Tell your students the following everyday
- Addition, subtraction, multiplication, and division is part of math but math is so much more! While algebra is a focus in many classrooms, don't forget about geometry, probability, or data analysis.
- You can't make understanding an algorithmic process. Students need to experience confusion, misunderstanding, and failure as part of the understanding process.
- Being good at math is not something that is easily explainable. Accuracy and speed in procedural mathematics is only one way. Other ways include being open minded, logical, discussing conjectures, and trying to answer "why?". Everyone can be good at math and we need people who can do math in those different ways.
- While a problem may only have one correct answer, it may have many different correct solutions. While some solutions are more efficient, or effective than others, it is more important that everyone understands at least one way of arriving at the correct answer. Math must make sense.
- We can get better at certain skills of math by practice, however talking and discussing with your fellow students will allow you to understand mathematical ideas deeper.
When developing algebraic thinking with students, we shouldn't force students do what is the most efficient (as sometimes this is objective anyways), but to do what makes sense. When presented with a set of data, or a pattern, Thorton (2001) argues that it is less important that students be able to find the algebraic rule than they recognize a rule can be represented in equivalent algebraic expressions. When "finding the rule becomes the focus" most mathematical thinking is lost.
Lets, for example, look at the following grid. I want you to tell me how many blocks are coloured in this 10x10 grid without counting the blocks individually. It is crucial you do this without writing, talking to a partner or counting, as it will force your mind to make generalizations which will be the basis of deeper learning to come.
As a teacher we should first ask students now to share answers, not strategies, with an elbow partner. This would then force some to reevaluate their strategy and possibly pick up any common errors he/she may have made.
Now how did you do it? Here are some strategies in Arithmetic form:
As a reader, are you able to explain how each of the expressions above arrive at the same answer? An important task in furthering one's mind into algebraic thinking.
Imagine if this was done in a classroom. Jason stands up and explains why he simply went 4x10-4. There might be some smiles, nods, confused looks. Some students will have seen the same answer provided a different way. Does this happen on a worksheet?
Of course now, you have realized that the correct answer is 36. Would then the expression "30+6" be appropriate? This is where conceptual differs from procedural. Yes 30+6 equals 36 but unless a child can create meaning to the 30 and to the 6, I would disagree that 30+6 would be an appropriate response in this context. This illustrates the idea of solution(30+6) vs answer (36).
What if the grid was 8x8, how would the above expressions change? Would a number change in every expression or only some? This idea of visualizing a problem and solving it, is crucial to the advancement of one's knowledge.
Going back to your strategy and stretching the square to an unknown length, could you create a verbal description of what would you do?
Now the algebra begins..but first...
Noss, Healy, and Hoyles (1997) point out that somewhere we stop seeing algebra as a tool but instead of the end point of a problem. The confusion starts when we see problems as a way to practice algebraic skills instead of using algebra to explore problems.
When bringing in letters into these expressions we must remember some powerful pitfalls:
- The largest misconception is that letters are labels or initials. (pg 25)
- Changing the variable to a different letter changes the solution. (Let students pick the letter, and encourage different students with different letters)
- The equal sign simply means "gets the answer". (Try writing 5=2+3 as often as 2+3=5 especially in earlier grades)
Taking these points into account, create a variable for the side length of the square. Could you create an expression using your variable to show the amount of squares which would be shaded? Knowing what you have done so far, how could you test your expression?
In a class at this point it would be helpful to ask the class "What will be staying the same? What will be changing? As we bridge the gap from numbers to a variable". For example if the strategy you are using is 10*10-8*8, then the multiplication and subtraction will remain the same while 10 and 8 will change to x and x-2 respectfully. I would not tell them this, but instead question it.
If you followed all of these tasks then you have now done the problem in 3 different approaches: Arithmetic, Verbal, and Algebraic.
The crucial part of this example, or if this lesson was to be developed into a classroom, is to ask questions and not simply give answers. Remind them we are not seeking the solution, but a solution that makes sense to them. Embrace students' wrong answers since learners who are given competing ideas, engage in cognitive conflict and such conflict promotes learning more than the passive reception of ideas that are always correct and seem straightforward (Fredricks, Blumenfeld, and Paris 2004).