This quarter our school established three cohorts in the 7th grade and three in the eighth that are based (primarily) upon academics. We also split our math blocks in half with one half dedicated to our grade level curricula and the other half dedicated to remediation of core number sense skills. The core skill blocks are further homogenized such that there are six distinct groupings that are independent of grade level. If you are in the seventh or eighth grade you are in one of three groups attending normal curricula for your grade and in one of six groups attending remediation, independent of your grade. These splits are the best we could do based upon scheduling, teacher, and room considerations. Our goal was to create the most homogeneous groupings possible.

After about five weeks with this schedule I've come to the conclusion that there are two kinds of learning disabilities. There are those that are inherent to the child and there are those that we have created. Each of my grade 7 cohorts are about a third of the class, with the highest being no more than one year below grade level, the middle group being 2 or 3 years below grade level, and the lowest group being more than 3 years below grade level. The really interesting feature of this schedule is that I see most of my kids in two entirely different academic settings.

One setting, the grade level curriculum, is fairly conceptual so you get to see kids working with new concepts and from that you can assess their prowess with connecting the dots in their zone. The other setting, core skills, is not big concepts or word problems. It is simply raw calculation of rational numbers in all their various forms. This skills component lets you see more of what they bring to the table from lower grades.

Here's the nut… My highest group is making progress in both grade level curriculum and their core skills training. My middle group is making progress in their grade level curriculum (subject to the limitations inherent with their lack of core skills) and little to no progress in the core skills block. My lowest group isn't making progress in either curriculum or core skills.

My observational shock is not so much with the lowest group as they have clearly identified, documented learning disabilities. The highest group is making progress across the board so they're not a big concern either. The real conundrum is the middle group. In their grade level curriculum they appear to have no problem attacking new material (as long as the computation is simple) but their core skills are every bit as resistant to improvement as those in the lowest group. For these kids in the middle, it's like they have two personalities, one of which has a learning disability.

One more relevant point of reference is that this middle group has a normal amount of enthusiasm and energy level in the grade level work but in the core work they have all the inherent joy of a glazed doughnut. They sit in the core class with obvious boredom and do not apply themselves at all. In this class you could easily mistake them for the kids in the lowest cohort.

I would argue (perhaps foolishly) that this middle group is capable, based on my assessment of their grade level work, but disabled when it comes to computation. I would further submit that this seeming disability is induced by their prior failure, i.e. we created it. Could it be that after enough exposure to 'failure' in a particular domain, kids simply give up on it, concluding that it is a skill that is beyond them? Remember that this skills stuff is what they've been getting for the six preceding years.

My anecdotal evidence is telling me that these kids have an externally induced learning disability. It's induced by too much early indulgence towards their early lack of mastery and the school's failure to address it before it has damaged them. As a result they level out at a place that is far below their full potential. Is it possible that at some point, the failure to master becomes a built in disability that impedes further progress? Is there a threshold, beyond which a lack of progress becomes viral, thereby blocking future attempts to improve?

Has anyone experienced this?

Am I drinking too much coffee?

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## 55 comments:

I think you are right, Paul. Learned disability - and it's almost impossible to correct in later years.

I see the same in my HS science classes. Elementary computations, numbers make them look like the deer in headlight...They ARE affraid. The ones that are not are either my ESL students who recently moved to the US or "math kids". And please, we don't do anything higher than what in Soviet schools would count as 6th grade... Maybe even 5th.

It's just that immediate "I don't get it" as soon as the numbers are involved.

Well my comment just got "lost" and I'm too tired to replicate it. Paul, I believe this is due to damage done in elementary school. Our older daughter could never catch up despite Kumon. No doubt Kumon helped her, but she couldn't apply what she learned in Kumon into her regular math class. It was like she was beaten before she started. We didn't realize she needed both Kumon and therapy.

If I had taken action when I first got upset about the math instruction, it would have been easier. Unfortunately, in retrospect, we waited too long.

I also think you are right, Paul. It is learned helplessness. I don't get too many students who are that weak in math, but when I do, I can always tell as they are the ones who reach for their calculator to get 10% of something, and when I encourage them to do it in their heads, panic.

We can induce the same kinds of issues in algebra, with a weird combination of too much mushy conceptual stuff and not giving them variations in problems. I will never forget working with a student and realizing she couldn't solve the problem because x was ending up on the right side of the equal sign, rather than the left.

I have found with mine that if they miss a milestone - including that 2+2=4 - then they never pick it up.

Even my tutor was looking at some of the fun exercises in the language book we have - there's games at the end - and told me "those are too easy". I told her all the pgs were being completed INCASE something had been missed. It's a Gr 4 book for a Gr 5 child. I'm not buying new until we've completed the old one's first. Redoing the Gr 3 math curriculum was the smartest thing I've ever done... amazing what learning the basics has done.

Chem Prof I tutored dh's cousin's in Math when we first moved here. First thing I did was change all the letters... OMG... You'd thought the world had come to an end. But they learned. Both redid Gr 9 math in summer school and did very well but first I had to teach the basics... x+y = a+b.. the letters are place holders ONLY.

Paul,

I was that kid. We moved around quite a bit when I was young, so gaps were everywhere. I could fake it in language based courses because I could read. But math started to fall apart around 5th grade. Some seriously bad teaching through those years made it even worse.

After limping along in high school and seeing my first Fs, my parents realized there was a problem. But the general attitude was that this just was something I wasn't "good at."

The older you get as a child, the less likely you want to revisit grade school. That's why grade schoolers are such marvelous fact collectors and memorizers while middle schoolers just roll their eyes.

It wasn't until college that a friend who planned on being a math teacher assessed me as having huge computation gaps. I assumed I had a disability. I often froze up in any math class and simply couldn't think straight. But now I was 21 and highly motivated to get through this blasted college algebra class I was required to take.

Seriously, I had to go back and clean up my arithmetic skills. I just asked that she not laugh at me. My algebra 1 skills were spotty and also had to be "cleaned up." After a few months of practicing every day, things started to click in the classroom. For years I just faked that I understood the teacher. Now it was making sense.

When SteveH talks about the linking of mastery and understanding, I can't tell you enough how true that is from personal experience.

At least you are taking the time to really assess the situation. I could have been saved some grief if someone had told my parents what the issues were and what to do about them. But the middle schooler,s horror at being considered stupid can cause the to hide, making it that much harder to figure out what's going on.

