Writing a paragraph to show how to solve a math problem???? Opinions?

[lynn]

My sister is a teacher 4/5th grade math.  The new common core math, which she embraces has kids writing paragraph telling why one number is larger than another?  If they answer problems correctly but do not include a written explanation it will be marked wrong??  

 

Would your child be able to write out why he/she arrived at a certain solution to a problem.   

 

In a way I think it's helpful to understand why and how and on another level it has me scratching my head....

 

I was always good in math but only an average "C" student when it came to language arts anything.  Writing was not something I was good at or took me a long time to gather my thoughts and get something on paper.

 

 

[LMA]

One of the CC Math Practice Standards is: Reason abstractly and quantitatively. So this type of question fits this standard and is a concrete way of showing that the standard has been met. Having the kids reason in their minds is not concrete. It has to be written down.

 

And this is nothing new. The textbooks have been putting these types of questions in for many editions. Now it's official.

[Arcadia]

Before common core was "invented", this style was implemented as math journals even in the lower elementary grades from Kindergarten.

[lynn]

One of the CC Math Practice Standards is: Reason abstractly and quantitatively. So this type of question fits this standard and is a concrete way of showing that the standard has been met. Having the kids reason in their minds is not concrete. It has to be written down.

 

And this is nothing new. The textbooks have been putting these types of questions in for many editions. Now it's official.

 

 

Interesting, I have not come across it in math with my kids who were in ps.  Was it something that was being tested over the years in some school districts.  Just curious?  

[lynn]

Before common core was "invented", this style was implemented as math journals even in the lower elementary grades from Kindergarten.

 

I know in lower grades they had to show pictures of how and why.     Here you would think it to be a new concept or people are just against the change to cc

[regentrude]

It is extremely useful. Putting math into words clarifies the process and helps detect when a student simply memorized a procedure without understanding.

I ask my children to narrate their solutions to me and to talk me through, especially when they got stuck.

Of course this is not feasible for a classroom teacher, so asking for a written explanation is the next best thing.

Naturally, the response should not be judged about beautiful writing or good language use, i.e. as an "English" assignment" ; it should be about logic of the argument and conceptual understanding, however poorly written.

[crazyforlatin]

AOPS Pre-algebra asks kids to explain concepts after solving a couple of problems.  DD doesn't always write it (there's only so much writing she can tolerate for one day) but she does narrate and sometimes I'll condense it for her in her notebook to show that she did actually do the work.  I find it useful for her to organize her thoughts into coherent sentences.

 

 

[bettyandbob]

Doing this helps the child actually understand the math. Many children fly through multiple levels of math without really knowing what they actually did to manipulate the numbers. Then they get to algebra or algebra 2 and nothing makes any sense. I teach algebra 2, so I see it all the time. 

[Tammi K]

I drives me absolutely insane! I hate it.

 

My youngest is just starting 7th grade and doing Algebra II, He understands math. He 'gets' it. But, can't 'EXPLAIN' it. That he can't verbalize what he knows to be true doesn't make him less of a mathematician. I agree, he probably would be a lousy teacher but he's an excellent manipulator of numbers and intuitively see how they work. 

 

I liken this to asking my piano major or my art major daughters to verbally explain why/how they play a certain piece of music or how they paint a picture. How do you write about something that just comes from somewhere inside of you?

 

I see the intent - if you can explain the process you understand the process. I just don't believe the revers is true...just because you can't explain doesn't mean you can't do.

 

End of my rant....

[Caroline]

It is extremely useful. Putting math into words clarifies the process and helps detect when a student simply memorized a procedure without understanding.

I ask my children to narrate their solutions to me and to talk me through, especially when they got stuck.

Of course this is not feasible for a classroom teacher, so asking for a written explanation is the next best thing.

Naturally, the response should not be judged about beautiful writing or good language use, i.e. as an "English" assignment" ; it should be about logic of the argument and conceptual understanding, however poorly written.

Yes, this, exactly.

[Mom0012]

I drives me absolutely insane! I hate it.

 

My youngest is just starting 7th grade and doing Algebra II, He understands math. He 'gets' it. But, can't 'EXPLAIN' it. That he can't verbalize what he knows to be true doesn't make him less of a mathematician. I agree, he probably would be a lousy teacher but he's an excellent manipulator of numbers and intuitively see how they work. 

 

I liken this to asking my piano major or my art major daughters to verbally explain why/how they play a certain piece of music or how they paint a picture. How do you write about something that just comes from somewhere inside of you?

 

I see the intent - if you can explain the process you understand the process. I just don't believe the revers is true...just because you can't explain doesn't mean you can't do.

 

End of my rant....

I do understand the benefits of having the kids write out explanations for math, but it would drive me insane as well.  My son has soooo much difficulty communicating, but understands very well.  Math is a pretty strong subject for him right now, but if he had to start writing paragraphs about it, that would change very quickly.

[Spy Car]

A critical ability IMO. We tend to do much of this orally (easier one on one than in a classroom) but being able explaining ones reasoning is vital and something that should be expected from the beginning IMO.

