tag:blogger.com,1999:blog-6570061087276796800.comments2015-01-31T14:16:14.140-05:00Out In Left FieldKatharine Bealshttp://www.blogger.com/profile/02838879769628392605noreply@blogger.comBlogger4077125tag:blogger.com,1999:blog-6570061087276796800.post-10326332223819128252015-01-31T14:16:14.140-05:002015-01-31T14:16:14.140-05:00My child with autism asks repetitive questions bec...My child with autism asks repetitive questions because of the two below reasons:<br /><br />(1) Difficulty verbally expressing her needs (she'll ask the same unclear question repetitively instead of clarifying what specifically she wants)<br />(2) She doesn't like the answer she's been given and thinks that if she keeps asking, eventually she'll get the answer she wants. Crimson Wifehttp://www.blogger.com/profile/03254830856234479999noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-8004027818132694612015-01-31T14:10:51.149-05:002015-01-31T14:10:51.149-05:00Singapore Primary Math includes tessellations so w...Singapore Primary Math includes tessellations so while I can't speak to what other countries are doing, the kids in high-scoring Singapore have been doing those kinds of problems for decades. Crimson Wifehttp://www.blogger.com/profile/03254830856234479999noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-43521897048275134902015-01-29T23:08:26.655-05:002015-01-29T23:08:26.655-05:00BG, Very well written with kudos to your ability t...BG, Very well written with kudos to your ability to share your experiences with sensitivity and humor.<br />Your passion for your subject material is exhibited clearly in your choice of words and your delivery.<br /><br />I interpret that your passion invigorates you and supports your strong desire for positive outcomes for your students.<br /><br />Exceptional teachers certainly have, among their numbers, those who themselves maintain a presentation-grade understanding of several alternate methods of instructing how to achieve a goal of communicating a sometimes difficult-to-understand concept during math class(or any class).<br /><br />Those instructors who, in addition, are also blessed as thoughtful, clear communicators as well as possessed of a robust and quick-witted sense of humor are truly a gift from Olympus, and those instructors will be recalled with some passion by their students throughout their subsequent brushes with "real-life", even long after their student days are past and they find themselves in the role of parent/instructor.<br /><br />This type of exceptional instructor, accomodating but still on-point(and on-schedule) could be(may I say should be) bottled, like fine wine, to be decanted(among age-appropriate-participants) unto future generations of math instructors who need only to enhance their teaching repertoire a bit beyond the "standard-allocation" of teacher understanding plus an added measure of humorous patience, like a spoonful of sugar, to make the "medicine" of sometimes intimidating math concepts become absorbed.<br /><br />To those advanced-degree students now graduating (Ed.D , Ed.M. and, of course, the less lofty math-teaching-credential), fully intending to become members of the math-teaching fraternity, Please take note.<br /><br />I have never regretted having both matches, flint-and-steel as well as magnifying-map-reader-sheet as options I may choose among when it comes time to build my campfire on a hiking trip.<br /><br />To build a bit upon the hiking analogy, we are here, student and teacher, starting out together at semester beginning in this math course described in the course catalog. Certainly we comprehend that there are multiple ways to arrive at our selected destination(skill-level)as anticipated and hopefully adequately described in the course-catalog. I recall vividly that the environment often helps(sometimes forces) us to select one method over another, one path over another, and will surely continue, over time, to affect our choices along the remainder of our successful hike and certainly also be instrumental in achieving success in accomplishing class-teaching goals. <br /><br />It's clear that this description is the description(brief though it is) of life itself.<br /><br />On a map, you must know where you are (you are HERE!) as well as your destination(goal) to enable you to make choices and allocate resources to be successful.<br /><br />As on any map, there are pre-printed routes that may be selected to move between starting location and destination and typically you may select from among several alternate travel routes. Of course, you may also determine that moving between several pre-printed routes may hold some advantage, with consideration given to changing environment, resources, skills, goals, and always, unanticipated events.<br />Flight Captain Sullenburger springs to mind. Maintaining some flexibility in your thought processes and execution can, indeed,be helpful, and intuitive, deeply-embedded skills are lifesavers, in this example.<br /><br />Thoughtful selection of alternate paths can be rewarding, conserve resources and save time. Practice improves performance. Rote or right? Labels at some point, aren't helpful, but may be polarizing and resource-sapping.<br /><br />Please, let's use all the gifts we are possessed of to enable upcoming generations of math students to grasp, to their greatest extent, the full palette of math concepts.<br />Chris L.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-51333991887481559712015-01-28T11:19:56.469-05:002015-01-28T11:19:56.469-05:00Completing the square is worth spending some time ...Completing the square is worth spending some time becoming comfortable with in anticipation of putting quadratic equations in 2 variables in standard form (comics sections) but I had a better use (not as good as getting a date however). I was in calculus class in college and was done with a problem provided I had remembered the quadratic formula correctly but I hadn’t used it for a couple years and wasn’t sure. I had time to spare and was able to derive it on the spot. I wasn’t sure that I was remembering how to complete the square correctly either but, when I got the same answer, I was sure of both and forever convinced that mathematics really does “work”.Wayne Bishophttp://www.blogger.com/profile/06572254540248429795noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-85518447282309986012015-01-27T11:16:54.772-05:002015-01-27T11:16:54.772-05:00Lots of interesting points.
