The Rookery

Can You Convince Me?

"Is this right?" Way too often that is the main question students ask in a math class.  So much of traditional elementary math curriculum is focussed on computation - it can seem as though getting THE RIGHT ANSWER is the whole point of mathematics. 

Of course, that's not the point of mathematics. I don't claim to have the right answer for what mathematics is - some of the Herons are engaged in an ongoing debate about whether math is invented or discovered, but that's another blog.  What I do know is that in the best math lessons, the point of the math isn't to come to a single foregone conclusion but rather to push and prod on our understanding of how numbers work, to make and test conjectures about patterns we think we see and to explain the validity of how we are using numbers in a way that can convince others.

Over the years, we have collected and created a large repertoire of lessons and routines that emphasize authentic math explorations and discussions.  Illustrative math, the curriculum we adopted more fully this year, has been the source of many of those explorations.  A typical lesson has only one or two deeper tasks in which students seek to use math meaningfully and then talk to each other about what they are thinking.

This past week, the fourth graders were engaged in just such a task.  They were given three solved problems and told that there were either two truths and a lie or two lies and a truth and their job was to figure out which problems were worked correctly.  Each group tackled a different problem.  They worked to understand what the situation was and then to figure out if the solution presented was correct.  They used diagrams and estimated; some groups solved the problem themselves then compared it to the solution presented.  All three groups decided that their problem was not solved correctly.  All three were lies. (Here are the problems)

"Wait a second, I thought there were supposed to be two truths and a lie!" one Heron exclaimed.  Hmmm...

Each group got up to the board and explained the math they had done to get their answer.  After each one we discussed and agreed that the group was right and that the solution as written was a "lie."  "Maybe they're trying to get us to think!" I said hopefully.  We checked the teacher manual...and made a startling discovery:

 

 

 

 

"They multiplied by 53 instead of 52!"  "They didn't take out the bottom floor that's not 11 feet." "They did the bottom floor twice!"  The students were jumping up and down and...dancing (this is a group that likes to dance!) "They're wrong!!"

I calmed them down (just enough...it's important to ride the wave a little in these kinds of situations) and asked them what we should do.  "Write them a letter!" was the response.  "O.K. when you do it, make sure you explain yourselves really well.  After all, you have to convince them that they are wrong so you have to be very persuasive, show them your solution and explain how it's different from theirs."

Here is the result:

This is the goal of math at Prairie Creek - to empower students to understand their world and to be able to explain their understanding effectively.

"Is this right?"  You tell me.

A Part of the Whole

In exploration math, we've begun to work with fractions, decimals and percents.  For many students (and adults) fractions are an enigma.  Indeed, they are sometimes taught as an inherently counter-intuitive subject.  I know that "the bigger the bottom number, the smaller the fraction" was taught in my elementary school as a way to remember the impact of the denominator.  Unfortunately, it often just solidifies children's confusion.

It can be easy to think of math and number systems as a set body of knowledge but they're not.  As humans, we are continuously expanding the ways we can express our world in numbers.  The development of fractions is very recent mathematically.  While many ancient cultures had unit fractions such as 1/2 or 1/4,fractions were primarily conceived of as a way to represent equal sharing.  Common fractions (such as 2/3 or 4/5) are much more recent - around 500 BCE.   Decimals are an even more recent mathematical development.  The Hindu-Arabic base 10 system was not adopted widely in Europe until 1300 CE (1300!) and the use of a decimal point to show fractional parts came after that.

Why do I share all of this?  Because it illustrates that it's taken humans a long long time to figure out fractions.  It's no wonder that children need a lot of different experiences to come to understand them and use them fully.

I've been re-reading Young Mathematicians at Work - Constructing Fractions, Decimals and Percents that dissects the challenges many students (and adults) experience with fractions.    Often, we move to the manipulation of pure numbers far too quickly.  Procedures and algorithms are taught independent of the context that makes them useful.  (I've actually read about a teacher whose mantra about fraction division was, "Don't ask why, flip and multiply!" - here's a blog about a fraction lesson I did in response.)  By proposing real world problems that necessitate the use of fractions to come to satisfactory answers, we give the concept of a fractions and decimals meaning and utility.  This leads to a deeper understanding and an ability to generalize understanding across a range of situations.