SusanS

SusanS....

Can you expand a bit on what you mean by the linking of mastery and understanding?

I'm sorta guessing that mastery means being able to manipulate the numbers and understanding is that the numbers really represent something?

Where I'm going with this relates to a team teaching experience my 8th grader is presently going through that combines both the geometry and algebra class. They are "understanding" fractions using manipulatives (triangles, hexagons and squares oh my!) and I don't think this is understanding.

To me, understanding would be applying the ability to use fractions with algebra to real world problems. Tn other words, the practice of taking word problems and translating that into an algebraic equation and of course getting the right answer. Adding "units" to the mix would further enhance understanding.

The same thing can be found in advanced foreign language classes. Students who never had to, in English or another language, memorize vocabulary or retain forms lose the skill quickly. When they get to the upper levels they love reading the new stories and will work hard on translating but, if you revisit older skills to check for mastery, they often seize up and even rebel when asked to produce vocabulary or are expected to display familiarity with common words. Whereas the skilled ones enjoy memorizing new words and learning how it all works. Since we get them in middle school around age 12, certain habits are already set and hard to reverse.

Poor teaching when they're young can absolutely lead to learned disabilities. It creates stress and angst and unhappy students, despite how smart or talented they are otherwise.

Chris using fractions. My eldest is doing equivalent fractions - finding the lowest common denominator.

5/10 = teacher says it's OK to guess, Mother says it's not.

How many different ways to make each of these numbers using multiplication.

5 = 5x1

______________

10 = 5x2, 10x1

Which number is the same. Cross out other multiplication. 10x1. Cross out the 5's. What is left 1/2.

How did you get that:

5 divided by 5 = 1

10 divided by 5 = 2.

That is understanding. Mastery is the guessing.

I need a concrete example. How are you able to give them core work?

Are you starting far enough back that they can be successful?

Are you starting with stuff they can ace?

If not, why not? If you are starting from a place where they are already shaky (like trying to teach multiplication when they don't KNOW that 4 + 8 is 12), they will never learn.

Don't get stuck in the trap of "they will never learn". They may have missed more milestones than you realize. They need to go back earlier. It may be that you have no materials for "earlier" that isn't insulting to them, so they see baby stuff for K-1 grades, and are mad, but you need to find a way to go back to where they got off the train. It does no good to start from anywhere else.

I think you are thinking too hard. Of course failure to master impedes further progess. If you can't connect sounds and letters, you won't be able to read. Of course you can teach kids that they are failures. Of course you can teach them not to try. So what?

If you want to teach them to succeed, you have to go back to where they can ACE things and get things RIGHT from the very beginning.

you get a person past the terror that "I don't know how to do this" by starting from someplace that they DO KNOW how to do it with 95% confidence or better.

"linking of mastery and understanding"

Since my name was invoked, I suppose I should go into more detail, since it's easy to spin it many ways.

I've made this linking comment many times. It's a reaction to the educational mixup I see between conceptual understanding and a mathematical understanding that comes from experience and practice.

Some claim that mastery just means speed. It doesn't add understanding. I disagree. The tools of math are not just black boxes. I think this misconception comes from those who treat the mechanics of math as a black box. Teachers make comments that kids act like mechanical monkeys. They can do problems fast, but they don't know what they are doing. This leads them to think that understanding comes from some magical place other than mastery of the methods. It reminds me of the comment by Sirius Black in the movie: "Harry Potter and the Prisoner of Azkaban."

"Brilliant, Snape - once again you've put your keen and penetrating mind to the task and as usual come to the wrong conclusion."

The understanding is right there in the methods. I'm not talking about some sort of superficial short cut to solving specific problems. I'm talking about understanding real mathematical techniques.

I remember being very discouraged (in the old traditional math days, no less) trying to understand mixture problems because the book we used approached it using tables and grids. When the problem changed a little bit, I couldn't figure out which numbers went into what boxes. I finally learned to approach the problems using governing equations and defining variables.

That understanding didn't come from solving one or two problems. I had to work at it. There were so many times when I thought I understood what I was doing only to feel completely lost when I tackled the homework set. That's when the real lightbulb goes on. Look at any proper math text book and you will see homework sets that give you all sorts of problem variations of the material in the section.

I also want to make a case for speed in helping understanding too. As you move along to more complex math, you need this speed or else you will be completely bogged down. In high school, I got really good at "seeing" right triangles in word problems, even if the triangles weren't explicitly drawn. I was very fast at finding any side or angle given "enough" information. I could state that a length was something like d*cos(theta) just by looking at it. I didn't have to draw a picture and stew over which leg is for sine and which leg is for cosine.

The mechanical monkey paradigm leads to all sorts of wrong conclusions. It also conveniently fits in with their predisposition to equate mastery with rote learning and drill and kill. When they talk of balance, they really don't mean it. They still think it's just for convenience rather than understanding.

This position might seem reasonable when it comes to the basic algorithms of arithmetic, but it falls completely apart as you head into algebra.

We faced this with our eldest child. In fourth grade, the teacher set 2nd to 3rd grade level work for her math group. This freaked us out enough to have an IQ test administered. That test agreed with our general impression. She was well above average in mathematical ability, although not so high that she could ignore instruction. At any rate, there was no reason she should be doing below-grade level work.

That was our first encounter with doubting the school math curriculum, which was (is) Everyday Math. Through this site, and others, I discovered Singapore Math. I ordered workbooks, and the supplementation was successful, in the long run. Now, in 9th grade, she's a strong math student.

We went back to the first place her skills were firm. For her, that was two-digit addition. We could do this at home. I do not know how one could tackle this in "public," in a classroom. We showed her the standard algorithms for addition, then subtraction, multiplication, and division. She needed practice "doing" the math, and she needed drilling on the multiplication tables. After that, she was fine.

We also used Aleks. It was very interesting, and germane to this discussion. At first she would shut down when she got a wrong answer. Absolute. We had to explain to her several times that, unlike in school, when she made a mistake in Aleks, it did not mean that she was "stupid." It just meant she had either made a mistake, or needed to relearn a concept.

Shutting down from a reluctance to appear stupid is a major stumbling block.

When educators look at the traditional multiplication algorithm (with carrying), they don't like what they see. In place of it, they often teach the Partial Products approach. It's easier to see what's going on with that method. You think that they would take the next step and show kids how carrying is easily added to that mathod, but they don't. Why bother when calculators can do the job.

