The images are then shared by hundreds of people who add their own comments that sound very much like "This is crazy! If a PhD in engineering can't do it, how can we expect children to?"
The most recent one was posted to my own wall and a discussion ensued that made me feel as though I needed to write about this.
This is crazy! ...So, he started at 427, then subtracted by hundreds? Ok, I can see that. So he went down on the number line by 3 sets of 100, then... It should have been a 10 and six 1's, but he went down by six 10's instead.
This entire process took her less than a minute. I did not prompt her to figure out what was happening or how to do it or what mistakes had been made. She came to all of those conclusions on her own. While my wife is a college graduate (with a better GPA than I had), she is not a mathematician or a math teacher. Her area of study was biology. Yet, when presented with the above problem, she worked through it to figure out what was going on.
Several things struck me that I'd like to lay out, but first I feel the need to clarify one thing:
Common Core is NOT a curriculum.
It's NOT a curriculum. It is a set of skills and concepts. When students are asked to "solve this problem using the Common Core method" there is either a misunderstanding of the teacher, or the Common Core standards.
There is no "Common Core method." Students are not being required to learn a new type of magic that no engineers, and certainly no parents, can possibly grasp.
But I'm getting ahead of myself.
Let me address the image above.
I have a Bachelor of Science degree in Electronics Engineering... Even I cannot explain the Common Core Mathematics approach, nor get the answer correct.
First, there is no "Common Core Mathematics approach." I will assume that what the author was trying convey was that he/she wasn't able to explain what was happening with the number line. This means that the author has no problems solving skills, or simply chooses not to apply them to determine the method being used which, while it may be unfamiliar to the viewer, is fairly easy to discern with some minor effort.
Second, I'm not sure what the author means by the inability to get "the answer correct." A quick glance at the directions shows that no answer is being asked for. What IS being asked for is for the student to examine the method used in the example and determine where it went awry. At no point does it require the student to use the method or reproduce it. It asks to explain the mistake. They are being asked to analyze the work of another person, examine it for errors and convey what those errors are.
This is the method of subtraction that has been taught for decades to countless children. While I (and Common Core) have no real problem with it, it doesn't convey an accurate understanding of number sense. If you did this method, you probably said something along the lines of "four minus three equals one."
This is true!
But it's not what's happening in the problem above. What's actually happening is "four hundreds minus three hundreds equals one hundred." This is a minor change that makes all the difference. The latter wording expresses a solid understanding of mathematical practice and numeracy where the first one does not.
To further illustrate this point, let's examine a slightly different problem.
Using the "traditional method" for this example requires "borrowing" from the next column. This makes it even more difficult for a child to explain how and why "one minus two equals nine" as well as why every other number in the problem must be changed. A person with a solid number sense would not have difficulty doing so. On a number line, starting at 1001 and moving two spaces to the left is a very simple concept that anyone can grasp.
The answer is solved in under 5 seconds.
Absolutely! This is an incredibly simple problem. That's why the student was not asked to do it. The student was asked to analyze the method used by another person and explain where a mistake was made. Since the person writing this claims to be an electronics engineer, perhaps they would have had a greater understanding of the problem if the directions had been written as follows:
Jack wrote a program to display a number line and solve 427-316. Examine the output and create a bug report.
All of this, however, is minor compared to the major issue in this letter. What it comes down to is this: A parent, instead of encouraging their child to face a difficult task head on, devise a plan and search for solutions, instead writes a complaint letter. Instead of modeling appropriate behavior when faced with a challenge, this parent demonstrates that it is acceptable to give up and blame their failure on a "ridiculous" process.
In the real world, if I asked an employee to complete a task and their response was to hand it back, incomplete, with a note about how the process was ridiculous, THAT would "result in termination."
I will conclude this rant with three things. First, the link to the Common Core Standards so that you can read them yourself. Please take the time to do so before you believe what you see posted on Facebook.
Second, the statement that I acknowledge that there are many things to object to in Common Core. There are legitimate concerns about training and implementation that should not be taken lightly. However, images such as the one above, which provides no source or context, do not serve the public discourse. They exist for the sole purpose of misinformation and praying on the fears of confused parents and teachers.
Third, in case you don't have time to browse their whole site, here are the Standards for Mathematical Practice. These are, in my mind, the most important parts of Common Core.
- Make sense of problems and persevere in solving them
- Reason abstractly and quantitatively
- Construct viable arguments and critique the reasoning of others
- Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated reasoning
Aren't these the things that students, and all citizens, should be able to do?
What am I, an English professor? I don't know how to make heads or tails of all this gobblety-gook that you've been doing here, professor, putting these "words" together to make "sentences" and show off your "literacy" and other high-falutin' common core nonsense.ReplyDelete
If anything, the question as it is screen-capped is in part confusing because the correct answer is written next to the number line, so I looked at it and was like, "wait a minute, did they make a mistake?" And after a moment, it becomes clear that the mistake is ONLY with the number line.
I think "catch the other person's error" is basically an okay way to set up a task, but here where it's combined with catching their error in using a non-traditional method, I think it's not surprising for the response to be, "well they shouldn't have used that stupid non-traditional method--THAT was their problem."
Which is not to say that the task is illegitimate or wrong-headed, and as you say, the way to respond is to battle through it. Not to immediately question the authority and legitimacy of the test. It's funny, though--the latter response probably came instinctually to me for many years. One of those bad habits of thought.
