Fraction a Day 17 - Sample Fraction Images

Kids would use cuisenaire rods to answer these questions:

Should add "if your unit is" after "1/2"

Fraction A Day - Day 16 - Rational Number Project

Started reading things from the Rational Number Project.  Two things I like from my initial readings:

"Children learn by having opportunities to explore ideas in these different ways and by making connections or translations between the different representations.

Image from Page 4 of their report

Thinking about how we can encourage and foster those connections seems very worthwhile.

I also liked the style of their assessment questions, where they ask the students to show how they would do a problem on the number line or using decimal boards or grids.


I was surprised when they said:

"that of all the manipulatives available for teaching about fractions, fraction circles are the most effective for building mental images for fractions."

Because by William Clevland (The Elements of Graphing Data) should that Pie Charts were the worst form of information graphic, because people had a hard time comparing area.  Cleveland's research had showed that horizontal lines were best for facilitating comparisons.  Of course it may be that children have more experience with Fractions in their Pizza and Cake form, so for them fraction circles make sense. Me I personally prefer the chocolate bar form ;)

Fraction A Day - Day 15 - Multiplying Fractions

Today I worked on an are model for multiplying Fractions in Etoys.

I created two transparent playfields one with gridding set to 30@300 the other set to 300@30. Then placed and resized rectangles within the playfield.  The created a simple "Multiply" script to move the one playfield to the left (I changed the forward direction to move left, try it, it won't rotate, there is another method, like most things ;) until it overlapped the first playfield.

I have been spending too much time in Etoys trying to do things, rather than simply think about what I want to do, then figure out how to realize it in Etoys.  I hope to spend the next few days looking at other manipulatives and thinking about how to design my own, so that they are simple and fun for teachers and kids.

I also need to think more about fractions and the operations using fractions.  If I find time I will head to Rutgers to check out the Robert B. Davis archives.

Fraction A Day - Day 14 - Numbaland

Check out Numbaland its Battleship Numbaline is a great program that reminds me a lot of the early Plato program Dart (Designed by Sharon Dugdale and David Kibbey 1975) as part of a program at the University of Illinois.  Plato was also used as part of the Madison Project.  It is discussed in chapter 3 of Cynthia Solomon's excellent book "Computer Environments for Children"

Fraction A Day - Day 13 - Randy's Polygon Challenge

Today we are lucky to have a guest Fraction from Randy Caton.  Randy has taught introductory and advanced physics courses and laboratories to classes ranging from 5 to 700 students and has used Peer Instruction and Just-In-Time-Teaching for 30 years. He has directed several science education programs for teachers and students from 1986 to 2008. He is currently retired and lives in northern Minnesota. Randy has used Etoys to develop web-based activities for NASA programs and physics courses.

Randy created a wonderful polygon challenge where kids modify a script and the number of vertices in a polygon to create and discover beautiful things.

Part of what I like about Randy's approach when designing lessons (at least what I can tell from seeing some of them), is that he has the kids discuss the ideas, predict what is going to happen, try it, then record what they observe.



Maria Droujkova of Natural Math joined us for our meeting today and had a wonderful idea to go along with this where the kids are the polygons.  She uses kids as vertices and yarn or rope as the sides.  See images of Maria in action with some parents and kids below:



Our goal is to come up with a set of materials so that teachers, parents and kids can use these activities to explore and learn.  One of the goals for the materials will be that "anyone" can use them, to run these activities and the fact that you are not a mathematician (Maria) or a rocket scientist (Randy worked with them) will not hinder you from using and getting a lot out of the materials and lessons.

Randy's project can be found here Polygon Challenge 1

Fraction A Day - Day 12 - Esteban's Fraction Cutter Design

So I asked the question, how would you create a program to "cut" the polygon in half so you have two polygons and can separate them (similar to Slice It!).

One person asked can we use the slope (she had been studying slope).  I responded its natural to use what you just learned and what you know and its wonderful when it works.  Can anyone think of another way?

