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Psychology of Math Learning
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Convergent vs. Divergent Thinking
Several class members mentioned in their opening math essay that they liked math because answers were "right" or "wrong" and always the same (2 + 2 = 4). While some math educators, such as my good friend Dr. Maria Droujkova, would argue that not all math is like that, certainly that is how we commonly think about math.
In psychological terms, this is called "convergent thinking"--the ability to find one single best answer to a question or problem. The opposite skill is "divergent thinking"--the ability to find as many possible answers to a question or problem. Studies have shown that convergent thinkers do better on typical intelligence tests (the fill in one bubble types of tests), while divergent thinkers do better on open ended exams.
In the 1960's, a psychologist named Liam Hudson did a study of convergent versus divergent thinkers and whether that ability influenced whether they decided to go into science fields or arts and humanities fields. While he did not include mathematics in his study, I think we can assume that math would be seen more of a science topic than an arts and humanities. Here is a table of his findings:
|
|
Null hypothesis |
Physical science |
Biology |
History |
|
Extreme divergers |
10% |
3% |
4% |
16% |
|
Mild divergers |
20% |
12% |
15% |
34% |
|
All-rounders |
40% |
32% |
54% |
39% |
|
Mild convergers |
20% |
36% |
15% |
7% |
|
Extreme convergers |
10% |
18% |
12% |
5% |
|
Number of students |
|
104 |
26 |
44 |
|
Significance |
|
0.1% |
>20% |
5% |
To quote Hudson's conclusion on this topic:
“Between three and four divergers go into arts subjects like history, English literature and modern languages for every one that goes into physical science. And vice-versa, between three and four convergers do mathematics, physics and chemistry for every one that goes into the arts.” (Hudson 1966, pp.56-57) quoted in Orton (1992, p.112)
(Note: I haven't read the original sources quoted; I got this information from Paul Ernest, a fascinating mathematician from the University of Exeter, from his website at http://homepages.which.net/~gk.sherman/mbbaaaaa.htm#1)
This convergent versus divergent thinking relates to several of the psychological topics we will be discussing later in this course. But what do you think about this idea? Do you think convergent or divergent thinking plays a role in your math experience? For example, I think I tend to be a divergent thinker, but my son is an EXTREME diverger. I don't know how much that played into my math experience, but I do think math's tendency towards convergence is one of the reason's he started off not liking math (although Maria has been working with him and has really helped him to enjoy math more).
What I would LOVE to see: math activities tagged by the target audience's degree of convergence. For example, yesterday I invited math club kids (ages 7-8) to draw infinity. This is a very divergent activity and more divergent thinkers dived into it with gusto. The convergent kids would suffer, except we quickly started to share first sketches, so convergers were remixing, organizing and systematizing ideas happily.
I think I am ambidextrous on this scale. Can convergence and divergence be trained?!
That is such an interesting idea. Im very divergent. I excel in finding more than one answer. Im most comfortable with trying to see things different ways. Although I do find it puzzling that math and music are so closely linked, because I see music as being divergent.
Mathematicians see math as divergent, too. But math educators often design very convergent and linear tasks for students, hoping to "simplify" their life. I think these efforts often backfire so badly!
I see math as very divergent. There are so many ways to go about getting a solution to the same problem, and so many problems that can be solved with a single type of solution.
I think a recurring refrain in this class will be "math" versus "the way math is generally taught." But I believe I was taught math in a very convergent way, and I think that is the way it is usually taught.
While I tend to see most of the things in a convergent way, if not almost everything (im sometimes called a squared head :D ) there are topics where i can get out from there and go to the divergency, things that i do, that i just do it, not thinking too much.
Maybe with the right method both points of view could be taught, but to some extent.
Can a convergent person learn faster from a convergent teacher, or a divergent one?
by the way, this is a very interesting group of people, i hope i can learn from all of you :D
I think the general assumption is that students learn faster from a teacher that is like them, but I don't know of any studies that demonstrate that, particularly in regards to convergent or divergent thinking. However, here is a great article about how BOTH divergent and convergent thinking are required for true creativity:http://www.newsweek.com/2010/07/10/the-creativity-crisis.html
I think this article speaks to Maria's statement that she is both. Effective creative people are. And I think that there can be that kind of creative thinking nurtured through mathematics. However,especially in the current focus in the US, at least, on test scores (extremely convergent thinking), most math teachers never have time to include many divergent activities in their curriculum. However, disciplines that AREN'T being standardized-tested-to-death, such as history, have more freedom to explore divergent as well as convergent thinking.
