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Paula Beardell Krieg
Math Revisitor
Math Revisitor


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This post is for Telanna and Sarah, with whom I've been chatting about what are derivatives. If you can read a graph enough to recognize when something is going in a postive or negative direction, then looking at a graph might be the best introduction to understanding what a derivative is.

Since I also write about dyslexia on these pages here at G+ I thought I'd sort of double-dip my interests in this post.Since you are teachers I thought you'd connect to my graphic here. This graph represents the progression of one of my children's reading skills between 2nd and 7th grade. The amount of pain and struggle embedded in these curves will hopefully make the derivative super vivid.

Now, here's something kind of magical to know: without doing any math or calculations at all, by only looking at the visual representation of functions, -the lines on this graph-, you can glean much of the same information that the derivative gives you.

You can see that in second grade my child's reading scores were in the low eighties. I thought this was like getting a B on a report card so I didn't panic. But over the course of years there wasn't much overall improvement. Look at trend of the blue between 3rd and 5th grade: notice how it's slowly dipping. It's not hard to predict that this trend will continue to go down. If you calculated the derivative of the function that the graph is describing during that time, it would come out as a negative number, indicating a downward trend.

This is what I most want to convey about the derivative: think about it as an adjective. It is a tool that is used to precisely, numerically describe the very same thing you can see on a graph.

You can see the red line on the graph changes from going in a positive direction to a negative direction at third grade. The derivative would be a positive number before the third grade testing. At some point the graph starts going down. At that place where is changes from positive to negative is called a maximum (or local maximum) and that where the derivative would be zero. The place where the derivative is zero is one of those ultra important things to know in calculus. It's good to understand when trend change direction, right? Oh, but look at the red line at fourth grade, it changes direction again. but this time it goes from negative to positive. At the moment of change the derivative again is zero. Now it's called a minimum. Minimums are really important too. So expect to have to find the when derivative equals zero.

I'd like you to notice that there are two different ways of thinking about the derivative here. First, the derivative can tell you the trend the function is going at any given point. You are using it to answer questions like, are we going up or going down?

But I'm mentioning a second way of thinking about the derivative. I'm asking it, when are you zero? And my answer to that question leads me to that maximum or minimum, which can be really helpful to know in lots of situations.

BTW, it wasn't until the end of 5th grade that I began to figure out that those 80's and 90's weren't like B's and A's. They meant that my child was reading at the level you would expect of someone whose IQ was in the 80's or 90's. Finally, Mama Bear made sure some pretty strong intervention were done. It made all the difference.


Considering Triangles

Suzanne von Oy and Mike Lawler have diverted my attention toward triangles with their recent blogs, and

I am still trying to sort out the why it is that math people have an affinity to geometric shapes. One of Susan’s students commented, while using triangles to create patterns, that, sure, this was fun, but how is it math? Then Mike and sons paired and rearranged congruent triangles to create wordless proofs showing that the shapes of two different looking triangles had the same areas, thus doing math in a way turned out to be fun.

I was heading into the Adirondacks yesterday morning, knowing that in my class with first graders we would be working with triangles. As part of a bookmaking project, the students would fold a double square (domino shape) in half to form a square, then cut this square in half along its diagonal. The students would then end up with three similar isosceles triangles, one of which is twice the size of the two smaller ones. We then use these triangles both structurally and decoratively.

I don’t talk about the math of the triangle with these first graders, but I like getting them to make and handle these shapes, encouraging the familiarity of the properties of triangles well before trig functions dominate their identities.
Triangles demand their very own rules. For instance, whereas we can pretty much determine the center of a square or a circle without much instruction or thought, finding the center of a triangle requires a whole new skill set.
But during yesterday morning’s drive I was trying to sort out why it is there are so many triangles in math.

It’s only recently that I connected the square to a generalized unit, and a circle to a generalized cycle. Now I was pondering, what does the triangle represent?

Why should we care if two different looking triangles have the same area?

What’s the significance of two triangles that are similar, but one is larger?

Here’s what I came up. First let me say, this thought is new to me, coming to roost in my brain less than 24 hours ago, and I am so eager to share it that, well, forgive me if my expression of this idea is awkward and incomplete and correct me if I am wrong!

