22 Jul 2005 - 21:35

I need tutoring advice

I'm probably going to spend some time working with a friend of Christopher's on his math.

They're the same age--both going into 6th grade--and my sense is that math is probably this boy's strong suit.

I just gave him the Saxon placement test, and he placed into Saxon 7/6, which is the 6th grade book.

That would be great, but here's the hitch: he has been taught almost nothing about fractions at all. (He had a good math teacher--he and Christopher were in the same Phase 3 class for the first half of this year--who left to have a baby. So it seems that the subject of fractions & decimals fell through the cracks.)

So.....if anyone has thoughts, I'd like to hear them. I'll probably go ahead with 7/6, but that means I'm starting a 6th grade book with a child who's been taught virtually nothing about fractions and decimals.

update

Here's the fraction worksheets site Carolyn J found.

whose job is it, anyway?

This is the kind of thing that I just don't get.

Why should I be the person figuring out that this boy hasn't been taught fractions & decimals?

Why shouldn't the school be figuring this out? (Yes, the school might say he was taught fractions and decimals, but didn't learn them. However, it's clear to me that there are certain topics he simply hasn't even heard of, because with some topics he'll say, 'I kind of remember that.' In other words, he can tell me which topics he failed to learn, or didn't learn well enough to retain, or whatever it is. With topics like adding fractions, he simply doesn't know anything about them, and has no memory of having been taught.)

So, yes, the school might say, 'He was taught, but he didn't learn.'

But so what?

If he was 'taught' and 'didn't learn,' then he wasn't taught as far as I'm concerned. It's the school's job to perform formative assessment to know what students have and have not learned.

Then it is the school's job to re-teach if a child has not learned.

Then, if the child still isn't learning, it's the school's job to figure out what else he needs.

common sense from The Education Wonks

I don't want to take this too far, of course. Parents & students are responsible, too:

That's one of my major concerns with NCLB. When students don't do their homework or study for exams, or even attempt to do classwork, it's still considered to be the teacher's fault if the students don't achieve their federally-mandated level of proficiency in reading, math, and science.

And yet NCLB doesn't give me, as a teacher, the authority to require student's who aren't even attempting the work to stay after school and complete their assignments. Unless the kid has committed some breach of the school's disciplinary policy, I can't keep them any later than the school regular dismissal time.

The No Child Left Behind Act holds me solely accountable for my students' academic progress but doesn't give me the authority to help make that happen, especially for children that are considered to be "at risk" of failing to meet minimal standards of academic progress.

Sadly, under the law as it is now written, a large number of children are going to be left behind.

He's right. If a student doesn't do his work, and the parents don't require him to do his work, that isn't the teacher's responsiblity.

But that's not the case with Christopher's friend. This boy has done all of his work; he's a serious student; his parents are serious parents.

It's the school's responsibility to know whether this boy has or has not learned how to add, subtract, multiply, and divide fractions, and to teach or re-teach the subject if he hasn't.

Back to main page.

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Go ahead and start 7/6 but make use of fraction manipulatives for short periods each day and pretend you are just reviewing (for your own son's sake) but you are really finding out what the other student knows.

Add and subtract fractions with fraction pieces and let him learn how to write the answer (numerator changes, denominator doesn't), first by seeing you write it and then let him write the answer. He'll catch on quickly.

Multiplying and dividing fractions will take longer but it's reviewed in 7/6 so you'll have some time to work that in.

Good luck.

Most curriculum start slowly anyway to allow students to catch up for the summer loss.

-- CarolynMorgan - 22 Jul 2005

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If he's added and subtracted with money, he's had an introduction to decimals. Most students quickly see the connection of decimals to money (dollars and cents) and I doubt that that will be hard for him.

-- CarolynMorgan - 22 Jul 2005

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-- DanK - 22 Jul 2005

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Oops. Bobbled my first attempt.

Of course I agree with the two suggestions by CarolynMorgan^?. I would add, though, that practice in factoring into prime numbers would be a good thing to work in. If you are comfortable in factoring numerators and denominators, then you get a better understanding of getting to a common denominator. I also like to break things down into factors to reduce fractions. Getting comfortable with the idea of the fraction as one product (the numerator) over another product (the denominator) leads to "crossing out" common factors to reduce the fraction. Once this makes sense, then we can work in dimensions, which will help with real world problems (e.g. distance = rate * time).

