Bonnie and all,

Sorry to be so slow in responding to the question about what I meant when I referred to critical thinking and math. In the interim, there has been some rich discussion related to critical thinking and math - looking at social issues through the lens of math, for example, issues of gender and race; making math relevant to adults' daily lives; and resources such as EMPower and others, some I know and some not, some of which address what was in my mind when I wrote my statement. Many of the discussion themes are related to each other and to what I was thinking.  

I was thinking of getting students to think about math critically or to think more critically in math rather than using math to think critically on some topic. Before I go further into what I mean, let me say that while I think it's important that we relate learning to adults' every day lives, in many cases, that should just be a starting point. Some, perhaps not all of our students need to get to the level of relating ideas to ideas and concepts to concepts in order to meet their goals or attain their dreams. The transition from thinking concretely to thinking abstractly is developmental but it is not one that individuals necessarily make. Often support is needed for this transition to occur. (Piaget discusses this.) 

So, what was in my mind when I said critical thinking and math and what might it look like in practice? (Again, some of what I am proposing is covered in materials that have been referred to and I am not suggesting critical thinking as it came to my mind to the exclusion of any of the other approaches that have been discussed on this list.) I can't lay out a curriculum; it's been too long since my practice was at that level, but let me give some examples, in no particular order. 

What's the area of a triangle? Is this something the student understands or memorizes? A very simple way to demonstrate this builds on the understanding of the area of a rectangle. I think most instruction develops understanding and critical thinking for the area of a rectangle by using graph paper, tiles, or some method of showing the square units needed to cover a rectangular surface. So for a rectangle that is 3 units high and 6 units wide, three rows that are 6 units long can be constructed, the units counted, and it is apparent that the area is 18. Then, the formula, length times width can be introduced and it makes sense. My experience from observing teachers is that, when it comes to triangles, there's often only a formula to memorize. But, if you take a triangle, you can fold it in half along the base, cut it, put it back together as a rectangle and see that the area is 1/2 the base times the height. If you want to explore the commutative property of multiplication, try folding along the height and seeing if 1/2 the height times the base works. (Why is this base times height not length times width? A discussion of jargon, insider language, vocabulary specific to a group or subject could be introduced here.) Then see if you can use this approach to figure out the area of a parallelogram and write a formula based on what you did. Do the same with a trapezoid. Formulas becomes demystified as students realize they are just someone's way of writing down the way they discovered how to solve the problem. Further, if you forget the formula, you can still figure out the answer. 

Another example might be multiplying with fractions. Often we show multiplication of whole numbers using a grid, so 3 times 6 is 3 units high and 6 units long, and when you count them, you get 18 units, just as in area above. What happens with fractions? Again, my experience in working  with teachers is that at this point, students are asked to learn how to solve the problem, not taught how to think it through. But if you create a shape that is 1/3 unit high and 6 units long, you can see that 1/3 times 6 is 2. While I don't want to do that every time I have a problem to solve, once I understand what the problem means, I can approach it with understanding. And I will remember which formula to use.

Many of our students don't have a basic understanding of the number system. Adding starting at the left most column really makes more sense. It also takes out the hocus pocus feel of carrying in addition. (Same is true in subtraction and borrowing.) Mental math is another good way to develop math thinking, and you start with the left most column for this, too. Using patterns such as thinking in tens works well. Estimating is really important. 

Once this lower level of critical thinking in math is laid, then higher level topics can be addressed building understanding in those areas based on understanding of lower level concepts, not just knowing formulas and how to perform operations. Also, we might want to rethink what we consider the progression of concepts. Probability, which is usually taught as a higher level concept, is really very necessary in our everyday lives and concepts (critical thinking) in that area can be developed fairly easily and in ways that are a lot of fun. 

A math teacher I had years ago said that the reason algorithms that we use for operations such as adding a series of numbers came into common use was because paper was scarce and it took less of it to solve the problem using the algorithm, which, for adding starts at the right most column, rather than solving the problem more logically, that is starting at the left most column, which uses more paper. I don't know if this is true, and certainly the algorithm is an effective method not just because it saves paper, but because it is often faster. But once there is understanding of the process, the algorithm is just a tool. 

Well, I've gone on probably for way too long. I hope that I have added something to the discussion of critical thinking, specifically to critical thinking in math.

Rose

 
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Rose, I'd like to hear more about that, since our Bridge program for at-risk (variously defined) students has a math course, and many come in expressing math anxiety. Could you talk a little more about how critical thinking works in math? and since I'm anumerate(illiterate in math) I may or may not understand what you mean, but will pass it on.

Thanks,

Bonnie Odiorne, Post University

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