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Mathematical Common Sense


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I’ve an 11 year old who needs to work on developing common sense when it comes to math. Mental math, estimating—that sort of thing. She goes full steam ahead & doesn’t always use common sense or logic to determine if her answer even makes sense. She’s been dealing with math anxiety for some time, which is most likely the reason for this ‘just get it done’ behavior. Do you have any sites or books that we can use to work through this & develop stronger reasoning skills? Thanks!

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I don't have any suggestions for you but I did wanted to empathize because I have an 11yo DD who does the same thing. She doesn't have math anxiety.....she just rushes to get it done. Yesterday her answer to 4 1/2 divided by 3 was 13 1/2. I wanted to question her as to whether that was a logical answer but refrained (because she'd get defensive) and just erased it and told her to do it again. I've wondered if it is a developmental thing to step back in your mind and consider if the answer is reasonable.

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This problem isn't confined to young people.  I had people working in my lab who had bachelor's degrees in science from highly ranked universities who would make calculation errors that were off by several orders of magnitude and couldn't see that there was anything wrong.  

I suggest simply working with her, saying things like "what will this be approximately?" every single time (it will take a lot of sitting with her).  And when she gets things wrong, having the first step be thinking about whether the answer was reasonable.

Another thing that can be fun is estimating things in the real world.  Once when flying over the Puget Sound, for example, we estimated the volume of water it contained.    

Edited by EKS
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I have an 11-year-old daughter who makes mistakes, too.

I go over problems with her after she makes a mistake and show her ways to think about them.

I also tell her sometimes I forget things and have to think things through.

She often does not completely understand a concept and is trying to rush through, and she just does not know how to think of things that would help her.

I sit with her and work out a problem different ways or show her how I do it.

Yesterday she missed .025 times 100.  She is supposed to be doing the thing where you slide the decimal point over, so she just does that, and doesn’t think about the numbers.

I go through and ask her questions like:  should your answer be larger or smaller than .025?  What is .025 times 10?

I also asked her if she knew what 2 cents times 100 was.  She did know that.  I showed her how that related to .025 times 100.  That seemed to help her a lot.

I go over and over things like this, I am sure I will go over them again.  She does have some intuition when it comes to money, so that is good, but she needs to relate it to decimals, and she doesn’t automatically do that.  
 

Overall I am being patient and telling her I forget things too and have to think about things.  

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I think my daughter also is in the process of seeing it’s worth it to think about if an answer looks reasonable.  I think she has to see it’s worth doing and not some stupid idea mom has.  I think she is starting to see that but it’s not like she sees it already.

With decimals too, she is towards the beginning with them and will have a lot more exposure.  I know for me it helped me when we would measure things with mL in science. Well, I don’t think I did that in 6th grade.  It’s something I know helped me to understand decimals.  
 

I try this with cm and mm on a ruler but — it has not extended for her yet.  

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34 minutes ago, HeighHo said:

The usual solution is to require them to show the check of their work. That check allows them to catch their mistakes.

Yes, we do that. She catches most of the errors, which is something. Perhaps this is something that'll take an insane amount of practice.

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One idea is to have her look at incorrect problems and try to determine the mistakes. Not necessarily her own problems, but each day show a few problems that are incorrect and have her look for what errors were made. Perhaps make it a fun thing, call it Teacher Time and add in a red pen. 

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This is an extremely common problem, and I see it in kids of all ages. I saw it just as much when teaching kids calculus in college. 

For me, the unpleasant discovery was that kids who had no "common sense" often had no robust mental model for what a question even meant. A question can only "make sense" if you can make some sense of what the question is, if you know what I mean. 

As a random example, a LOT of my 7 year old kids would fill in the blank in the question

_ - 6 = 5

with 1. Does that "make sense"? Actually, it depends. If your mental model of subtraction is "take the smaller number from the bigger," then it's an extremely logical answer. 

So when trying to figure out why she's not using common sense, I'd make sure she has access to models that allow her to make sense of the questions. Then practice those models with her. That's what will allow her to figure out whether her answers are reasonable. 

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9 hours ago, Earthmerlin said:

Thanks, folks! It sounds like this might be a typical issue at this point. I'll continue to work with her on this, showing as much patience as I can muster (LOL). 

And remember to stay positive and model the behavior you want to see.  Sometimes after reams of calculations and producing an answer for x, I'll go back to the problem statement and plug in the answer and see if it meets the conditions of the problem statement.  

Yesterday I was reviewing a MathCounts problem with some students, and it involved finding the fraction of an equilateral triangle that was shaded in some weird way.  Finally, after much calculation and multiple pairs of similar triangles I proudly produced the fraction: \frac{14}{15}.  Then I looked at the equilateral triangle and actually said, "Yep, that looks like \frac{14}{15} of a equilateral triangle to me!"  

ETA:  The triangle is the equilateral one in the middle.  (Tell me you can't clearly see it's \frac{14}{15} shaded!)

image.png.6c4afd51bcd314ca50663e72f6a8e835.png

Edited by daijobu
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"What would this look like with objects?" and "Can you turn this question into a story/word problem?" might be helpful questions, to help with showing what the possible parameters of the answer could be.

 

For example, someone sharing 4 1/2 pizzas between 3 people can't possibly give anyone minus pizza, nor can they give anyone more pizza than existed in the first place. If a student establishes this, they know there must be some pizza for everyone (a positive number) and that it can't be more than 4 1/2 (because one cannot share pizza one does not have). This gets more difficult with some of the more obscure branches of mathematics, but at this point, getting a feel for what problems are like outside strict numbers are a good step to getting estimation learned.

 

There are specific resources for estimation included in some curricula (MEP has it integrated in several contexts), but it's also fine to simply take whatever your student was going to do next and provide one or both questions with a few questions each day. It will take some months, if not a couple of years, for the sense to fully generalise.

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