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Mastery vs. Perfection

I receive many questions from homeschoolers about math. This one is about mastery vs. perfection when it comes to math.

Mastery vs. Perfection



Dear Lee,

Thank you very much for sharing so much with the rest of us homeschoolers! I have a question regarding carelessness.  Using math as an example, my 12 yr. old son is working out of Saxon Algebra 1 currently and is understanding everything very well but gets a few wrong (2-4 on average, occasionally up to 6) due almost always to carelessness. He will work the incorrect ones over again and get them correct and truly understands what he did wrong. However, I have decided to cut him back to working only half a lesson in the hopes that he will take his time and strive for 100 percent and if he does not score a 90 percent or higher then he has to re-do the half-lesson over again until he achieves this. What are your thoughts on this?? He is definitely heading in the direction of engineering and I know how important it is for him to be careful.

Thank you for your thoughts,
Theresa

math

Carelessness, especially with math, is a complicated subject.  Homeschool parents want their kids to do well, understand a subject and achieve mastery. We want our children to learn about hard work and the benefits of doing a job well. But there is another side to carelessness that we have to consider.

When I subtract a purchase in my checkbook, I don't always get the answer perfectly right. Sometimes I slip up, get the answer wrong, and have to search until I find the arithmetic error so it balances again. Yet I do believe I have achieved mastery over subtraction!

Mastery is different than perfection. Your child may demonstrate mastery by scoring 90% on an assignment (you may consider they have mastery with less than that). They shouldn't be required to be "perfect" though. To be honest, I did have my children correct all their math errors in their daily work.  When they got it wrong, they corrected it. Like you, I was hoping the tediousness of correcting would encourage them to be more careful in their daily work. Just don't jump from requesting being "careful" to expecting "perfection." Striving for perfection can cause strife and possibly rebellion, because we all know, intuitively, that we simply can't be perfect.

I understand this situation, because I have a math and engineering loving son. I like the idea of holding your child to a high standard of 90%. I like the idea of assigning half the problems so he has time to be more careful. I like the idea of re-doing any problems he misses. Just make sure you steer clear of perfection, and strive instead for mastery. Tell your son that 90% is what you consider mastery.

Do you strive for mastery in your homeschool? Are you frustrated by a child who is careless in his work? Please share!

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Please note: This post was originally published in March 2009 and has been revamped and updated for accuracy and comprehensiveness.

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Comments 12

Guest - Suzanne (princapecos) (website) on Friday, 08 May 2015 12:28

When we used Saxon, we began also using the DIVE CDs. While you may not need that extra help, one thing the DIVE teacher encouraged was giving partial credit for problems if the child did the process right but made an error in part of the problem. Of course, this requires the child to show his work, but that's the only way to know whether or not they truly understand the concept. Marking off only 1/4 or 1/2 for missed problems when possible was a real moral booster for my kids (and for me, too). It's a little more tedious (use the solution manual rather than the answer key), but I think it's worth it. If it's too much work, try just grading odds or evens -- that should be enough to catch any ongoing problems. Or let the child grade daily work himself, while you just grade tests (using the same partial credit system).

When we used Saxon, we began also using the DIVE CDs. While you may not need that extra help, one thing the DIVE teacher encouraged was giving partial credit for problems if the child did the process right but made an error in part of the problem. Of course, this requires the child to show his work, but that's the only way to know whether or not they truly understand the concept. Marking off only 1/4 or 1/2 for missed problems when possible was a real moral booster for my kids (and for me, too). It's a little more tedious (use the solution manual rather than the answer key), but I think it's worth it. If it's too much work, try just grading odds or evens -- that should be enough to catch any ongoing problems. Or let the child grade daily work himself, while you just grade tests (using the same partial credit system).
Guest - Assistant to The HomeScholar on Monday, 11 May 2015 22:45

That's a great idea, Suzanne!
Kids will find that process more encouraging, and certainly more enlightening!
Robin
Assistant to The HomeScholar

That's a great idea, Suzanne! Kids will find that process more encouraging, and certainly more enlightening! Robin Assistant to The HomeScholar
Guest - Jill Pike on Thursday, 03 January 2013 09:09

I do have a son in engineering, and college has encouraged him to take the time (on tests, anyway!) to check and recheck answers. We all make mistakes. You can't recheck everything, there just isn't time. But when the thing is important, it is worth taking the extra time to make sure there were no mistakes.