Dan K.

-- DanK - 22 Jul 2005

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You might want to look at this site www.keymath.com. The "Key to" series has workbooks on individual topics such as "Key to Fractions"

-- LoneRanger - 22 Jul 2005

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A lot of homeschoolers like the "Key to" series. I almost got it a couple of years ago. It looks very thorough and systematic with lots of practice. And like Singapore, it's affordable.

Wow Catherine, what a great opportunity. A sharp kid with a "gap." You'll be great at this! You must tell us how it goes.

I'm gradually starting on fractions with one of mine. I look forward to all of the advice you get, I need it, too.

I totally had to re-learn fractions in college, so I feel a little shaky teaching it. I'm re-reading the Liping Ma chapter on fractions for the umteenth time.

-- SusanS - 22 Jul 2005

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Go ahead and start 7/6 but make use of fraction manipulatives for short periods each day and pretend you are just reviewing (for your own son's sake) but you are really finding out what the other student knows.

Great advice.

Let me ask one further question, before I get to the other pieces of advice.

His mom would like him to learn fractions quickly, before the fall. I'm going away from a week or two, he's going away for a week.....we'll have only a couple of weeks at most.

This is exactly where I was with Christopher at the beginning of last summer: I thought I should (and could) teach him fractions quickly, so he could catch up.

I then discovered that I couldn't teach him fractions quickly, which was fine because he's my child & I became obsessed with not only catching him up but pushing him ahead.....so I started my Quest.

This is not my child, and his mom is pretty intimidated by math (his dad's a math guy, though--has a Ph.D. in computer science, I believe).....so here's my question:

Are there any shortcuts?

The more I sit with this, the more I think this boy needs to spend a year with Saxon Math, or with the Russian Math book I'm working my way through now (or with Singapore Math).

His mom may well be amenable to that, although it is a big commitment. I worked Christopher pretty hard this year, and SAXON is a big, huge book.

You can do it in just a short time each day, but you have to do it.

So:

• are there any shortcuts?
• are there any semi-shortcuts? (meaning, can I get him in some kind of shape for fall with the recognition that this is going to be fragile & almost entirely procedural knowledge)
• would most of you think that he should spend a year with a coherent curriculum, a la Saxon, Russian Math, or Singapore?

Susan

What is the 'Keys to' series?

Somehow I've missed that.

Carolyn M

Yes, indeed: money. Thanks for the reminder.

Although....I think he's got addition and subtraction of decimal numbers down just fine. The issue is multiplication and division of decimal numbers and, I'm sure, expression of decimals as fractions & fractions as decimals.

oops--Christian (works with Jimmy & Andrew) wants to take the Nerd test.

Dan--bingo! I'll be back shortly, and tell you the first thought that came to mind--thanks!

-- CatherineJohnson - 22 Jul 2005

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If mom is the one who wants him to know it by school time, MOM should be willing to invest her time to help him also.

Intensive work now for two weeks can do a lot, but you can only do so much in two weeks. If the two weeks are consecutive that is better, but it must be continued after the two weeks is over. Anybody, not only a student, has to do something about 34 times for it to become a habit.

The procedures you use for addition are not the same for multiplication and it takes time to get those procedures down pat. If you rush him through all the different things that he is going to have to learn to cover all operations of fractions, his brain will be on overload and he won't have time for the procedures to become a habit.

(That's what I love about Saxon. Saxon introduces the steps in increments and students have opportunity to master each part withou being overwhelmed. Even so, I have students who get procedures for one operation confused with another.)

Frankly, I think it's too much to expect in two weeks time.

Do what you can; have his parents continue to reinforce what you cover, no more. Then after school starts, mom should continue the tutoring in fractions, either with you or with Kumon or something similar. His new 6th grade probably won't introduce all of the operations with fractions right away.

Maybe he'll prove to be a whiz and catch on to a lot of it, but my experience of 13 years teaching 5th grade tells me trying to cram it all into 2, or even 4 weeks, is going to be frustrating for this student.

Good luck.

-- CarolynMorgan - 23 Jul 2005

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Carolyn What's your sense of how long it takes to gain some basic procedural fluency?

