tag:blogger.com,1999:blog-7691251033406320222.post1108170858411831492..comments2019-09-17T22:19:31.277-07:00Comments on kitchen table math, the sequel: Frederick Hess: Common Core tests were the fix for NCLBCatherine Johnsonhttp://www.blogger.com/profile/03347093496361370174noreply@blogger.comBlogger19125tag:blogger.com,1999:blog-7691251033406320222.post-46488205851458003182015-01-24T06:42:20.122-08:002015-01-24T06:42:20.122-08:00Good morning,
I ended up posting a little more ab...Good morning,<br /><br />I ended up posting a little more about "invert and multiply" on my blog, here:<br /><br />http://advancedmathyoungstudents.com/blog/<br /><br />Then I got curious. I realize that I don't actually know how elementary schools teach this topic. (I taught a lot of math to my children in an informal way, but forgot to closely monitor how they were taught in school.) So what I am curious about is: is this kind of step-by-step intuition building part of the process?<br /><br />One reason I am asking is that some of the anti-common-core material that I have seen seems unfair. I am talking about the kind where they point at a series of exercises designed to build understanding and then mock it by showing how much quicker the standard algorithm is. I am in favor of the algorithms. But mindless application does nothing to build number sense or understanding.<br /><br />PhilAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-13461090536450044082015-01-08T15:08:30.290-08:002015-01-08T15:08:30.290-08:00I agree with Glen that sometimes, showing the proc...I agree with Glen that sometimes, showing the procedure and then going back and showing how it works later is valuable. <br /><br />I audited calculus in high school and took it for real as a freshman in college. With that experience, I have always thought that teaching students the quick-and-dirty way of taking a derivative, spending a little time telling them the power of it and what you can do with it, and then going back and showing how and why it works would be a better sequence.Auntie Annhttps://www.blogger.com/profile/05777983027361603449noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-14699537232640501602015-01-06T09:35:58.644-08:002015-01-06T09:35:58.644-08:00I think Mr. Novak's proof is completely approp...I think Mr. Novak's proof is completely appropriate for pre-algebra students. And you only have to replace the numbers with letters to have a general proof. <br /><br />I also would like to see some exercises that increase the intuitive feel for the result. For example, 20 cookies are used to provide a bunch of students with servings. How many students can you serve if each gets 5 cookies? 4 cookies? 1 cookies? 1/2 a cookie? 1/4 a cookie? 1/5 ? 2/5? <br /><br />But also, I agree with Glen's post about adding the reasoning at a later look -- as long as that actually occurs!<br /><br />PhilAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-89546913913810685972015-01-06T06:14:00.215-08:002015-01-06T06:14:00.215-08:00What is understanding? Is it a formal proof? Clear...What is understanding? Is it a formal proof? Clearly, that's not possible or necessary for students just learning math. Here is a "proof" of "invert and multiply", which really isn't a proof, but it's what I used when I taught math.<br /><br />http://www.moveitmaththesource.com/realfractions/mrnovaksproof.html<br /><br />However, does this explanation tell you what to do when you see 3/(7/5)? Do you need a different "proof" for that? Understanding is not one thing, and there are different levels of understanding.<br /><br />How about 3/4/5? One of my pet peeves is order of operation. There is no such thing. It's only a construction needed for equations when they are defined in a linear text format. I only learned about it when I got to programming. Proper math is a beautiful two-dimensional construction. There is no order of operation. There is only what you can do and what you can't do. This leads to understanding issues when students are constantly told to "simplify" as if that form is correct and any other form is incorrect. I liked to mess students up by moving factors up and down by changing the signs of their exponents. I would also write things as X2 instead of 2X. Understanding is a on-going process at all levels.<br /><br />Is understanding a process whereby you can solve problems with some sort of vague Polya technique - draw pictures, label variables, think backwards? In those terms, I see understanding as the ability to know which governing equation applies to a problem and solving it using developed skills with that equation on a number of basic variations. In most cases, one will not have to "think". If the problem is a new variation you haven't seen, then you are starting at a base camp, not some blank Polya slate at sea level.<br /><br />When I was in 8th grade algebra, we were required to explain what rule (identity) we used for each step. We found it annoying for really simple things. We wanted to do two or three steps at one time. So each of our problems had to be "proved" as in the link above. There is understanding to be had in each new problem you do. Understanding comes from mastering homework sets, bottom-up, not top-down with a vague thinking process.<br /><br />However, one might want to have some sort of big picture understanding like when to use explicit, implicit, and parametric equations, but that's not the sort of vague trivial understandings I normally hear about in educational pedagogy where mastering skills is only rote. There is no way to pass a math course with rote skills unless the teacher is incompetent in testing and grading. Doing problem variations proves understanding. There is no understanding that is compatible with lack of skills. Skills (understandings) might be limited, but the solution is still bottom-up, not top-down.