The Rookery

"Use what you know to figure out what you don't know."  That phrase lies at the heart of our current math focus:  algebra.  Mathematicians are puzzlers – they try to put pieces together to figure things out.  They use numbers to make sense of the world.  The fourth and fifth graders are using these skill constantly during this theme.

Time was when "algebra" was the stuff of middle and high school.  But algebraic concepts and thinking are now used by students much earlier and they embrace the challenge.  Here is a great article about algebra in elementary school by Marilyn Burns, one of our favorite math writers.

We are having students study real world situations to find patterns, graph the patterns and then create a number sentence or expression (or rule or equation or function – all words which can be interchanged) that can be used for any variable.  For example, my group helped me figure out how many tables I needed to rent for my friend's wedding.  At first she wanted a long table of squares pushed together.  We found out how many seats were at different sized tables (one table had four seats, three tables pushed together had eight seats.)  Then we put our numbers into a table and made coordinate pairs of the vairalbes.  We graphed the coordinate pairs and discoverd that they made a line (and we could use the line to find out how many seats we would need for any number of guests!)  

By the time we had done all of this work, the students were almost bursting – they needed to tell the class there was an easier way to figure out the number of seats than building the tables with blocks (which was the strategy most started with).  They discovered that if you doubled the number of tables and added two more seats (for the ones at the end) you got the total number of seats.  IT ALWAYS WORKED!  The emphasis is theirs, not mine.  They then worked to explain their "short cut" using symbols.  Some explained that T*2 +2 = S others that T+T+2=S.  We also worked with triangles and hexagon tables and students began to discover they could find a pattern, create a graph and write an expression for each kind of table.  

As we discussed our findings, students shared some wonderful questions and observations – they were extending and playing with the concepts.  "Do the coordinate pairs always make a straight line?"  "Can their be a third number in a coordinate 'pair'?" "The more sides the shape has, the steeper the line it makes in the graph!" "For this table, you always go one over, two up." and "Wait, why are we graphing four different coordinate pairs – you only need two to make a line."

That sense of discovery is very important in math class.  When we as teachers present math as a finished product, we are doing students a diservice.  They begin to believe that math is just a giant series of unconnected facts that must be memorized. Instead, if they develop the concepts themselves, they are building from what they know and the new concepts are richly connected to their previous knowledge.  Students are learning math and they are learning to think like mathematicians.

This table problem is just one example of the type of work students are doing in our algebra unit (although "unit" is a bit of a misnomer since many of these concepts are woven into work we do throughout the year.)  Students will also play with "in and out" tables which are puzzles in which students try to figure out the rule for what happens for each number that goes into the "function machine."  Students learn to test their initial hypotheses for the rule since it must work for each pair.  Students will also explore rate problems and use their new graphing and equation writing skills to represent these rate relationships in new ways.  Many students will also play with "pan balance" puzzles.  These are logic puzzles in which you try to figure out equivalencies using some beginning algebraic tools.

I highly recommend Dragon Box, an algebra puzzle app that Gabe found this summer.  It uses game play to introduce basic algebraic concepts.  The same company makes an app called Dragon Elements that teaches kids intuitively about geometric proofs.  The apps are the rare gems in which the game play actually enhances students' understanding.

Whew.  That's a lot to read (and that's only 2/3 of the math that's going on right now.)  If you have questions about the math we're doing or would like for me to write about something in a blog please let me know.  We tend to write more about theme or writing because of the way we structure instruction – not every Heron has me as a math teacher right now so not every Heron has had this exact experience.  But we certainly do math every day and as a staff we think about math a lot, too – if you let me know what you're interested in, I'll be certain to address it.