Then Esteban suggested:

"You fill the square with little turtles until it all fills up. Then you count how many are on each side to get the percent."   

(this was for the part of the game where it reports the fractional parts as percentages.)

I asked then said that's an interesting idea, how could you tell which side of the line a turtle is on? To which he responded:

"So lets say ou have a tool that measures color and a tool that measures distance. then you merge those tools and make a script that sees two colos and measures the distance between them and from that you can say, what if I do it to measure the whol box and have that tool tell you the area of something.

I then asked: How would you program that?

"Take two basic tools and build a script that uses both of them then build a script that uses two other basic tools, then combine them and keep going so you have like a ladder to the ultimate goal."

 Now, I ask myself, what basic tools could I provide that kids could use to build a "slicer" script.  Dr. Geo and Geogebra come to mind (thanks Karl for the idea).  I like the idea of Dr. Geo (not only because a version is embedded in Etoys, but it allows you to script).

FYI, I already have figured out a way to do this (I think) just have not had the time to program it.

Fraction A Day - Day 11 - Classifying Fractions

This post was inspired by a paper simply titled Fractions  on the  site.

In the paper they had a section titled "Classifying Fractions" and which had an image similar to these I created from my Etoys project:


Changes that I plan to make to this (or volunteers who want to join the team can make, hint, hint, nudge, nudge, wink, wink, you'll get free Etoys training!!!) include:

• embedding visual fractions (which could be easily made with other tools from this project)
• embedding pictures of fractions kids find in the "wild"
• animated feedback options when they properly and improperly classify

One goal of this project is to make it easy, for teachers and kids, to create content.  So if you have a lesson idea you would like to create let us know we'd love to have you (plus your lessons might be used by kids around the world ;)

Fraction A Day - Day 10 - Fraction Cutter

This is a first draft of a tool to let teachers and students create fractions and fraction problems by "cutting" an object.  The amazing thing is it only took 11 scripting tiles in Etoys (could have got it down to 7, but went for readability).  The first page lets you make only 1 cut, the second lets you make multiple cuts. I had to put a playfield on top of the "unit" (aka rectangle, which is in a gridded playfield) so that I could allow you to "clear the cuts" by simply using the Playfields "remove all" tile from the collections category. With only 1 playfield the remove all would have removed my "unit" as well.

Next I am thinking about using polygons as my "units" so I can visually show the cut by duplicating the polygon and modifying the vertices to points where they intersect the cut. Then I will "animate the cut" by moving the polygon's apart. This should scale to multiple cuts without code changes (using the Playfield's "tell all contents" tile from the scripting category). Well at least I think it should testing and the mandatory debugging that occurs anytime I write code will tell ;)

Here is the project:

Cut it up

Fraction A Day - Day 9 - An Extremely Hard Problem

As mentioned yesterday I said I would work on the problem using Etoys (the above images are from the that project).  The image show notes (using "text box" from the connectors category of the supply bin) about questions I have and some design issues.

In doing this I am thinking I may want to switch things around.  The study interview started with what is 1/2 + 1/3, but that was to assess understanding, I am thinking I should create a version where the experiences lead to this.

This problem was based upon the problem on page 40 of: The Development of the Concept of "Fraction" from Grade Two through Grade Twelve. Final Report. Part One, Part Two and Appendix. - Robert B. Davis 


Also saw an article in the NY Times  Brain Calisthenics Help Break Down Abstract Ideas which mentioned  a 2010 study, researchers at UCLA and the University of Pennsylvania had sixth graders in a Philadelphia public school use a perception-training program to practice just this.  (more detailed version here)

Fraction A Day - Day 8 - Mapping the Developmental Evolution of a Child's concept of Fraction

On pg. 40 of The Development of the Concept of "Fraction" from Grade Two through Grade Twelve. Final Report. Part One, Part Two and Appendix. - Robert B. Davis it states:

"In the case of fractions, the developmental evolution of each concept is important. At first, one probably defines a/b by taking a candy bar, or a pizza, or something else, dividing it into b pieces, and taking a of them. ..."
"When one encounters 'improper fractions,' this meaning fails. One cannot use this meaning to speak of (say) 5/4" We must introduce the concept of unit. We divide each unit into b equal pieces, and take a of them. Now, with these new definitions, we can easily deal with 5/4, although we shall need two units in order to do it."