Triangles that are created with a vertex radiating from the center of a circle are the visual representation of accumulations during a slice of a cycle.

When we say that two different looking triangles have the same area, it’s the same as modeling that my husband can rake the same amount of leaves in one hour as it takes me 3 hours to do.

Two similar triangles of different sizes model the proportional similarity of how much money I earn getting paid the same rate, but working different number of hours.

A representation of accumulation over a period of time.

Does this seem right? If so, then my next question is, what is the absolute youngest age that this connection between the shape and what it symbolizes be introduced to children? Keeping this kind of info under wraps makes no sense to me. The awareness that shapes are symbols which are embedded with ideas has so much richness that I’d really love to have some clue of how old a child needs to be before they could grasp this concept. Would appreciate some insights from anyone who has something to say!

For the longest time f(x) baffled me.

When I see f(x) I still have to silently repeat the translation that works for me.
I silently mutter f(x) means that there is an expression somewhere nearby (maybe hiding in the wings, maybe never to be explicitly identified) that has an x in it.

Feels like there’s a bait and switch going on here.

The fact is, I’ve liked solving for x. It’s a puzzle to figure out. Solving for x seems like, well, isn’t that what x is for, to be solved for?

But put x in a parenthesis with an f in front of it, well what am I supposed to do with that? I mean, even if there was an equal sign and an expression like 5x + 2 next to it, what can I do with that? It’s not like there’s anything to solve for.

Except there is. But I’m not supposed to solve for x any more. Bait and switch. All of a sudden I just am supposed to arbitrarily pick values for x. Except it’s not even really arbitrary. I kinda figure out that it’s just values between negative three and positive three (be sure to use that zero, too!) that I’d generally be using.

Using for what? Using to get an answer. Not just one answer, though, but a series of answers. When there’s an f(x) I’m not supposed to be thinking of one point, but a series of points that connect to create a meaningful line. Ok, I’m good with that.

Oh, but then it gets worse again. Some teacher type will draw a curvy line on the board and start a sentence with the words, “For some f of x…..” and the world as I know it goes blank as my mind scrambles to figure out what that even means. Some f of x? Seriously?

Okay, I’ve got it figured out now. What the teacher type means is, I want to talk in general about some expression that graphs out as some sort of curve and the ideas I want to convey can be made through visuals without having to talk about a specific expression, aka polynomial.

Ok. Got it. Whew.
But this is not the end of my bafflement with f of x.
It gets so much wilder. To be continued.

(I have struggled to understand the most basic beginning of math. I can't help but think that the nature of my befuddlement is a widely shared experience, so, in no particular order, I'll be writing about details that I found so difficult to sort out.)

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Multiplication Sundae and One Child's' Story

She's twenty and doesn't look like a child anymore, but she's my child.

We drove down to visit Grama this past week, which means we were in the car together, just the two of us, for a total of eight hours. Using my best negotiating skills (ie agreeing to let her play her music for as long as I could bear it) I was able to get her to talk to me, for about 11 minutes, about her experience learning her multiplication tables.

Me: Do you know your multiplication facts?
Her: (pause.) yes
Me: What's 6 x 7?
Her: (pause) um42

Me: Do you like knowing your multiplication facts?

Yes, she was bombarded every morning for years with doing her multiplication tables for one minute. I knew this was always problematic for her, but what she said next came as a complete surprise.

Her: And I especially hated the Multiplication Sundae

A week or so ago Christopher Danielson referenced "Multiplication Sundae" on twitter, bemoaning it. I didn't understand the reference and didn't think about it again. Now, I am finding out, a full decade after the fact, that my daughter had strong feelings about the Multiplication Sundae.

Her: Depending on how far you got with Multiplication you got scoops, whipped cream, toppings and a cherry. I knew I never a chance to ever get to the full sundae. It was humiliating to kids like me. It made me feel retarded.

NOTE TO READER: the word "retarded" is almost never used in my household. The dyslexic gene is strong in my family, manifesting in three people across two generations, one of whom has never been able to read two sequential sentences, so, with this info, I will leave it to the readers' imagination how highly charged, and how sparingly used, the word "retarded" is within the culture of my family,.

Me: If you were running the classroom, how do you think you would get students to learn the multiplications tables?