I do have a son in engineering, and college has encouraged him to take the time (on tests, anyway!) to check and recheck answers. We all make mistakes. You can't recheck everything, there just isn't time. But when the thing is important, it is worth taking the extra time to make sure there were no mistakes.
Guest - Rachel Grebe on Thursday, 06 September 2012 05:14

After a couple of years of stressing out over this same issue with my oldest daughter, I finally consulted with Mr. Art Reed, who has made a very helpful set of instructional DVD's for students who use Saxon. He gave the same advice I later found in his outstanding book, "Using John Saxon's Math." (Mr. Reed actually knew John Saxon, and so is familiar with his philosophy about teaching math and how he intended the curriculum to be used.) He insists that parents should not even grade the daily assignments, much less have their students rework missed problems. (I was doing both, and was causing my daughter to become overwhelmed and to hate math!) Instead, Mr. Reed recommends looking over the daily work to make sure it is completed and that there are no consistent problems with understanding concepts, and basing most of the grade for the course on the test scores (he says that with Saxon, 80% or better reflects mastery, and an occasional 75% can be overlooked) Once I started using that approach--quite reluctantly, I must add--my daughter's test scores began to go up and her math anxiety plummeted! Now in Advanced Math, she is doing well and enjoying math again. I am so thankful that I received and followed this excellent advice, and hope that it will be helpful to others! The Advanced Math course, by the way, is intended to be used over two full school years, using the half-lesson per day method some other readers suggested.

After a couple of years of stressing out over this same issue with my oldest daughter, I finally consulted with Mr. Art Reed, who has made a very helpful set of instructional DVD's for students who use Saxon. He gave the same advice I later found in his outstanding book, "Using John Saxon's Math." (Mr. Reed actually knew John Saxon, and so is familiar with his philosophy about teaching math and how he intended the curriculum to be used.) He insists that parents should not even grade the daily assignments, much less have their students rework missed problems. (I was doing both, and was causing my daughter to become overwhelmed and to hate math!) Instead, Mr. Reed recommends looking over the daily work to make sure it is completed and that there are no consistent problems with understanding concepts, and basing most of the grade for the course on the test scores (he says that with Saxon, 80% or better reflects mastery, and an occasional 75% can be overlooked) Once I started using that approach--quite reluctantly, I must add--my daughter's test scores began to go up and her math anxiety plummeted! Now in Advanced Math, she is doing well and enjoying math again. I am so thankful that I received and followed this excellent advice, and hope that it will be helpful to others! The Advanced Math course, by the way, is intended to be used over two full school years, using the half-lesson per day method some other readers suggested.
Guest - Edjumpoff (website) on Thursday, 24 November 2011 09:05

This was right on time, my 12 year old is doing the same thing. He is also using Saxon and I never thought about cutting the lesson in half. I used to have them redo problems they got wrong, but realize that a lot of it is pure carelessness and stopped. I look for the mastery, if I know they have mastered a concept and just made a mistake I blow it off. However, if I see that answers are wrong involving a concept I address the situation.

This was right on time, my 12 year old is doing the same thing. He is also using Saxon and I never thought about cutting the lesson in half. I used to have them redo problems they got wrong, but realize that a lot of it is pure carelessness and stopped. I look for the mastery, if I know they have mastered a concept and just made a mistake I blow it off. However, if I see that answers are wrong involving a concept I address the situation.
Guest - Linda on Sunday, 31 October 2010 22:53

Good comments. I also have a 12 year old that sounds a lot like Theresa's son. Like her, I only require half a lesson, and he always must correct his mistakes. The other thing that helps is that I now require him to show his work. He has always been good at doing the work in his head, but I believe that at the higher levels, it becomes more difficult, if not impossible. It also leads to those "careless mistakes." Some of those careless mistakes can be avoided, or at least correctly more easily when it's down on paper. It also helps me see what he's thinking. Hope this helps.

Good comments. I also have a 12 year old that sounds a lot like Theresa's son. Like her, I only require half a lesson, and he always must correct his mistakes. The other thing that helps is that I now require him to show his work. He has always been good at doing the work in his head, but I believe that at the higher levels, it becomes more difficult, if not impossible. It also leads to those "careless mistakes." Some of those careless mistakes can be avoided, or at least correctly more easily when it's down on paper. It also helps me see what he's thinking. Hope this helps.
Guest - Pam on Sunday, 22 March 2009 19:28

Thanks, Lee and JW, for the encouraging words. She wasn't like this when she was younger, and has always been a "Math Head". Especially mental math. But, she also used to like keeping her room neat. Now, not so much! I'm really hoping it's an age thing. It just feels strange that she'll be starting geometry soon when she doesn't care if she's accurate with 7x8 and such. It's oddly comforting to know that others are having similar experiences. lol!

Pam

Thanks, Lee and JW, for the encouraging words. She wasn't like this when she was younger, and has always been a "Math Head". Especially mental math. But, she also used to like keeping her room neat. Now, not so much! I'm really hoping it's an age thing. It just feels strange that she'll be starting geometry soon when she doesn't care if she's accurate with 7x8 and such. It's oddly comforting to know that others are having similar experiences. lol! Pam
Guest - J W on Friday, 20 March 2009 08:01

Oh, Pam, that was *so* "me" at age 12/13 in pre-algebra! I'm not a math genius, but I did take math up to calculus, and got good grades thanks to my father's tutelage. Hang in there.

My husband has always been gifted at math/logic and didn't know this problem even existed until he started teaching my daughter :-) Bit of a surprise to him :-) but he's been very patient.