So one step in our process to "map" is the developmental evolution of each concept (okay its a BIG step, but someone must have done this before). The challenge is to understand where the child is and what questions and experiences you can provide to help them progress. It is unrealistic to expect elementary school teachers, who must teach all subjects, to know let alone be masters of the concepts and maps in all subjects. So, how can you do this if the teacher may not understand certain concepts or the map?

How can we help students move from their existing mental models to more powerful and more sophisticated models? (please post answers as comments ;)


Obviously it depends on their models, any mis-conceptions they may have, and other factors.


The following problem is from the interview excerpt starting on pg 45.  The The student in the interview was described as "a generally bright resourceful 5th grade girl" who reported that "in her previous school there was very little instruction in arithmetic". : 


"I want to show you what most people think is a really hard problem: 1/2 + 1/3"


She was stumped and did not know how to solve the problem. Later the interviewer then brought out cuisenaire rods and asked: "If we want to talk about "one half"  and "one fourth" which rods do you supposed we want to call "one"?


Once she had the physical model she was able to answer "three fourths."  She did this rather quickly it seems and without using the rods completely. She had simply identified which rod was "one half" and which was "one fourth" and then "solved the problem just by thinking about it.  The concrete model and "enabled her to build up an appropriate representation in her mind".


Today I will start working on an Etoys version of the problem, perhaps with a pre and post assessment.

Fraction A Day - Day 7 - Development of the Concept of Fraction

So today I decided to do look for and review some research.

The Development of the Concept of "Fraction" from Grade Two through Grade Twelve. Final Report. Part One, Part Two and Appendix. - Robert B. Davis


What I like about this (besides the author) is that it includes a lot of excerpts from interviews with the kids.  


I also am reading Developing Effective Fractions Instructions for Kindergarten Through 8th Grade completely different, but good to know what the Department  of Education thinks :)


One thing I like in this report is that it lists "Roadblocks" which, in the kids cases, are what I have seen others call mis-conceptions, but to me seem like "I am applying an existing mental model that has worked fine up and has worked fine up until now and makes sense to me."


In the adults cases, they seem to me more political problems. That happens when you have a large group of humans.  These also need to be understood and dealt with if you want to engender change.


Finally an interesting way for kids to discuss and communicate their ideas about fractions: (thanks to Maria from Natural Math for pointing this one out).

Fraction A Day - Day 6 - Area Geo boards

So this Etoy uses a quiz format and shows the relationship between area and multiplication (hopefully in an easy way the kids can discover without being told.  We get to fractions when we change units on pages 3 and 4.

Also it has free play areas which are virtual Geoboards.  One of my goals for this project is to develop a set of playthinks and a set of tools that teachers and kids could use without much special knowledge or programming skill (yet thanks to the beauty of Etoys, they can still program them and "look inside" to figure out for themselves how they are programmed and if they want modify them).

Here is the Etoy Area Geo Boards.  The original version did not when entering fractions, this has been fixed.  Hopefully it will be updated soon on the squeakland site.