 

Bill

[TechWife]

I think it's a great skill to have - being able to explain the why behind something as well as the how. However, from the curriculum samples that I've seen and from my conversations with teachers who are actively involved in aligning their school curriculum with the common core, it seems that they are asking these types of questions (at least some of them) at developmentally inappropriate ages. 

 

Of course, this is JMHO and YMMV. 

[Ellie]

I think it is stupid.

[Word Nerd]

I doubt they have to write a paragraph to explain every answer or that this practice will continue indefinitely. I had to show my work in school many years ago, so that aspect is nothing new.

[Melissa in Australia]

Just another way to make dyslexic children feel dumb.

 My ds19 couldn't even show workings out for his math until  16. Psychology tests show the reason why the answer just magically appeared in his head is because he uses long term memory for mathematics, not short term memory.

[Tammi K]

Just another way to make dyslexic children feel dumb.

 My ds19 couldn't even show workings out for his math until  16. Psychology tests show the reason why the answer just magically appeared in his head is because he uses long term memory for mathematics, not short term memory.

 

I agree. While my son isn't dyslexic, he has a much harder time communicating in written or verbal form and he computes/calculates most of the answers in his head.

 

When he was 2nd or 3rd grade we would rework problem so that they contained fractions or decimals just so he would have to write out the steps. No luck, he just did those in his head as well.

 

Having to 'explain' is torture for him.  Actually, when he was younger, he one day asked me if I knew how to do a problem and when I said yes, he wanted to know why I had to ask him!

[Tammi K]

A critical ability IMO. We tend to do much of this orally (easier one on one than in a classroom) but being able explaining ones reasoning is vital and something that should be expected from the beginning IMO.

 

Bill

 

Don't you think that being able to understand the process and being able to explain the process are two vastly different things? Not everyone will be a 'teacher' and need to the ability to explain the process.

 

Would you mind explaining WHY you think it is vital and should be expected? (No snark intended. I'm really curious.)

[FloridaLisa]

My sister is a teacher 4/5th grade math.  The new common core math, which she embraces has kids writing paragraph telling why one number is larger than another?  If they answer problems correctly but do not include a written explanation it will be marked wrong??  

 

Would your child be able to write out why he/she arrived at a certain solution to a problem.   

 

In a way I think it's helpful to understand why and how and on another level it has me scratching my head....

 

I was always good in math but only an average "C" student when it came to language arts anything.  Writing was not something I was good at or took me a long time to gather my thoughts and get something on paper.

 

 

Before common core was "invented", this style was implemented as math journals even in the lower elementary grades from Kindergarten.

 

My first reaction: will this adversely affect boys? Will is affect their math grades more than girls'? My hunch (and years of reading about learning/classroom differences between boys and girls) is that across the board, these types of assignments would be tolerated better and performed better by girls.  And while it should be graded on content, there are teachers that grade everything based on neatness/handwriting.  Ask my neighbor who (though super pro-ps) pulled her son after he continued to receive F's for perfect spelling tests that weren't neat. 

 

Lisa

[matrips]

I'm sure my kids would hate to write that. But I do have them walk me through problems, and tell me how they've solved them. Or I'll even have them walk me thru a mult or division problem and explain the steps. It makes them think and process it in a different way to explain and put it into words.

[regentrude]

Don't you think that being able to understand the process and being able to explain the process are two vastly different things? Not everyone will be a 'teacher' and need to the ability to explain the process.

 

Would you mind explaining WHY you think it is vital and should be expected? (No snark intended. I'm really curious.)

No, I do not think those are really that vastly different. Actually, being able to explain something to another person is the highest level of understanding. If somebody REALLY understands a concept, he can explain it.

I do not teach math, but physics, and I have found a very strong correlation between a student's ability to explain what he is doing and his conceptual understanding.  In students and professional scientists, you only very rarely come across genius types who "see" their physics but have a communication disorder like Aspergers that renders them incapable of communicating their findings. In almost all cases, for neurotypical students, an inability to explain is related to a lack of conceptual understanding.

 

I can think of many reasons why talking (or writing) about math is important:

 

Being able to put a mathematical process into words strengthens logical thinking. Logic is first, algorithm is second.

Conversely, mathematics problems typically have their origin in a problem that is first a "word problem " (I hate the expression!) and usually do not occur in the form of a neatly isolated equation (unless in a math textbook). In order to put an actual problem into an equation, the student needs to translate between "words" and mathematical concepts.

Also, talking through a math problem is a great tool to resolve questions. Very often, once the student has been made to formulate a question in plain words, he will notice that he does no longer require the answer, because the act of organizing his thoughts into coherent sentences has made him see the pattern more clearly.

Lastly, communication is an important skill. Even the most gifted mathematician who intuitively "sees" math will have to communicate his findings to other mathematicians; a solution or proof only he understands but can not communicate is useless. (This is not only true for mathematicians, but for all sorts of related STEM disciplines. I single out STEM, because for humanities people the need to write about their findings is well accepted and undisputed). Not just teachers need to communicate.