I disagree, but OK.

What happens when this philosophy is applied to more critical skills, like manipulating fractions? They really, really don't like invert and multiply, but they don't have a calculator or Partial Products to bail them out. It's not like they are making value judgments about whether computer tools can replace certain manual skills. They are translating their conclusions about simple arithmetic algorithms to all levels of math. They think that general ideas of understanding drive problem solving.

At best, they struggle with explaining how pieces of pie can represent fraction manipulation, but kids become completely lost when they have to tackle complex rational expressions. To go to the next level of understanding, you have to have skills, not vague concepts.

Many of the algorithms EM teaches, such as partial products, or the lattice method for multiplying, would be really useful to explain what in the world's going on during the operations. In my opinion, schools should use them in class, as examples to answer, "why does 56 x 93 = 5208." Then, they should make certain the kids can DO the standard algorithms.

I understand the principles behind the internal combustion engine. I couldn't repair one.

Chris,

The cobwebs started to clear when I worked the problems over and over again with my tutor. She could catch the moment when I really didn't know what I was doing and didn't understand and force me to go back until I got it. But she really had to go all the way back to fractions with me. I'd know this, but not that. It was always murky, but I would never ask for help for fear of looking stupid.

The whole manipulatives in middle school thing seems like a total waste to me, too, and here's why: When my oldest son was in the 7th grade I realized that he wasn't going to be sent home any worksheets that didn't involve telling time on a clockface, so I grabbed Saxon 6/5 and started teaching him from that most days of the week.

The thing is he has a borderline IQ and yet he still understood how to solve problems using fractions when taught properly, albeit slower than regular kids. Since he had never really been taught them properly we did start with the manipulatives, but we moved past that pretty quickly.

A similar story to yours--when my other son was in the fifth grade he watched as the teacher pulled out manipulatives for work on subtraction...for two weeks! Luckily, he was taking algebra at the junior high at time and could quietly sit and read during class, but he told me how some of the weakest students were bored out of their minds. He also knew that those same students didn't know their math facts or how to divide.

SusanS

I've gone all the way down to place value with these kids. Even this fundamental concept remains resistant to improvement. I did a simple repeatable measurement with single digit addition, capturing their rate per minute. The strongest kids were in the 50-60 problems per minute range. The middle group was predominantly 20-40 per minute and the lowest group was mostly under 10 per minute with some as low as 2 per minute.

The middle and low groups were treated to alternative strategies for figuring out their sums, like regrouping and Chisan Bop techniques (regrouping with fingers and thumbs). The interesting thing is that their is no movement with these kids. Their numbers are rock solid stable, wiggling around a mean but not sloping upwards at all.

Worse, you can't do these things with these kids for more than a few days or they completely zone out and do no work at all. This is what makes me lean towards the presumption of a learning disability. It's not like you can find a gap, fill it, and move on with some assurance that you've actually plugged it for good. It's more like you pour it on for a week, then find it accomplished nothing a week later. The things you use for the little ones, learning addition for the first time, don't work on the bigger kids in a remediation setting.

The sweetest strategy in the world isn't going to have an effect if the kids are tuned out or if they've erected barriers to letting it in.

So education works. The more math-phobic you are, the more likely you are to teach the youngest grades. As I understand it, to qualify to teach middle school and high school, teachers must pass exams which contain some math.

I suspect the authorities, if they thought of it, assumed that not requiring advanced math knowledge of elementary teachers made sense. After all, "it's easy" in the early grades.

But, what if you can pass on a phobia? It would be very interesting if some school system assigned high school math teachers to teach 1st - 4th graders math. Not all subjects, just math. Would that group do better in later grades than a control group? Of course, the control group would need to have as much class time devoted to math as the experimental group.

I wonder if other schools would find this pattern, if they looked?

In my district we have teachers who only teach math down to fourth grade and it's moving down to third in future. There's complete awareness that you can't teach it if you don't know it. For us, at least, the days of the all-in-one elementary classroom are over.

I think, at least in our case, it's more the case that there is no mechanism for early detection, and subsequent course correction for those kids who aren't getting those low level skills. The root cause is partly curriculum/text choice (Investigations) and partly structural. There is no infrastructure to focus correct levels of instruction. Each child gets the median lesson irrespective of what they're in need of.

After five or six years of neglect, they're damaged and I'm not sure there is a simple way to repair them once things have reached this stage. Susan might be the poster child for this last conjecture. She only overcame the damage by her own awareness and subsequent actions that pulled her out of it.

It's possible, isn't it, that the only fix possible has to come from within as that is where the roadblock lives? Of course you can teach these kids but not, I would guess, in a typical classroom (for me) with 26 kids in it. There is just no way to give the kind of direct one on one remedial attention they need to get out of it.

My daughter has a learning disability. She was tested as having a low working memory. I can tell you what did and didn't work with her. Kumon didn't work, it only made her a faster finger counter. She would try to hide it but I could see her fingers twitching ever so slightly.

What worked for her with math facts was to start very small. She would do one minute speed drills on two math facts until she could do them really fast. Then we would do two more. Then all four together. It seems like it would take forever this way but I was surprised at how fast she made progress. It didn't take long before she was the fastest multiplier in her 5th grade class.

The other part that I think was important was that she needed instant feedback. Doing a page of problems and then grading them is not 'instant' enough. We achieved this instant feedback by having her do her drills on the computer. We used a program called Practice Mill, which is a precision teaching software program. It's not all that user friendly but it worked for her. For a classroom setting where not everyone has their own computer, you could do paired drills. Otter Creek Institute has a Math Fact Mastery program that is good and uses this method. We only did the speed drills for learning the basic math facts and we only did 10 min. max each day. Short and intense. I used bribery to get her to do this. She loves chocolate.

For the procedural math we did workbooks. She did a whole workbook on multiplication, then division, and then fractions. We actually used Kumon workbooks for this. (Not at a Kumon Center, these workbooks you can just buy and use at home). She needed a lot of practice on one skill before she could move to another. This helped to 'cement things in'. If she tried doing them all at once she would just get confused.

She has made a lot of progress but she still has some of those 'barriers'. I attribute a lot of it to her elementary school math program which was a fuzzy, spiraling one. Thank goodness we are finished with that nightmare of a program!