Since you did a numb...Lots of interesting points. <br /><br />Since you did a number of cases of (x+y)(x-y), then they should have "discovered" that it produced a "pattern" of X^2 - y^2. However, this leads to my problem with student-driven class discovery - that just a few might discover anything and then directly teach it (badly) to people like Cindy. They will say things like: “Because it works out that way; just follow the rule and figure it out later.”<br /><br /><br />"In my opinion it is a large part of what mathematics is about. I wanted to give at least a few students that same epiphany."<br /><br />"Give?" I can hear the pedagogy alarms going off. I agree with you, however, and would say that discovery is too important to leave up to the students. A few might discover not quite the right thing and then directly teach it to other kids badly. Fixing misunderstandings is much more difficult than getting it right the first time.<br /><br />SteveHhttp://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-13469147434304865262015-01-26T15:15:19.219-05:002015-01-26T15:15:19.219-05:00Very upsetting.
All of it.
btw, C. heard that ki...Very upsetting.<br /><br />All of it.<br /><br />btw, C. heard that kind of "Gail" talk on the bus a lot, from one student in particular.Catherinehttp://kitchentablemath.blogspot.comnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-53933773862830163712015-01-26T03:58:17.618-05:002015-01-26T03:58:17.618-05:00“Just like a first grader who memorizes the multip...“Just like a first grader who memorizes the multiplication tables, he will never come to understand. It will be plug and crank for him, all the way.”<br /><br />I take issue with this statement. It seems as though educators have a view of memorized knowledge that is only negative. It is as though they imagine all children who produce such knowledge to be pale and forced to sit inside by parents who make them do repetition for hours on end. Their performance of this knowledge being only a parlor trick with no depth. First, why will the child “never understand”? It seems inevitable that in fact even if he is just chanting or singing the facts, in the coming months or years he will continue to apply this knowledge to math problems and develop the same understanding as his peers. Likely he will go faster through the material as he can do problems without struggling to access the facts. Second it seems as though as much as this is denigrated, teachers know that this must happen at some point. A first grader who has this memorized is a victim of his parents distorted vision, but a fourth grader who has this memorized is just an ordinary child. Certainly as an adult when I think of 3x5 I rely on my memorized knowledge and never consider the concept of how that is achieved.<br /><br />Also I don’t think it is true in all cases that a child could not understand. Multiplication is not that difficult to grasp. 2 groups of 3, equals 3 groups of 2, equals 2+2+2 or 3+3, all of this can be shown and explained fairly easily. We are abroad this year in a new school system and I noticed looking ahead in the first grade textbook, that multiplication is introduced toward the end of the year.<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-30877545740562342122015-01-25T09:06:00.377-05:002015-01-25T09:06:00.377-05:00Before there was writing, the traditional cultures...Before there was writing, the traditional cultures that we all are descended from accomplished most of their learning by demonstration, repetition and practice. Writing added the ability to catalogue learnings and have them for reference, but it did not invalidate the older forms of learning.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-71784699867403621282015-01-24T17:17:47.479-05:002015-01-24T17:17:47.479-05:00Well-said Steve. I'll only add this, that I t...Well-said Steve. I'll only add this, that I think the very word "rote" has been abused and tormented into something that it never was before, by modern discourse.<br /><br />Today when one says "rote learning" they generally mean "without understanding". But that is not what the word means, although some dictionaries now include that meaning. "Rote" means, essentially, "by repetition". That is, rote learning is that which is learned through repetition of some task. Think of words sharing a common root: "rotation", "rota", "rotary". It means to repeat something over and over.<br /><br />Repetition ... is the basis of most memory-building. And memory is the seat of learning. So where there is no rote, there can hardly be much learning. An actor learns his or her lines by rote. Does this mean they do not understand those lines? You'll find few in the business who would agree with that conclusion. Athletes learn their competitive skills by rote ... repetition. Does this mean they have no understanding of the game? No, in fact most would tell you that "understanding", in a vacuum of those skills, is of little value. Same in mathematics.