Thanks for sharing.
The public arguments against Common Core seem to me to be analogous to claiming smoking is bad because it makes you allergic to gluten.Delete
The argument is non-sense and unrelated to REAL problems.
Not quite how I put it in my post to another article but spot on. Maybe the traditional subtraction method is the fastest way of solving this problem as posed to Jack, but the number line method gives students the opportunity to see the problem in a different way, and, for some students, this may give a "Eureka!" moment that was lost by traditional methods. I'm not a big supporter of common core, but the original "Mom" and many posters are displaying their narrow mindedness and inability to see that there is more than one way to skin a cat.Delete
There's none so stupid as those who think that their own way of looking at something is the only way it can be looked at. "Mom" is closing her mind to other possibilities that might, or might not, be appropriate here. Basically it is the old "Training" (You will do it this way!) as opposed to Education (Here are some ways of thinking about this).
The ONLY thing I can agree with in regard to this stupid math "problem" - is that children learn differently. Some learn by doing, some by hearing, some by seeing.... most by some combination of 2 or 3 of them. To insist that a child use a method that is counterintuitive to himself and those who attempt to assist him is BAD TEACHING. Any teacher who relies on a method that a majority of the students are unable to grasp - when a perfectly good and time-proven alternative method exists - is an idiot and should get a job driving the school bus or passing out McBurgers. You don't do subtraction from left to right. You subtract from right to left and use the "borrowing" method. It's easy, it makes sense, it always works, and it doesn't require sloppy diagrams.Delete
Well put. May I also suggest the delightful 1001—2 for really driving home the point that the number line strategy is quite a bit more efficient than the standard algorithm for some classes of problems?ReplyDelete
You're absolutely right! That would have been a MUCH better one to use!Delete
Editing now! :-)
You do not need a number line to take 2 from 1001... This is useless study of math which could be better spent getting our children to higher education math level earlier, pretty soon where going to be adding years to high school because we need a quarter to teach Line Subtraction/Addition etc... The parent did show him what is wrong, the entire method used and showed him the proper equation + method.Delete
Steve, you have missed the point entirely and your reply leads me to believe that you didn't read the post, since everything you stated was addressed above.Delete
Steve, I have been an Algebra and Geometry teacher for 13 years, and the students who struggle with positive and negative numbers are the ones who don't understand the number line and who can't visualize it. These are usually the students who also struggle with memorizing an algorithm and can't remember the rules to adding and subtracting signed numbers. I also learned recently that a very well educated friend of mine, a scientist at a pharmaceutical company, uses this sort of number line method of subtracting, albeit mentally. Steve, if you have the good fortune to be able to understand higher level mathematics easily, then you have lucked out in the genetic lottery. There are many people who find it a daily struggle.Delete
I agree whole heartedly with both Justin's post and mathypeople's observations above.Delete
Steve, do you genuinely believe that using the traditional algorithm for 1001-2 is a more efficient method than using a mental number line? I'm very surprised that anyone would see that as preferable.
I don't have a problem with kids being taught number lines. I do have a problem with kids being taught that a number line is the only or best way to solve any subtraction problem. If you are going to teach them more than one method, then you are obligated to teach them when it is appropriate and best to use each method and when it is not. The problem in question is clearly inappropriate for solving with a number line. The 1001-2 problem would probably be ideal for a number line, but then the student misses out on practicing the borrowing method of subtraction. The author should probably put down the Koolaid and really think about this stuff. Do we want our kids to be weighed down with 27 unique ways to do something simple, or do we want them to be able to move swiftly through the easy stuff and spend their time on the hard stuff? It really seems as if the goal of Common Core is to turn out adults who do nothing but navel-gaze over the simplest problems. This is not the kind of teaching that put Americans on the moon, or that landed freakin' robots on Mars.Delete
Lowflyer, apparently, you missed the entire point of the question depicted above. The question asked the student to critique a hypothetical student's unsuccessful use of the number line to solve the problem 427-316 and determine where it went wrong. This question is an example of how to teach a student appropriate use of a method, clear understanding of a method, how and when it works or does not work.Delete
Lowflyer, we don't know at what point in the instructional progression that this was being taught...this may not have been a teaching lesson. It was most likely a "we have been looking at this for a while now and I want to see if you really understand how you can use a number line to solve subtraction problems, so that next I can teach you how to do harder problems in your head so old folks don't say how the younger generation can't do anything in their head because they need a calculator to do everything" lesson.Delete
I doubt efficiency is the point of the technique. It seems more likely it's intended to instill some depth of understanding of what subtraction really is beyond the memorization of the more traditional algorithm. It's a lot like working with Riemann Sums and graphical integration in the first part of an integral calculus course. It's unlikely one would ever solve a real problem using these methods (although they are the basis of some computer algorithms to do the job). Rather the purpose of the exercise is to increase understanding of the principles involved.Delete
What parents and those outside education (and some inside for that matter) need to understand is that the CCSSM is intended to promote flexible thinking and making connections, not just the most efficient way to "solve' a problem, though that will most likely be a byproduct of the former.ReplyDelete