Since I've worked only with Saxon, I have no idea how much you can shorten the time frame by.

I'd say his mom just has no idea what's involved in learning fractions, the same way I had no idea. I was shocked when I read Liping Ma saying fractions are THE hard subject in elementary math.

Then when I met Carolyn, and she started telling me about college students who couldn't go into nursing because they couldn't do algebra because they never learned fractions in grade school that blew me away.

I just had no idea.

I'm sure that's part of the reason why everyone thinks it's a perfectly fine thing to remove fractions from standardized tests. Some of the people overseeing the tests don't realize that fractions aren't just Another Thing.

Dan K

I love it that you said this!

My FIRST impulse with Christopher's friend, even before he'd finished his test, was to scan in the prime factor lesson from Russian Math!

That is a brilliant, brilliant little gem of a lesson, and it's the base of the book. Everything you do after that lesson rests on it (I think, off the top of my head).

-- CatherineJohnson - 23 Jul 2005

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Here's my thought:

• do the Russian Math lesson(s) on prime factoring and all the prime factor problem sets (and have Christopher do them, too)

• then move to Singapore Math and do the sections on fractions sequentially, starting with the very first lesson in Primary Math 3B

• find the fraction worksheets Carolyn J & Susan came up with (I think they're on the 'math supplements' page) and print out a bunch for him to do every day, day in and day out, to give him the advantage of procedural speed and fluency, which will hold him for a while (Now that Carolyn's made the terrific search engine options, I'll be able to find all the worksheets people have mentioned.

Any thoughts?

-- CatherineJohnson - 23 Jul 2005

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Catherine,

I haven't been following your links to Russian Math, so I don't know what they've got. I'm happy, though, to take your word that it is presented well.

The little point that I would add here is that factoring is an excellent point to reinforce the idea of writing down every step. Reassure your student that paper is cheap, and we should feel free to use a lot of it. I hope your worksheets don't skimp on alloting space to work out the factoring. Too often when dealing with factors (not so much when PRIME factoring is emphasized), students are simply asked to list all the factors of a number. This leads to haphazard listing of factors in no particular order, leaving a great possibility that a pair might be left out. I think the student should write line after line, restating the number as a product of an increasing string of shrinking factors, eventually becoming a product of all prime numbers:

72 = 6 * 12 = 3 * 2 * 12 = 3 * 2 * 3 * 4 = 3 * 2 * 3 * 2 * 2

It's okay to do two expansions in a single step:

72 = 6 * 12 = 3 * 2 * 3 * 4 ...

But keep things organized and WRITTEN OUT in a line-by-line progression that is easy to follow. Kids that are good at math or at least motivated to try to impress the teacher will want to solve the problem without writing out any steps. That will bite them somewhere down the line. Factoring is a great place to learn the practice of using lots of paper to show every step.

-- DanK - 23 Jul 2005

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I agree with Dan K. 'No skipping steps' has got to be the rule. When you are in a hurry to cover much material, that is the temptation. Students must include all steps, in all procedures of doing fractions.

Catherine, I would say that for the average student it will take a minimum of 1 month for a procedure to become second nature. Of course, you will have students far out on both ends of the spectrum who gain fluency faster or slower. But you want mastery and comfort at each stage before you add another.

Then, you must keep doing the previous procedures while you are learning the second and the third. Never, never stop one operation to teach and practice another.

Understanding factoring and multiples plays a huge part in properly performing fraction operations. I've never taught them at the same time, as you've been writing about this summer, but students eventually must use them at the same time. For a student who is having to digest many new fractional concepts in a short time frame, skill in these two areas must be a part of his overall regimen. How well he handles identifying and using factors and multiples and how well he differentiates between their uses will determine how quickly he can manipulate numerators and denominators in fractions.

It's a lot to learn in a month. Aim a little lower and go for mastery of skills to build confidence. My recommendation for what it's worth.

-- CarolynMorgan - 23 Jul 2005

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I'm jumping in before reading full comments (I shouldn't do that!), but the skipping-steps issue is HUGE for me.

At some point I finally figured out that the reason there are so many procedures in math that I don't understand is simply that steps have been left out--as they should be, at some point.

Part of the elegance of maths, it seems to me, is the incredible efficiency and paring down of the procedures, steps, operations, & so on.