<br /><br />Rote is not "understanding", or certainly not "complete understanding". But understanding is not complete even after years of excellent learning. Even as a professional mathematician I am continually learning my subject matter at a deeper level, even the elementary topics. I rankle at the suggestion that there is some royal road and down which teachers should supposedly lead students directly to understanding, prior to mastery of requisite skills. Plato understood well that there was no such road, and the situation is no different today. R. Craigenhttp://wisemath.orgnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-42981573205198580792015-01-24T15:36:55.010-05:002015-01-24T15:36:55.010-05:00ime This is a typical full inclusion activity. Bri...ime This is a typical full inclusion activity. Bring the pattern blocks up from kindy, but call it tessalation to disguise that reg ed has already figured out tiling. lgmnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-21323038562139487462015-01-23T14:20:39.583-05:002015-01-23T14:20:39.583-05:00Subway Finally Agrees to Tessellate Cheese:
http:...Subway Finally Agrees to Tessellate Cheese:<br /><br />http://gawker.com/5551263/subway-finally-agrees-to-tessellate-cheeseAuntie Annhttp://www.blogger.com/profile/05777983027361603449noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-80815923664815461172015-01-23T14:19:49.585-05:002015-01-23T14:19:49.585-05:00It's helpful when working your post Masters-de...It's helpful when working your post Masters-degree job at a sub shop to know how to tessellate triangular cheese slices.Auntie Annhttp://www.blogger.com/profile/05777983027361603449noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-23371238413361872692015-01-23T13:58:05.491-05:002015-01-23T13:58:05.491-05:00I really don't understand the fascination with...I really don't understand the fascination with tessellations. Has manipulations with tessellations become a 21st century job skill?Niels Henrik Abelhttp://www.blogger.com/profile/00554447042962336254noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-92134119182999226142015-01-23T12:50:32.047-05:002015-01-23T12:50:32.047-05:00Our kids did tessellations in about 4th grade...IN...Our kids did tessellations in about 4th grade...IN ART CLASS!Auntie Annhttp://www.blogger.com/profile/05777983027361603449noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-20978299502316240072015-01-22T23:00:22.486-05:002015-01-22T23:00:22.486-05:00I think the biggest myth is the idea of rote knowl...I think the biggest myth is the idea of rote knowledge. Rather than see it as partial or incomplete understanding, many mischaracterize it as lacking any understanding and use that to justify flipping the education process around and approaching it from the top-down, where they hope that some vague understanding will drive the development of "rote" skills. This in turn justifies student led discovery in class.<br /><br />Real mathematical discovery happens individually when one consistently, day after day, year after year, works hard on homework sets - problems carefully crafted to increase in difficulty and explore a range of variations in the material in the current unit. Proper mathematical development requires a carefully controlled feedback loop that tries to ensure mastery of the material. Detailed understanding comes from this work on skills. As annoying as they might seem. weekly quizzes work well. <br /><br />My son suffered through MathLand and Everyday Math in K-6, but when he got to pre-algebra onward starting in 6th grade (with proper textbooks), life became much better. His high school math teachers did not do discovery-driven math. They all directly taught the class, but some left time at the end of class to allow them to start doing their homework. They could collaborate, so one could claim that they were doing group "discovery" in class. What this did do, however, was to help the students get over the initial hump generally encountered when doing homework. There might not be enough time to do all of the homework in class, but all of the students were off and going in the right direction. This required less time going over homework in class the next day, and it was more effective.<br /><br />Talking about understanding never works as well as doing understanding individually. When I tutor students in math, I can see that they "understand" when we go over a problem step-by-step. However, I always tell them that they have to prove it by doing the exact same problems themselves - alone. Too many students breeze through homework sets and end up with all sorts of partial and incomplete understandings. It's not rote, just incomplete. If you want better math students, you have to enforce a proper homework feedback loop year after year.<br /><br />One cannot be successful with rote skills in math unless the teachers do not know how to create proper tests. Students would love to see the exact same homework problems on the tests, but just with changed numbers. It never works like that. These are variations that won't be solved by general logic or understanding. They are solved by the understanding that comes from time spend on homework sets.<br />SteveHhttp://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-65345365738222087112015-01-22T20:07:01.846-05:002015-01-22T20:07:01.846-05:00Hi Steve. You say