That may be what we are led to believe about CCSSM, but when the tests are multiple choice, that is not what they actually will get. Don't get me wrong, I LOVE the flexible thinking and making connections goal, but am totally against CCSS for other reasons. I want to throw it all out, and look to what the NCTM and NCTE and all the other national subject organizations have come up with....they are the education experts and had already done the hard work of coming up with standards that were open to discussion and debate, compromise, best practices...unlike CCSS. What was wrong with those? Why did so much money have to be spent to produce a crappy product? According to ANSI, the international body responsible for overseeing standards set in hundreds of industries, CCSS were created outside any of the parameters of how standards are to be set for international recognition, so really, CCSS are not even standards at all. What they are at best, is a bunch of topics that are easy to test on cheap multiple choice tests at much profit for those who own the copyright to the CCSS and the tests, AND at worst, they are a set of taggable pieces of data that can be used to gather data about evry student in the US from birth to work life for the profit of those who hold the copyrights. I don't see anywhere where the ones profitting are students.Delete
Actually, you got it wrong. It went down by 100s, 3X, then by 1 six times and skipped the 10 'jump'. The answer the original person got was 121, not the handwritten 57. If you can't see this, I'm unsure about the rest of your 'explanation'.ReplyDelete
I'm unclear how your third statement relates to your first two.Delete
I do not see what you are seeing. I see 6 jumps of ten each. But it is difficult to ascertain with certain since I can't tell what was written on the original assignment and what the student/parent added on to it.Delete
Those are 6 jumps of 1 each. The handwritten 117, 107, et cetera, are the frustrated parent's attempt to figure out what is going on.Delete
Justin, if you can't get the basics correct in the problem, how can you get the more complex situations solved? Once again, you've showed your inability to grasp concepts now not once, but twice.Delete
Agreed regarding the problem, but the explanation is still spot on. The parent never understood that the question wasn't asking the student to solve the subtraction problem, it was asking the student to analyze the use of the number line method. Also, Common Core is still not a method, which is the main point of this article.Delete
Additionally, JamiDanielle, you honestly can't tell that everything typed was original to the assignment and everything hand written was written by the parent?
Though ultimately I think you're correct (type is original to the assignment and handwriting is "Frustrated Parent"), it is not really clear how to read the worksheet. These kinds of problems often have printed "faux" handwriting to represent the hypothetical student's work. The poster's wife used that interpretation when she said, "It should have been a 10 and six 1's, but he went down by six 10's instead."Delete
Depending on how you interpret the picture, the error(s) could be that Jack skipped subtracting the 10, or skipped from 127 to 107, or subtracted by six 10s instead of six 1s. Even without Frustrated Parent's handwriting, the problem is difficult to interpret. It could be a useful tool for in class discussion when led by a competent teacher, but seems like a poor choice for homework.
Rachel, you're absolutely right! Since we don't have a source for this picture, we can only speculate about what the original assignment was. If, as you say, the written numbers were there before the parent, then that is simply another error that Jack made that needs to be identified.Delete
This comment has been removed by the author.Delete
I get the point of your post - the question isn't asking the student to solve the question, but to explain what "Jack" did wrong. But the worksheet problem makes no sense and is actually not solvable. The question is 427-316, which should equal 111. Let's see what Jack did: 427-100-100-100=127. Jack is fine so far. Now, Jack has only SIX "humps" on the number line left. If he subtracts six ones (which the typeface numbers imply is what he did), 127-1-1-1-1-1-1=121, which is incorrect. If Jack instead subtracts six tens (which the frustrated parent seems to have done), 127-10-10-10-10-10-10=67, NOT 57 as the frustrated parent wrote. If Jack subtracts one ten and six ones (which the original poster's wife did), 127-10-1-1-1-1-1-1= 111. Correct answer, BUT WHOOPS! What the wife did requires SEVEN "humps" on the number line which is one more than allowed. There are only six humps, which limits what the wife wants to do to 127-10-1-1-1-1-1=112. And that is the wrong answer! So Jack, the frustrated parent and the poster's wife have all got it wrong. In fact there is actually no way to arrive at the correct answer of 111 using the number line shown and only the numbers 100, 10 and 1, so there is either an error in the problem as published or an instruction missing. I would be happy to be corrected, but given all that I am inclined to cut the the frustrated parent some slack. If I was trying to help Jack, I would have to tell him not to let well-meaning but careless adults spoil the beauty of math for him.Delete
Matt, check out my post from 3/24 to see how my students did it.Delete
I will, but on my way home from work I realized what the problem was: The instructions say that "Jack used the number line..." I usually interpret the word "use" or "used" to mean that a tool has been supplied for me to use and I cannot change it. I think many others interpret that would that way, too. So on seeing this question I instinctively assumed that the number line shown in the problem was the tool that had been supplied to "Jack" and that he had to use it as-is to solve the question, as well as the real-life students assigned this problem. By contrast had the question read, "Jack made the number line below..." (implying that it was not a supplied tool that had to be used, but something Jack created) or had an additional instruction been supplied stating that the number line shown with the problem could be altered, I think all the brouhaha over this problem would have never happened.Delete
The worksheet shows Jack going from 427 to 121 (306). Wife is explaining what Frustrated Parent did wrong, and Frustrated Parent's work should be discarded - because Frustrated Parent gave up.Delete
Since we are asked to describe what Jack did wrong, it is clear that we are free to think outside the box and show the path to the correct answer. Jack forgot to count down the tens digit, that's all. "Jack, remember to count down all the digits being subtracted in the number line."
A further clue is the size of the humps. 3 equal "hundreds", 6 equal "ones".
