Algebraic Thinking
What is it? Where does it start? How do I facilitate it in my classroom? When will I have the time to add something new to my already crowded day?
Research is telling us that Algebra is the gatekeeper of success, and that our elementary curriculum must begin to address the precursor skills necessary to algebraic understanding. But if you’re like many teachers across the USA, questions such as those above are rattling around in your brain. November’s little tip is written to help you take the lessons that you are already teaching, and with a little tweaking, begin to nurture algebraic thinking in your students.
Of course, structuring specific mathematical tasks to promote algebraic thinking is important; but the higher level questions that we ask during almost any task in the math class can help students to begin to think algebraically. Let’s look at the strands and suggested questions that can be used at any level. These questions are certainly not all inclusive, but rather are meant to provide a springboard for your thinking.
Patterns
How did you make that pattern?
What will
you place next? Why?
Look at these patterns (patterns made with different
materials or using
different numbers in the sequence). How are these patterns alike?
Can you
make a pattern using a different (material, number sequence) that is similar
to this pattern? Describe how the patterns are similar.
Explain how you know how to generate a pattern using different materials
that will follow this pattern’s rule.
What rule could you use to find a specific term of the pattern?
Number
What makes the number odd (even, prime, composite, a factor, a multiple)?
How do you know when a number is a multiple of 2 (3,4,5,6, etc)?
Describe what fractional parts mean.
What makes a fraction (decimal)
different from a whole number?
What is the relationship among (coins). What
does that mean? How can you describe the relationship so that you can compare
any number of coins?
What happens when you combine numbers? What patterns do you see? How
are the patterns alike in all number combinations?
What happens when you
separate (compare) numbers? What patterns do you see? How are the patterns
alike in all number separations (comparisons)?
How is subtraction related to addition? Division to multiplication?
Subtraction to Division? Addition to multiplication?
What are some patterns
in addition (multiplication, subtraction, division) that can help you remember
basic facts?
Geometry
How is this polygon alike/different from this solid?
What are the attributes that make this polygon (angle, line, solid)
that polygon (angle, line, solid)?
What will you need to do to change this
polygon (angle, solid) into another polygon (angle, solid)?
Explain how you know the difference in the transformations. What are
you looking for when identifying a translation (rotation, reflection)?
Measurement
Why did it take fewer crayons than pennies to measure the length of
that object? What is your estimate? Why do you think the object will measure
that many units?
Explain how you know that 12 inches (or any other measure) is 1 foot
(or any equivalent measure).
Can you generalize a rule to convert any number
of inches (or whatever unit of measurement) to yards (to equivalent measurement)?
Probability and Statistics
(Graphing)
Explain the patterns that you see in the graph. What story does this
graph tell (or could this graph be telling)?
How do you know that the relationship this graph is representing is
(or is not) a proportionate relationship?
Look at the chart (or table) and describe the graph which the points represent. How do you know what the graph will look like?
What is the relationship of the x axis to the y axis for this data?
How is the scale of each axis related to the data?
When we ask students to think beyond the original task to the how and why of the strategies or representations, we are asking them to begin to analyze relationships, rules, formulas, and functions in order to formulate a general understanding that can be applied in other related tasks. Not only are we helping students to make the connections necessary to become expert problem solvers at their own levels; but we are helping them to see that math is a wonderful series of connected patterns within and outside mathematics. To help students to this revelation is to open the whole world of mathematical possibilities to them.
We hope that this month’s little Tip will prove a catalyst for you in raising your everyday mathematics to powerful algebraic understanding by asking questions throughout your lessons. Everyday you can add to your students’ facility in higher level thinking and the richness of their mathematical understanding. And the only thing you need is a handful of probing questions!! What a deal!
Happy teaching to all of you, and
Happy Thanksgiving!
Ms Fritzie
PS: Just a reminder -- All of the products on our website are written to encourage algebraic thinking.