Her: By teaching them to see the patterns first.
Me: Would you want them to memorize?
Her: Yes. I don't want to talk about this anymore. I'm putting on some music now.

Me: (days later, on the ride home from Grama's, after negotiating a new deal) What's 7 x 5
Her:( crosses legs.) 35.
me: You know your mulitpication facts well?
Her: Yes, but every time you ask I have to do something to give me more time, like shift position or say something so I have time to remember, that way I don't panic.
Me: So each time you hear a question about multiplication you first have to deflect panic?

Further conversation reveals that she is either experiencing panic or deflecting panic when she hears a multiplication question. But, still, she knows her facts and is glad to know them.

I turn the conversation towards a strategy she would like to see used to support learning multiplication facts. We discuss and discard numerous ideas. It's interesting that she keeps trying to find a way to use the Multiplication Sundae concept.

She does recognize that knowing her facts made much of other math easier for her, and she is glad to have the skill, She expresses the sentiment that the repetition that builds familiarity is something that needs to happen, but not within the Mad Minute Multiplication Sundae competition.

Finally we arrive at a solution( and this is the last she will converse with me on this topic). She says that every morning the class could be given those drill sheets, students could do them at their own pace within the timed constraint, then hand them in to the teacher, who would then count up the total number of right answers that the class has produced. When the class as a whole reaches a certain amount of right answers, then everyone gets a Sundae. That way, she explains, it becomes a team effort that unites the class towards a common goal.

And, with that, she turns her music on.

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Looking at Integral Calculus through the Lens of the Unit Square

I lost my way into mathematics during my first year in college, during calculus. I’ve been finding my way back, but this time around I get to search for the conceptual anchors that could have kept me from drifting away.

From my current vantage point I’ve come to realize that the foundation of the ideas explored in integral calculus are closely related to what I learned in elementary school about the unit square and the area of a rectangle.

I’m writing this for people, who, like me, are attracted to math, but who missed the memo about the contents of calculus. I’m not talking here about the techniques for solving calculus equations, rather I’m looking at the questions integrals answer, and how they connect with the math that I learned when I was ten years old.

This has to do with two ideas: the area of a rectangle and the more mysterious ideas about what that area represents. The connection between area and integrals exist, inherently and directly; all I want to do here is to make this connection explicit and meaningful for people like me who didn’t grasp these concepts while we were in school.

For anyone who needs a review of the concept of area, I am going to quote from Christopher Danielson’s Common Core Math For Parents For Dummies book, page 91.

“Area is the number of unit squares that fit inside a figure without gaps or overlaps. A unit square may be any size –a square inch, a square centimeter, even a square mile. These squares may be cut into fractional pieces, and ultimately students will be able to think about area without imagining counting squares. But at the heart, area is about covering a figure with unit squares.
At first, third graders count these squares to find areas. But soon they develop a formula for the area of a rectangle. Students learn that the area of rectangle can be found by multiplying the length and width (both measured in the same units) which the formula A = l x w summarizes. “

Now, after many passing years, I’ve discovered that, besides counting square units or solving the length times width formula, I can also choose to learn calculus to calculate the area of a rectangle --though this would be something like learning to fly a helicopter just to check the mail at the end of my driveway. My point here, however, is that it turns out that the unit square looks exactly the same in third grade as it does in college calculus.

Integral calculus, generally speaking, looks to calculate the number of square units inside of something that, like a rectangle, has a bottom , a top and two sides, although, unlike a rectangle, one of the edges can be curvy. A curvy edge makes finding square areas more difficult, but calculus makes this doable: this is not, however, what I found to be gloriously illuminating. Finally understanding what the big deal was about area was my aha! moment, and this is where I could find meaning in connecting my elementary education to grown-up math.

In grade school I never got beyond the rule of measuring the sides of a rectangle with anything but the same units for length and width, thus never tapping into the depth and breadth of what mathematics could tell me about the world.

“But!” I have said to myself, “Measuring length times width with the same units is one of the foundational skills we acquire for discovering area.” True enough, but the secret is that big kids (eighth graders) learn how to use different units for length and for width: the trick is that these units have to be different enough: such as, the width can be an abstraction , such as hours of time, while the length is a measurement of something real, not just of inches, but inches of something real, like falling snow.

