# Math Fact Practice ≠ Fluency

Discussion in 'Debate & Marathon Threads Archive' started by TeacherShelly, Mar 9, 2014.

Not open for further replies.
1. ### TeacherGroupieModerator

Joined:
May 13, 2005
Messages:
29,555
1,061

Mar 17, 2014

There's no point contributing to the jerkishness, even by those members who wish to remake A to Z in their own obnoxious images. That's a hint, children: the moderators are getting unhappy.

Joined:
Oct 9, 2006
Messages:
3,565
0

Mar 19, 2014

Over the past week my students have been really enjoying Dot Cards. They are an 8 1/2 x 11" paper with a number of dots arranged in some pattern. They are shown the paper for a few seconds (prompted not to count the dots, but try to see a pattern), then tell how many dots there were. Finally they explain how they saw them. I draw the patterns they saw on the board. The ones who mis-saw the dots (e.g., saw 6 when there were 9) also describe the pattern they saw which is really insightful. The main goodness of this approach is to get kids to see multiple patterns. It opens them up to seeing patterns in more and more places, too.

This is hard to explain. Here is a .pptx that you can look at and modify for your use if you want to.

3. ### a2zMaven

Joined:
Sep 16, 2010
Messages:
5,628
1,513

Mar 19, 2014

Thanks for sharing that ppt. It was interesting how I could see different patterns within the sets to help figure out quickly how many dots were being displayed.

Joined:
Oct 9, 2006
Messages:
3,565
0

Mar 19, 2014

a2z, you have flexible thinking!

5. ### teacherbatmanCompanion

Joined:
Jan 7, 2014
Messages:
121
0

Mar 20, 2014

ajr, I think you make great points, and I agree completely. I even favorited your first post. If it makes you feel better (or worse), this happens not only in math education, but many other subjects as well, including the one I teach. I know there is a much better way to teach math (and many other subjects) and a much better way to create curriculum for schools, though like you, I don't claim to have all the answers. Still, I try to have some answers.

What you are suggesting is bound to meet plenty of conflict. I am similar to you in that I argue for a moderate to severe upheaval, redesign, and replacement of current curricular practices. You have proof in your experiences that many college-educated adults don't understand math. I have my own proof in my experiences that many "educated" people don't really understand much at all. This problem irks me, as it does you, and I do my best to modify the system on a micro level, in hope that it contributes to a change on the macro level. I have a huge appreciation for not only my own subject matter, but many other subjects. I want others to share this appreciation. Well, you can hardly appreciate something if you don't understand it.

I know how to calculate and balance equations, but I will never claim to understand math on a high level. And yet, I probably "understand" math better than most people who made better math grades than me. While I was exploring, visualizing, theorizing, and thinking in math class, the people who made good grades were simply memorizing "facts" and following instructions.

6. ### PashtunFanatic

Joined:
Jun 17, 2013
Messages:
2,985
435

Mar 20, 2014

Why were you able to "understand" math better, but not be AS successful than those that just memorized facts and followed directions?

If you truly understood it better, wouldn't multiple choice math tests be far easier for you?

7. ### teacherbatmanCompanion

Joined:
Jan 7, 2014
Messages:
121
0

Mar 20, 2014

See ajr's points. The curriculum determines the assessment and therefore what is defined as success in that class. That's not to say I didn't do well on multiple choice tests -- I did. But that doesn't matter -- those kinds of tests hardly check for understanding (as opposed to fact memorization).

I had a girlfriend in high school who got better grades than me, but she would usually ask me for help on her work. When test time arrived, she would have the formulas and techniques memorized, and I wouldn't. Of course, she usually got a better grade on the test. Did she understand math better than me? I highly doubt it. She was lost without specific instructions. I could look at a problem and have an idea how to solve it, perhaps create my own formula to solve it -- without ever seeing instructions. Sometimes I would get the answer wrong, but at least I had ideas... and parts of these ideas would lead to further ideas, and sometimes even "correct" answers. Few if any other people in my class would do that. To them, algebra and geometry were about plugging in formulas that were given to them by the teacher, then finding the "correct" answer through the given technique. Very little deep understanding going on. Sadly, this is how most people are trained to operate in life through their relentless schooling.

