This course will become read-only in the near future. Tell us at community.p2pu.org if that is a problem.

# Getting started

The initial premise of this course was to remix some existing materials from an open math textbook (see especially chapters 5-7) into a Moodle course, adding more multimedia and interactive materials along the way.

Also, once you get signed in, you might check out the "builder's forum" discussion group there to see what's been done and some ideas of what needs to be done.

Then perhaps we all could post some initial thoughts here about:

• How the premise of this sound to you?
• What parts might you like to focus on?
• What else should we be thinking about for this project?

•

As a fifth grade teacher in New York State, I am interested in aligning my fraction instruction with the Common Core State Standards. With 45 states having adopted the standards, I feel that it is important to keep the new standards in mind. I have been working my way through several relevant readings on the topic.

I look forward to working on this project in general.

Steve

• I agree that we must keep in mind the common core standards.  And in fact the standards themselves about fractions are reasonable. But I notice that the Draft Progressions is brand new (August 2011). When the first drafts of the Common Core came out 4-5 years ago, there were pages with examples and then these pages were abruptly removed from the internet. (I had written several responses with details about some of the inadequacies of the examples for the algebra: expressions and equations so I still have that set.)

Now I am looking at this Draft Progressions and I see several similar problems. It seems to me that the people who write these things have never taught kiddies in the classroom (I myself have never taught young ones - I get the older kids who are the result of this teaching and I simply start over.)

My complaints about the Draft Progressions after a brief look at it.

(a) There is no verbalization. Teaching is done verbally - particularly fractions. Any explanation should include how to actually talk about fractions.  Two years ago, I made a video about my pet peeve about people talking fractions: http://youtu.be/xOQWmLGPmWQ , i.e. how do you say "1/4". Many people say "one over four". What does that mean? "One-fourth" is a picture in my head; "one over four" is 3 words.

(b) I also looked at the pictures in this draft progressions. They take fractions out of context. We teach kids at a very early age that we cannot compare an apple to an orange. They understand this and they understand that if we have 2 apples and 2 oranges that we DO have the same number of each type of fruit but we DON'T have the same fruit.  But apparently a child will compare 1/4 of a giant pizza to 1/2 of a small pizza and conclude that 1/4>1/2 (see page 4). I simply cannot believe this. (I might believe they think 1/4>1/2 since 4>2, but not from a silly pizza example - please correct me if I am wrong.)

From this - in the very next paragraph - it goes on to explain how important it is to compare 2/5 > 2/7 (which indeed is a much, much harder concept) and use this example in SPITE of the fact that the standards specifically state: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.

Then this draft continues onto 4th grade and immediately they go on to the stuff that (I think) Sue was talking about: complicated expressions "over" complicated expressions.

There is no way to understand this reasoning in terms of defining fractions as parts of a whole and it is easy to see exactly where we lose them. First we lose the teachers, who then lose the kiddies.

(The draft progression actually mentions presenting to 4th graders

$\frac{28}{36}=\frac{28\div 4}{36\div 4}=\frac{7}{9}$

which (I think) is ridiculous at that age and again in SPITE of the fact that the standards specifically state: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.)

So while keeping the standards in mind, we need to focus on teaching methods that really communicate what fractions are and how they help us function; teaching methods that work in the classroom and have been used by real teachers teaching grades 3 and 4.

• I understand your concerns with the draft progressions. That being said, they are drafts (In fact the blog in which it was posted referred to the fractions progression as being a drafty draft) . Let me address some of your points.

(a) There is no verbalization. Teaching is done verbally - particularly fractions. Any explanation should include how to actually talk about fractions.  Two years ago, I made a video about my pet peeve about people talking fractions: http://youtu.be/xOQWmLGPmWQ , i.e. how do you say "1/4". Many people say "one over four". What does that mean? "One-fourth" is a picture in my head; "one over four" is 3 words.

I agree correct verbalization is critical in math. It may be that the writers of these progressions make the mistaken assumption that practitioners are disciplined in their use of proper mathematical language. This is a hazard of the standards having been largely written by those removed from the classroom. Imprecise and inaccurate use of terminology is rampant in elementary math. Teachers must be conscious of the dangers of slipping into the vernacular.

Hung-hsi Wu's (One of the contributors to the progressions) Understanding Numbers in Elementary School Mathematics gives a great deal of attention to precision in speaking in mathematics instruction.

(b) I also looked at the pictures in this draft progressions. They take fractions out of context. We teach kids at a very early age that we cannot compare an apple to an orange. They understand this and they understand that if we have 2 apples and 2 oranges that we DO have the same number of each type of fruit but we DON'T have the same fruit.  But apparently a child will compare 1/4 of a giant pizza to 1/2 of a small pizza and conclude that 1/4>1/2 (see page 4). I simply cannot believe this. (I might believe they think 1/4>1/2 since 4>2, but not from a silly pizza example - please correct me if I am wrong.)

I agree that the pizza illustration may not have been the best illustration of the importance comparing the same whole. I think most students in grade 3 could tell me why this illustration is a fallacy. (I can test this by asking some third graders).  This illustration appears to be an over simplification of the pitfall discussed by Wu in pp. 193-195 of Understanding Numbers

The use of examples that are outside of the domain of denominators reflects a certain inattention to the standards (this is a draft); nonetheless, I still believe there is value to what is being presented. I think that examples may have been pulled from other works without revising them to comply with the common core.

I cannot speak to fourth grade, but I can certainly say that examples such as 28 ÷ 4/36 ÷ 4 are within the realm of what is done currently in (my) fifth grade, and I am positive that that similar examples using multiplication (and perhaps division too) to create equivalent fractions are routinely used in grade four.