<br />SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-64868334415108013702015-01-05T16:05:59.020-08:002015-01-05T16:05:59.020-08:00One thing I've seen repeatedly is that it is o...One thing I've seen repeatedly is that it is often more efficient to go back and deepen understanding of a previous topic than to require a deep understanding at first encounter. Learn things to a limited depth, memorize what you don't understand (and what you do), and come back later to deepen understanding. You must not omit that last step.<br /><br />The key is the proper sequencing of topics. Do enough explanation, memorization, and application of a topic to push it over a certain threshold of usability, then move on to topics that require its ongoing use in ever-expanding ways. As you proceed through the sequence of topics, you both move forward and methodically revisit and deepen the understanding of previous topics using your new skills. (You don't just "spiral" over unusable skills in a random sequence trusting that some of them will eventually "stick".)<br /><br />For example, I take a kid whose operations on algebra I equations have become habits, and have him look at his old "invert and multiply rule" in terms of his new algebra. I show him: A/B = A(1/B). Okay, that's now obvious to him. And what does it mean? Since he now knows that the reciprocal of X is 1/X, it means: "A divided by B equals A times the reciprocal of B". And if B is a fraction, N/P, what would 1/(N/P) be? We write (N/P)(P/N) =1, divide both sides by (N/P), and 1/(N/P) is obviously (P/N). Whatever N and P are, the reciprocal of the fraction is the inverted fraction. So A divided by (N/P) equals A times (P/N). Invert and multiply.<br /><br />When we look at an old idea through the lens of the new skills, it becomes an "Oh, I see, of course!" experience replacing the original, arbitrary "rule". Some teachers seem to think that if you don't fully understand invert and multiply at first, you will be handicapped, but trying to gain this degree of understanding prior to learning the later ideas is a waste of time. Other teachers seem to assume that a kid in algebra who has thoroughly memorized invert and multiple, uses it correctly, and is doing well in algebra must understand invert and multiply algebraically, but that's usually not the case. Without explicitly revisiting it and examining it using algebra, it will probably remain fossilized as a memorized procedure. You have to explicitly revisit old topics with your new tools.<br /><br />The sequence is what matters. Get enough understanding to aid memorization, get enough memorization to make it usable, keep using it until habits form, keep adding new skills, then go back with those skills and deepen the foundation of understanding. Eventually the memorization will become just a convenient summary of the deep understanding applied to common problems.<br />Glennoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-13884620658788031342015-01-05T08:50:33.491-08:002015-01-05T08:50:33.491-08:00Understanding is only shown by being able to do al...Understanding is only shown by being able to do all sorts of variations of problems. Do we expect students to do a proof to justify how to divide fractions? Does a proof mean that the student will be able to divide 2.34 by 7/16? How about 9/13 divided by 2 1/2? What words prove understanding in a way that would show that they can solve any problem? What about the transition to dividing 5/(x-3) by (x+3)^2/22? This requires a good understanding of factors and basic identities? Everyone can spit back the "understanding" that a/1 = a, but do they understand what that means in a situation where you have to divide 5 by 3/16? <br /><br />When I tutor high school students and back when I taught college math, it's easy to get them to say things in words and I can tell that they "understand" when we go over something like synthetic division. However, the proof is when they do the homework assignment all by themselves. In a properly taught math class, it is not possible to pass with just rote understandings. They might have partial understandings and gaps, but there are no understanding words that will fix the missing understanding that comes from a lack of practice.SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-91093564834031926122015-01-05T07:19:16.667-08:002015-01-05T07:19:16.667-08:00This is going to be a "it depends" answe...<i>This is going to be a "it depends" answer. If the kid can solve that problem after having be shown how to solve that same problem explicitly, and maybe having been shown a bunch of examples of that problem, then I would say "no" - if he or she can't explain, most likely it is just a memorized procedure which will disappear once no longer practiced. Unfortunately, this is the way a lot of math is taught.</i><br /><br />No. If a kid understands what's going on with the problem so that he/she knows that division is the way to solve it, but cannot explain why the invert and multiply algorithm works, so what? And yes, many things disappear if you don't practice them; again, so what?<br /><br />Students are usually shown fractional division via a whole divided by a fraction. Then gradually to fraction divided by fraction. <br /><br />The invert and multiply action becomes more obvious in 7th or 8th grade when they've had algebraic equations and can also know what reciprocals are and that a number multiplied by its reciprocal equals 1.