The authors, Catherine Twomey Fostnot and Maarten Dolk, propose that a child's understanding of fractions works like a landscape in which the understanding of one skill exposes new skills on the horizon which, when arrived at, open up further skills.  It is not a linear journey and different children reach epiphanies from different directions.  It's a good description of the work that is going on in all three classrooms as we explore fractions.  These are the concepts that are on the landscape.  Your child is moving among them and solidifying his or her understanding of them through carefully constructed experiences:

• Part/whole relations - the whole matters when one is comparing fractions.  This concept is revisited often as students extend their use of fractions.  Finding parts of wholes, extending ratios, and eventually working adding, subtracting, multiplying and dividing fractions.
• Equivalency vs. congruency - fractional pieces do not have to be identical, only equivalent.  1/2 + 1/4 = 3/4
• Extending concepts of equivalency to decimals and percentages - students see the relationship among these different ways of recording fractional parts and can apply them fluidly
• Connecting multiplication and division to fractions - children understand fractions as "fair sharing" and see the relationship between fractions and division (splitting apart) and eventually multiplication 5/8 = 5 * 1/8.
• Relations on relations - when multiplying and dividing fractions there are two wholes to keep track of.  For example, when I divide 2 1/4 cups into half cup scoop measures, I end up with 4 and 1/2 half cup scoops.  My whole for the dividend is cups.  But my solution's "whole" is the half cup measure.

These are the big concepts on the horizon. They enable students to identify fractions, order and compare fractions and decimals, find equivalent fractions, find equivalent decimals (and percentages), combine fractions with like denominators, combine fractions with unlike denominators, and eventually multiply then divide with fractions.  The conceptual understanding enables the skill work which becomes more efficient through experience.

Whew!  If you've made it through to this point in the blog, you are very dedicated to your child's mathematical development and might be wondering what you can do to support understanding of fractions and percents at home.  Here are some ideas:

• Cook!  Read recipes.  Then pretend you can't find some of your measuring cups.
• Connect computer downloading bars to fractions and percents.
• Play games of chance and talk about probability.
• Compare things.
• Encourage "rate" thinking - i.e. "We've gone 40 miles since we got gas and we've got half a tank left, how much farther can we go before we get gas?"
• Talk about time as fractions - quarter of an hour, 3/4 past.
• Count down using decimals or fractions instead of just whole numbers.
• Play fraction dice games - in the simplest you try to make the largest (or smallest) fraction by rolling a die and deciding if you want to use the roll as the numerator, denominator or one of your two throw away rolls.  Works with decimals, too.
• Put your child in charge of distributing anything fairly - goldfish crackers, screen time, hot dish.

"Mathematizing" is a word in education which makes me cringe a little but the idea behind it is very, very important. It means finding ways to explore situations mathematically - seeing relationships and using numbers to explain the world around you.  Students who do it regularly feel like math makes sense and, importantly, that working with numbers is a creative endeavor. Students who don't mathematize see math as a set of arbitrary rules one follows to reach an answer that is correct.  Math topics are not related and don't make sense.  

The more you and your child use fractions, decimals and percentages to describe the world around you, the better the chance your child will begin to see numerical relationships and begin to mathematize her world.

 

A Date with Rates

Our new math unit focusses on the critical concept of "rates."  A rate (or ratio) is a change that occurs at a constant pace.  Some examples:  I can write my name 18 times per minute.  The Herons can put out 180 chairs in 3 minutes.  Three pounds of grapes cost $1.80.

The key tenant of rates is that we can extrapolate them.  We can use what we know to find out what we don't know.  This is the definition of algebra that we use with students and it certainly applies here.  For example.  If three pounds of grapes cost $1.80, then 6 pounds is $3.60 and 1 pound is $.60.  This relationship can be graphed on an axis.  It can also be written as $.60 x p = total price (where p is pounds of grapes).  Graphing and equations are tools we can use so we don't have to compute all of the "in-betweens" as seen below:

Of course, initially, students do need to figure out the inbetweens to build their understanding.  Otherwise, once they use the tool of a graph or equation, it seems disconnected from the real world.  They only know it works to find an answer but they don't understand why it works.

Before we began this unit, we assessed students' current understanding of rates by asking them to answer a few questions and share their confidence with that type of problem.  This helped us see if a child had little experience with the concept, was developing their understanding or was ready to tackle the idea of rates in a more complex way.  Amy, Cathy and I set goals for each group of learners together and then individually develop and teach a curriculum to help students reach those goals.

As you can see above, a facility with basic multiplication and division facts makes this work much, much easier.  We use this as an opportunity to explain to students, once again, why we feel strongly that they should master their basic facts.  It's an authentic application.  Kids who know without thinking that 3x6=18, instantly see a relationship in the above grape example that is missed by students who don't know that fact.  It's a lot easier to use multiplication to extrapolate than to use repeated addition.

The next few weeks are a great time to explore rates at home with your Big Bird.  Allowance, getting ready for school, cooking, gas mileage, and speed all afford great opportunities.  I highly recommend The World in One Day by Russell Ash -- it uses stunning visual images to share rates for everything from the world's daily banana consumption to the heart rate of a blue whale.  It's a great addition to any home library.  As a staff, we've been discussing ways to make home numeracy work as easy as home literacy work -- rates is a great way to incorporate "math talk" into your family life.