Maths; The Way It Should Be

Last night marked the last class of probably the best group of graduate students I have ever had the pleasure of working with. The course, Math and Diversity, focused on teaching maths to children with diverse needs including ELL, Special education, poverty, maths disabilities such as dyscalculia, and working with mathematically talented students. The course focused on developing the students' relational understanding of math as well as a growth mindset. Here's a sample journal from Bria, one of the students in the class that is pretty typical of how all the students in the course saw their growth as math teachers this semester.

 "I am so glad that I took this class. My mathematics confidence has skyrocketed as a result of this semester. I used to be told that I was good at math, but I didn’t believe it because I was grouped with people who were notably good at math and did not compare to them. I was one of the weaker links in Mathletes, for example, and I felt that I struggled in my college calculus class. I know that I will never be as strong in math as many others, at least as long as it is not my academic or professional focus. But what I am learning is that I am better at mental calculations and quick consolidation of numerical information than many other people I come across in my everyday life. I was able to comprehend and work with new concepts presented in this class fairly quickly, and enjoyed being able to experiment with my new knowledge. By gaining conceptual understandings of things I had learned procedurally growing up, I am able to approach new math problems that come to me in life more thoughtfully, and I understand those thought processes. My new hobby is doing mental calculations of various operations and then analyzing exactly how I came to my answer. Feeling in control of math is a fairly new feeling, and I love it. It makes me more confident in my mathematics skills, and makes me more comfortable when I find myself out of control of math.

In the past I have not been able to be comfortable with accepting that something was challenging for me; I would admit defeat at the first sign of struggle. I understand that this is a common ailment of the person who has spent their childhood floating through academic requirements and being told that they are smart. But through the education I am receiving in the graduate program at Saint Michael’s, I am finally learning to practice what I preach. I am learning that finding something challenging does not put a blemish on my intelligence, and asking for help does not signify weakness or make the person asked think I am less intelligent than they are. I believe these things wholeheartedly when they come out of my mouth as a teacher, but I continue to struggle with it personally because I always prided myself on being “smart” growing up and worried that people would find me less so if I asked for help. But during the second half of this course, when a concept or problem presented in class was challenging I began to actually feel alright about admitting it and getting help from a neighbor, rather than chastising myself for not understanding something as quickly as my classmates. I think that the confidence I built from taking this class has allowed me to get over any math-related anxiety I used to feel, such that I now know I am “good at math.” I understand that people were not lying when they told me this in the past. With this newfound knowledge, I am comfortable with struggling. Finding math concepts that are difficult for me are not something to avoid, but something to tackle head-on. It’s fun now, and I have this class to thank for it."

Another student in the course felt that, as a child,  she was "a conceptual person trapped in a procedural world". I will miss teaching this course. 

Maths Isn't Difficult

There's absolutely no reason why learning maths should be any more difficult that learning anything else. The main reason why it is seen, almost universally, to be difficult is that for generations it has been so poorly taught. Ever since the first math classes were held the focus in math education, especially at the elementary school level,  has been on learning and memorizing procedures that have no conceptual basis and personally significant meaning.

The analogy in the English language arts would be to teach children to decode words so that they could read them without giving them any sense of what they mean. For example, what does "Splad the yuricles into the bundecloupus ando tpzig" mean? We can decode words in any language so that we can pronounce them but without knowing what those words mean it is pretty much a fruitless exercise. So in math, when we say "carry the one" or "invert and multiply" or six fours are twenty-four" without any sense of what these phrases actually mean we might just as well be using a foreign language or gobbledegook.  By the way, David Pimm called this "teacher patter". The result of this is to develop dis-empowered learners. If we know something but don;t understand it there's not a whole lot we can do with it. If you know that pi is 3.14 and goes on for ever without recurring but have no idea what pi is then you can do little more than reply 3.14 when asked what pi is on a test.

If we teach math with understanding then learners are empowered to use the knowledge they are developing. If you understand that pi is the ratio between the circumference and diameter of a circle then you can apply this understanding to all kinds of situations involving round things. The advent of technology has significantly improved out ability to teach math with understanding as these Math Gifs my daughter Marie recently sent me show. Once you understand the visual/spatial relationships of a particular mathematical phenomenon it then becomes much easier to abstract or "mathematize" it through the use of a symbolic relationship such as an equation or set of symbols.     

Thinking Fractions

If you think about what you are doing when working with fractions everything becomes crystal clear. The worst thing to do is to visualize the algorithm you learned in elementary school; something like 1/2 + 2/3 or 1/4 x 4/5 or, heaven forbid, 1/2 divided by 1/4. If we do this then we are confined to thinking instrumentally procedurally about something that is comprised of only procedural knowledge. We fall back on senseless rules like "find the common denominator" or "cross multiply" or "change the sign and flip the second fraction". learning tricks like this will probably get you a better score on a traditional math test (like those used to compile the TIMMS report)  but will do little to help you understand what fractions are all about.

So for 1/2 + 2/3 think about different ways you could make 1/2. You could say it's the same as 2/4, 3/6, 4/8 and so on. Now do the same with 2/3. It could be 4/6. Aha, no need to go any further if you know that you can add fractions that have the same denominator. So 3 sixths plus 4 sixths is 7 sixths. We can count sixths just like anything else. Now you can see how you can get sixths by multiplying 2 by 3 if you need to do more difficult ones. But at least you now understand why.

1/4 x 4/5 is more challenging because the x (multiplication symbol) and the whole concept of multiplication of fractions can be different from using it with whole numbers. Visualize 4/5 using the model in the picture above. Now take 1/4 of that 4/5 and you end up with 1/5. The key to understanding this is seeing how the size of the 1 to which each fraction refers changes. The 1 of the 4/5 is the red 1 above whereas the 1 being referred to by the 1/4 is the 4/5. Really we're asking what is 1/4 of 4/5? Once you start to see the pattern, the relationship between the numerator and denominator then things get easier. Try 1/3 x 3/5 or 1/2 x 2/7 or 1/4 x 4/9. If there is not relationship between the numerator and denominator you can do it another way. For example 1/2 x 3/8 is 3/16. This can be done conceptually by halving the size of the fractional pieces. 1/16 is half of 1/8 so 2/16 is 1/2 of 3/8.

1/2 divided by 1/4 is really asking how many 1/4s are there in a 1/2. This is easily conceptualized by thinking about a football game; how many 1/4s are there in the first 1/2? Clearly there are 2.

3/8 divided by 1/4 is a little more difficult because the referent of the fractions changes. Again, think how many 1/4s are there in 3/8 by visualizing the fractional pieces. Compare 1/4 (a yellow piece above) with 3/8 (three dark blue pieces and you'll see there are 1 1/2 quarters in 3/8. The referent for the 1 1/2 is the 1/4.

from this conceptual understanding it's easier to "mathematize", to use Bob Wright's term,  what is going on by working out how the procedural knowledge of operating with fractions works.