 

 

 

[abba12]

I think it is absolutely vital to be able to explain how you reached an answer in math. I am relearning some math at the moment and finding I recognize the formulas but have no idea how or why they work which is why I forgot them so easily. To me, they need to be able to manipulate the numbers, and know how to manipulate them, in what way, and why. Just throwing a formula on it shows no understanding and will make it hard to look at abstract math later where you don't know straight away which formula to use.

 

However, writing a paragraph about why two is larger than one? I don't know what I would say... because two is higher on the number line? That's not a paragraph. It is an inappropriate use of the method.

 

Also, a child needs to be able to show they understand it, but they shouldn't have to write it, or necessarily explain it in a way which uses the correct vocabulary if that is strongly against their personality. I couldn't imagine writing multiple paragraphs each math lesson. I would do it orally, once in awhile, just ask why they know that, how they got that answer, talk it through a little.

[Tammi K]

Don't you think that being able to understand the process and being able to explain the process are two vastly different things? Not everyone will be a 'teacher' and need to the ability to explain the process.

 

 

 

I can think of many reasons why talking (or writing) about math is important:

 

Being able to put a mathematical process into words strengthens logical thinking. Logic is first, algorithm is second.

Conversely, mathematics problems typically have their origin in a problem that is first a "word problem " (I hate the expression!) and usually do not occur in the form of a neatly isolated equation (unless in a math textbook). In order to put an actual problem into an equation, the student needs to translate between "words" and mathematical concepts.

Also, talking through a math problem is a great tool to resolve questions. Very often, once the student has been made to formulate a question in plain words, he will notice that he does no longer require the answer, because the act of organizing his thoughts into coherent sentences has made him see the pattern more clearly.

Lastly, communication is an important skill. Even the most gifted mathematician who intuitively "sees" math will have to communicate his findings to other mathematicians; a solution or proof only he understands but can not communicate is useless. (This is not only true for mathematicians, but for all sorts of related STEM disciplines. I single out STEM, because for humanities people the need to write about their findings is well accepted and undisputed). Not just teachers need to communicate.

 

 

I think it is absolutely vital to be able to explain how you reached an answer in math. I am relearning some math at the moment and finding I recognize the formulas but have no idea how or why they work which is why I forgot them so easily. To me, they need to be able to manipulate the numbers, and know how to manipulate them, in what way, and why. Just throwing a formula on it shows no understanding and will make it hard to look at abstract math later where you don't know straight away which formula to use.

 

However, writing a paragraph about why two is larger than one? I don't know what I would say... because two is higher on the number line? That's not a paragraph. It is an inappropriate use of the method.

 

Also, a child needs to be able to show they understand it, but they shouldn't have to write it, or necessarily explain it in a way which uses the correct vocabulary if that is strongly against their personality. I couldn't imagine writing multiple paragraphs each math lesson. I would do it orally, once in awhile, just ask why they know that, how they got that answer, talk it through a little.

 Absolutely, I agree with all of the above. I never meant to imply that I am satisfied with 'plug and chug'. We work a lot of word problems and situational math - a  "Here's a real life problem figure out how to solve it." type of thing.  I've always felt the purpose of learning math was to be able to use math. I've always believed that if he can USE math in a meaningful way, he is under no obligation to explain his thought process to me to convince me of his understanding. Although I do see the point of being able to communicate ideas especially in scientific fields - an area for us to work on for sure.

 

But, like abba12, my experience is that kids are asked to explain ridiculous things in paragraph form.  For example, on a placement test last year my son dealt with this: " Tom's classmates had several siblings. One had 2, one had 3, and one had 4 siblings. What was the average number of siblings Tom's classmates had? Explain how you arrived at  your answer. Show your work."  Mind you, this was a 5 or 6th grade test and my son was doing Algebra I at the time, so he was several years beyond simple averages.  He wrote: 1) The answer is 3  2) I looked at the problem and could see that the answer was 3 because it's the average. I didn't need do any work.

 

It was marked incorrect. The only problems he had incorrect were the ones requiring a written explanation. He didn't need to 'do the math' because he knew the answers. But, he was really upset that he didn't 'do well' on the test. Every math problem was correct but he didn't pass the test to their satisfaction because he did it  in his head.  How does a child explain what he knows in his head?

 

I believe there is great benefit to being able to coalesce your ideas at higher levers of mathematical thinking but, as abba12 , explaining why 3 is bigger than 4 on a number line is, in my opinion, not crucial to mathematical achievement. 

 

I appreciate your replies and have some new ideas to consider. It appears written explanations won't go away, so he better learn how to do them!

[Arcadia]

My first reaction: will this adversely affect boys? Will is affect their math grades more than girls'? My hunch (and years of reading about learning/classroom differences between boys and girls) is that across the board, these types of assignments would be tolerated better and performed better by girls. And while it should be graded on content, there are teachers that grade everything based on neatness/handwriting. Ask my neighbor who (though super pro-ps) pulled her son after he continued to receive F's for perfect spelling tests that weren't neat.