I wholeheartedly disagree with you that "it is clear that we are free to think outside the box." I personally think this problem is as clear as mud. The size of the humps is not remotely helpful unless you grasp that Jack drew them himself while trying to solve the problem. But that is not made clear. If you quite reasonably interpret the humps as a tool supplied to Jack which he was told to use to solve this problem (instead of something he drew himself), as I explained above, the math cannot work. I don't object to what this problem is trying to teach, which the original poster described as, "The student was asked to analyze the method used by another person and explain where a mistake was made." I do object to how the poor wording of the question vis-a-vis the supplied number line as printed on the page obfuscates the method "Jack" actually used. Basically, the idiots who wrote the instructions failed to write clear ones and turned this simple math problem into a complex grammar one. If this had been a homework assignment and I was trying to help my child after coming home from a long day at the office, I too would be a frustrated parent.Delete
Matt, the assignment clearly says, "this is what Jack did, tell us what he did wrong" to paraphrase...so the imaginary Jack wrote the imaginary number line, and one of his mistakes, there are several, was that he did not write enough "humps" as you call them. As part of the student's explanation, the missing "hump" would /should be included. If I were the teacher, I would never have officially graded this and sent to parents, this would have been for me to see where a student was in their understanding only. Teachers are under such pressure to document and gather data to be input somewhere, that that was very likely the motivation for grading it and sending it home.Delete
Great Post. My thought is that many of the parents who are upset with questions like this, are upset because they themselves have memorized algorithms and truly don't understand the concepts.ReplyDelete
One solution: flip the classroom, allow the students and parents to learn from teacher-made videos together at home, and then let the teachers help the students with question like this at school.
That is an excellent idea! and not videos from Khan.Delete
Or, you know, they could open lines of communication with the teachers and the school.
More unpacking of that worksheet here.ReplyDelete
Justin, what I like best about your post is your reiteration of the point that the Common Core standards are not curriculum, but rather a set of guidelines of what students should know and be able to do upon completing certain levels of education. And the Standards for Mathematical Practice, as you point out, reflect most clearly the spirit of the Common Core, which is that students learn how to think, act and problem-solve creatively and with a rigorous depth of understanding. I've done a lot of work reviewing curriculum materials that are allegedly aligned to the Common Core standards, but the area in which most things fall short is in this shift to the conceptual understanding which needs to underpin algorithmic fluency. And anyone who appreciates the Three Act Tasks, Mathilicious lessons, Estimation 180, Math Talks, Visual Patterns, Nix the Tricks (and on and on and on) is already embracing that spirit.ReplyDelete
Sadly, most parents (and many teachers) are only experiencing the Common Core as a series of new and different assessments, which is an indicator of an entirely different problem. Your post is a great step in the direction of clarifying that state of things.
What I love about this worksheet most is that it shows how a good task can start conversations. A healthy debate about the most efficient way to solve a problem can be very useful to students for learning new ways to view a problem. As others have pointed out, the trouble is that adults who aren't teachers often miss that kids see things in different ways. For some kids (and this "frustrated parent"), it is obvious that the 4 - 3 is really 400 - 300. But for other kids, it is the furthest thing from obvious. For these kids, a visual "jump" on the number line is nice because they immediately see the difference between the 7 - 6 and the 400 - 300.ReplyDelete
I agree with hermathness that one cause of the problem is that parents are experiencing the assessments, but not the lessons. They don't get to see the discussions we have about methods and how different kids are successful with different ones. They also never see the "aha" moments in class when a kid's face lights up because they figured something out using a number line.
The way I see this is that the PARENT is the one that did the handwritten numbers on the page. She/he subtracted 3 100s. Then why did they subtract 20, 10, 10, 10, 10, 10? Why the 20? Then why 10 five times? And then got 111? The type written numbers are what the student, Jack, did. He subtracted 3 100s, then subracted 6 1's as shown by the resulting 121. That's the error. He didn't subtract the 10.ReplyDelete
Who says that little jump is an increment of 10, or 20? The type written numbers indicate that those little hops are 1.
The author of this blog post then says that -- Jack or the Frustrated Parent -- subtracted 10 six times. But that is not what is indicated by the numbers handwritten on the page.
Not to flog a deceased equine, but there is, in fact, no "Jack" here. Whether there's really a "FRUSTRATED PARENT" with a degree in "electronics [sic] engineering" makes no real difference to me, since the butchering of understanding of the POINT of the task can easily be done by any average American idiot (see some of the less insightful comments here and in many other places where this and similar problems circulate online). It's all propaganda meant to "prove" that there's a single, time-honored, eternal Right Way to solve "mathematics" problems (this is, of course, an arithmetic exercise, hardly a problem, and not exactly what mathematicians mean by mathematics), and that any other approach is part of a vast conspiracy by your favorite set of those trying to ruin Amuhrika.ReplyDelete
People like "Rizon" have minds so utterly narrow and closed that trying to show them anything at all is a fool's errand. Justin's take, like Chris Danielson's on his blog, like mine in comments I've made on various blogs, including those by the utterly closed-minded (on this issue) Mercedes Schneider, are attempting to open eyes of those willing to consider so doing. Those who can't or won't, I've pretty well given up on. If they come around, it might be as the result of a sharp blow on the back of the head with a Zen master's stick, but I don't carry or use those. Nor am I a Zen master.