The Sieve of Brandon

We are taking a break from the Fraction a Day posts so we can bring you "The Brandon Method"
My nephew needed help to prepare for his Algebra test (which was only two days away) so I was fighting the competing goods of helping him succeed on the test and actually having him learn math.  Along the way the need to teach math won out for a while (well most of the time). We were doing prime factorization and greatest common divisors and I asked him what the primes were, then I brought out my handy dandy number grid and here is what he created:

I "colored in" 0 and 1, then asked him what was the first prime #, he said two, I then asked him to color every multiple of 2 (except 2 of course), then asked him what was the next prime #, (he said three) then asked him to color in every multiple of three and repeat.  The "Brandon technique" was to change colors each time, so he started with green for multiples of 2, then went to red for multiples of 3, etc.  After he got up to 8 I asked him to stop because we were running short on time, and we had a lot of material to cover, but his response was "but I'm having fun, I want to finish".  He persevered and stuck with it (a great habit, so why kill it just to pass a test).    I told him that was the first time I had seen anyone use that technique and that there may be one right answer, but there are multiple ways to get there and the important thing was to find the ways that worked for him (and understand different routes as well).  He was actually having fun doing math (we spent four hours doing nothing but Algebra and I finally had to tell him I was not going to teach any more and he had to go to bed.  This from a kid who usually does not like to do a lot of homework and tries to finish as quickly as possible.

Another teaching idea that came up was a better way for them to discover interesting things while doing factor trees, but we will save that for another post. Hint: think visually and you can move around parts of trees you already created, which leads from this will take forever (when he was asked to list all the prime factors for numbers up to 75) to an aha moment and a big smile.

Fraction A Day - Day 5 - Frustration and not much progress

Trying to make a "Teacher/kid" friendly version of the Fraction maker where can simply "choose" two graphics to put in the fraction box (for example an image of a Boy and an image of a Girl).  See examples below of how the fraction boxes could be used:

Kids would then create their own 1/2's by clicking on the Boy/Girl to get them to change and if you gave them the fraction tool they could create many more.

Looked at using a Player in Etoys,  as it is easier to create an interface to select those using a Player variable by simply clicking on them (actually two clicks, one on the player variable and the second click on the object).

Okay as I write this another method comes to mind, I could use two holders (or one) and place the objects to switch to (on MouseDown or MouseEnter)  use those two objects (aka Players) for the fraction box.  Ran into some problems and created some issues in tracker (fortunately we have Bert and Karl working hard to fix issues).   Another thing I tried (my original idea) was to have one object that was programmed to  switch costumes (the graphic that is displayed on the screen).  I think the an interesting set of lessons to develop would be one around the programming/development process but that's much harder and besides folks much smarter than myself are working on it (ex: Mark Guzdial and his excellent site for teachers, one of my way too many side projects is to create an Etoys version of his course, have one or two projects but a lot of work still to do).  One of the key lessons if I ever finish the Etoys course should be that frustration is normal and to be expected, if your not getting frustrated and failing your not trying hard enough (well it's late and I am frustrated so I thing I have tried hard enough for tonight ;)

Comments and suggestions always welcome.

Fraction A Day - Day 4 - Tool Building

So today (and actually for a number of days) I have been working on a "Fraction Maker" tool, to help create curriculum and playthinks for kids.  I have been going through multiple iterations (a good habit to teach the kids).   

Fraction Box w/Controls

The user can specify the object to use (in the above case I used a rectangle, but you could use any shape or graphic), the user can specify the # of rows and columns.  The object used has information about the color changes (or other changes such as changing graphic) and whether the event occurs on MouseDown or MouseEnter.  Hmmm, I think I should move that MouseDown or MouseEnter to the Fraction Object and just ensure each object used to "fill" the fraction has a "toggle" script.