<br /><br />And there is no shame in being shown how to solve a problem via examples. If the problems then become harder by variants of the same problem, then they are properly scaffolded. Sweller, Kirschner et al talk about how the worked example effect can and should be used in teaching math.Barry Garelickhttps://www.blogger.com/profile/01281266848110087415noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-50922922721083539922015-01-05T07:04:46.375-08:002015-01-05T07:04:46.375-08:00"Today's Common Core approach to teaching..."Today's Common Core approach to teaching STEM is at least superficially appealing."<br /><br />PARCC does NOT do STEM. We can't get very far in a national debate when people get the basics wrong.<br /><br />NCLB and CC do NOT deal with how to get to honors or AP or to STEM. With NCLB, nobody expected that. I'm not sure why CC leaves people with that impression. Is it because they talk of "college readiness"? Do they just gloss over the fact that the highest level in PARCC only means that one is likely to pass a college algebra class?<br /><br />So now we get backtracking from all sorts of people to say that it's not "college readiness" - that CC's goal is really much more modest. But we still don't have a national test. CC actually defines very little. There are several players in the test mix. ACT is appealing because they claim that they will use the same sort of college-ACT numbers (up to 36) so that students and parents can track their progress from the earliest grades. Unfortunately, this is where the funny business of nonlinear calibration happens. How does an ACT of 30 in fifth grade translate to an ACT of 30 at a junior in high school? Is it done via help at home or with tutors?<br /><br />With PARCC, your child might get top-level "distinguished" ratings all through K-8 and then find out that he/she is completely not prepared for a STEM program in college no matter how many summer courses or doubling up is done.<br /><br />CC is a one-size-fits-all curriculum where there is no common national test or calibration.<br />SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-64739627068666877532015-01-04T13:50:27.660-08:002015-01-04T13:50:27.660-08:00I agree with Anon @ 1:38 that it's not necessa...I agree with Anon @ 1:38 that it's not necessary for young kids to be able to explain in words but that the older kids who would have been on the 60s-version of the college prep track should understand the whys and hows. I don't think it's necessary for all HS kids. Many, if not most, would be better off if they could actually DO what used to be k-8 arithmetic, without calculators; basic facts, operations, std algorithms, fractions, decimals, percentages, mean, median, mode, std deviation etc. The whys and hows are important for higher-level math, but not everyone is able or willing to operate at that level of abstraction. The one-size-fits-all approach doesn't work for many kids. momof4noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-88120969494806708442015-01-04T13:38:22.134-08:002015-01-04T13:38:22.134-08:00But, froggiemama, it's entirely possible to un...But, froggiemama, it's entirely possible to understand why an algorithm works without being able to express is efficiently in words. I agree that students (at least those who expect to go to higher levels in math) need to understand why/how; I just think it's unrealistic to think they can begin to explain that, in words, and especially at an early age. So then the question becomes, how do we provide chances to demonstrate that understanding?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-51934307938871917582015-01-04T08:53:28.498-08:002015-01-04T08:53:28.498-08:00And there, I fundamentally disagree. yes, students...And there, I fundamentally disagree. yes, students should be graded on whether they understand the reasoning. That kind of mathematical thinking is important. <br /><br />Students who memorize the procedure without understanding it will lose the procedure unless they are doing it over and over. Generally, that doesn't happen. This is also a reason why many bright kids hate math so much. They learned it as a set of memorized procedures that made no sense to them. As soon as those kids no longer have to do invert-and-multiply, they quickly forget. That is how we get so many students in college who seemed to have had good grades in math, and who seem bright, but who can't do basic math. They learned a class of problems, like the one posed above, by memorizing a bunch of similar examples, then aced the test by just doing the same thing. They quickly forget, though they may have to rememorize in their SAT prep course. But as soon as that SAT is over, bye-bye.<br /><br />That is also why so many of my CS students insist on memorizing examples of programs rather than understanding them. They get upset because we don't teach CS the same way they learned math. They want us to give them a program, which they will memorize, and then give them an assignment which is exactly the same with different numbers. It is very upsetting to many students that we don't do this in CS, and that we want them to understand the reasoning behind the algorithms.froggiemamanoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-49559775362714280182015-01-04T07:08:39.180-08:002015-01-04T07:08:39.180-08:00I was shown why invert and multiply works; it was...I was shown why invert and multiply works; it was interesting, and I think the teacher wanted all of us to be able to follow why it worked. But that is a FAR cry from having each student be required to be able to call up that reasoning on demand, and describe how and why it works as part of their graded seat work or homework, or on a test.