 

Lisa

When my older was doing envision math in public school, common core was not created yet. He has 1 or 2 multi-step word problems that requires him to write out the steps daily. I'm sure that would vary by classrooms as teachers implement the textbooks and workbooks their way.

He writes neatly but slowly and gets accomodated. I am pretty sure our assigned school's secretary always assigned kids that need accomodations to certain teachers.

I do encounter lady teachers who teach K-3 that have a fear of math and think boys do better in math and girls in language arts. It is unfortunate that your neighbor's son had a teacher who cares more for aesthetics.

Writing in math is very procedural and there is vocabulary that goes with it. Kids are taught to use words like add, take away, times (multiply) to describe what they are doing. They can explain pictorially by drawing the objects, drawing bar diagrams or whatever means they can think of. I have not encounter any teacher who expect a grammatically correct, perfect spelling paragraph from a student for a word problem.

[Catwoman]

I think it is stupid.

:iagree:

 

I'm right there with you, Ellie. :)

 

I am all in favor of making sure a kid has a clear understanding of the concepts, but I think it is an incredible waste of time to make them write paragraphs about it.

[regentrude]

 But, like abba12, my experience is that kids are asked to explain ridiculous things in paragraph form.  For example, on a placement test last year my son dealt with this: " Tom's classmates had several siblings. One had 2, one had 3, and one had 4 siblings. What was the average number of siblings Tom's classmates had? Explain how you arrived at  your answer. Show your work."  Mind you, this was a 5 or 6th grade test and my son was doing Algebra I at the time, so he was several years beyond simple averages.  He wrote: 1) The answer is 3  2) I looked at the problem and could see that the answer was 3 because it's the average. I didn't need do any work.

 

It was marked incorrect. The only problems he had incorrect were the ones requiring a written explanation. He didn't need to 'do the math' because he knew the answers. But, he was really upset that he didn't 'do well' on the test. Every math problem was correct but he didn't pass the test to their satisfaction because he did it  in his head.  How does a child explain what he knows in his head?

 

By being trained to write out problems in a systematic manner, beginning from the definition of a quantity. To some degree, this is a question of training and habit. So, an explanation for the above problem could simply have included the definition of the quantity "average" - and then the answer would most likely have sufficed.

(As an aside: "Show your work" and "write a paragraph" are two very different requirements. Requiring a verbal explanation probes much deeper into conceptual understanding than merely requiring mathematical work to be shown, something students without thorough conceptual understanding and mere procedural memory may be able to accomplish.)

 

The "seeing answers" and not showing work will eventually become an issue for students, and I see great benefit in training them early on to document a math problem while they are still in the stage of not absolutely needing it, so that they have the skill developed when the math gets so hard that they must write all steps in order to be able to solve the problem. That point may be at different stages in different students' math careers, but it will come. I can see at my college students' work whether they have been taught these skills well back in middle school math. Those who have not been taught how to write out problems and who have gotten away with half-mental and undocumented problem solving struggle.

As a practical incentive for the students, it helps to remind them that, should there ever be a mistake, an instructor can only determine whether it was a simple arithmetic error warranting a small deduction or a major conceptual mistake punished by a large deduction, if the student documents his problem solving process. (Of course this is not the purpose of the documenting, but it seems to be an argument that gets students to do it.)

[Spy Car]

Don't you think that being able to understand the process and being able to explain the process are two vastly different things? Not everyone will be a 'teacher' and need to the ability to explain the process.

 

Would you mind explaining WHY you think it is vital and should be expected? (No snark intended. I'm really curious.)

 

Regentrude made most of the points I might have made most ably, so I won't repeat them.

 

I would just add having a child (especially in a one on one situation, where it is feasible) explain their reasoning as a matter of standard course achieves a couple other purposes.

 

It lets the parent/teacher know that the child really understand the mathematics behind their answers. nothing can slip past just because they got "the right answer." I was famously tough at home, giving "zero credit" for correct answers that could not be explained or justified with reason and/or an informed discussion of how the problem was solved. If there is/was a problem a parent/teacher spots it straight away, and can work to make sure the lack of understanding is addressed. This is no small thing, as compounding "illusions of competency" are (in my estimation) a prime source of ultimate failure and problems. if you have to explain the math, you can't fake it. You can't just "memorize," you have it prove it.

 

This also leads to a far deeper sort of learning,as the student is motivated to really understand the mathematical axioms and how they factor in the problem-solving and explanations of such.

 

Language, vocabulary for ideas, and articulation all cement learning and understanding. And to repeat one point from above, I consider material mastered when a child is able to successfully teach others. And that means teaching for understanding, not just teaching procedures. So starting that training from the beginning is vital to education IMO.

 

Is this always well implemented school math books? In my limited experience, no. And it is harder for young children to write than it is for them to discuss (an advantage of home education). But the importance of being able to explain the work remains vital to the learning process IMO.