In fact, the more math I learn the more common it has become to see width appearing as a measure of time, and length appearing as something that is bound in a relationship with time (such as plant growth, distance, the spread of disease, changes in population and CO2 emissions.) The area, then, comes to represent the accumulation of that which changes in relationship to time.

Now, here’s an example that will, hopefully, make a connection between elementary school math and integral calculus. The two images below measure width with hours of time. The length (think HEIGHT) is the accumulation of falling snow . If snow is falling at a constant rate of 2 inches per hour for 4 hours, a rectangle can be drawn to represent this relationship.

The area, of the rectangle, then, represents how much snow has accumulated in 4 hours. You can multiply length time width to find out that eight inches of snow have fallen in 4 hours, or you can count the unit squares, or you could do a calculus calculation, but the idea is all the same: you are finding the area of the rectangle in unit squares and the area of the rectangle represents an accumulation of snow in relationship to time.

The bottom image shows a different representation of falling snow: the difference is that now the rate is not exactly constant. You can see that the rate of snow increases at about the 2 hour mark, then starts to decrease again, but, other than that, the two images look remarkably alike. The only difference is that the top of the second shape is now curved, so we need the tools of calculus to compute the exact number of unit squares that make up the area of the shape. That said, I can still just count the unit squares and say, confidently, that in four hours a bit more than 8 inches of snow have fallen. I might even look at the extra fractions of squares and decide that they add up to about a half a square, so maybe I’d say that 8.5 inches of snow have fallen. Even though I’m not stating a perfectly precise measurement of the unit squares that make up the area of the rectangle, I can still use my third grade counting or multiplying concepts to get pretty close to finding out the same thing that calculus can precisely tell me.

The thing about real life is that much of what we want to measure doesn’t conform to the regular shape of the rectangle. The tools of calculus are used to discover the area these irregular shapes, which in itself is stunning and magical, but, what I’ve loved learning and want to convey here is more about the realization that, though their methods are different, both integral calculus the third grade geometry equally care about the area as defined by the unit square, and that the unit square can be a generalized representation for real things in our world that accumulate in relationship to something more abstract.

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I took me years to learn enough math to be sure of the veracity of this way of thinking about the area of a circle, but now it seems so obvious. I am just going to say this but I will probably edit it out later: I love the ideas in this video.

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In a recent conversation with Andrew Gael about supporting  people who are interested in how to support dyslexic students, it occured to me to post  the quotes I collected from Dr. Sally Shaywitz's book "Overcoming Dyslexia." I found my way to this book during the long time I spent flailing around, trying to figure out what was going on with my children. It put the pieces together for me in a way that was life-changing for my family. I don't know if it's okay to quote a book so extensively (I've copied about 1400 works) but I guess if it's not okay someone will tell me. For me, the line in the book that was the most powerful is on page 116: "  Dyslexia inflicts pain. " Well said. Thank you Dr. Shaywitz. 


Page 88. It is possible to diagnose dyslexia early and accurately, to provide effective interventions, and for children who are dyslexic to reach adulthood feeling confident and able to achieve their potential. 

Page 93.  Applying our model, the essential ingredients of a successful program for the dyslexic child are identifying
• A weakness in getting to the sounds of words
• accommodations to help access the strengths
and then providing
• early help for the weakness
• accommodations to help access the strengths
I cannot emphasize enough the importance of focusing on the strengths as well as the weaknesses.  The goal is to make sure that the strengths and not the weaknesses define the child’s life.

Page 99.  Not surprisingly, in cases where a child is identified as dyslexic and his parents are then evaluated, in one-third too one-half of the cases a parent turns out to be dyslexic, too.

Page 116. Furthermore, the necessity of devoting all his attention to decoding the words on the page makes a dyslexic reader extremely vulnerable to any noises or movements.  Reading for him is fragile, and the process can be disrupted at any moment. Any little sound that draws his attention away from the page is a threat to his ability to maintain his reading.  He needs all his attention to try to decipher the printed words….The practical consequences of this fragility are that dyslexic readers often require an extremely quiet room in which to do their reading or to take tests. 