Here are a couple of things I want the Fraction Box to be able to do:

1. When a user clicks on (or mouse's over) a box it will change color
2. Let user (teacher or kids) specify the colors 
3. Button to hand the user a copy of the Fraction Box
4. Button to "lock" the Fraction Box (so the mouseEnter or mouseDown events stop changing colors)
5. Button to hand the user a copy of the Fraction Maker (Fraction Box + buttons)

Some challenges/design issues:

1. When "handing the user a copy of the box, unless I put a "border" (not an etoys border) around the "Fraction Box" it is not obvious how to "pick up" and move it. Ideally you could easily get the halo, but that doesn't work too well as I currently have the size of the playfield set to the size of the fraction box, so the only way to "select it" (get its Halo) is to "shift drag" select, which is easy to teach, but not obvious until you are shown how to do it (hmmm, sounds like I need another Etoys minute).
2. The scripting to do all I want to do (lock/unlock Fraction Box), etc. not hard it just takes time and some figuring out

A Fraction A Day - Day 3 - Color Blind/Dyslexic Fractions

During dinner at a friends birthday party, one of the folks there mentioned that she had dyslexia and had problems with the greater than and less than signs. She also talked about some of the tricks her father taught her to help deal with her dyslexia. Another friend talked about how his son had eye tracking problems and how hard it was to detect. So I started to think about how we could embed various "tests" into lesson plans. For example perhaps you could distinguish if a child understood which fractions are larger and smaller but had problems with the > and < symbols because of dyslexia. One possible way to do this would be to ask them to place the the larger fractions in the bigger box and then run a similar test asking them to choose the correct symbol (> or <).

You could also use "place them in a bigger box" as a starting task, then introduce the symbolism after, by asking them to choose the correct symbol, while keeping the larger and smaller boxes on the screen to give an additional visual clue as to the meanings of the symbols.

Here are some sample pictures to give you an idea of what I mean:

FYI, I am not a doctor nor have a I ever played one on TV, but I think this may have some merit and seems easily testable.  We would need to connect to experts in the fields to come up with good tests.  In any case while far from perfect these could act as early indicators of possible issues.

A Fraction A Day - Day 2 - Cut the Cake Riddle

Day 2, here is a fraction embedded in a riddle.

Below is a link to an Etoys project which has hints as to how to solve it and lets you "play with" the fraction to try and discover the answer yourself.  Try it, you never know what you might learn.

Cut the Cake Riddle

A Fraction A Day - Day 1

I have heard some writers talk about writing every day, I have decided to take this approach with curriculum development. And will be creating a Fraction a day, to support an OER (Open Educational Resource) for mathematics education I am working on with some folks from OLPC (Randall Caton and
Edward Cherlin).

Rather than a complete curriculum/textbook/OER for each grade (which should be an end goal) I would like to tackle things one bite at a time, with the first bite being Fractions. Why? Its a subject kids struggle with and therefore teachers and parents worry/care about.  I do not want to re-invent the wheel here.  Lets use good existing ideas and it is possible to teach first graders to solve problems like: (1/2 x () + (2/5 X 10) -(1/2 x 8) = ? (see Lore Rasmussen et al - Lab Sheet Annotations and Mathematics for the Primary Teacher, pg 140).  If we can provide good materials to help kids understand fractions and get evidence that they work we will create a good entrance path for the rest of our materials.

So a couple of tasks:

1. Identify XO Activities we can use to teach fractions (I spend most of my time in Etoys, but we should use Turtle Art and others)
2. Identify existing lessons we can refer to or re-render (ex: http://illuminations.nctm.org/LessonDetail.aspx?ID=U152)
3. Identify virtual manipulatives needed (I have cuisenaire rods and pattern blocks in Etoys, here is a nice one at Illuminations)
4. Identify physical manipulatives that can be used (and id how can be easily and cheaply created by kids/teachers)
5. Identify no computer, concrete activities kids can do (ex: 1) Fold a paper in half and cut, 2) Label as 1 piece of X needed to make a whole sheet; repeat)
6. Create/Identify a lesson, manipulative, blog pos or video each day. Okay ambitious, but do-able (for short sprints). It helps me keep the perfect from being the enemy of the good and gets me focused and moving.  I look at it as the equivalent of some writers practice of writing something every day.
7. Identify Models of Fractions and how we can present these different models in concrete forms as a kind of Fraction Exploratorium
8. Identify common mis-conceptions learners have about fractions and using them and identify ways of addressing these mis-conceptions.