliar way to pose such a question. Does the student have confidence he’ll know when he’s complied with the requirement? I’m not sure the teacher even has a clear idea. I think he has in mind that those numbers will appear somewhere on the page. But, a stickler for meanings of terms, I expected to see a sequence of numbers separated by commas. (Ideally accompanied with sufficient explanation to indicate what those numbers represent). Alas, it is evident that what the teacher clearly intends is that the student labels two of the sides of the rectangle. Sorry, that’s not a “list”. Not in my dictionary.

    4. I dislike that the teacher “leads” the student throughout, including leading him astray. You can hear the student listening for cues in the teacher’s questions. The teacher’s voice signals approval when the student writes “24 units” along one side: “okay … ” (signalling “correct so far” so the student believes they are on the right track) “…so how long are the other sides?” (Now the student can infer that the teacher believes by putting that number there, the student was indicating that was the length of the labelled side. And the teacher said “okay”. So if he didn’t think so already the student now “knows” that it is correct to understand that this side has length 24).

    5. Now using good Socratic technique, when the student is now apparently lost, the teacher prompts “so this side is 24 units long?” Student: “Yeah”. The teacher has now effectively reinforced the misconception and signalled the student to use that as a starting point for finishing the problem.

    6. The teacher asks how long is the opposite side — the student correctly replies (using obviously formulaic knowledge about the properties of rectangles — what ought to be recognized as “understanding”) “24” But this display of understanding goes unremarked.

    7. “And how long are these [other] sides?” The student, evidently now completely lost (who wouldn’t be? he is now trapped in a world of internal contradictions! Not only does he not know how to proceed correctly, there isn’t one. Because of the ill-posed nature of the question, there wasn’t one in the first place, anyway. The best I can do is, if the perimeter is 24 and two opposite sides are both 24, then this must mean that the other two sides add to -24, so they must both have length -12 units!)

    8. So the student, in good progressive-education style, resorts to a combination of guess-and-check and “using sense to estimate”: “I’d say about 18”. In my view he’s eyeballing the diagram badly. Looks more like about 7 or 8 units, relative to the 24 unit side, to me. But anyway, he’s doing “guess and check” so why not?

    9. [Teacher:] “Can you label them please?” [Student, to himself:] “whew, I guessed right — he’s asking me to write that answer down. No idea why, but that’s a relief!” [Teacher:] “Okay [there it is again!] than you very much” [Student, to himself:] “Thank goodness. He likes my solution. This stuff isn’t as bad as I thought it would be. I’ll try to figure out how I got the answer later…” And he comes away now with the belief that to “list” dimensions means to label them on a diagram. NOW I KNOW WHO’S RESPONSIBLE FOR DESTROYING OUR INCOMING STUDENTS’ GRASP OF STRAIGHTFORWARD ENGLISH INSTRUCTIONS!

    Aside: One reasonably intelligent student on our recent combinatorics midterm answered each of the 10 true-false questions with which I began the exam by writing TRUE or FALSE to the right of the question. The instructions for the question read (in boldface):

    “Answer (T or F) in the left margin”

    Sigh. It’s not just one student. Every year I get these. And this was an honours class of 10 students. Don’t get me started on my first year calculus classes of 140 non-majors!

    It is really easy to manufacture whatever you want with simple misdirection techniques like this. Nice staging and banter. I wonder if Kaplinsky moonlights as an up-close illusionist?

    The ill-posed problem includes a bad constraint implied by the definite article “the” indicating dishonestly that there is a unique correct “list” of dimensions comprising “the [only possible] dimensions” of the rectangle. Ignoring this incorrect wording, the correct solution to a corrected version of the problem is, where x=width and y=length: (x,y) = {(t, 12-t): 0<t<12}. I would not recommend this problem for grade 8 students unprepared to give something like this parametric solution. Give them a well-defined problem so they can concentrate on the immediate intended learning outcome.