"Just thinking with no to...Hi Steve. You say<br /><br />"Just thinking with no tools is anti-math."<br /><br />As a career mathematician who lives and breathes the subject I agree 100% with this statement. Very well put. With this statement you've nailed one of the most disturbing trends in contemporary "math education" -- the stripping out of actual math in favour of generic and poorly defined "thinking skills" at the expense of systematic development of generalizable, established tools of the trade.R. Craigenhttp://wisemath.orgnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-73076934985062537272015-01-22T12:05:10.808-05:002015-01-22T12:05:10.808-05:00Hi IGM. You say "Providing an activity witho...Hi IGM. You say "Providing an activity without an instructional objective is baby sitting, not teaching"<br /><br />I almost agree with this. I would say providing an activity without HAVING an instructional objective is ... not teaching. The difference is that I do not think that one must "provide" the instructional objective (i.e. to the students in advance). It is often good for teachers to have a hidden agenda and students to believe they are doing something as a lark or getting away with wasting time. This notion that teachers must explicitly tell students the objectives of every lesson is pedantic nonsense. The teacher is a craftsman who crafts learning experiences. Sometimes good teaching is stage magic, where in the end the rabbit comes out of the hat and everyone gasps. Students should often be pleasantly surprised at how much they have learned over time.<br /><br />You say "Your last paragraph states the reason that the student should not have been shown the algebra. He does not have the preliminary understanding . Just like a first grader who memorizes the multiplication tables, he will never come to understand. It will be plug and crank for him, all the way."<br /><br />This is absurd. You are talking about knowing "why" without knowing "what". Understanding without knowing? I encountered this years ago as the ministry folk here were pressing this understanding-first business. It is essentially a doctrine saying that you can build a house from the roof down. No, you can't. There's a reason why basic skills and knowledge are always put at the bottom of Bloom's taxonomy -- they are foundational.<br /><br />Mathematics is relentlessly hierarchical. Every lesson has a foundation, has precursor skills and knowledge, which ought to be in place, quite often before the lesson begins. And understanding quite often (not a firm rule -- I only say "often") is far up the ladder. Higher skills generally are built upon lower ones, not the other way around.<br /><br />Your last statement that someone giving students basic skills dooms them to "...never understand. It will be plug and crank for him all the way" is the old false dichotomy between understanding and skills. I've got news for you -- far from getting in the way, facts and skills REINFORCE understanding. Ask any cognitive scientist.<br /><br />Further, you want to know what "plug and crank" looks like? Guess and check! -- it's almost exactly that. Guess and check only barely registers in my view as a "problem solving skill" -- for many it's a problems-solving-skill-AVOIDANCE procedure. Here's a problem ... rather than approaching it systematically and analytically, let's ... uh ... try 17. Nope. How about 23. Nope, -5? No good... That's not problem solving! And it's an impoverishment where a good teacher could and should be helping students to apply systematic reasoning and organized methods.<br /><br />You are probably a very good teacher -- I liked your analytical approach to the bicycle problem and believe you would prefer to see students doing this than guess and check. So I know you "get" this. But you seriously need to divest yourself of this mythology about the relationship between facts/skills and understanding. They are not enemies.<br /><br />Recommended reading: Daisy Christodoulou's "Seven Myths about Education". In particular, Myth #1: Facts prevent understanding.<br /><br />Daisy has posted an excellent summary of the seven myths on her website, here:<br />https://thewingtoheaven.wordpress.com/2014/03/09/seven-myths-about-education-out-now/<br />R. Craigenhttp://wisemath.orgnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-20836780505360868172015-01-22T11:09:03.082-05:002015-01-22T11:09:03.082-05:00"Just like a first grader who memorizes the m..."Just like a first grader who memorizes the multiplication tables, he will never come to understand."