Fractions and My Brother

Man changing stock quotes on a chalk board. NYPL Digital Gallery

My brother, Zack, was here visiting the other night. My 12 year old son had a question about his fraction homework, and this sparked a conversation between Zack and me about how fractions are taught in school. 

Zack said, “Fractions are a useless implementation of decimals.” 

Hmm. Zack worked in finance for many years as a derivatives trader. I never really understood what he did besides the fact that it entailed making split second decisions involving lots of money and using insane spreadsheets. Zack is still my go-to person when I need help writing a really complicated spreadsheet formula or macro. Not only does he understand what I’m saying, but he has the answer in .9 of a second or less.

Here is a video of Zack sharing his thoughts on fractions. I hope it sparks some interesting conversation for readers of this blog. I brought this up with my colleague, Penny Stearns, who expressed a completely different perspective from Zack. Perhaps she will star in a video rebuttal at some point. Professor Tim Whiteford weighed in with still more to think about. What an excellent topic for students to question and debate in order to deepen their understanding of the application of fractions and decimals in different contexts.

Hyperbolic Space

Poincaré disc model of hyperbolic space from fractalsciencekit.com

Margaret Wertheim gave this interesting lecture called "The Beautiful Math of Coral".

  

She does a thorough job of explaining the math of hyperbolic space here on her website, the Institute for Figuring. There are photos of the gorgeous crocheted coral reefs, too.

In her words...

The Crochet Reef Project was inspired by the technique of hyperbolic crochet originally developed by Dr Daina Taimina, a mathematician at Cornell. In 1997 Dr Taimina discovered how to make models of the geometry known as "hyperbolic space" using the art of crochet. Until that time many mathematicians believed it was impossible to construct physical models of hyperbolic forms; yet nature had been doing just that for hundreds of millions of years. It turns out that many marine organisms embody hyperbolic geometry in their anatomies - among them kelps, corals, sponges, sea slugs and nudibranchs. Thus the Crochet Reef not only looks like a coral reef, it draws on the same underlying geometry endemic in the oceanic realm.

There are very good reasons why marine organisms take on hyperbolic forms: this geometry is a marvelous way to maximize surface area in a limited volume, thereby providing greater opportunity for filter feeding by stationary organisms.

An unidentified folded coral in Flynn Reef, part of the Great Barrier Reef, near Cairns, Queensland, Australia. By Toby Hudson.

Crochet works because it is an easy way to increase stitches in each row to produce ruffling. I haven’t crocheted since childhood, but I do remember increasing stitches when I wasn’t supposed to. Maybe I should try a coral reef. 

Brahmagupta and Fibonacci

I am reading a book called The Man of Numbers: Fibonacci’s Arithmetic Revolution, by Keith Devlin.

So far, it is fascinating. I didn’t know anything about the history of mathematics, so it is all new and exciting to me. I liked Chapter 1 so much, I cajoled my kids into letting me read it aloud to them. There was some concealed eye-rolling, but they were interested once I began reading.

First, we learned about how the earliest known tally marks date back to Swaziland, circa 35,000 BC. Twenty-nine notches were carved into a baboon’s fibula. It hadn’t occurred to me that Roman numerals are really a more sophisticated version of tally marks. Devlin points out that, while it is easy to add Roman numerals, multiplication becomes problematic. Multiplying is best done with repeated addition when using Roman numerals, so anything involving two large numbers is impractical. Calculations were done using complicated finger arithmetic systems and abacus. It sounds like only a few people could do that sort of thing, and you just had to trust that the answer was correct (ahem).

The Hindu-Arabic system we use today dates back to 700 A.D. in India. My kids were surprised to hear that the symbols chosen to represent numerals may have been designed so that the number of angles match the quantity represented. You have to write the numerals in a certain, Flintstonian way for this to work.

The best part of our reading was learning about the invention of zero. The other nine numerals were well-established when an Indian named Brahmagupta entered the scene. Brahmagupta (fun to pronounce) created the number zero in the year 628. Before Brahmagupta and his gigantic book entitled Brahmasphutasiddhanta (which means “the opening of the universe”), folks were simply circling the blank spots in numbers. Brahmagupta wrote about zeroes in elliptic verse, which appeals to my sense of rhythm and brevity:

A debt minus zero is a debt.

A fortune minus zero is a fortune.

Zero minus zero is a zero.

A debt subtracted from zero is a fortune.