    • “…a rich problem is almost always a classical word problem with some information taken away. In other words it is an ill-posed problem, such as “The difference of two numbers is seven. What are the numbers?” ”

      And if a student has difficulty coming up with answers to the ill-posed and dishonest question (i.e., the word “the” implies only one right answer), this is viewed as evidence that the student lacks true understanding of addition and subtraction. Interestingly, if you ask an adult this question there will be some confusion,(i.e., “There’s more than one answer here”) and come up with some answers. Evidently they gained the understanding despite being taught in the traditional manner that supposedly results in rote memorization and no understanding. In the meantime, the poor 6 and 7 year olds presented with this so-called “rich problem” feel they are bad at math, which is the opposite of what the purveyors of “rich problems” wanted.


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  5. “Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)”

    How do they test for this? We have seen many of their silly examples, like the perimeter question. Traditional math is always taught by “connecting those ideas to what they already know.” There is a lot of understanding that comes from mastery of sequentially scaffolded units in a traditional textbook. That scaffolding and building of skills requires a lot of understanding at many levels. This develops proper understanding a level at a time from the bottom up. There is no magic top down understanding that makes doing P-sets simple for each individual. In-class group projects hide individual fuzziness and allows them to ride the coattails of those who most likely are getting help at home or with tutors. We NEVER hear how these educators support individual success on homework. They only care about what goes on in class. They talk about conceptual understanding, but don’t have a clue how to create it for STEM students.

    So the question is – after 20++ years, where are their success stories and curricula for students who have gotten into STEM degree programs? Why did my son graduate from college with a degree in math when I didn’t care one bit about their definition of conceptual understanding? I had to support him in K-8, but then let the high school traditional AP Calculus sequence take over – where I didn’t have to help him one bit. Why is their blather ignored by all traditional IB/AP math tracks in high schools. Why is their blather ignored by all STEM programs in college? Phillips Exeter might use a Socratic Harkness table approach, but it’s opt in and goes far beyond the low expectation silliness they promote.

    Vacuous – and harmful.


  6. Slightly off-topic comment:

    You address the complications that arise in student learning that occur with progressive math instruction. It is sad to see these ill-formed, open-ended questions waste student learning time.

    When my oldest daughter, DD, was still in public ed, she had this type of question, except in terms of area. It was so bad we ran through the proofs with her and created a problem that had actual content. The problem wanted the student to maximize an area for a park given a limited perimeter of fence. The perimeter was NOT given. And nowhere did the problem state the shape must be rectangular or even a quadrilateral or if the fencing could use an existing building edge. They were called low-floor, high-ceiling questions.

    She explained her work well after we taught her exactly what was wrong with this problem and how to change it to a useful question. She showed why the maximum area is a square for rectangles and explained why circles minimize perimeter and maximize area. She had the area formulas for a number of different common shapes and used a perimeter she set. She presented her work in her group and was told by the teacher circles or other shapes weren’t allowed. The reason? The class had not covered area of circles or other shapes and the group needed to agree on the answer. However, one group decided they liked the rectangular park they designed to play soccer and have a playground and so chose those dimensions – which was an accepted answer. All this took 2 and a half days of class!

    Progressive, open-ended, inquiry-based learning is a problem in science too. It is only after kids have a strong foundation in principles and technique that they can innovate effectively.

    DD will be ready for advanced science next fall (AP Chemistry, though I no longer believe AP is as “rigorous” as they claim and may go another route).

    The public school (WA) was happy to lend equipment support until I gave them the course description I created and list of required labs (every lab found in the text “All Lab, No Lecture” as well as a few others from various resources). The current AP Chemistry teacher emailed me back, stating she could not recommend my course because the labs were “old-fashioned,” too numerous, did not create enough inquiry since the student had a lab manual with instructions, and used chemicals that might not be green. Requiring the student to create a triple beam balance, their own bent-glass apparatuses for various experiments, designing and running their own labs from the lab manual, and creating their own fuel cell and calorimeter is apparently not the right type of inquiry.

    Her links went to a digital simulation lab website and “Beyond Benign.” She stated that her class runs fewer but “deeper” labs that the kids create themselves, practice micro-chemistry to be green, and model “best practices” with simulated labs.

    Does anyone have recent experience with the AP chemistry course? It could be a local problem but the teacher’s description of AP Chem does not seem as rigorous as my old advanced chem class in the 1990s. I can get approval from the AP Board to create my own homeschool course but I’d like to make certain it is a useful endeavor. The AP lab manual is called “Guided Inquiry.” Sigh….


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