Page 116.  There is one final clue to dyslexia in children and adults alike: the fact they say they are in pain.  Dyslexia inflicts pain.  It represents a major assault on self-esteem.

Page 120.  No one wants to be an “alarmist” and put her child through an evaluation for trivial or transient bumps along the road to reading.  Evaluations can take time, and those carried out privately can be expensive.  But I think we have to remind ourselves that our children are precious, one-of-a-kind individuals and have only one life to live.  If we elect not to evaluate a child and that child later proves to have dyslexia, we cannot give those lost years back to him.  The human brain is resilient, but there is no question that early intervention and treatment bring about more positive change at a faster pace than an intervention provided to an older child. And then there is the erosion of self-esteem that accrues over the years as a child struggles to read.  

Page 121.  Most parents and teachers delay evaluating a child with reading difficulties because they believe the problems are just temporary, that they will be outgrown.  This is simply not true.  Reading problems are not out grown, they are persistent.  As the participants in the Connecticut Longitudinal Study have demonstrated, at least three out of four children who read poorly in third grade continue to have reading problems in high school and beyond.  What may seem to be tolerable and overlooked in a third grader certainly won’t be in a high schooler or young adult.  Without identification and proven interventions, virtually all children who have reading difficulties early on will still struggle with reading when they are adults. 

Page 140. In developmental dyslexia the phonologic weakness is primary, other components of the language system are intact, and the reading impairment is at the level of decoding the single work, initially accurately and later fluently.  Intelligence is not affected and may be in the superior or gifted range.  The disorder is present from birth and not acquired.  

Page 149.  Rather than age or maturity, it is reading instruction that leads to better reading.  The evidence seems clear:  Delaying a child’s entry to school does not help him become a better reader.
     On the other hand, early identification, when linked to effective programs of intervention, can make a difference.  Such intervention can ensure that the vast majority of today’s children will never have to experience reading failure.  Good help is available to them now as never before.

Page 165. Last, I often hear that the diagnosis of dyslexia is somehow vague and lacking precision.  As a physician I am always amused by these comments.  The diagnosis of dyslexia is as precise and scientifically informed as also most any diagnosis in medicine. … Now I want to turn to the highly effective steps we can take to teach reading: first to the typical reader, and then to the dyslexic reader, young or old.

Page 172.  Principle Two. Remediate the phonologic weakness and access the higher-level thinking and reasoning strengths (through accommodations).  This is important because it places emphasis not only on the child’ s reading difficulty, but on his strengths.  It reminds everyone that the isolated phonologic weakness on only one small part of a much larger picture.  

Page 173.  But you can brighten her future by maximizing the two factors over which you have some control:  the reading program your child receives, which will be addressed in the following chapters, and how she is treated by the adults around her. 

Page 195.  Above all, do not keep the child back a year in school.  Research indicates that retention is not effective.  These data come from studies that compared two groups of children for whom the only difference was that one was retained and the other went on to the next grade.  Students who were not retained were better off academically and emotionally.  Staying back did not help the children in their learning and seemed to carry an additional negative psychological burden. This should come as no surprise.  Earlier I discussed data which indicated that there is no such thing as a developmental lag.  If a child is experiencing a problem in learning, it is not going to get better by allowing more time to pass before he receives the appropriate help.  

Page. 196.   It is critical to identify a child’s reading problem before he fails.  

Page 233.   From extensive review, the NRP concluded that guided repeated oral reading programs “help improve children’s reading ability, at least through grade 5, and they help improve the reading of students with learning problems much later than this.”  Still, the proven effectiveness of guided repeated oral reading to increase fluency is too often ignored.  That is unacceptable.  In fact, the evidence is so strong that I urge the adoption of these reading programs as an integral part of every school reading curriculum through primary school.

Pages 257-258.  And happily, as said earlier, we have strong brain imaging evidence that scientifically based interventions can rewire a child’s brain so that it is virtually indistinguishable from that of a child who has never had a reading problem.  

Page 258.  Reading instruction for the dyslexic reader must be delivered with great intensity.  This reflects the dyslexic child’s requirement for more instruction, which is more finely calibrated and more explicit……Optimally, a child who is struggling to read should be in a group of three and no larger than four students, and he should receive this specialized reading instruction at least four, and preferably five, days week. A larger group or less time will greatly undermine the possibilities of success.  