A fortune subtracted from zero is a debt.

The zero revolutionized the Hindu-Arabic system, making it possible to perform calculations much easier than before. I was definitely taking zero for granted before reading this book.

Now, on to Fibonacci. There was no one really named Fibonacci, but there was a guy named Leonardo de Pisa who lived in Italy in the 12th century. He brought math to the masses. I haven’t gotten to the part of the book yet that tells the history of the famed Fibonacci sequence, so I’ll have to keep reading. In the meantime, Vi Hart has released a new video entitled Doodling in Math: Spirals, Fibonacci, and Being a Plant which does an excellent job of explaining Fibonacci numbers and simultaneously making math seem young, cool, and beautiful even for those who didn't previously think so. It is worth finding 6 minutes to watch it with your students!

Math and the iPad

 

I have a new iPad and I’m looking for good math apps.

Lee Orlando has a new iPad, too. She loves trying new things and is already way ahead of me on using the iPad in math class. This is from a recent email:

This weekend I bought an adapter for my iPad so I can hook it up to the LCD projector, and I also got a wireless keyboard.  Today, I went into school to try it out.  It was so cool to see the iPad screen projected and to sit at the back of the room (or any place in the room for that matter) and see the text appear.

This was soon followed by another email:

I just found some awesome free apps for the iPad.  All are from "Mathtappers" and the three that I downloaded involve placing numbers (including rational numbers) on a number line and finding equivalent fractions.  All games are designed for three levels of play.  I've been trying them out at each of the levels, and they get pretty challenging at the highest level.  However, the easiest level is well within the ability of fifth graders.

...these apps look like a great way to engage students in a whole-class warm-up activity/discussion.   I had made up my own number line activity using a sketching app that I have... the kids' attention level goes WAY high when they get to come up and draw on the iPad!  Getting that adapter for the LCD projector may have been the best investment I have made in a long time.

Later, I ran into Lee at school and she showed me how she’d photographed a piece of graph paper to use as the background of her sketching app, so that students could draw arrays and fractions with the aid of the grid. What a great idea!

I hunted around a bit and found some other useful apps. My favorite so far is Sketchpad Explorer. If it’s been awhile since you contemplated the Pythagorean theorem, you’ll enjoy the Getting Started screen, which allows you to drag right and non-right triangles around to see the theorem in action.

The real fun, though, comes when you touch the little book icon in the lower left corner of the screen. Choose “Elementary Mathematics” and Sketchpad Explorer presents you with a suite of eight activities involving symmetry, triangles, fractions, decimals, multiples, and volume. There is even a logic game which gives less than and greater than clues to find an unknown number. Sketchpad Explorer’s creativity and nice, clean graphics are appealing. At first glance, there seems to be a wealth of resources and lesson ideas for teachers on the website. I can’t wait to try these with students.

Student Conferences

From The city and country builder's and workman's treasury of design, Langley, Batty (1696-1751). NYPL

Many teachers are in the practice of holding individual conferences with students during literacy blocks. Teachers sit with students to converse about the book they are reading, asking questions like “Is this a just-right book for you?”, “Have you made any connections to the book?”, “What questions came up while you were reading?”, etc. Leah Mermelstein, a literacy consultant, suggests regular, one-on-one writing conferences with students.

 
What about math conferences? Over the last year, I have used an assessment tool called the Primary Number and Operations Assessment to help me interview students about mathematics concepts. I enjoy the opportunity to sit with a single student and spend time listening as they perform various math tasks and explain their thinking. I always learn important things about the student’s strengths and I gain insight about appropriate next steps for instruction.

Here is an email from my colleague, Sally Hayes, fourth grade teacher extraordinaire, in which she shares her experience conferring with students during math class:

Karyn,

I have to tell you about something I tried recently. It came to me several years ago when I was correcting one of our Bridges unit math assessments. I asked myself, "What is the most important outcome of this assessment?" The answer, of course, is student learning.  As a teacher, I must figure out where each child is on the continuum of learning for a domain so I can help them move forward. Grading the assessments and entering scores into the database is only part of the work. For years I had gone over assessments with the entire class, but it never seemed very efficient or terribly useful for individuals. I realized I should try going over the assessment with each student individually. This would allow both me and the student to gain a clearer understanding of their strengths and areas for improvement as a mathematician.