Her first reaction every night when she starts her math homework is to say, 'I don't get it!' If I take my time getting around to helping her sometimes she figures it out on her own. If not then I explain it or do the first problem for her and she is usually fine with the rest of the problems. She can do the math just fine, it's understanding the directions that is tripping her up right now. I think that it's partly her language processing difficulties showing themselves, but I also think that she has gotten into the habit of just telling herself automatically that she won't understand it.

I'm starting to have some hope though. Her teacher this year has made a point of repeatedly telling her how great she is at math, something she's never believed before. Just a few nights ago she told me that she 'kind of likes math'. Hallelujah!

The roadblock stays as long as the fear is present.

It is a rational decision by a student to avoid trying, to not appear stupid, to not take a risk. If half of the class time is spent moving "forward" and the other half is spent on remedial work, why improve the remedial work? It isn't apparently preventing you from moving forward, is it? who wants to go back to the beginning of where they fell off the track?

It may be impossible to fix in a classroom. i'm still not sure what is gained by calling it an LD.

I'd call it a risk-disability. They will not be "learning" anything until the risk goes away. If you don't have single digit pair addition down cold, you will never improve their rates on anything else. They have no reason to improve. you may really need to go back to that level, all of the time, for the whole group, and not try to move past it after a week.

First, to Steve H: they do have a trick to avoid invert and multiply now. One of my colleagues taught prealgebra at my cc this semester. When she got to dividing fractions, none of the students had seen invert and multiply. It took almost the entire class, but she realized that they have been taught a cross multiplication technique. (i.e. leave the problem as division. Bottom right times top left, put in numerator; bottom left time top right, put in denominator) Not only do the students not understand what they are doing, but then they get confused when you teach cross multiplication in proportions.

I'm a big believer in teaching basic skills with games. I know that it is hard in classroom with 26 middle schoolers. But kids love games and will work hard to win. (Look how much time they put into their video games.) I even use games in my community college classes because it teaches them to generalize specific skills.

Anne Dwyer

As long as you pass Math (whatever number of classes there are based on whatever you major in from sociology to engineering) in your Univ program here - and you can do it more than once - you can teach K to 6 math. Teacher's programs are Univ degree (any kind) plus 1yr Teacher's college. Therefore K to 6 are taught by those that don't know the math themselves. My eldest son's Gr 3 and 4 teacher admitted to me I knew the curriculum better than he did.

My youngest uses a calculator but he has Autistic disorder and the biggest difficulty we have is making him understand he can memorize math facts as easily as he does words. We are up to x+0=x, and a few x+1's but I need to only work on this skill next summer. Flashcards, and as someone above said, a couple at a time, over and over and over again. The other is the language used - one of my gripes that the SLP isn't working on - so IMO, where he is in a classroom is irrelevant, he needs modified instruction.

The elder memorizes. He has poor short term recall = tested psychometry and S/L testing. But excellent long term memory. So the Saxon books of unending rote practice worked the best for him. He told me last night that fractions didn't go well so I've left a note demanding the school/test work - none of which came home. If it doesn't come home today... actually I'm going to email the VP and make certain it does... I will either find materials or make my own.

If you can do basic fractions (Invert and multiply), cross multiplying is easy, but you need to be able to do the first to understand the second. Without understanding the "rule", you never know if you've done it wrong. It is also necessary if you change units. It saved my backside in Engineering since knowing what the end units needed to be at times I could work backwards simply doing the math.

Lastly, if someone wants to learn math badly enough, they will be willing to go back to the beginning. Those that are beyond frustrated won't. But coddling students isn't the answer, they need to be shown and told that if they don't understand, they can be taken advantage of. What's a 30min of a quick lesson in finance... to make them understand that what they don't know, can do them harm - credit cards, cell phone rates etc. Many may decide to try it again, from the beginning.

SteveH said

>>You think that they would take the next step and show kids how carrying is easily added to that mathod, but they don't. Why bother when calculators can do the job.

The explanation I'm getting from the middle school math teachers here is more in line with the belief that intelligence is fixed. There are no problem sets that would lead a child to figure anything out or develop any insights - it's all memorization & monkey see my algorithm, monkey copy on the homework. Then the unit test is given with problems that have to be reasoned out. If the child succeeds, he's deemed intelligent and can stay in Regents or Honors math. If he fails, he's not intelligent enough and he'll be dropped into below grade level math. No longer is the study of mathematics seen as a way to develop thinking and reasoning skills.

Mad Minute has saved many a kid. Of course, our Mad Minute was more like Mad Three Minutes, but pounding away at it made a difference. I would use it for everything--basic math facts, conversion of fractions to decimals and vice versa, everything that was a short answer and required some fluency.

Also, another trick that surprised me by its effectiveness was the skip counting at the beginning of many Saxon lessons. I just didn't see how that would do any good, but after a week or two, things seemed to loosen up in his head and he was seeing patterns quicker. It was nothing more than making him skip count out loud up to 50 and back down with other numbers besides 2 and 5.

I will say that I learned an interesting lesson while daily afterschooling my son in math. I figured I was good enough to teach basic arithmetic, but when I would babble on and on he would just get this glazed look on his face. Of course, I blamed him because I'm such a natural teacher and all. Or so I thought.

Well, I got tired of that look, so I decided to read the Saxon lessons like a script and voila', the lightbulb came on. That was another epiphany. Good curriculum has targeted language. I was probably using words for older kids, forgetting that he really couldn't follow me.

Good curriculum has far more precise language, so I was careful to reinforce using Saxon's language and not my own. What may appear boring to teachers (scripts) might be exactly what the child needs, particularly a struggler. I would recommend anyone who is nervous about their math skills (or who gets that glazed look from their child while they're teaching them) to try a good script.

SusanS

I never understood how "invert and multiply" came to be some kind of boogie man. Nonetheless, I think it has something to do with a reluctance to use equations and variables early on. Dividing by fractions makes sense when you introduce division explicitly as the un-doing of multiplication. For example, what number is 3/7 of 12? You need to know that the English word "of" corresponds in arithmetic to multiplication. So find T such that

(3/7)*T = 12

How to get at the T? You need to know that 3/7 = 3*(1/7). So find T such that

3*(1/7)*T = 12

We don't want the 3 on the left, so divide it out. Then we don't want the 1/7; cancel that by multiplying by 7. When all is said and done, T turns out to be (7/3)*12. Wow.