I don't carry a Zen stick, but I do carry a yard stick wrapped in masking tape. Does that count?Delete
I was asked how I can keep engaging with people who miss the point of this post entirely and my response is something along the lines of "I'm a teacher. It's what I do."
Justin, I'm pushing 64, been arguing this stuff for several decades, get tired dealing with people whose skulls were long ago replaced with iron pots. But I do still write my thoughts, even if I don't expect to penetrate the metal of those most thoroughly shielded from contemplation.Delete
Well, personally I think this question is just begging to be misinterpreted. The student is being asked to evaluate how someone else answered a question, which is a great exercise in analytical reasoning (one that a teacher has to figure out on a regular basis) but the presentation is seriously flawed. The student has to figure out what Jack saw, how Jack interpreted what he saw, what he did about it, and what he should have done about it. First. What Jack saw is a number line with three big jumps backward representing 100's and then 6 little jumps backwards representing what? Clearly he didn't draw the jumps or the printed numbers. The fact that the little jumps end up at 121 seems singularly unhelpful. Where is the intermediate jump to 117? That seems the logical approach. No wonder the poor guy floundered and tried to come up with other values for the jumps. As for his error. Well, depending upon your perspective he made several errors, but frankly he was set-up. And so is the guy who is supposed to figure out what was going through Jacks confused brain at the time.ReplyDelete
Leo, please check out my posts for yesterday and today where I put exactly this question to my students.Delete
Leo and others: way too much conclusion jumping here, with less than complete information. We really know so little, including for whom this exercise is intended. Young kids? Teacher education students? In-service teachers? We don't know. But if you come at it with the assumption, as apparently did the "PARENT" who is in high dudgeon (such an easy state to reach in the Math Wars era and now the Common Core era, I'm afraid), that there's one single RIGHT point to be made here, then, well, the internally-imposed pressure may be on. And few characters are more notorious for feeling that sort of pressure when it comes to math/science/application questions than engineers, in my experience. They just get so obsessive with the notion that there's a single point to be made and that if they don't get it, either the question/questioner is an idiot or they themselves have failed (sounds of sharpening sepaku knives in the background).Delete
I never look at pedagogical questions (for that surely is what this is, or can be) that way. Here's a scenario. Here's some (by its very nature incomplete) evidence. What do you think? What would you say/do? Period. No civilizations rest upon the "correct" solution to this puzzle. It's not a "gotcha!" It's a prompt for - dare I say the word? - reflection. Not what engineers and ideologues are interested in or particularly good at.
So to those here and elsewhere with knotted shorts: re-fucking-lax. It's just something to think about. We don't know all the facts. And in fact, there may be no ACTUAL facts, in that this is probably just a hypothetical. I used to give those to ed students all the time and most were bright enough to get that there was nothing real in what I asked, though there COULD be. Sheesh.
This example serves an an inkblot test for so much. The biases people bring to this "problem" and how much they assume, add, misread, misinterpret, etc., speaks volumes about why it's so hard to teach the average American anything about mathematics education. And much else.ReplyDelete
What would be very interesting is to know the political persuasion of each of the responders. It would be my suspicion that, in far to many cases, that would have a pretty good correlation with people who agreed with you, and people who did not.ReplyDelete
linlyons at yahoo
I was thinking not just political persuasion, but also whether or not they are teachers...Delete
Doesn't this problem, and ones like it found on the web, just show us that "the math wars" are still alive and kicking? I was arguing about this stuff in the early '90s with the"old school" teachers at my school. Haven't we reached a point where we can say that just algorithms is not good, and just concepts is not good, but a balance of the two is? I think the real problem is that we need to spend much more TIME in the early years really laying a good foundation of mathematical understanding of the basics than we currently are doing. The countries that score very well in math spend much more time early on, so that they can move faster in the higher grades. The US is so concerned about "getting it all in" that we rush through everything...and we pay for it later on when a straight "A" student suddenly realizes they don't really understand what is going on in Trig or Math Analysis, or Alg I, for that matter. But CCSSM do no better about ending the rush, or going deeper with fewer...and the rush to produce material is the same old same old...a mile wide and an inch deep so that the published material has something for everybody.ReplyDelete
Great response! I agree and also blogged about this same image:ReplyDelete
I would like to make a comment. As a parent, who can see the other parent's frustration, I see another problem here. You seem to think the Engineer Parent made a "mistake" by thinking the small marks were 10's markers, but when I first looked at it, I too assumed it was a 10's marker, as 1's markers would need to be (in my mind) much smaller, if they were in proportion to the other lines on this number line ;-). The lines on your number line, shown in the picture, look like they would be proportionate to 10's markers, and if one assumes that, the problem becomes inordinately difficult, as the diagram stops matching all reasoning. One is told there is a "mistake" in Jack's math, so one tries to reason out what the markers represent, as it becomes obvious they don't represent 10's. I'm not sure then that assuming they are 1's is really logical, is it? Why not LABEL the lines, admit it's confusing (the the parent) and move on, if one wants to clear up any confusion. After all, the point of the problem wasn't to determine what those smaller lines represent but to understand how one could use a number line to do the same math one can do using a more traditional approach. Why not use this kids paper (and the resulting comments the parent made) as a "teachable moment" where you put it up for the students to READ and understand that some kids will have a "different" but very VALID way of solving that same problem quickly. Always take the higher path and rather than taking offense at the suggestion that the use of a number line would mean "termination" make a point of saying, that while we should expect kids to be able to use more than one method to solve a problem, note that there are fast ways to solve problems and that speed is 100% necessary, as one progresses in math. I don't know very many educators that would value a high school student using a number line on his SAT! So, a good teacher can use this parent's comment/feedback as a way of saying, "Hey, sometimes, no matter our intentions, as teachers, we can present a problem in what can be a confusing way to some kids--or even their parents." Tell kids that if they "think" differently or had problems, they weren't alone! (and ask them to raise their hands to observe if others had the same problem as the engineer!) Admit to kids that there are more ways to solve a problem than with number lines and that to the engineer's point, they won't be able to use them their entire life. At some point, speed and accuracy need to be practiced in order to really do well in high school. Obviously, if one is still using a number line, in upper elementary classes, that poses a problem, as it can encourage a child that they don't need to memorize basic math facts in order to move fast--something that might sound nice, but won't get you very far on the SAT. Building a strong foundation is important (for sure) but so is building upon speed and accuracy, once the concept is formed.