 

Bill

[regentrude]

It lets the parent/teacher know that the child really understand the mathematics behind their answers. nothing can slip past just because they got "the right answer." I was famously tough at home, giving "zero credit" for correct answers that could not be explained or justified with reason and/or an informed discussion of how the problem was solved. If there is/was a problem a parent/teacher spots it straight away, and can work to make sure the lack of understanding is addressed. This is no small thing, as compounding "illusions of competency" are (in my estimation) a prime source of ultimate failure and problems. if you have to explain the math, you can't fake it. You can't just "memorize," you have it prove it....

 

Is this always well implemented school math books? In my limited experience, no.

 

Because doing it would require the math teachers to have thorough conceptual understanding themselves. A teacher who has memorized procedures, can teach only algorithms, but does not truly understand (and hence also can not explain) why, for example, division by a fraction is multiplication with the inverse, will be incapable of evaluating students' explanations. Teaching math by discussing math requires subject expertise and not just a solution manual. Sadly, that is asking too much of many math teachers.

[Arcadia]

As a practical incentive for the students, it helps to remind them that, should there ever be a mistake, an instructor can only determine whether it was a simple arithmetic error warranting a small deduction or a major conceptual mistake punished by a large deduction, if the student documents his problem solving process. (Of course this is not the purpose of the documenting, but it seems to be an argument that gets students to do it.)

Actually that is how my kids school math tests are marked. The final answer gets about 25% of the marks for that question.

Giving marks for working also acts as a deterent to cheating. It is hard to copy both working and answers of your exam hall neighbors be it for math or science exams.

[Tap]

19yo son was taught this method all the way back in 1st grade.  WASL testing (Washington state's former state testing) used this method.    The had to use pictures and/or words in some problems and just words in others (some problems didn't  really work with pictures)

 

Definitely Not new. 

[NilaGroveSchool]

Math = Language Class  ????????????????? 

 

Horrible idea!  This idea is a stupid and dangerous use of time.  Teach the kids / future adults number skills, PLEASE!  It is sorrily needed!    

 

Why would we waste number skills "time" on language skills????????? 

 

 

 

I do have an alternate idea... the schools accept a Math essay INSTEAD of a writing or literature essay...

 

Literature / Writing Class = Essays on Math instead, ...awesome idea!!!!!!!!!!!!!!!!!!!!!!!!!!!!

 

 

[bettyandbob]

If a student is actually really good at math and pursues a STEM field he/she will need to communicate what he/she is doing at higher levels. 

 

When I was working as a patent attorney one thing that got a lot of billable hours was just talking to inventors on the phone and translating engineering-speak into English. A patent is to be written in English. Knowing how to explain in English would have saved their corporations a lot of money. Since my undergrad is in Chem, it is fairly easy for me to go from engineering-speak to English. I was a fairly inexpensive attorney, but my time would have been prohibitively expensive for an independent inventor or small corporation--so those of you who want to encourage your dc to be entrepreneurs need to encourage good communication in English.

 

A decade ago my neighborhood high school became an IB school. A few years back the dd of a friend transferred there from the districts nationally recognized tech high school (recently rank the no. 1 public high school nationally). She was chosen to speak at presentation to parents about at meeting regarding redistricting boundaries for the neighborhood high school. Many people were against attending -- one argument was IB was too liberal arts. She transferred in the middle of 10th grade and had excellent grades at both schools. One point she made was that at the tech high school, which has AP courses and very high level math and science she learned how to solve differential equations. At the IB school she learned not only the formulas for solving differential equations, but she could identify when use of a differential equation was the best approach. Because she could really explain the math she understood the  best applications of the math. That was something she felt was not emphasized at the tech high school. This young woman is set to begin a PhD in Engineering next year and she has been working through her college years at a start up engineering firm in the northeastern city where her college is located. At that parent meeting years ago, several parents told her, she had "ruined her life" because she had transferred from the tech high school. Ironically, she transferred because she was not getting the strong understanding of engineering and math she wanted and felt she needed to pursue engineering. 

 

In homeschool, you do not have make a child write out his/her explanation. You can go over this orally. Unless your child has LDs that affect communication, there is no reason not to expect your child to be able to explain what is happening. In the classroom, a teacher cannot go through the explanation one on one with each student, so written explanations would be a requirement. Most teachers do not grade heavily for spelling and grammar outside of language arts--in math they are looking for the content of the subject, not the mechanics of the writing. 

 

It is true that many elementary teachers do not understand math and  often give formulas with little explanation. I've only seen teachers' manuals to a few of the nationally used math curricula. Most of them do provide some of this type of explanation in their notes to teachers. So, a teacher of elementary students who has poor math skills would have something to look back at to compare student explanations. (another topic would be my pet peeve that every time I've applied to recertify as a teacher, I am required to take a writing exam and write essays, but I have never been asked to solve basic Algebra, which I think even a first grade teach should be able to do. ). 

[mom2bee]

Because doing it would require the math teachers to have thorough conceptual understanding themselves. A teacher who has memorized procedures, can teach only algorithms, but does not truly understand (and hence also can not explain) why, for example, division by a fraction is multiplication with the inverse, will be incapable of evaluating students' explanations. Teaching math by discussing math requires subject expertise and not just a solution manual. Sadly, that is asking too much of many math teachers.