So, last Friday, the day after correcting our Unit 2 math assessments, I decided to try conferring with each student. After lunch, I gave the class some options for independent work during the next hour, which included practicing math skills, writing in writer's notebooks, and reading. While the class was quietly working, I met with each student to discuss their math assessment. First we looked at strengths and then areas that needed work. Amazing! The conferences were so much better for the student and incredible for me; I finally felt like I was getting something really useful out of a unit assessment. I only wish I had allotted an hour and a half, which is a lot of time, but would be well worth it. I need to refine the way I spend time with the students, but I think it will get better and easier each time that I do it. I was thinking that it would be nice to create small groups of 2 or 3 students that had similar challenges with certain problems. I am glad I finally tried math assessment conferences!

Sally

Who Has?

Sidewalk card game by Lewis Hine

Who has played the Who Has game? I have. It’s a really good one, because it keeps all students on their toes.

Basically, you print up a bunch of index cards and hand them out to the class. If you’ve got 30 cards and 22 students, give extras to some of your stronger students. Choose a student to read their card. The game will go around the room until you are back to the first card.

Who Has? can be played with cards for any topic or subject area. There is a nice collection of Who Has? cards for different math activities on the excellent Mathwire website.

I’m going to specifically recommend two of them: More or Less and Place Value. I’ve encountered many students, even in the upper elementary grades, who do not fully understand place value. I think these Who Has? decks might help. Let me know if you try them.

Finding Balance

How do we teach students about the equal sign in math?

Professor Tim Whiteford brought this up in a meeting recently. Says Tim: “Traditionally we have used language like ‘three plus four makes seven’ or ‘three and four are seven’. We now know that both these forms of language actually develop in children a misconception about what is happening in this piece of procedural knowledge. Children tend to think that the equals sign makes things happen.” (see Tim’s full blog post on the equal sign)

I remember having this misconception as a child, and children in the U.S. continue to struggle with it today. I was looking at the 3rd grade NECAP released items last year and noticed lots of students got this question wrong: 1+4+?=6+14. (Many students incorrectly chose 1, which makes sense because 1+4+1=6.)

Researchers at Texas A&M University found that 70% of U.S. middle school students lack understanding about the equal sign. Students in other countries like Korea and China do not have the same misconceptions. When students begin algebra in middle school, understanding the equal sign is critical for their success. (full article here)

On the bright side, this seems like a relatively easy thing to fix. I visited a second grade class the other day and watched the students excitedly working with a number balance scale. Their teacher used this tool to help them develop their concept of equality as a relationship, as opposed to an operation. If you don’t have a number balance scale, here is a very nice virtual pan balance scale from NCTM Illuminations, and a virtual number balance scale.

We can also mix up the way we write equations. I could decide to write 7=9-2 instead of 9-2=7.

At what age do students need to learn the correct meaning of the equal sign? Why wait? This is a Mathematics Common Core State Standard for first grade: 1.OA.7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

The Powers of Ten

Lee Orlando, fifth grade teacher, kindly contributed this article.

Our Bridges November calendar focuses on decimals and "base-ten fractions," and we've been having some good discussions around the nature of decimal numbers.

Yesterday, we were ordering some numbers that included whole numbers and mixed numbers (expressed as decimals)... one student confidently announced:  "3 is greater than 33.45 because any decimal is smaller than a whole number."   The salient feature for this student was the decimal point, and she was working under the misconception that whenever you see a decimal point, the number is automatically smaller than any whole number.  Just that comment alone kept the students talking and debating for quite a while!

I also did a quick formative assessment: Instead of giving students decimal numbers already arranged appropriately (in a column) in order to add/subtract, I simply gave them the numbers and had them arrange them in order to add (there were four numbers). Perhaps you can guess what many students did. They applied their understanding of place value of whole numbers (ones, tens, hundreds, etc, lining up the numbers from right to left) and totally disregarded the decimal point.  Here were all these decimal numbers, neatly lined up as if they were whole numbers, with the decimal points totally misaligned. These were students who have been adding and subtracting decimal numbers in our weekly math computation practice, but when given the numbers separately - not pre-arranged in a column - their lack of conceptual understanding about decimals and their values was completely transparent.

All this has led to lots of discussions about the power of zero which we had already explored in our Great Wall of Base Ten, but which was now coming back in light of decimal numbers.  I have been digging up some cool resources around this, including this awesome video which you may already know: "The Powers of Ten."

http://www.powersof10.com/film