I don't mean to act like I am teaching anyone anything new, and this is somewhat off-topic, but, the procedure of multiplying by the reciprocal is in no way an arbitrary, unmotivated bit of voodoo.

"I never understood how "invert and multiply" came to be some kind of boogie man."

I don't either. I remember the first time I came across this feeling. I wondered what they wanted to do instead. I found out that they didn't want teach it. They want kids to have some sort of conceptual or pictorial understanding. But this, of course, does not translate into the skills needed to divide complex rational expressions.

"I don't either. I remember the first time I came across this feeling. I wondered what they wanted to do instead."I'll hazard a guess that the original objection to 'invert-and-multiply' was that the kids (and probably the teachers as well) didn't understand why this made any sense. But, it was a "rule" so the kids merrily ground away at the problems.

So ... a different approach might have been this.

Teach the kids the following rules/identities/whatever:

A) N × 1 = N

B) N ÷ N = 1

C) N ÷ 1 = N

D) N × reciprocal-of-N = 1

Now when the kids run into problems like:

(3/5) ÷ (2/7)

[This would look better arranged vertically, but lets not fret about HTML formatting]

the approach would be this:

1) We want to get to a final form with a denominator of 1, so that we can toss the denominator.

2) If we multiply the denominator by its reciprocal, we will have that one.

3) We can multiply any number by 1 without changing it, and a number divided by itself is one, so ...

4) Reciprocal of 2/7 is 7/2 ...

5) (3/5) ÷ (2/7) =( (3/5)×(7/2)) ÷( (2/7)×(7/2)) [applying A and B]

6) ( (3/5)×(7/2)) ÷( (2/7)×(7/2)) = ( (3/5)×(7/2)) ÷1 [applying D]

7) ( (3/5)×(7/2)) ÷1 = ( (3/5)×(7/2)) [applying C]

8) ( (3/5)×(7/2)) = 21/10

In short, the objection to teaching 'invert-and-multiply' is that 'invert-and-multiply' is a short-cut to much more basic algebraic operations. Teaching the short-cut without teaching the underlying principles leads to the ability to grind the answer, but no understanding of why it works.

That's my guess as the origin of the objection.

What people who today object to teaching invert-and-multiply object to, I cannot guess.

-Mark Roulo

I remember reading that there was a big difference in perception between American kids and Chinese kids. For the Americans, doing well meant they were smart, but for the Chinese the critical component was hard work. The American kids saw a poor grade as risking or invalidating their smart status, but the Chinese saw it as an indication that they had not worked hard enough. BTW, the Chinese kids were much less likely to say they were good at math, even though they scored much higher than the Americans. Maybe we need to stress the idea that serious effort is needed in school, but it's an uphill fight.

In the 60s, a college classmate asked me to go with her to some social event. When I said I needed to study instead, she said "I don't know why you spend so much time studying; you know you always get As." Bang head firmly against wall...

I think invert and multiply became a problem because most American math teachers, even the ones who enjoy math, do not understand how it works.

See Liping Ma.

I also love it that Ma points out that invert and multiply is always an effective way to divide. It's just that only with fractions is it the most efficient way.

Maybe playing with that a bit could help understanding.

I like Mark's approach. It's not too formal and it breaks it down into other rules students should know. Most students don't have any problem understanding that you can multiply a numerator and denominator by the same number. They only have to realize that a fraction is just a number.

Many educators want to claim the high ground when it comes to mathematical understanding, but when you look closely, it only amounts to a pictorial or manipulable lowest level of understanding. It never amounts to abstract understanding, like the importance of the simple identity:

a = a/1

This is what I mean when I talk about the linkage between mastery and understanding. You don't fully appreciate what this means until you work with fractions and rational expressions. This is true for all of the basic identities. I don't think I fully understood algebra until my junior year in high school. Only then could I manipulate any expression in multiple ways without the tiniest bit of doubt.

Would teachers like "invert and multiply" if they understood Mark's simple explanation? I don't think so. Didn't we (KTM) find a quote once where Everyday Math claimed that most kids don't ever need to know this when then grow up?

(My fear is that after all of these years at KTM, I am just repeating myself.)

That's my general impression of all modern reform math curricula. Lower expectations. Understanding is only conceptual and gives up when things start getting abstract (mathematical).

So, K-6 math dumps the kids off at pre-algebra, where abstraction and skills matter. If you can't make this leap, it must be your fault. The lower schools happily pump the kids along into the big algebra filter when it's too late to do anything about it. Better yet, schools point to the kids who do well and get the rest to blame themselves.

The best students have helicopter parents.

I had to pinch myself last night when the Math teacher called after school :) :) My son has 2 teachers for Gr 5. I had requested all the fractions stuff home... Teacher said it was there, with blank copies and the kid forgot it. WOW!!!!

We talked for a few minutes. He had given up on equivalent fractions until after Xmas and had started the next unit over the last few days. The kids couldn't grasp it.

I wanted to tell him you had to teach them their multiplication facts first, all the way to the 9x's tables first.

But I kept my mouth shut. I simply explained how I'd worked on it with my son and he thought that breaking it down like that was too much work. The kids should be able to guess the smallest fractions.... HUH!!!! So again, I closed my mouth and thought we'd do it ourselves.

I also bought one of those Gr 5 math workbooks you can find at Walmart for a couple of dollars, yesterday. It had a decent fractions section in it so over Xmas we'll work on it too.

I make a point of not telling them how to teach. I dislike the curriculum and that's no secret btwn them and I. But as I told the teacher, if the kid is struggling... send it home. My biggest complaint is the fact these teachers think the parents know less about math (education in general) than they do... many probably do... but there are a lot of us that know more and can help our own kids.

Here is some unsolicited advice for teaching equivalent fractions. (First of all, the terminology is bad -- they are equal, not equivalent; equal as numbers, i.e. points on the number line; equivalent as numerator-denominator pairs.)

I hate to see the "multiply by 1 = 3/3" approach that Saxon and other curricula pull out of a hat before actually teaching fraction multiplication.

Better, draw out subdivisions. Easy example, to show that 2/3 = 8/12. This actually is easier to see if you don't use unit fractions (numerator of 1). Two ways.