Page 309.  In addition to providing the loving and nurturing that comes naturally with parenting, parents (and teachers, too) of children with reading problems should make their number one goal the preservation of their child’s self-esteem.  This is the area of greatest vulnerability for children who are dyslexic.  Teachers and parents often hold high expectations for a child who is harboring a hidden disability and then are surprised, disappointed, or even angry when the child does not perform will in school. If he is accused often of not working hard enough, of not being motivated, or of not really being that smart after all, the child soon begins to doubt himself.  The enormous effort and extraordinary perseverance he must expend just to keep up seems to have no payoff.  That is why it is critical for parents, teachers, and ultimately, the child to understand the nature of his reading problem in order to help him develop a positive sense of himself. 
  An unwavering commitment to the intrinsic value of a child with dyslexia is essential. Every child with a reading difficulty is invariably going to endure ups and downs in his school experiences.  So, early on, each child needs to know that no matter what, he can always count on his parents for unconditional support.  All dyslexics who have become successful by any account share in common the unfailing love and support of their parent’s) or occasionally, a teacher or a spouse.

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I've had a couple of days to think about the post I wrote about some strategies  that worked with my children in relationship to their challenges with dyslexia. I have couple more things I want to add at the moment. 

The first thing I want to add is this image of a cat. This was done by the same hand as the one that filled out that  scary looking math sheet in my last post. My intent in showing this is to make the point that you can't judge a dyslexic child by their handwriting. But people do make these judgments, and they only serve to disregard the talent and the intelligence of the child.

I want to offer a couple more thoughts and resources here. For anyone who is wondering about dyslexia, it would be helpful to become familiar with Dr.  Sally E. Shaywitz's work. An excellent place to start is 

I also recommend The Dyslexic Advantage (Eide). especially chapter 3, especially from the middle of page 28 to the middle of page 29 and especially the sentence, "Individuals with dyslexia and procedural learning challenges also tend to forget skills that they have appeared to have mastered more quickly than others if they don't practice them." When my son heard this sentence while listening to in on audiobook he became furious because this is something he would try to tell teachers and they would never believe him, and would make him feel like he was just bad or lazy or lying or something else negative. He wanted every teacher he ever had to read this.

I spoke to my daughter last evening, by phone from her college dorm. She said very succinctly what her experience was with math: it was an issue of COMMUNICATION (her word). If a concept was explained to her in a way she could understand it, she would grasp it quickly. The trick was in the communication. She never could understand what her teachers were talking about, yet she did well in math...because, as she would readily admit, she would come home and ask me to explain things to her. I knew enough about the way her mind worked that I knew where she would need more explicit explanations. 

Both of my children ended up doing well enough with math to score high enough on the ACTS to get really good scholarships at the colleges that they wanted to go to. For anyone who thinks that this should not be the goal of math education, consider this: rather than EACH accumulating about $40,000 in student loan debt, they have no debt at all. This is a big deal.

So one last thing about test scores. Dyslexic children need more time to do the same amount of work as others. Not only should this accommodation be an official part their school records, but, (planning ahead) the only way the a student can get extra time on the SATs or ACT's is to have a record of their extra time accommodations from their schools AND fresh testing to  prove their need for extra time. What I mean by fresh testing is that the student needs to have testing done within a couple of years of applying for extra time for SAT or ACT, or else it will not be granted. Having this extra time made a life-defining difference to my children (no debt!) so don't be tempted to shrug this off! 

The link to my last post, which mostly talked about the mechanics of math support for the dyslexic child is

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I'm putting down some thoughts here on some ways to offer math support to a child who has dyslexia. Every child who has dyslexia will have their own unique issues, but I think that some of things here might be helpful. Both of my children are dyslexic so I have had a ringside seat to their struggles, failures and successes. Everything I say should be prefaced with "it's been my experience that...."

First, this is written with the dyslexic not dyscalculia, child in mind.

Many of the problems that my dyslexic children had with math were purely mechanical.