It's true that kids ALWAYS want to skip steps if they know how to do a problem.

In one sense, they're right about that. They have become efficient, and it's good to be efficient.

My job as a teacher is to know when to insist they take the long way 'round, not the short way.

-- CatherineJohnson - 23 Jul 2005

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The Russian Math mode of factoring is so pleasurable I wanted to sit around all day just factoring numbers.

No idea why!

I may be able to show it here.

Shoot. I can't do it. I'll figure out some way to get it posted online.

Do you like the standard factor trees?

I like them, but I'm curious what mathematicians & math teachers think:

-- CatherineJohnson - 23 Jul 2005

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Dan

The Russian method does go line by line.

Maybe I'll just describe it step by step, and you can draw it yourself (you may get the point instantly, but I would need step by step).

Here's how you would prime-factor the number 12:

• draw a vertical line on your paper
• write the number to be factored - 12 - on the lefthand side of the line
• write a 2 on the same horizontal line as the 12, and to the right of the vertical line (2 is the first factor; I think they tell you to start with the smallest factor you can think of larger than 1)
• divide 12 by 2 and write the quotient - 6 - underneath the 12 (on the lefthand side of the vertical line)
• write a 2 to the right of the 6, on the righthand side of the vertical line
• divide 6 by 2 and get 3
• write the 3 below the 6, on the lefthand side of the vertical line
• write a 3 to the right of the 3, and to the right of the vertical line
• divide 3 by 3 and get 1
• write 1 below the 3 on the left

You now have a complete, organized list of prime factors on the righthand side of the vertical line.

You also have--and it's possible to see this without much difficulty, I think--all the other factors.

If you 'stop' after the first 2 on the righthand side of the line, you can see that 2 & 6 could be factors.

If you 'stop' after the second 2 on the righthand side of the line, you see that 4 and 3 are factors.

And so on.

I haven't looked at this super-closely, but I'm pretty sure you can figure out all of the different factoring permutations from this simple factor column in a systematic way.

-- CatherineJohnson - 23 Jul 2005

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This is terrifically helpful--I'll try to get it pulled up to the front page, but in the meantime I'll direct people here--

Thanks so much.

-- CatherineJohnson - 23 Jul 2005

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Catherine,

I think I get it. If I did it right, it catches all the primes, but it might miss some other factor if you were interested in finding all the factors.

As you described it, I would know that 4 * 3 =12 because where 3 is on the left side, I would "multiply down" the right column 2 * 2 to get four. Do I have that right?

Anyway, I took 240 as an example. I tried to follow your steps, getting a left column of 240, 120, 60, 30, 15, and 5. My right column is 2, 2, 2, 2, 3, and 5. Does that agree?

My problem is that I never see 10 and 24 as factors.

I tried again in a different order, making sure to write only primes on the right side. My left column is 240, 48, 16, 8, 4, 2. My right column is 5, 3, 2, 2, 2, 2. So I think it works to get the primes whether you start with the smallest factor or not. The value in using the smallest is that you group all the 2's, then the 3's, etc. This is helpful if you want to represent the factorization using exponents: 240 = 2^4 * 3 * 5.

Still, I don't see 10 or 24 pop up. I don't think that's a hardship. Getting to the primes is the most important thing. If you really want all the factors, you just need to combine the primes in all different combinations.

-- DanK - 23 Jul 2005

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Everyday math did prime factorization with factor trees. I think it's a good way to learn.

-- CarolynJohnston - 23 Jul 2005

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Anyway, I took 240 as an example. I tried to follow your steps, getting a left column of 240, 120, 60, 30, 15, and 5. My right column is 2, 2, 2, 2, 3, and 5. Does that agree?

My problem is that I never see 10 and 24 as factors.

You're probably right.

I never tried to see if I did get all the factors.... I just noticed that you get more than just the prime factors. (And the Russian Math book doesn't teach this method as a way of getting all the factors, I'm pretty sure.)

I love factor trees!

But there was something riveting about the Russian factor columns.

I was so drawn by them that I'd be sitting around thinking, 'I want to factor something.'

I'm not kidding.

-- CatherineJohnson - 24 Jul 2005

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Hey!

I was stimming on factors!

-- CatherineJohnson - 24 Jul 2005