(1) Line. Draw a line segment. The length represents 1. Partition into 3 congruent segments. Each represents 1/3. Now partition each third into 4 congruent segments. Don't be embarrassed to use colored pencils! The end result is that [0, 1] has been partitioned into 3x4 pieces, each 1/(3x4). (I know that 3x4 is 12, but make clear what is happening, so write as 3x4.) Thus we pick up factor of 4 in denominator. The number we are working with is 2/3; now the numerator is 2x4. Counting small subdivision, we have 8/12; counting the big subdivisions, 2/3. You naturally see 2/3 = (2x4)/(3x4).

(2) Unit square. Choose a length for side length of 1. Then the area of the unit square represents the number 1. Partition with vertical lines into 3 congruent rectangles, each of area 1/3. Shade in 2. The shaded area is 2/3. Now partition going the other way into 4 congruent pieces. The whole square is now a 4x3 array of rectangles. Each of the 3 of the original 1/3 's is now shown as 4/12. That's how the denominator picks up the factor of 4. We have shaded in 2 of them, which is 2x4 little rectangles, thus the numerator picks up the factor of 4.

Do an example or two a day, drawing it all out with line and area representation. The student builds up intuition for it. Also work with numerical examples where, assuming they know multiplication tables, they find common factors top and bottom.

I don't like to teach "canceling" because kids tend to think there is some magic in crossing stuff out, and the next thing you know you will see (3 + 4)/(7 + 4), cross out 4s, and voila, 3/7. Instead, use the equal sign. So if the fraction is 8/12, rather than crossing out the 8 and writing 3, etc, just have them write out

8/12 = 2/3

without show any "canceling". Stone-faced instructor says, "Nothing is 'canceling', it's just that these two numbers are the same." Then, when they fully know what they are doing, they can cross out factors top and bottom, "cancelling".

I have gone through what farmwifetwo talks about; teachers who go through the motions and are afraid of really getting to the heart of the problem. They aren't interested in comparing notes on the best way to teach fractions.

"But as I told the teacher, if the kid is struggling... send it home."

They will just give up and you might not know there is a problem until the quarterly report card. Then you're left trying to figure out what the hell is going on. Our schools even hide everything in portfolios that stay at school so that you can't make your own judgment. It's all unbelievable. I'm not asking them to do more work. I just want them to send the graded homework and tests home! They claim to teach critical thinking, but they just parrot back ed school philosophy in a rote fashion.

Invert and multiply: See Wu's explanation in the fall 2009 issue of American Educator.

Very simple to understand.

Singapore Math, by the way, does not explain why invert and multiply works, but leads up to it starting in 4th grade. 1/4 of 12 is the same as 1/4 x 12, which is 12 divided by 4. So studnets are seeing one aspect of the inversion principle early on without realizing it.

They build on that, showing that 1/2 divided by 4 is the same thing as 1/4 OF 1/2 which is 1/4 x 1/2.

Finally, in sixth grade, they get to dividing a whole number by a fraction. How many quarter pieces are in an orange? 4. So 1 divided by 1/4 = 4. How many 3/4 pieces are in 3 oranges, and so forth? A pattern is established but no formal explanation. Students make the leap to 2/3 divided by 4/7 and so on. The formal explanation comes in 7th grade when they've had some algebra and can understand it in terms of symbols. Richard Baldwin, a mathematician and one of the authors of Elementary Mathematics for Teachers, and who is a big proponent of Singapore Math sees nothing wrong with Singapore's approach of letting students work with a procedure before getting into why it works at a later time. It's done all the time with limits and continuity in freshman calculus classes. The deep theory about limits and continuity comes later. (Except perhaps for honors calc classes.)

The big thing about fractional division is not it's a big waste if kids learn the invert and multiply procedure before understanding why it works. No. What is important is that students understand what it means to divide 3/4 by 2/9. What is really being asked? Word problems that use fractional division are key.

Once students get the concept, you can explain invert and multiply when they have enough tools to work with it. Or use Wu's approach; it's very good. Straightforward and simple. Similar to what Mark Ruolo demonstrated above.

The boogey man about invert and multiply is the notion that teaching students that rule is associated with giving students division problems and telling them to solve it with invert and multiply and not connecting the problems with what is really going on. I've looked at a lot of math books. Most I've seen make some effort to explain what is going on when you're doing fractional division. The boogey man is very often a straw man.

In my case, I didn't know why it worked til I had algebra. But prior to that I knew what fractional division was and when it was to be used.

How is the push going to get (at least) grades 4-8 taught by teachers trained in math? Our state now requires this for grades 7 and above, and I credit this for our change from CMP to real math textbooks. That and the fact that many parents complained that their kids had a curriculum gap when they got to high school.

Our older daughter could never catch up despite Kumon. No doubt Kumon helped her, but she couldn't apply what she learned in Kumon into her regular math class. It was like she was beaten before she started. We didn't realize she needed both Kumon and therapy.That's interesting.

I have a friend with a super-smart son who scored 800 on reading & something like....high 500s on math. They paid for beaucoup tutoring, bringing his math score up to the low 600s, as I recall.

He, too, went to lots of Kumon, which didn't help.

I never quite understood that.

I wonder if he had a learned helplessness issue?

Through all the math mishegoss around here, C. developed some minor math test anxiety and his sense of 'self-efficacy' where math is concerned isn't great.

BUT Hogwarts is such a cheerful place that he's basically OK. We've done no tutoring or reteaching & he's now in sophomore year, earning a B+ in Honors geometry. He's only slightly off from an A (no minus grades at the school), and we've told him he needs to just go ahead and get an A.

That seems to be fine. He has at least 'half' an idea that in fact he can do math.

I tell him whenever it's necessary that the goal is for him to do math as well as Carolyn (& a lot of folks here) did writing. Math isn't going to be his primary focus, but he needs to be fluent.

A friend told me something interesting.

Her son, who probably is a 'math brain,' has had a terrible time making the transition from Trailblazers to regular pre-algebra in the middle school.

She says that he spent so many years "doing it more than one way" that he never developed a 'feeling' for WHEN HE HAS THE ANSWER RIGHT - because there's always something else you could do.

He has absolutely no confidence THAT HE IS FINISHED.

On math tests - and this is a very bright kid - he'll just keep going and going on one item.

So now she's ordering math books to start teaching him at home.

I walked her through my collection.

One of the books he's ordering is the Mad Minute book Susan S just mentioned.

Her son, who probably is a 'math brain,' has had a terrible time making the transition from Trailblazers to regular pre-algebra in the middle school.Oh, now there's a shocker.

Hey, after a couple of years with Trailblazers in our district they got rid of the 6th grade pre-algebra class.