<br /><br />No, it's not just like memorizing the times table. There is no magic connection between trying to figure something out with no help and with understanding the beauty and power of defining variables and equations, and making sure that you have M variables equal to N independent equations. <br /><br />Besides, students able to tackle this kind of problem should already have experience with creating simple equations from words, like "Mary is three years older than Jane and Jane is 12 years old. How old is Mary?" Students start to write equations when they can still do it easily in their heads. You run a big risk to wait until problems get too hard before you learn the basic skills of algebra. But even at three equations and three unknowns, you think that there is some math understanding that will never happen because they can't first do this without math? At what point does this stop?<br /><br />Understanding in math is understanding the tools, not understanding some vague thinking process that can leave you high and dry if you don't see the missing pieces. Algebra, however, gives you the tools for organized thinking.<br /><br />I used to tell my students to start defining variables and finding equations that they know are correct. They don't have to "think" to find the best or minimal set of equations. It's usually easy to find at least one equation. Then that will help you find the others because you can more easily see missing relationships. If you create too many variables, then it's easier to see when some combine or drop out. This is not a mechanical times table rote process.<br /><br />Just thinking with no tools is anti-math. <br /> <br />Guess and check-based discovery is neither necessary or sufficient. Besides, one cannot do that for everything and none of the discovery educational pedagogues do that anyway. It's just an excuse to push student centered learning in class.<br /><br />SteveHhttp://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-27297666003853524732015-01-22T08:32:09.262-05:002015-01-22T08:32:09.262-05:00And, R. Craigen, let me also state that I am oppos...And, R. Craigen, let me also state that I am opposed to, the current practice of omitting problem solving techniques from the instructional, offerings. Throwing a POD or PaoW out and expecting more than guess and check is the equivalent of throwing the kid in the deep end of the pool and hoping someone else discovers how to tread water or swim or demonstrates what they learned at home and the kid can copy before he drowns...or the bell rings and the water level is lowered. <br />Providing an activity without an instructional objective is baby sitting, not teaching.lgmnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-37284654263112904032015-01-22T06:57:21.641-05:002015-01-22T06:57:21.641-05:00Your last paragraph states the reason that the stu...Your last paragraph states the reason that the student should not have been shown the algebra. He does not have the preliminary understanding . Just like a first grader who memorizes the multiplication tables, he will never come to understand. It will be plug and crank for him, all the way.lgmnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-59184307962744177622015-01-21T21:24:01.484-05:002015-01-21T21:24:01.484-05:00You're quite right, Igm, and a gold star for y...You're quite right, Igm, and a gold star for you.<br /><br />Nevertheless, the student did not ask our hero to show him how to reason through the problem -- he knew that they were in prep for algebra, correctly fathomed that this was probably an algebra problem, and requested to be shown how algebra may be used to solve it. He wanted the door opened a crack, to see what was on the other side.<br /><br />Now the student could have been shown your seat-of-the-pants method (which I don't mean to denigrate -- it's perfectly good!) and come away with some idea of how to solve similarly-posed problems that are elementary enough to solve in a series of very easy one-step deductions like this.<br /><br />But instead, the student -- evidently already highly motivated for this -- came away with the rudiments of understanding how algebra is done, how to use it to model a large class of problems, and some of the first techniques therein. A valuable hour of learning for that one child, I'll say, and a lesson that will pay back much in later educational dividends.<br /><br />While in class the student generally ought to be focussed on the problem in front of them, the teacher should always have an ulterior motive: getting the most educational bang for his or her chronological buck. You have only so many hours with those children, and it is your job to give them as much academic shove as you can, so their educational bikes will roll as far as possible after they walk out one day and never return.