But maybe that is just my bias show...:D
interesting, but I need to read a second time to assimilate.
While reading for the first time, I thought about "thinking outside or inside the box".
It seems like maths is taught to be done "inside the box" while the greatest and most beautiful proofs are done "outside the box" and the difficulty is to show students the way outside where there is no directions at all. It is easier to stick to the rules but this does not work all the time even in maths.
Although convergent and divergent are good terms that psychologists have formulated and taxonomized (as so many things are in the world) to understand a system that we have created. Unless you are at some extreme, most of us are a combination of the two convergent-divergent thinkers (perhaps weighing more heavily towards one or the other).
We as human beings want to believe in our own uniqueness but to some degree we are pretty much the same in our behaviours which are pattern-based.
The problem for the educator is how to use standardization to its maximum effectiveness but also to be able to address the "uniqueness" factor of each individual person. I don't think this is simply a problem of public school systems-- private schools and charter schools have these problems as well.
No one has really solved the problem of addressing (what I call) the uniqueness factor effectively. For one, it's too costly. The ideal situation would be to have a teacher who is best suited to the learning style of a particular student.
Of course, this goes beyond mathematics and touches upon almost every field of study.
Exactly. I think this is why homeschooling is so effective. However, it is WAY too costly for a societal solution to the problem.
I think this is the exciting potential of technology--to individualize instruction in a group setting. However, most of the educational software we have so far doesn't really do that effectively. And outmoded educational structures and procedures don't allow us to do that effectively, even if we would. But I think there is great promise for this in the future.
Back in the Signup Post thread, Oula, Maria, and I were talking about the need for teachers to share their thinking (and sometimes their mistakes) to increase student understanding. I mentioned that I was involved in a research project that demonstrated that one of the biggest problems in math and science education was that teachers taught problem-solving techniques (generally, equations) without explaining the thinking behind those techniques/equations. So students memorize the equations, but are at a loss to apply them correctly.
This isn't something that we looked at in the study, but now I'm wondering if this was an effect of attracting convergent thinkers into teaching in those fields. That is, if the teacher were a convergent thinker, it may not occur to them that the approach they were teaching wasn't obvious to their students, and/or that their divergent thinking students might have multiple other approaches to solving the problem.
I never thought about this before, but it makes sense to me now that we're having this discussion.
I love this discussion about C & D thinking, think what it really captures is the big picture idea what all this comes down to is that Math is a way to understand our world. It is such an amazing human trait and a testament to our consciousness. Its both C&D and what think what gets lost (at least it did for me) was this idea. I learned my math tables (all rote learning) but the cool factor was missing. Not saying I needed to be entertained to learn - just saying I didnt make the connection untill it was too late to realize math was bigger than 12 x3 = 27. Okay that was a joke....I really know it's 42.
We are a combination of convergent and divergent based on the requirement of the task at hand and the parameters thereof. For example, in a work situation where backup and fail-safe strategies are required, we have to be divergent. Similarly, there are times when we require just 1 solution, convergent. We need to learn/know/apply both ways of thinking based on the situation.
I remember a movie a long time ago where a teacher used an apple to teach fractions to students in the "Bronx" (I think that is the term used) and engendered a love for that subject by explaining the concept in a method that was easy for those particular students to absorb and understand. Divergent/out of the box at that time (not sure about now). In NYDailyNews article (http://www.nydailynews.com/ny_local/bronx/2009/11/17/2009-11-17_a_fabulo...), November 17, 2009, "A fabulous four: Bronx teachers honored for making science, math fun" - each of the teachers had their own way to achieve "making science and math fun".
Yet there are more engineering students in India (can only speak for India as I do not have information about other countries) who excel in Maths and they are taught in the same conventional way.
Both seem to work.