Dyslexic children often have difficultly with writing, and this can play havoc with doing calculations. Dyslexic children are challenged with organization, so keeping track of homework papers and assignments are an issue. Dyslexic children have a hard time transcribing, so taking notes from the board, or even transferring their work from one piece of paper to another, can be a real problem. Dyslexic children are notorious for completely, thoroughly and absolutely forgetting math that they understood very well just two weeks ago. And dyslexic children need things to be explained  explicitly.  

Given that students with dyslexia have difficultly with writing, having them squeeze their calculations into the smallish space of worksheets can make things very confusing. One problem runs into another, sometimes problems are skipped over because the page has already become visually cluttered. Writing the numbers down is as stressful as doing the calculations. You tell these students to be neater, and they can be neater, but only to a point. They need more space to work, which means using more paper or giving them less problems. They work slowly, but they are working hard, so I would vote to give them less work. What a typical child can do in 20 minutes will take the dyslexic child three times as long. Giving these students more problems to solve will not result in greater retention. The only thing that helps with greater retention is frequently revisiting concepts. So, do less more often with more writing space.

Single sheets of paper go, well, I don't know where they all go, but they go. Keeping track of single sheets of paper is a huge challenge for the dyslexic child. This item, about twenty bucks, from Staples, helped both my children a great deal  Trying to keep a dyslexic child organized by using any more than one folder is a recipe for disaster. One sturdy clipboard case that holds all papers was our best organizational tool. Keeping track of exactly what to do for homework can be especially challenging because if the assignment is written on the board, well, remember these students are challenged with being able to copy from the board. Put it on a piece of paper, it gets lost. What worked best for my children is to have all assignments written in an assignment book. Dyslexic children do well with lists. Teaching this skill should be a priority. And the assignment book goes into the clipboard case. If the homework can be listed, and even retrieved from on-line, great. 

Do not assume that the dyslexic child remembers anything. They will remember certain things but never assume. They need a clear map of the concepts they need to know, with  concrete examples. They will not remember that x is the same as 1x, they will forget all sorts of learning and they will do their calculations slowly. Make sure they have some sort of quick reference resource which shows concrete examples of concepts they've learned. Maybe this can be laminated on to the back of their clipboard case. Let them ask the same questions over and over again. Don't let them do a whole group of problems before checking their answers. Check them as they work, and once they have the concept let them stop working and let them do the same problems tomorrow. 

Encourage dyslexic students to do math in their head. It so helpful for a dyslexic child to be released from the burden of writing while doing math. This kind of mental math may be unacceptable in the must-show-your-work culture, but it will be helpful in  the actual real life the the child. 

I would love to hear other suggestions from people about supporting a dyslexic child in math. I am happy to continue this dialogue.

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Second Derivative

I've been thinking about the point of diminishing returns aka is-the-pain-worth-the-gain point. This notion of diminishing returns has always been a bit suspect and foggy to me. It's when your'e gaining but not gaining?  I would look at graph and have trouble seeing just where that inflection point happened. What I'm mostly writing about here is being able to see the point of diminishing returns by looking at a curve that has a second derivative, and imagining the circles that the curves could be part of.

Sometimes an inflection point is the maximum or minimum point of a curve, and really easy to see, like when it's the curve ion a parabola.But that's not what I'm writing about.

A second kind of inflection point can be a curve that is going up, and then it is still going up, but there's a change that's tucked in somewhere, and it means something. This second kind of inflection point, the point of diminishing returns, is what I've been struggling to see and make sense of.

A couple of nights ago I was looking at a book, Visual Design by Jim Krause. On page 88 he had created wave forms from circles. It struck me that the circles he used to make the waves was the clearest illustration I have ever seen of the elusive inflection point: it happens exactly at the point where the two circle meet. Look at the SMALL SPACE BETWEEN the circles in my drawings, imagining the curve of the function is the arc of the first circle which continues on to the next circle at the point where the two circles meet. (This would be so much easier to see if I could just point to this for you.)

These circles that form a wave is a great way to think about this kind of inflection point because we know how a wave rises then begins to fall, falling even as it is rising higher. Sorting out where the curve leaves the first circle and goes to the second circle is, to me, a great way to see this point of diminishing returns. I want some of my math minded friends to weigh in on this. First, do you agree? Second, is this way of using circles an already typical way of looking at this concept?

(The 2 columns of drawings are the same, except that the second column highlights the line of the function by filling the area under it with color)
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