Here's a story for your friend. A friend of mine had a son in the 95th percentile in math right before his year of Trailblazers. On the next state test he dropped to around the 65th. She was in shock. I just used our standard line about kids that age. I said, "it's probably fractions." She said he was fine with fractions, but she decided to test him anyway with the Saxon test, and yep, it was fractions.

She might want to use the Singapore or Saxon test to get a better view of what's going on.

SusanS

Someone asked (Allison I think) about what's the big deal about calling this condition I posted about, a learning disability. For me, the importance is that this condition is much different than simply being behind your peers. Being behind implies a fix via simple things like staying for extra help or perhaps some in class differentiation.

I think a child with an induced learning disability needs all of the tools available to kids with 'real' learning disabilities. The problem is that these kids are in never never land. They don't have an ed plan, they aren't behavior problems in the same sense as lots of the kids with ed plans. Oh they may have behavior issues deriving from their inability to stay focused on tasks where they are completely lost, but they basically fall through the cracks. They are severely undermotivated, listless, and convinced of their helplessness.

They're salvageable given the appropriate effort and focus by the school but the system is rigged against them in ways that prohibit them from getting exactly that. They just continue to pull poor grades, sink further behind, and feel ever more overwhelmed and hopelessly lost in math.

Here is a good example of what happens with these kids that ties into where this topic has drifted (into fractions).

We're working on a unit that is heavily based on scaling polygons. The concept isn't particularly difficult. You multiply dimensions of an original figure by a scale factor to obtain the dimensions of the scaled image you're creating. Lessons focus on things like what happens to the perimeter and area when you scale by a known scale factor.

My bubble kids get this pretty well.....

As long as the scale factor is a whole number!

If you use a scale factor that is a fraction, decimal, or percentage they are lost. They just don't have the computational horsepower to extend their new found scaling knowledge into the very real world of scaling by all the different forms of rational possibilities. They will even say things like, "You can't scale down because their is no number you can multiply the original by to get a smaller number." Or say you have an image dimension of 6 and an original dimension of 4; they'll say this isn't possible because they can't find a whole number scale factor (it's 1.5 but this will not occur reasonable during a frantic search for a whole number).

These are kids who can obtain exact knowledge about the concept of scaling without being able to apply it in all of its myriad applications. You may get two weeks to work the concept but there's no way you get the three months they need to revisit elementary school and that is what they really need.

Conventional wisdom at this point is to shove a calculator in their hands at this point but the calculator is of no use in the hands of someone who thinks you can't make small numbers from big ones by multiplying.

Hmm, the comment didn't go through...

Well, learned helplessness or not, what do we do? As a parent, I know what to do: - get that gap filled. Tutors, myself etc. Identify and eliminate as soon as I can.

As a high school science teacher - what do I do? I have 5 classes, that must move pretty fast (and better be on the same page or I will go nuts). I teach so called "Geophysical Science", which is a mix of everything taught to students in preparation for further Bio, Chemistry, and Physics. I have to make sure they get the lab skills, and that they can understand and apply basic concepts in chemistry and physics. Such as ions, atoms, formulas of compounds. Basic. But - we stumble upon the elementary things, like understading the charges, because a quarter of my students had missed on positive/negative numbers... I can't reteach and ensure mastery of every single math skill involved in science. Simple brief reviewing does't help to those in need of mastery and annoys those who are fluent.

Many science teachers therefore avoid even touching on topics involving math (I was the only one this year teaching gas pressure with formulas and through problems). They leave it to teachers in higher courses. And the teachers in higher courses cannot do anything with it either - except dumbing courses down to "conceptual".

I told my classes today (after reteaching the charges on ions again, "When you cannot understand what I explain, when you that mathematical explanation do not make sense for you, you should have the red light flashing: A MISSING KNOWLEDGE! In this course, all matematical reasoning involves things you must have learned up to grade 6. So mark that missing concept for yourself, and find the way to learn it before it's too late. Go to math tutoring (we have it in school) and ask to teach you that specific skill or concept. Go online and find tutorials. I can help you identify what is missing, but I will not teach you that." Because if everything is in place, the science concepts and math behind them should be cristal clear...

Do your students have the chance to take any woodshop classes? I know it isn't popular now, but some school still have them in middle school. For all the enthusiasm about "real world" applications, many schools don't realize that woodshop can help practical skills.

In the practical arts, cooking, sewing, and shop, one does need to convert quantities proportionately. I do not agree with the current push to separate schools into "vocational" schools and "academic" schools.

The closest my kids get to wood shop is sharpening pencils :>}

Seriously though, your point is on target. One of the things that does not get enough sunshine poured on it is the relentless push towards curricular bloat in the 'core' subjects, math and ELA. That bloat has not only pushed out things like drill and practice. It has also served to eliminate things like woodshop, real music, real art, and home economics.

I'd love to see the 'research' that shows how expanding math classes from 45 minutes to 100 minutes improves the knowledge transfer. I suspect it's more consensus fog.

"They just don't have the computational horsepower to extend their new found scaling knowledge into the very real world of scaling by all the different forms of rational possibilities."

"Computational horsepower"

Another good one.

It's more than just rote or speed, it's horsepower. I keep trying to find clear ways to link back up what they have broken. They never get out of first gear. When they do, they have no momentum; no mass. They don't have an engine.

Perhaps one way to explain it is that they give up after teaching about whole numbers. Students understand the concept, but they have to know more than that. They have to know the variations. I understand that it's not much help to do 50 problems with whole numbers, but that's not what proper textbooks do. They do progressive variations. Start with whole numbers, then to fractions below 1, then to any number on the number line, then to percents, and then to word problems. One of my biggest complaints with EM is that there are few variations. Just cover the concepts and spiral on. Repeated partial learning.

It's not just rote practice. The variations teach understanding. (This is so obvious that I can't believe I'm saying it.) It has led me to believe that they are either really ignorant or they just don't like math. They want to change math into an image of themselves.

When you are dealing with percentages, like with the Christmas present problem thread, you have to tackle the problems in every direction with all variations. Many kids can take 30 percent off of a retail price, but can they find the original price given the sale price and the discount? Can they play store owner and find how much money they will make if they mark up a product by 100% and then give a discount of 30%?