<br /><br />A child would have a perfectly good educational outcome by solving this problem with your method -- I dare say a much better outcome than he would gain by an hour of guess-and-check that finally hits on the correct answer. But for that one student, we saw a supercharged lesson, the sort that good teachers lie in wait for, hoping to spring on students who show readiness. I think that's the point of this chapter. Or it's one of the points. Creative breaking of the rules, I suppose, is another point.<br /><br />I thought I'd also point out (something I'm sure you noticed) that your solution is the same as our hero's except for the lack of symbols and equations. The steps can be mapped from one solution to the other -- because what you are doing is a classic elimination-of-variables as is taught in starting algebra, only informally in words, without the variables and equations. It is a nice observation that the extra clothes are unnecessary. However, those clothes are where the power of the method lies, and it is good to learn how to put them on.R. Craigenhttp://wisemath.orgnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-22993388844821734472015-01-21T11:52:33.786-05:002015-01-21T11:52:33.786-05:00Let the math do the thinking as I like to say. How...Let the math do the thinking as I like to say. However, many educational pedagogues seem to believe that the goal of math is a thinking process. Yes. You think about the governing equations that apply and whether you have mn. Math is not about thinking your way to a solution when you don't have a clue. Besides, they can't have all kids discover all things. Even in group learning, only one or two "discover" anything, and it could be wrong. Teachers in most classes, student-centered or otherwise, do a lot of "telling."<br /><br />SteveHhttp://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-33726727685142834892015-01-21T11:14:16.378-05:002015-01-21T11:14:16.378-05:00Please change typo...2nd trike in sentence 3 shoul...Please change typo...2nd trike in sentence 3 should be tandemlgmnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-49473714222323665782015-01-21T11:01:21.050-05:002015-01-21T11:01:21.050-05:00Your students do not need algebra to solve this, n...Your students do not need algebra to solve this, nor do they need bar models. They need logical thinking, a key insight, and the ability to do a multistep solution.<br /><br />First, notice that 118 front handlebars means there are 118 total bikes, tandems, and trikes.<br /><br />The bikes and trikes have one seat each, while the trikes have two. There are 135 seats. Key insight: if we give one seat to each vehicle, there are 135-118=17 seats leftover. They cant belong to the bikes or trikes, but can belong to the tandems, so there must be 17 tandems. <br /><br />There are 269 wheels. Key insight:all 118 vehicles have 2 wheels. Trikes have more. So 269-2 (118)= 33 wheels left after passing out two to each vehicle. That is the number of trikes, since each needs one more wheel.<br /><br />You have reasoned that there are 17 tandems and 33 trikes. That means of the 118 vehicles, 118-17-33=68 are bicyles.<br /><br />No algebra required, despite that red herring teacher manual statement. Just plain old algebraic thinking, of the type found in elementary school, extended to more steps. <br /><br />My kids had the two step version in 5th grade with Houghton Mifflin 10 years ago. Use the 'draw it out strategy', not ' guess and check'.<br />The point of my kids lesson was to stop g and c, and use Polya's first step...understand the problem.<br /><br />lgmnoreply@blogger.comtag:blogger.com,1999:blog-6570061087276796800.post-78009633737666161252015-01-18T15:53:22.798-05:002015-01-18T15:53:22.798-05:00The academic content and delivery methods that are...The academic content and delivery methods that are understood and appreciated by cognitively able, well-prepared and motivated students (at any educational level) are likely to be very different from those which are understood and appreciated by students cognitively incapable, poorly-prepared and/or unmotivated. Colleges have far too many of the latter and are constantly seeking ways to "engage" them and create the fiction that they are doing college-level work. The reality is that they are taking on huge debt while doing little to nothing to enable them to repay. The "college premium" doesn't apply to "college-grad" baristas, retail clerks and bartenders.<br /><br />The k-12 system does this also, under the pretense that all kids are capable of, and interested in, learning the same things, in the same classroom, in the same amount of time. In HS, most kids are now pushed into a (pretend) "college prep" program, under the delusion that college is the only viable path to personal worth and financial success. momof4noreply@blogger.com