Let's just simplify this for what it is. High school math teachers are people who were not good enough at it to get into engineering. To pull the "my type of math is different" is laughable. To pull the passive aggressive "Well, I don't understand what you're talking about" and patronizing a bunch of engineers is pretty typical projection. If your math skills aren't good enough to work at Bombardier as an engineer, you don't have the background to argue the point. The wonderful thing about numbers is that the end answer is either right or wrong. Anyway, I'm sure teachers or their defenders will quite happily keep going in circles and continue to refuse to acknowledge that this is the case, and continue to fingerpoint with the same misplaced confidence that resulted in this article. Here's a thought: learn your math correctly. The rest is just pedantics. If you want to make a right ass of yourself, keep "correcting" an engineer. Further proof that those who can't do, teach...ReplyDelete
Further proof that those who can't form viable arguments or engage in meaningful discussion, post anonymous comments on the internet.Delete
"This entire process took her less than a minute."ReplyDelete
---It should take a couple seconds, schmuck.
"Yet, when presented with the above problem, she worked through it to figure out what was going on"
---This is your wife, if you ask her to do it, do you expect her to throw it back in your face? You can't use this as proof of anything.
"What IS being asked for is for the student to examine the method used in the example and determine where it went awry. At no point does it require the student to use the method"
---It is requiring the student to use the method as it asks the student to examine where the error in the application of said (Common Core-endorsed, untraditional) method is to be found.
"While I (and Common Core) have no real problem with it, it doesn't convey an accurate understanding of number sense."
---Common Core is trying to teach what is innate in those with natural mathematical talent. It is a waste of time - those without natural talent have genuine trouble in bringing their understanding of numbers and math to a more abstract level and those who already do are weighed down (and graded down) by tedious requirements to "show your work". It is like substituting basic piano lessons with music theory.
You have to learn first by rote, then separate out those with more academic abilities in a subject.
"But it's not what's happening in the problem above. What's actually happening is 'four hundreds minus three hundreds equals one hundred.'"
---You are not making a point! In the 'traditional' method, nobody pretends that 4 - 3 = 1 does not represent the hundreds category and in the 'new' method, you cannot escape the fact that the student is still essentially thinking in single digits within the hundreds category.
"Using the "traditional method" for this example requires "borrowing" from the next column. This makes it even more difficult for a child to explain how and why "one minus two equals nine" as well as why every other number in the problem must be changed. A person with a solid number sense would not have difficulty doing so."
---Yes, a person with a solid number sense has no difficulty understanding immediately, by simply viewing the problem, that 1001 - 2 is 999, based on the relationship of the difference to 1000. A person without this needs to learn to borrow from next column so that he may complete problems like 8,231 - 4,378 in under five minutes.
"A parent, instead of encouraging their child to face a difficult task head on, devise a plan and search for solutions, instead writes a complaint letter"
---The parent's point was precisely that you made a difficult task out of something that can be solved in under 5 seconds. Instead, for the purposes of learning, perhaps the student could be given difficult tasks that are truly difficult, not artificially made so by crap pædagogical methods.
"Instead of modeling appropriate behavior when faced with a challenge, this parent demonstrates that it is acceptable to give up"
---Instead of teaching children, you are wasting their time (and your own). Stop. You are not supposed to create pointless challenges. Basic math is not supposed to be a philosophy class.
"In the real world, if I asked an employee to complete a task and their response was to hand it back, incomplete, with a note about how the process was ridiculous, THAT would 'result in termination.'"
---In the real world, if you asked an employee to do a simple task in a most idiotically complicated way and graded them on the process rather than the result, and barked that if you did not follow the idiotic process just right you would be terminated, the employee WOULD GET A NEW JOB.
And, perhaps, so should you.