So when they complain about "drill and kill", they are denying the reality of math. Is it enough to just tell a student that a scaling factor can be any number on the number line? Can they then tell you what the decimal scaling factor is if you want to reduce something by 28%? What if you tell them that a scale model is 23.5% of the original? Can they find the size of the original? Can they immediately tell you that you divide the scale size by .235? For number sense, can they quickly estimate the size by multiplying by 4? This is not just speed. It's not just concepts. It's variation and practice.

With EM, you can keep very busy but never develop any horsepower.

The devolution to whole numbers is exacerbated by differentiation. Every differentiation strategy ever handed to me for CMP involved changing decimals to whole numbers, changing oddball percentages to benchmarks, or changing fractions to benchmarks.

We teach struggling students a modified number line with bits and pieces and favorite places.

"favorite places"

Please write a paragraph about your favorite place on the number line.

One!

It's easy to write. It's easy to multiply by. It's easy to divide with and fantastic to add with. I was going to pick zero but zero has that pesky division problem when it's the divisor.

Now can I have my cookie?

Learned helplessness is the LD created by an education system that doesn't actually teach.

LH is what occurs when there aren't any consequences or responsibility to learn by either Teacher or student. Student's learn in K they can't be failed and Teacher's learn they don't actually have to teach it b/c they can push students forward.

My son's fractions homework from the school is a joke at best. There's "describe this" all over it. "draw this picture" all over it. There's 5 actual equivalent fractions to do out of 6 pages. In total out of 6pgs there are less than 20 questions. There is more work, better explanations in the 2 pages (one side) in the little work book I have. It was UGLY (the LH was in full swing), but I think we got the jist. Now while they are at swimming lessons at noon, I'm going to make up my own sheets. We're going to do them OVER AND OVER again over the next 2 weeks... then I'm sending them to the school with a snark (polite), but the Teacher'll get the point. "Dear Sir, these are my son's homework on equiv fractions. I have included blank worksheets for you to copy for the rest of the class and a sheet with the answers on it. I didn't feel the homework was helpful so I made up my own. Please feel free to copy these for the rest of the class to work on."

We've raised a generation of children that think they are owed. That think someone is going to do it for them. That don't think they are responsible for their own education.

Last Sun I quit helping my youngest with his rote sections of math (saxon Gr 2). It took 30min of steady argument to do 20 questions with a calculator. He knew how to use it, he knew what needed to be done. Today, it was 10min and I only had to redirect him approx once/line. The school does it for him - Learned helplessness. Mother makes him do it himself - responsibility for his own work.

Zero. It must be zero! So often, if you find a zero, you're home free. It's either undefined, or solved!

Oh, please, yes -- bring back sewing and cooking and wood shop! I was an academic kid, but took an extra year of sewing in 8th grade, and it was incredibly valuable. The problem solving and learning patience when it was better to rip something apart than to try to fix it has been so useful in thinking about research. Of course, I'm in a field where a lot of what I do day to day looks like plumbing, but still. When I think about homeschooling my daughter, part of it is that I want her to do things like sewing and typing, which could be taken in school when I was in Junior High, but which are now replaced by watered down academics.

Partly, I wish they would see that these classes are great for "academic" kids as well as "vocational" ones. In my area, these classes were killed at the middle school level because of the sexism issue -- girls took a year of cooking and sewing while boys took a year of basic shop -- but instead of having everyone do some of everything, they just dropped it. Sigh.

In NY, the middle schoolers take a Home & Careers class each year that includes cooking & hand and machine sewing. Technology class covers metal shop and very basic wood shop. Please remember these are fully included classes so safety means many things that were done in the past cannot be done now due to budgetary reasons relating to the necessity of adequate supervision.

I understand why schools have chosen to move away from shop and home ec; can we call these the practical arts? It's in part, I think, the association with the high track/low track type of tracking which did occur. I think that's throwing the baby out with the bathwater. I was a "high track" kid, but the practical arts taught me skills which weren't available in the more abstract classes.

There's a current obsession with STEM courses. My husband's an engineer, and all of his friends were disassembling and rebuilding machines through childhood. There was a recent article in the Wall Street Journal about the rise of tinkering (http://online.wsj.com/article/SB125798004542744219.html). The practical arts are deeply linked with STEM.

It may be that there are kids who need to work with their hands in order to grasp certain concepts. Not only manipulatives! Scaling is a physical concept. So many parents work these days, the skills which were once passed on at home may not be. Cooking, home repair, and such, all are necessary skills, no matter what one ends up doing as an adult.

It's simple. Your students were NOT taught their core skills conceptually. Therefore, your bottom groups are trying to apply broken methods of arithmetic to find solutions. Look at RightStart (ESPECIALLY!!!!!) or Singapore to remediate their poor approach to numerical manipulation, and miracles will occur. If your students aren't learning, there's something wrong with the teaching. My mother taught self-contained mentally retarded students and routinely got them to advance 2.5 grade levels in the space of two years--supposedly an impossible feat, moving toward closing the gap between them and other students. There is NO excuse for your students not learning.

>>>The same thing can be found in advanced foreign language classes. Students who never had to, in English or another language, memorize vocabulary or retain forms lose the skill quickly. When they get to the upper levels they love reading the new stories and will work hard on translating but, if you revisit older skills to check for mastery, they often seize up and even rebel when asked to produce vocabulary or are expected to display familiarity with common words. Whereas the skilled ones enjoy memorizing new words and learning how it all works. Since we get them in middle school around age 12, certain habits are already set and hard to reverse.

NONONONO!!!! That is NOT how to properly learn a foreign language. That is how to make good grades and utterly fail to actually be able to be able to use a foreign language--AKA, the American way. We have students taking YEARS of a foreign language, and they can't maintain a simple conversation not because they're stupid but because IT'S BEING TAUGHT WRONG. The class is stupid, not them. Almost every hearing baby learns to speak--and older kids can just fine, too, if given the opportunity.

I was a very good student and could manage an IRL Level 3 proficiency through the horrible traditional methods used to teach foreign language in my school, but I achieved IRL Level 4 only through living abroad for 6 weeks--and by the time I left, people were taking me for native, accent and all. I can assure that no one, and I mean NO ONE, achieve IRL Level 4 or 5 through grammar drills and vocabulary lists, and precious few can make 3 because it requires an immense ability for rote memorization and very strong raw processing powers.

It's not the kids who are broken. It's the teaching method. I say that as the strongest language student in my class of 1000 and an incredibly strong math student who could make almost every kid understand the material.

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