I don't know exactly who wrote this progression. It is part of a project from the University of Arizona:  http://ime.math.arizona.edu/progressions/  , and Wu is a writer. There is interesting active discussion of this document where it was posted: http://commoncoretools.wordpress.com/2011/08/12/drafty-draft-of-fractions-progression/. You might consider adding your legitimate points.

In any case, I now view these materials more critically, and I am spending more time with Wu's Understanding Numbers in Elementary School. If anyone is interested, the book is currently discounted at the American Mathematical Society's website: http://www.ams.org/bookstore-getitem/item=MBK-79

Steve
• Linda, after reading your post and viewing the video, several points come to mind:

1) I don't think there is enough verbalisation by students in math classes PERIOD. They get used to sort of distancing themselves verbally from what's on a math page and float on or swim in or drown in a sea of numbers bereft of language. I think that problemtic way of referring to a fraction comes from trying to make up a language that works inside weird kind of 'math think' that they get into. The concepts never get anchored in real language.

I used to do something called 'words first' in which students had to be able to say a process for solving a problem without using any specific numbers. They had to be able to generalise processes and mix math vocabulary and real language to express themselves. So given a question like the one in your video, they would have had to explain they why's and how's without numberical what's. At first they felt and acted like mathematical 'Helen Kellers' -- struggling to get out even the simplest of real sentences. Eventually they started to turn their ability to verbalise back on and the language flowed more easily. New neuro-research shows that math learning and language are closely associated in the brain --so I think I was on the right track.

I think that to get people to be fluid and flexible in math, they have to be able to talk themselves through experiementing with different problem-solving approaches. This can't happen easily if they are not used to conceptualising math in real language. People talk about 'number sense' as being intuitive, but I believe that a lot of our intuitions come in bundled in language. Unless one is so steeped in deep understanding of math that one lives in a math landscape, thoughts, flashes of ideas, the abilty to switch streams are partly a function of the ability to give language to insights. We keep dream journals. We write down ideas to come back to at another time. How can kids get close to being comfortable inside math if they leave their language sklls at the door?

2) Your unique perspective (you see the end result of years of math learning) is very helpful. Math mislearning it seems has a coninuum as does math learning.

3) Trying to create a math course for students I will not be teaching is a challenge that is bringing a lot of these kinds of insights to light.

4) Pushing children into doing overly complex questions too early in a way forces the teacher and the kids into memorizing processes before the concepts are fully grasped. We hope that as they get older their understanding of why they are doing the processes will magically mature, but the early 'why' learning is just let go by brains concentrating on what is the apparent higher priority -- storage of rote learning.

The brain prunes neurons that contain learning which is not used and reused and added to, so if we think that ensuring understanding of size and number of pieces (denominator and numerator) in an early grade ensures continued knowledge of that relationship, we're fooling ourselves. This is where 'following the standards' can defeat math understanding -- because early learning that is not deemed as useful by a brain is pruned. Recylcling through the fundamentals is the only thing that will keep them in kids' heads (literally).

Also if a concept is understood at one time but then incorrectly or only partially reactivated at a later time (eg.during review), the neural architecture that housed the original correct learning will be physically destroyed and new structures that embody the wrong or incomplete learning will grow instead. Unless the fundamental understandings are revisited SUCCESSFULLY (for every kid; not for the teacher) several times in a year and then many years in a row, they will be lost or replaced and show up in your adults as so much mathematical swiss cheese.

• 1. Warm regards to all!

2. Neither deep nor profound, but I try desparately to get my kids to use decimals whenever possible. However, I have yet to find a way to solve word problems involving "work" without fractions. But I don't call them fractions. I call them "the part of the work done per unit time" and I never require LCM just ACM (any common multiple). Does anyone know an easier way to explain work problems?

It should be visible on a smartphone and for my amusement I will try to insert a QR code (Unfortunately in the video I forgot to change my 1's for US l's  - They all are "flagged" like this type :) )

.

• Some of the reading I've been doing lately has to do with

(1)  learners being able to transition from relying in meaning and metaphor to being able to accept fractions as numbers in their own right an manipulate them with ease. Little kids learn that 1 + 4 = 5 by counting concrete objects, but eventually the quantities turn into number objects that can be maniputated without the kids having to constantly count or reconnect with the meaning.

(2) teachers confusing dressing up traditional problems in real world clothes (along the line of the Emperor's New Clothes) with creating authentic problems that are real to the students.

I want to create metaphors and learning experiences to help students connect wiht the concepts and skills on a more intuitive level, but I also want them to make that transition to being able to handle the procedures flexibly. I believe that the first can lead to the second, but that for all kids this is not an automatic leap. They have to be guided across that chasm.

I'm interested in exploring the metaphors and authentic experiences that can give meaning to concepts many of of take for granted.

Interesting article --  http://math.berkeley.edu/~wu/NCTM2010.pdf

• I would like to focus on deep, profound and modern definition of "fractions." I see it as closely connected to ratios and proportions. Fraction operations need to be de-emphasized, and proportional reasoning (with fraction notation) expanded.

• I'll be interested to hear how this fits in with the open math textbook materials that are there now. Is this an added section or does it call into question the whole structure of the unit?

•

My Initial Thoughts:

1. Would like to focus not just on multimedia and interactive materials, but concrete physical things kids can do/construct and play with as well
2. Also ideas for students and teachers on how to use the "interactive materials" like a lesson plan and questions to get kids engaged and thinking, so it is more about kids discussing and exploring, rather than us delivering our vast fraction knowledge from on high :)