# Math Fact Practice ≠ Fluency

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

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Mar 16, 2014

ajr, I disagree that inflammatory comments are the only way to have a productive debate. Personally, I respond to interesting or relevant comments that add something to the conversation. If you aren't getting responses, I'd consider your content, not the inflammatory nature.

2. ### SF_Giants66Cohort

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Mar 17, 2014

In fourth grade, the way I memorized multiplication tables is each time we misbehaved, forgot to write our name on our homework paper, were talking in class, we had to write out the tables two times each.

I wasn't the most well behaved child so I memorized them all very quickly. One day I had to write the tables out 8 times for several different infractions.

3. ### DrivingPigeonPhenom

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Mar 17, 2014

Wah, I'm sad that this thread became about how to not be a jerk when posting. I just ignored said post, and enjoyed the math discussion we were having.

Oh, well.

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Mar 17, 2014

Me too.

5. ### PashtunFanatic

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Mar 17, 2014

Stop talking about the "jerk" part and only respond to the substance of the topic which he wrote about.

6. ### TeacherGroupieModerator

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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.

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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.

8. ### a2zVirtuoso

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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.

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Mar 19, 2014

a2z, you have flexible thinking!

10. ### teacherbatmanCompanion

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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.

11. ### PashtunFanatic

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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?

12. ### teacherbatmanCompanion

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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).

13. ### orangeteaConnoisseur

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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.

14. ### MathCohort

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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.

15. ### MathCohort

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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.

16. ### orangeteaConnoisseur

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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.

17. ### orangeteaConnoisseur

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Mar 20, 2014

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

18. ### MathCohort

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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?

19. ### ajrRookie

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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.

20. ### MathCohort

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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?

21. ### ajrRookie

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Mar 20, 2014

Making up formulas

I'd argue that learning to make up formulas is one of the single most important things a person can learn to do in mathematics, and is something that just isn't taught. It's not math, per se, but it is an essential skill. It is the first stage in learning to write mathematics, which is a big topic. Ironically, early childhood education is probably the best at this, because it's what someone is doing when they learn to represent the size of a collection of objects with a symbol - four apples.

Writing your own formula is like writing a short essay in English.

A formula contains someone's full understanding of some part of their situation, with every assumption they're making. Physics is rife with examples of this; if you stop and read the formula as though you were reading English, you will come to an understanding of the situation as though you were reading someone's description (because you ARE reading a description). However, people get flustered because of how dense a formula is; it's one line, and yet contains a huge amount of explanatory power. You have to be willing to do a close reading of an equation in the same way one does in a literature class. People are used to natural language, where they expect to invest a particular amount of time to read a long passage.

A short equation can be a very, very long passage in terms of information content, but it has none of the familiar markers of expected time requirement.

Another difficult judgement call is knowing when to write your own formula, versus looking one up.

22. ### orangeteaConnoisseur

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Mar 20, 2014

No, because there is a point in coming up with your own formula. It's about learning how to think rated than just repeat what everyone else has already discovered.

Tests don't always test how well students understand something. Math tests often test his well a student has memorized a formula or followed a process. They don't always test how well a student actually understands something.

23. ### MathCohort

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Mar 20, 2014

So lets say the teacher says factor a polynomial... however, the quadratic formula is the next step. You are going to take time and waste it on trying to be the next genius because apparently you are so smart? Just because other people have used it you are not being unique enough? Then just let all teachers teach what ever when ever. That makes no sense. Tell the students that is quite productive and who wants to sit down and say oh well this is a great formula. Well I hope theirs has been tested and works for the different types of quadratic equations there are.

24. ### MathCohort

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Mar 20, 2014

I would say they have some questions on the math test if you want to check for a more thorough understanding. All of my tests have something to explain on it usually the main concept. To see do we understand what we are doing and why. I do not see that as hard or difficult. If the case is most math tests do not check for something. If it does not it is not the student's fault but instead the teacher's fault. All teachers should definitely check for an understanding.

25. ### ajrRookie

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Mar 20, 2014

I'm pointing out the "simple" examples because I want to illustrate my encounters with innumeracy in the college-educated adult population at every level of mathematics. I don't mean that to say one 'level' or another is better than any other.

I feel most levels of math are relevant, but every field of mathematics is not going to be broadly relevant to every field or every problem. I used graph theory the other day to solve a game I didn't want to play, and I use basic addition frequently. It just depends on what tools a person finds interesting (or doesn't find interesting).

I think the difficult thing to get across is how wide mathematics is as topic, and we just can't lump everything in together.

We usually present it as a linear progression, from one subject to the next, but there's no such progression in the field as a whole. Logic and set theory are the closest thing to a starting point we have, but there's a ton of variation even in those fields. Note, this is not me advocating we teach some kind of formal set theory or logic as an introduction to math.

26. ### MathCohort

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Mar 20, 2014

Plus instead of thinking of it as just using what someone else has discovered... you can simply explain why the formula works.

27. ### orangeteaConnoisseur

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Mar 20, 2014

I'm not sure if I understand what you're saying.

When I teach the quadratic formula, I help my students derive the formula themselves rather than just making them memorize it. They do memorize it, but they also keep the satisfaction of having "discovered" it themselves.

This doesn't work for all formulas, because there are definitely some formulas that high school students can't derive by themselves. But, as a math teacher, I still try to help my students discover as much as possible for themselves. I still do lecture, because I think this aspect of teaching math is important, but I also treat my students as mathematicians.

Teachers can still follow the curriculum while giving students the freedom to explore.

28. ### teacherbatmanCompanion

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Mar 20, 2014

This is true for pretty much all fields. The idea that learning must be linear and each concept must be "mastered" before going on to the next step, gives students a false impression of the subject, and of a good learning process.

In reality, a good learner will forever revisit concepts with a new perspective and build upon those concepts.

Personally, I never had a linear learning style. Students like me are set up from the very beginning to do poorly in school. If I come to a greater understanding of math a year after I'm done with the class, I still got a C in the class. :help: Worse, it is now assumed by everyone who looks at my grade that I "don't know math" as much as the A students.

29. ### PashtunFanatic

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Mar 20, 2014

I understand. My point is the curriculum is asking for very low level knowledge. Shouldn't someone with a deeper level of understanding be able to fully comprehend and excel at the lower fact memorization level?

I totally agree about teaching math for depth of understanding. My point is, if we are teaching for depth of understanding, I would think a typical multiple choice test would be a walk in the park.

30. ### PashtunFanatic

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Mar 20, 2014

Why are you thinking you have great depth in understanding of math that you are getting Cs?

Can you tell me what this looks like in the 4th grade? What does a 4th grade student with a deep understanding of math, yet get Cs in 4th grade curriculum looks like. Some examples if you have time.

31. ### PashtunFanatic

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Mar 20, 2014

Agree 100%.

Do you believe, that students who are allowed freedom to explore and make sense out of math should have an easier time with the low level standardized test type multiple choice questions?

32. ### ajrRookie

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Mar 21, 2014

Just out of curiosity, what kind of low-level questions and at what grade are we asking about?

33. ### PashtunFanatic

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Mar 21, 2014

4th grade. Say multiplying 2 digit by 2 digit numbers. Adding fractions with unlike or like denominators, adding or subtracting numbers in the ten thousands.