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-36158244829760795532015-01-04T06:20:34.861-08:002015-01-04T06:20:34.861-08:00I don’t think procedures “lend themselves to under...I don’t think procedures “lend themselves to understanding” or not. We have to actively take steps to understand them and to communicate that understanding. Invert-and-multiply is a concise rule with a lot of math behind it. But it’s not the only one like that. Math education is filled with such things for many years to come. <br /> <br />In general, we have these two tasks: teach the algorithm and teach the reasoning. We want students to be able to use the algorithms with ease so that they can apply them to problems and also use them as building blocks for the next algorithm. But we should also teach where the algorithm comes from. I know that students are not going to hold that reasoning in their heads every time they apply the algorithm. But if we don’t teach this part, then we are turning math into what it was never intended to be: a collection of unconnected factoids (that some would call “knowledge”). <br /><br />Also, don’t we hope that at least some of our students will reach the point where they can create their own algorithms, come up with their own mathematical ideas? That is not likely if you have not been teaching the reasoning as well as the algorithms. It can be done in either order, but I agree that complete understanding includes the reasoning behind the algorithm. Like many people, I don’t think I was ever taught why invert-and-multiply works.<br /><br />Phil<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-8959497132483278372015-01-04T05:37:17.523-08:002015-01-04T05:37:17.523-08:00This is going to be a "it depends" answe...This is going to be a "it depends" answer. If the kid can solve that problem after having be shown how to solve that same problem explicitly, and maybe having been shown a bunch of examples of that problem, then I would say "no" - if he or she can't explain, most likely it is just a memorized procedure which will disappear once no longer practiced. Unfortunately, this is the way a lot of math is taught.<br /><br />All procedures lend themselves to understanding. Remember, someone had to INVENT those procedures - I sure hope that person understood them. Also, if we want to be able to solve these problems by computer, developing algortihms to do this, we better understand the procedures!Froggiemamanoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-68217749637245748142015-01-02T16:23:24.923-08:002015-01-02T16:23:24.923-08:00Similarly, a kid who memorized the rules for divid...<i>Similarly, a kid who memorized the rules for dividing fractions, but who doesn't understand why it works (usually because he or she doesn't really understand fractions in general) does not KNOW how to divide fractions. In math, you have to understand the concepts before you move on to something else.</i><br /><br />So if a kid can solve the problem of how many 2/3 oz servings are in 1 3/4 oz of yogurt by knowing that you divide 1 3/4 by 2/3 and does the procedure correctly, but cannot explain why the invert and multiply rule works (as was the case for me in 6th grade), do you conclude the kid lacks understanding? How are you defining this holy grail of "understanding"? Sometimes procedures lend themselves to understanding, sometimes they do not. J.D. Salingernoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-80657180386046981042015-01-02T13:45:12.165-08:002015-01-02T13:45:12.165-08:00"But as with any new approach to teaching, th..."But as with any new approach to teaching, the Common Core builds on the culture that's already there."<br /><br />Yes!<br /><br />That's why we should be using the Iowa Tests (ITBS)! They are tried and true and offer everything we need.<br /><br />Has anyone made a coherent argument against using the ITBS as our common school accountability test?Katharine Bealshttps://www.blogger.com/profile/02006668809174569929noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-4870383371944243212014-12-31T13:10:18.564-08:002014-12-31T13:10:18.564-08:00If they had decided to make everyone use the NAEP ...If they had decided to make everyone use the NAEP instead, trust me, people would be screaming just as loudly.<br /><br />As for knowledge vs understanding in math, I don't think there is a difference. If you don't understand what you are doing in math, you don't know it. In the simplest case, a kid who can tell you the times tables, but has no idea that 3x3 = 3 + 3 + 3 certainly does not know multiplication. Similarly, a kid who memorized the rules for dividing fractions, but who doesn't understand why it works (usually because he or she doesn't really understand fractions in general) does not KNOW how to divide fractions. In math, you have to understand the concepts before you move on to something else.Froggiemamanoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-53197971085619638842014-12-31T12:21:38.323-08:002014-12-31T12:21:38.323-08:00And the other, other question is: Do we currently ...And the other, other question is: Do we currently have a problem of copious knowledge paired with insufficient understanding (such that fixing this would improve matters somehow)? I think the answer to that one is also "no."Hainishnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-46754684165818889122014-12-31T12:14:32.256-08:002014-12-31T12:14:32.256-08:00The question "Is it possible to have a great ...The question "Is it possible to have a great deal of knowledge but limited understanding?" is a good one to ask. The answer is an obvious yes.<br /><br />Of course, you also want to ask the other question, "Is it possible to have a great deal of understanding but limited knowledge?" That is a more difficult question to answer well, but I think that the readers of this blog will agree that the answer is "no".Anonymousnoreply@blogger.com