 

I agree with the majority of whats been said in most posts. I don't agree with the "Teaching math by discussing math requires subject expertise..." part, simply because I can teach the majority of the K-8 math sequence (statistics is my weakest area and I'm no good at geometric proofs.) by discussing and illustrating via examples, analogy and "applied" problems. But  I am no expert. I can also teach/tutor PreAlgebra - Calculus 2* and Differential Equations* and I can discuss the work fairly well but I'm still no expert. I'm an amateur at mathematics, many of my colleagues can do a much better job than me. Most of us are just high scoring math students, there are a few graduated math majors (though one guy had a BS in Math and couldn't assist with anything above College Algebra and the first couple of chapters of PreCalc and claimed to have never taken Diff. Eq.) and a couple of adjunct professors.

 

*Actually, I know that I'd need an answer manual for Differential Equations and parts of Calculus 2. I understand it, I just can't explain it well...(yet).

[In The Great White North]

 It makes sense to be able to explain the steps of a problem in real words. The example, however, had no steps.

 

 I would have no problem with a picture demonstrating this, or even a few words, but a whole paragraph?  By 4th or 5th grade, the students should know that a paragraph has a thesis sentence and several supporting sentences.

 

Would someone please give a sample of a paragraph explaining why 5 is bigger than 4? 

[EKS]

I agree that explaining one's reasoning is very important.  However, as a parent of two boys who are very good at math but who each have learning disabilities that interfere with writing output, having to write a paragraph to explain their reasoning at age 9/10 would have been extremely difficult and emotionally charged.  Which is why we homeschool(ed).

[Jen500]

Good grief.  The kids are not writing page long essays for every math problem.  A paragraph? My guess is that for these problems, 2 or 3 sentences will suffice for their paragrph. I don't understand the uproar. If an elementary student can not put a few words down on paper, then there are bigger fish to fry. For those who have documented learning disabilities, since this is ps, they will probably be exempted. I see the opponents to this as contributing to the dumbing down of education.

 

I agree. For the past few years, the math instruction/practice in our school district has been very heavy on multiple choice questions for 'test prep'. Many of them poorly worded. I am glad to see they are now replacing many of the multiple choice problems with actually solving math problems and some 'explain your answer' in a few sentences problems.

[Dmmetler]

I think part of the problem is if the person scoring it isn't used to how math is written. AOPS, even in Beast Academy and Pre-Algebra, is pretty heavy on the explanation and expecting kids to explain. But they're also defining properties and terms very deliberately, and then using these in explanation, so that the kids have the language to use, and modeling breaking a problem down into steps as you explain. That's appropriate math writing for that age group, and it's something that most kids can do, at least orally, in the elementary/middle school age group.

 

But I have indeed seen papers that kids are doing in the waiting room at tumbling or dance and some of the kids are indeed trying to write essays on how to solve a math problem. Which often DOES seem to lead to frustrated kids who know exactly how the math works, but are having trouble putting the idea that adding zero to a number doesn't change the number's value into words.

 

 

[Catwoman]

But I have indeed seen papers that kids are doing in the waiting room at tumbling or dance and some of the kids are indeed trying to write essays on how to solve a math problem. Which often DOES seem to lead to frustrated kids who know exactly how the math works, but are having trouble putting the idea that adding zero to a number doesn't change the number's value into words.

:iagree:

 

I think it can also lead to kids hating math class, even if they are good at it and would normally enjoy it. Understanding the concepts is important, but having to write paragraphs about them seems like overkill to me. I always liked math, but if I'd had to write little essays about it when I was a kid, it would have sucked the fun right out of it for me -- and I was a child who also enjoyed writing, so I can only imagine how a writing-averse kid would feel about it.

[regentrude]

I agree with the majority of whats been said in most posts. I don't agree with the "Teaching math by discussing math requires subject expertise..." part, simply because I can teach the majority of the K-8 math sequence (statistics is my weakest area and I'm no good at geometric proofs.) by discussing and illustrating via examples, analogy and "applied" problems. But  I am no expert. I can also teach/tutor PreAlgebra - Calculus 2* and Differential Equations* and I can discuss the work fairly well but I'm still no expert. I'm an amateur at mathematics, many of my colleagues can do a much better job than me. Most of us are just high scoring math students, there are a few graduated math majors (though one guy had a BS in Math and couldn't assist with anything above College Algebra and the first couple of chapters of PreCalc and claimed to have never taken Diff. Eq.) and a couple of adjunct professors.

 

*Actually, I know that I'd need an answer manual for Differential Equations and parts of Calculus 2. I understand it, I just can't explain it well...(yet).

 

Actually, I was not talking about a math degree. With your math background, you definitely have the expertise to teach K-8 because the math you studied goes far beyond the material you'd teach. (You probably have a better math understanding than many math teachers). And, while you have taken Calc 2 and Diff Eq, you are not sufficiently far beyond those courses that you can teach them without resorting to solution manuals, which precisely illustrates my point.