8. ### orangeteaConnoisseur

Joined:
Jan 12, 2012
Messages:
1,600
0

Mar 20, 2014

This is an interesting point. What level of mathematics are you speaking of? Do you think high school math teachers need more research experience? What about elementary teachers?

Proof writing is huge in college math, and I try to introduce paragraph style proofs to my students. I think this gets them to think more like mathematicians. The high school I last worked at prided itself on it's math program, but the kids were good at doing math rather than understanding math.

9. ### MathCohort

Joined:
Mar 13, 2011
Messages:
594
3

Mar 20, 2014

I mean but wouldn't the logical thing be to use the formulas provided. Why would you try and use your own formula for something that is proven to work? Sounds like the girl was taking advantage of what she could beating the multiple choice test by using answers to work backwards. I see nothing wrong with that because a math teacher shouldn't give multiple choice. Asking the teacher about why may have helped because if you understand you would get the A.

10. ### MathCohort

Joined:
Mar 13, 2011
Messages:
594
3

Mar 20, 2014

If my math teacher has taught me anything this year it is the why. Thinking about why we are doing the problem this way. I never really looked at the why before this year. I do find it interesting to be able to approach problems more logically.

11. ### orangeteaConnoisseur

Joined:
Jan 12, 2012
Messages:
1,600
0

Mar 20, 2014

Using formulas isn't difficult and doesn't require much though. In math, it's important to learn how to follow steps and use formulas, but also just as important to be able to think creatively.

12. ### orangeteaConnoisseur

Joined:
Jan 12, 2012
Messages:
1,600
0

Mar 20, 2014

I thought he/she had really good points. I didn't see the post as trolling. At all.

13. ### MathCohort

Joined:
Mar 13, 2011
Messages:
594
3

Mar 20, 2014

I agree and that is what I am telling the person I replied to. There is no reason to try and make up your own formula. When it comes time to test your understanding should show through your grade. You can not say oh I completely understand and end up with a C or lower. That grade says you clearly do not. Does that make sense?

14. ### ajrRookie

Joined:
Feb 13, 2013
Messages:
62
0

Mar 20, 2014

On multiple choice (replying to Pashtun's post tangentally)

When a test is administered, usually two things happen:

The question requires some set of facts to be recalled and used in a calculation, and a time limit imposed. This may be something like two minutes per question on a 30-question test, to give you an hour long test. There's little difference here between long form and multiple choice.

This whole premise and format is completely different from my experience with what actually happens out in the world when people try to use math to do things, at all levels. Getting people to pass a test like this is a non-indicator of how well they perform in applying mathematics to their lives and work. Might as well be asking questions on Sanskrit. As a corollary, some excellent mathematicians do really poorly when given impromptu highschool-level math tests - because math tests do not gauge mathematical ability.

A fairly easy real-world math problem might take a few of thinking to solve. Skill in this area requires research and the analysis of information, and often requires programming (something like Python or R), which adds to the length. Most of the math-work time is spent building an accurate mental model of the situation and trying to find the most appropriate mathematical objects that can encode the required behavior. Lots of experimenting, and the solution tends to emerge over time. Facts of all sorts don't usually need to be memorized, because they're looked up, used immediately, and you move on. Formulas, shortcuts, lemmas, and theorems are all either on Wikipedia or in reference books and can be rapidly found.

It's not all that different for "average adults," either.

Most of the very simple problems I encounter business folks having trouble with are smaller versions of this. I had an accountant come to me and ask, "I have the number of people in our state, and I have the number of people who live in the municipality. I need to find the people who live outside the municipality." This person understood calculation, and they understood what their physical problem was. What they didn't have was the confidence to map their problem to a calculation, because rote calculation is not representational in their mind.

Note the core thing here is a simple subtraction problem, not algebra, geometry, or tensor analysis. Elementary school level math.

Another person was wondering why rounding to two different decimal places in Excel produced two different numbers. This was serious, as an off-by-one error represented either \$650,000 too much, or \$650,000 too little.

15. ### MathCohort

Joined:
Mar 13, 2011
Messages:
594
3

Mar 20, 2014

I am not sure I understand ajr... are you saying higher level math is irrelevant? Basically since it is not used outside in the real world?