P.S. - I am almost completely incapable of doing mental arithmetic involving anything more than two one-digit numbers. I reverse digits and lose track of which step I'm on.

Oh, Pam, that was *so* "me" at age 12/13 in pre-algebra! I'm not a math genius, but I did take math up to calculus, and got good grades thanks to my father's tutelage. Hang in there. My husband has always been gifted at math/logic and didn't know this problem even existed until he started teaching my daughter :-) Bit of a surprise to him :-) but he's been very patient. P.S. - I am almost completely incapable of doing mental arithmetic involving anything more than two one-digit numbers. I reverse digits and lose track of which step I'm on.
Guest - Lee (website) on Friday, 20 March 2009 05:51

Hi Pam,
I don't know if your daughter will outgrow it or not, but we parents have to stick together and cope somehow. This was how I coped.

And yet... I haven't outgrown my imperfect math in my own checkbook or when I'm estimating the cost of groceries. I sometimes forget to concentrate and miss something. I don't think adults are always perfect at math either. Although we are a bit more likely to laugh about our own math mistakes.

Just musing... I have a nephew that is a calculus teacher at high school. I'm pretty sure he isn't perfect at math either.

Good luck, though!
Blessings,
Lee

Hi Pam, I don't know if your daughter will outgrow it or not, but we parents have to stick together and cope somehow. This was how I coped. And yet... I haven't outgrown my imperfect math in my own checkbook or when I'm estimating the cost of groceries. I sometimes forget to concentrate and miss something. I don't think adults are always perfect at math either. Although we are a bit more likely to laugh about our own math mistakes. Just musing... I have a nephew that is a calculus teacher at high school. I'm pretty sure he isn't perfect at math either. Good luck, though! Blessings, Lee
Guest - Pam on Thursday, 19 March 2009 23:18

I was SO glad to see this! I have EXACTLY the same problem with my 12 yr old daugter. She can miss 20% daily on careless arithmetic errors, when she clearly understands the Algebra. I'm at the end of my rope. I have her finding every error and correcting them. I think I'll try the 90% idea. Is this a 12 yr. old thing that they will outgrow

I was SO glad to see this! I have EXACTLY the same problem with my 12 yr old daugter. She can miss 20% daily on careless arithmetic errors, when she clearly understands the Algebra. I'm at the end of my rope. I have her finding every error and correcting them. I think I'll try the 90% idea. Is this a 12 yr. old thing that they will outgrow
Guest - J W on Thursday, 19 March 2009 20:57

I have a daughter almost the same age who is definitely *not* going into engineering, but who has the same carelessness issues in math. With her, reducing the quantity of work helps tremendously. I'm sure too that she would be a lot more careless if I were still teaching her math. She was absolutely struggling a couple of years ago, and what she needed most was someone who LOVES math (my hubby) to take over her instruction. Enthusiasm is contagious, and I'm sure that helps a child stick to a task. I don't love math. I'm sure my attitude didn't help my daughter. Also, showing how math is relevant to her life keeps it fresh. Adjusting the work to fit to the child's level of ability might also help cut down on carelessness. In homeschool, a child who is ready to move on to the next concept may move on immediately, and if a child needs to linger, he or she may linger. Boredom from spending too long on one concept could very well be at the root of many carelessness issues! I also think carelessness could be normal for 12 years old. I shudder when I think back to when I was that age. For me at age 12, carelessness was just that - "I couldn't care less about this." We want better than that for our kids! That's where delight-based learning comes in. I'm not sure what delight-based learning looks like for math, but I'll bet my husband does!

I have a daughter almost the same age who is definitely *not* going into engineering, but who has the same carelessness issues in math. With her, reducing the quantity of work helps tremendously. I'm sure too that she would be a lot more careless if I were still teaching her math. She was absolutely struggling a couple of years ago, and what she needed most was someone who LOVES math (my hubby) to take over her instruction. Enthusiasm is contagious, and I'm sure that helps a child stick to a task. I don't love math. I'm sure my attitude didn't help my daughter. Also, showing how math is relevant to her life keeps it fresh. Adjusting the work to fit to the child's level of ability might also help cut down on carelessness. In homeschool, a child who is ready to move on to the next concept may move on immediately, and if a child needs to linger, he or she may linger. Boredom from spending too long on one concept could very well be at the root of many carelessness issues! I also think carelessness could be normal for 12 years old. I shudder when I think back to when I was that age. For me at age 12, carelessness was just that - "I couldn't care less about this." We want better than that for our kids! That's where delight-based learning comes in. I'm not sure what delight-based learning looks like for math, but I'll bet my husband does!
Guest - Lisa (website) on Thursday, 19 March 2009 17:03

Good advice. I have my boys rework the math problems they get wrong and expect them to score at least 80% on their work. Working with them every day is a good way to determine whether they have mastered a concept or not.

Good advice. I have my boys rework the math problems they get wrong and expect them to score at least 80% on their work. Working with them every day is a good way to determine whether they have mastered a concept or not.
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