As a certified teacher, as a teacher with a doctorate in education, it is clear that concepts and their ability to be understood and applied in order to perform and meet the expectations of ones' career are what's important in living a successful and productive life. Complicating the simplistic in order for students to "understand" the "ideas of others" does nothing to further competence or creativity in individuals in order for humans to advance as a species. If yiou have a "math mind" the method of thinking about this "problem" is easily understood. If you're a math teacher and some of your students don't "get" this, after various approaches to teaching it, then you need to understand that they are meant to excel, to deeply understand, other areas of expression/intelligence and contribute to humanity in that way. Did Mozart, Picasso, and Shakespeare "get" this? Probably not. Yet according to Common Core they should have, or they'd be considered failures. The Common Core Standards neglect to acknowledge the developmental nature of the ability to learn and understand various concepts, that multiple intelligences exist and that for many, being able balance a checking account is enough to be self-sufficient and successful in life, given their inherent talents in other areas. Their "lack of ability" to understand math teachers and their presentation of these concepts exampled has nothing to do with the advancement of the human race. Oh...and nothing like supporting your position with a 15-year-old article based on the research of two authors who published their findings in a $23 dollar book. And I'm certain a mechanical engineer could apply "common correct, alternative ways of performing and showing computations" when faced with solving 427-316. In addition (haha, pun intended) there are numerous grammatical errors in this article, Mr. Aion. If you're going to publish your philosophy on teaching, at least get a proofreader before posting. And your wife who had a "better GPA" than you solved this problem by "working through it" did so because she is an adult with the capacity for conceptualizing such processes. This problem is the homework of a third-grader.ReplyDelete
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Thank you so much for your thoughts and comments.
I disagree with your initial premise about "what's important."
And note, Mr. Aion, that I am not posting anonymously. This is my name and I'm willing to form viable arguments and engage in meaningful discussion. I'm on FB and I will gladly "friend" you or discuss this here on your blog.ReplyDelete
Mr. Aion...or if I may address you as Justin...I am absolutely open to your explanation and experience when you say "you disagree with my initial premise about what's important." I WANT to hear your justification and support of the opinions/beliefs you've presented. I am willing to admit that I may be completely off base.Delete
We can do this privately. I feel no need to make myself "the competent one who is right" in public. I'm a teacher with 25 years of experince and I'm smart enough to know that I have much to learn.
@Tracey Harding wrote: "Oh...and nothing like supporting your position with a 15-year-old article based on the research of two authors who published their findings in a $23 dollar book."ReplyDelete
I missed that. What article and book?
And I refuse to bring Michael Goldberg or anyone else into our discussion. I don't feel the the need for this to "go viral" or be a debate on a public forum. I just want us to be able to discuss this freely.ReplyDelete
I assume by "Michael Goldberg" you mean me. Did I spit in your soup somewhere or other? All I did here was ask what the article and book were to which you refer. I can't find them in Justin's piece or his comments. Was that a crime on my part to ask you to name them?Delete
Interesting that you don't want the conversation out here in public after you were rather snarky with Justin above. I particularly liked this comment: "In addition (haha, pun intended) there are numerous grammatical errors in this article, Mr. Aion. If you're going to publish your philosophy on teaching, at least get a proofreader before posting." I really liked it in light of your comment below: " Please feel free to discuss this and know it will be between only you and I." Like a huge number of Americans, you miss the fact that "between" is a preposition. Thus in that phrase, it should be "you and me," since the first person pronoun is the object of a preposition. Knowing that you teach elementary students literacy does not fill me with feelings of unmixed delight, to borrow from Oscar Wilde. Knowing that you do so and are so snotty to Justin about his alleged grammar mistakes, seem to have referred to invisible articles and books, can't be bothered to get my name spelled correctly when going out of your way to mention me by name, all add up to my not wanting you to teach my children or grandchildren or anyone else's children. It's not just your ignorance: we all make mistakes and have areas of ignorance. It's your arrogance, your nastiness, and your fraudulent attempt now to portray yourself as open-minded. You blew that when you jumped in with hobnailed boots. Were I Justin, I'm not sure I'd want to be alone in a room with you, even electronically.
Good luck trying to come off as "the competent one who is right" outside of your classroom, filled with little children who can't stand up to you. Your Ph.D probably has them all cowering in a corner. Or would it be your attitude?
Michael Goldberg, I'm not even going to argue with you. I didn't pay attention to who I addressed my post to-and if you or any other had done the same, I'm quite certain I would have called you out on it. . And I completely agree that my grammar isn't perfect by a long shot. In my defense, however, I did not create a blog and publish my posts on it. I'm just responding to what's been put out there by Mr. Aion.Delete
I am sincerely sorry if I came across as arrogant. This was not my intention at all. I can only say that my stance as an early childhood educator and as a parent of two teenagers who are in high schlool and faced with understanding-and exhibiting that understand by their abilities to "pass" PAARC have most certainly influenced my position and have upped my passion for guiding and encouraging the various inherent talents of every individual.
The reason I said that I don't want this debate posted in public is exactly the opposite of what you've written. I want this dialogue to be honest, open and not subject to public scrutiny by those who have not followed the discussion from its onset.
What I mean is "discuss this freely" is between your stance and mine. I am not in any way attempting to present my stance that opposes yours as "right." Much of what you've posted and stated connects to my own beliefs as an educator. I just want to more fully understand your position as a middle school teacher and mine as an elementary school teacher. Please feel free to discuss this and know it will be between only you and I.ReplyDelete
Michael Goldberg, in Justin Aion's original blog post, he referred to this (link at the bottom of this post) in order to defend his stance. It is a book review published in 2009 by two researchers who published their findings in a $23 book.ReplyDelete
I reread my post and I agree that it came across as "snarky" and self-promoting. I acknowledge that I've let my passion as an educator and a parent surpass the professionalism I should exhibit on a public blog (or in any forum of debate) affected my ability to express myself as a professional. I sincerely appreciate that your response forced me to examine my post and the way I expressed myself.ReplyDelete
Yes, I know that sentence was grammatically incorrect. :-) I will never refute your statement that my grammar is not perfect. I am completely willing to admit that. My point is that I'm not publishing a blog without a proofreader. But yes, as a respondent, I am far from perfect. No argument from me.