Expertise does not have to mean a math doctorate. It means thorough mastery and understanding not just of the subject currently taught, but a few levels above what one is teaching, to gain perspective. (Which is why it is important that elementary math teachers study more than elementary math.)

[wapiti]

Obviously it's a great skill to develop.  "Proficiency" in this skill should not be expected overnight.  With many elementary teachers being more language-oriented than math-oriented, I might be concerned about the expectations or even that they're looking for the wrong things.

 

My big issue is the implementation - how often this would be required, how much actual teaching will be done on how to do this - giving them the language, etc., and really helping them connect their logic to words.  Mainly, I worry that kids who are very strong in math but weaker in language (not an uncommon combination at all!) will be prevented from moving forward appropriately in math based on their written explanations.  If this skill is not taught and developed, the results could be disastrous for such kids, taking their favorite subject, the only thing they do well, and making their success depend on their weakest skills.

 

For example, suppose my 2e-ish ds10 were in a traditional school.  Very good at math, relatively reluctant with the handwriting - his writing and handwriting are at grade level now but he does have significant processing speed issues (note, NOT quite enough for an IEP at this point and even getting a 504 plan would be questionable - he needs fresh private testing as the PS testing was insufficient). Not only would he be stuck at grade level in math (rather than in AoPS Intro to Alg as he is) but explaining what, to him, seem extraordinarily simple concepts might be torturous and make him want to avoid math.  At home (we afterschool his math entirely), I have him explain his reasoning orally (often due to the paucity of written math, LOL) and his reasoning is sound.  His language skills are slowly improving with age.  His teacher (who he has been with for years, long story) sometimes has him explain things to the other kids in his class and she totally gets him.

 

In general, I think it's a great idea to practice this.  I just worry that the devil is in the details.  It must be implemented very carefully so that they're really developing mathematical reasoning skills and actually connecting the math side to the language side rather than purely dealing with language skills.

 

Eta, pre-CCS, this isn't going to come up much in PS until the student hits geometry.  I wonder what the CCS for geometry look like - my impression is that public schools have been moving away from proofs a bit.  Will the CCS move back toward more proofs?  I'm all in favor of that.

[Library Momma]

My children's experience with this now that the common core standards are being looked at hasn't really changed.  They need to show their work, not write out long complex paragraphs explaining how they arrived at each solution.  95% of the time they are just showing the various equations they have used to determine an answer.  The only time I have seen actual paragraphs written were for complicated word problems, the type where a chart is needed to determine the answer.

[LeslieAnneLevine]

I agree. While my son isn't dyslexic, he has a much harder time communicating in written or verbal form and he computes/calculates most of the answers in his head.

 

When he was 2nd or 3rd grade we would rework problem so that they contained fractions or decimals just so he would have to write out the steps. No luck, he just did those in his head as well.

 

Having to 'explain' is torture for him.  Actually, when he was younger, he one day asked me if I knew how to do a problem and when I said yes, he wanted to know why I had to ask him!

 LOL

 

My son has always loved math and is good at it. He would hate it if he had to write out explanations all the time. He can do it here and there, but what he is being asked to explain has to make sense. 

 

If he were asked how do you find the average of 5 numbers, he could answer that (in a sentence, not a paragraph), but if he had been given a problem as easy as the one you cite in a later post, he would have answered it like your son did. Especially since they asked him specifically about that problem.

 

Math-loving kids everywhere will think they are bad at it. No kid should be asked to explain how they know that 54 is larger than 36. 

 

Good grief.  The kids are not writing page long essays for every math problem.  A paragraph? My guess is that for these problems, 2 or 3 sentences will suffice for their paragrph. I don't understand the uproar. If an elementary student can not put a few words down on paper, then there are bigger fish to fry. For those who have documented learning disabilities, since this is ps, they will probably be exempted. I see the opponents to this as contributing to the dumbing down of education.

 

That's quite a leap to the island of conclusions, there. I think all of us here care deeply about education.

[regentrude]

Are they being taught this?  How are they being taught this?

 

Well, obviously any skill kids are tested on should be taught. Whether it is performing the algorithm itself, or documenting  the process. (I can dream, right?)

 

How they should be taught? Through modeling by the teacher, through being  given clear expectations, and through practicing this skill on their daily work and homework. Just like any other skill they are taught to acquire.

[wapiti]

Well, obviously any skill kids are tested on should be taught. Whether it is performing the algorithm itself, or documenting  the process. (I can dream, right?)

 

How they should be taught? Through modeling by the teacher, through being  given clear expectations, and through practicing this skill on their daily work and homework. Just like any other skill they are taught to acquire.

 

Come to think of it, maybe that's another one of my fears, that the teachers won't be able to teach this skill due to weaknesses in their own mathematical understanding, turning the exercise into one of form over substance.