I'm sorry, the article was published in 1999. Yes, I continue to make mistakes in my posts. I get your point. I need a proofreader, too!ReplyDelete
Michael Goldberg, your comment that my classroom is filled with "little children who can't stand up to you. Your Ph.D probably has them all cowering in a corner. Or would it be your attitude?" is also snarky. You do not know anything about me and my career, personality, and presentation as a teacher, about me as a person because you read a post I made in response to a position presented on a public blog. I am not attempting to start an argument. My belief is that we both feel passionate about education and we've let our passions surpass our professional experiences and ability to express ourselves as such in certain parts of our posts. I don't want to invite enemies...I want to create dialogue.ReplyDelete
I'm sorry...Goldenberg. I was incorrect in addressing you as "Michael Goldberg." I read your name while I was thinking about my response to you. And I did it more than once, even after you pointed it out to me! Yes, I'm wrong in not reading more closely. It's not the first time I've not this and I'm sure it won't be the last. I apologize...you deserve to be addressed correctly, as do all people. I'll admit, I make plenty of mistakes in my grammar and spelling. I admit that my position on Mr. Aion's post is just my opinion based on my experiences and my education. I've been teaching in the same district for 25 years. I know there's much more out there than what I've learned and experienced.ReplyDelete
I've clearly made grammatical errors and I acknowledge that I came across as "snarky." I absolutely understand why that's been addressed. Yet the fact remains that no one has addressed my position on the CCCS and what students need to be taught and assessed on based on their abilities to be successful in this life and to make contributions to society in order to advance the human race and ensure its progression. This is what haunts me as I get into bed and attempt to fall asleep.Delete
Additionally, I never said that "I don't want this conversation out here in the public." as you stated (verbatim) in your response. I said that "We can do this privately. I feel no need to make myself "the competent one who is right" in public." in response to Mr. Aion's position. My grammar may not be perfect, but my comprehension of this discussion has not be based on twisted interpretations of comments you've made.Delete
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I made a long comment earlier, probably in the middle of your posting some of your recent ones, and like several others I've tried to make in the last week, it got "eaten" by some glitch in this blog's interface. Very frustrating and enervating. So I'll say a couple of things.ReplyDelete
I appreciate the correction to my name and your various conciliatory remarks. Thank you for those things.
I don't support the Common Core and in fact have blogged repeatedly against it, as a political force for conformity and corporate control of public schools. It's a huge mistake, and I've said so for several years. I particularly despise the high stakes testing, but that's not all I dislike.
That said, my apparent support of CCSS-Math has to do with very specific issues in the math standards, particularly the process standards. And like Justin, I resent how people are picking problems from various publishers, attacking them, acting as if those problems are: a) directly correlated with anything in particular in the math standards, which to my thinking they aren't; b) acting as if these problems ARE the Standards, rather than a particular publisher and author interpretation of what is called for (or just what they felt like putting out there and CALLING "Common Core-aligned math"; c) necessarily something new that was invented just for the Common Core Standards.
Many of the ideas under attack have been around for a long time. And many of them are things I've supported and fought for over the last 25 years. They have nothing to do with Common Core. They're not wrong-headed. They make sense if you stop to really think about them. But now they have become falsely identified as "Common Core Math." Twenty-five years ago, they were called, variously, reform math, NCTM math, fuzzy math, and a lot of much nastier things. Constructivist math has been a particularly intriguing name given to these things. All of that is highly political. I've written about it for a quarter century and likely will do so as long as I'm alive. There's much to discuss about all of it, but when the conversation turns into screaming and epithet hurling, I tend to rather quickly respond in kind. Twenty-five years of dueling with people will do that to you.
I saw the Common Core Post on Facebook and made the rebuttal below. It's not everything you mentioned, but it does align. Thanks for posting.ReplyDelete
...As an academic advisor for Engineering Technology and Engineering Physics students at ISU (and the daughter of an elementary school teacher) it's clear that this problem is very different from advanced differential equations (theoretical track math). This is a complex form of liberal arts math which promotes critical thinking and problem solving skills. I remember doing problems like this when I was in 2nd and 3rd grade. We were required to use the short form of addition and subtraction (that the frustrated parent did) to check our work. If my memory serves me correctly, this would be the lesson that follows counting, understanding units, and addition and subtraction (not the addition and subtraction lesson). I understand the parent's frustration, because it took me 10 mins to figure out the logic process of the solution (which is NOT reflected correctly)…but the parent's argument (through the picture) presents a lot of missing information which can be misleading. Unfortunately, I'm led to assume the context of the argument is based on what the frustrated parent was required to learn while in grade school, which is not a reflection of lesson plans the teacher built, the length of time the instructor stayed on the subject or the homework that Jack had to complete to be able to solve this problem (for all we know this could be extra credit on a test). I'm all for supporting and encouraging our children when they fail but there's a fine line between supporting and enabling them to get back up. The note to Jack was made out of frustration before the facts. Without having all the facts, this post is disappointing to me as a teacher, and a perfect reflection why it's important for parents to have a positive relationships their child's teachers and a little more investment in the learning process (teacher to parent) when their is a discrepancy in the teaching method.
Very useful comment, Brittany Stokes.Delete