[TravelingChris]

I totally disagree with requiring paragraphs to explain math problems.  If the child will be doing typical math, in geometry, they will be doing proofs using at least some English and showing that they understand.  SHowing steps in a math problem is another issue= I can see that being required.  But involving language use, when the beauty of mathematics is that it goes beyond language, is horrible.  ANd I have no idea how you explain in a paragraph that five is bigger than four- you can give examples but that doesn't negate the case where four might be bigger than five.  It is a philosophical argument not intended for grade school kids to consider.

[SMRB]

I have to ask - what would you write to explain why one number is greater than another? With 5 and 3, for example, I would have to draw a set of 5 circles and a set of 3 circles, and then cross off the 3 in each set leaving the 2 extra showing - I would be hard pressed to write sentences to explain that. Would I have to say "If you put a set of 5 counters next to a set of 3 counters and take one at a time away from each set, there will be 2 left in the set that originally had 5, so 5 must be more than 3?" I'm just curious if that's the type of explanation they are expecting, and what do you do with a child who can draw a picture but not necessarily write out a prose explanation?

[Joker]

My dd are both in ps for middle school. They do have to explain their answers in writing, but they do not write paragraphs. Some end up being one sentence, while others may be several sentences. They take notes in math class with the teacher modeling how to write out the reasoning. They also explain their answers on homework and go over it with the teacher. So, by the time they need to do it on a test they have a good understanding of what is expected. It's been very helpful for us in helping them because when they miss one we can see exactly what they were thinking and why it was wrong. Their understanding and abilities have also improved.

[Spy Car]

We've got two separate (but related) issues going on here. One involves "practical" concerns (will the teacher's be able to teach it?, etc). And the other is whether the ability of students to clearly articulate their reasoning is a skill woth cultivating.

 

On the first point I think Wapati is correct, the devil may be in the details. And lack of teacher education in mathematics is certainly an area of valid concern.

 

We can't be blind to the "realities" of what happens in schools (nor, indeed, in homeschools); however, the "practical problems" one might encounter pulling it off doesn't mean that students having the ability to clearly explain their reasoning in mathematical terms isn't highly valuable. It is.

 

Doing this Socratically with a child one-on-one is a lot easier than in a classroom full of kids. But for it to work in a classroom a teacher does have to teach the mathematical properties in ways that reach children, teach the vocabulary that helps allow children to explain their thinking (and here the "correct" terminology is preferable to while might be called "dumbed-down substitutes), and teach in a style that exposes the mathematical reasoning behind all the work that is being encountered.

 

Are these things a "tall order?" Sure!

 

Will it all falling place because someone wrote it into a math standard? No.

 

This does not change the fact that this ought to be the standard of an outstanding math education. Anything less falls short. Most of you home educate, so you have the option of what happens in your math programs.

 

I know what I want for my child. I won't really get it is I leave it to the school math program. Our school uses enVision (a program mentioned by a previous poster). enVision includes quite a few problems were children are asked to explain in words (or sometimes using pictures or models) their reasoning. Is it perfect? No. Sometimes the wording is confusing and (without a good teacher in the classroom) some of the preparation seems less than adequate. It is not horrible, but not perfect. And I bet sme students (and their parents) get frustrated at times.

 

I understand the practical downsides (and how difficult they are to overcome) but reman convinced that the ability to clearly express ones reasoning is the educational ideal. Anything less is inferior.

 

Bill

[regentrude]

I have to ask - what would you write to explain why one number is greater than another? With 5 and 3, for example, I would have to draw a set of 5 circles and a set of 3 circles, and then cross off the 3 in each set leaving the 2 extra showing - I would be hard pressed to write sentences to explain that. Would I have to say "If you put a set of 5 counters next to a set of 3 counters and take one at a time away from each set, there will be 2 left in the set that originally had 5, so 5 must be more than 3?" I'm just curious if that's the type of explanation they are expecting, and what do you do with a child who can draw a picture but not necessarily write out a prose explanation?

That would depend on the grade and level, should be practiced beforehand, and the teacher should give clear expectations.

If I were teaching 1st grade, I would expect an answer along the lines of:

"If I have five blocks and take away three blocks I will still have two blocks leftover, so I know it five is bigger." 

"5-3=2 which is more than zero, so 5 is larger than 3"

"5=3+2, so 5 is bigger because I have been adding to 3 to make 5"

or a picture. Again: the teacher should model what kind of answers he is expecting and should state clearly whether drawings, charts, equations or only words may be used.

 

Also, the question should of course be age appropriate! The above question would be appropriate for K and 1st grade and ridiculous for 6th grade. OTOH, the question whether -5 is greater or smaller than -3 would be an appropriate question for a 6th grader.

[In The Great White North]

 

"If I have five blocks and take away three blocks I will still have two blocks leftover, so I know it five is bigger." 

"5-3=2 which is more than zero, so 5 is larger than 3"

"5=3+2, so 5 is bigger because I have been adding to 3 to make 5"

or a picture. 

 

Great examples of explaining it in words but not really paragraphs.

 

 

 

 Again: the teacher should model what kind of answers he is expecting 

 

This is the detail I expect to be missing.