kitchen table math, the sequel: Math Education: An Inconvenient Truth Video

## Saturday, January 20, 2007

### Math Education: An Inconvenient Truth Video

Have you seen this video that presents an introduction to the problems with reform math? It includes specific examples from TERC & Everyday Math. I learned about it from Linda Moran’s Beyond TERC Yahoo group. It’s about 15 minutes long.

I would like to share this with some local parents because I want to inform them of what’s in store for them if our school district decides to adopt TERC (currently being evaluated here). In this era of YouTube appreciation I thought it would grab people’s attention in a positive way. (Although I would have preferred another title.)

Video embedded by Rory:

Nickd said...

I homeschool and use Singapore Math, so I am watching this debate from a different perspective.

However, I sent this to my friends with kids in our local schools (we adopted EM a few years ago) as a heads up for "math clusters."

I'm sure they will defend the alternative methods philosophy if their kid has passing grades in math. They've swallowed the kool-aide so to speak.

None of their kids currently using EM have made it to middle school. I am waiting for them to get a reality check in a few years...

and thinking of opening a Kumon center franchise.

SteveH said...

Great introduction to parents new to the controversy. I especially liked the part where she talked about what EM was doing with its "precious" class time and the part at the end where she plugs Singapore Math.

As nicksmama says, you have to be ready for the counter arguments, such as clustering teaches understanding. You will end up in a vague argument of generalities and come out a loser because they are in charge and you aren't. Plausible deniability. And, by the way, many schools "supplement" TERC and EM. End of discussion, but you don't know exactly what that means. It doesn't matter. They just want you to go away.

You need proof, but how do you do that? Unfortunately, many of the anti-TERC and anti-EM arguments focus on basic math and their algorithms. This argument is limited, but it can be potent as a starting point. Most of the problems of TERC and EM show up later on, especially in mastery and the use of fractions as a lead in to algebra. EM looks like it has all of the pieces when you look at the workbooks. What can you do?

The best solution I have is to put two math textbooks or workbooks side by side. What good is understanding if the fifth grade TERC student hasn't a clue how to solve a fifth grade Singapore Math problem. The school is then put into the position of explaining why this is OK. Unfortunately, you will never get the chance to do this. They won't let you.

The only hope is that groups like MathematicallyCorrect, NYC-HOLD, and KTM can achieve a critical mass for change. The problem is that this is not just about math. It's about fundamental assumptions of K-8 education.

Me said...

I watched the first part of this video and thought it excellent.

It would, of course, be helpful to parents who understand math but are unaware of the controversy and the details of the "new" methods.

I wonder, though, how parents who themselves have difficulty with math would react to the video? They may not wish to admit that they themselves have difficulty with even the standard algorithms. They may think that their children are getting some sort of advantage by learning the impressive-looking stuff like the lattice method.

TurbineGuy said...

I embedded the video to make it easier to view.

My girlfriend just watched this and got pissed off. It was funny watching her jump around, because she had no idea that this sort of stuff was going on in the schools.

Nickd said...

If you read the comments section on Youtube, you will notice that there are many people in love with the lattice method and comments about the WASL being a "terrible test."

Is their a vendor for blank lattice paper in Washington state? If not, forget about the Kumon franchise, I'm in the lattice paper business!

SteveH said...

"They may think that their children are getting some sort of advantage by learning the impressive-looking stuff like the lattice method."

Math teaching was not a bed of roses in the "old days". The convenient reason is that the content and teaching methods were bad, not the implementation. This plays well with many parents who didn't do well in math. Schools play the "understanding" card to the hilt with parents. It's quite astonishing that schools are so ignorant or arrogant.

PaulaV said...

Here is an email I received recently from my third graders school regarding multiplication:

Dear Parents,
We are on our way into the great world of multiplication. The unit involves multiplication in a variety of ways- through real life investigations of things that come in sets, models, and patterns. The children will solve problems, explore with concrete materials, write multiplication stories, and play games.

Can someone explain what a multiplication story is? Our school uses TERC.

--PaulaV

SteveH said...

"Can someone explain what a multiplication story is? Our school uses TERC."

It's a word problem, like this:

Jane has 5 boxes of cookies. Each box contains 15 cookies. How many kids can she give a cookie to at school.

The problem is not really about the Lattice Method versus the traditional method for multiplication. It has to do with practice and mastery. Hard work. My son's fifth grade EM workbook can be characterized by the term "spoon feeding". They would call it "understanding".

PaulaV said...

"It's a word problem, like this:"

Okay. Why not just call it a word problem then? I guess I'm just knew to the lingo...I have never heard of a multiplication story. Word problem, yes. Story, no.

--PaulaV

Instructivist said...

What a nicely done video! Thanks, Tex.

I hope it riles up a lot of parents. Like Steve, I was wondering what EM would do with "precious" class time. If there is a silver lining, at least EM has an awareness that class time is precious, even if they go off the deep end.

I went on a sentimental journey and revisited KTMI to read up on Trailblazers. Now that KTMI is a classic, I found the categories holding collections of writings extremely useful (Many thanks to Catherine and Carolyn for this enormous labor of love).

I found one entry that compares a fifth grade problem in Trailblazers (comical) to problems in Saxon and SM. This is the SM problem:

from the last page of Primary Mathematics 5B (U.S. Edition):

18. A fish tank is 2/5 full after Sara poured 14 gal of water into it. What is the full capacity of the tank in gallons?

I can solve this problem using proportions. I was wondering if the math people could show different ways to solve this problem?

KDeRosa said...

Here's an interesting take on the video from Don Crawford (from the DI Listserv):

I enjoyed this video. They did an excellent job of skewering the odd ways of teaching computation.

Unfortunately this is not the strongest argument against these curricula. It is not hard to make a case that those computational algorithms aren't essential. (I think they are essential, but these computational algorithms certainly are not the MOST essential things to learn in math.)

The bigger problem with TERC and Everyday Math is that the students with average and below skills and talent in math --don't learn ANYTHING from those programs. They don't learn any useful math skills because instruction is missing--there's just activities. They don't learn an odd way of doing division--they don't learn ANY way of doing it. There's no instructional wording, there's no practice, there's no tests, nothing that is needed to master math skills of any kind.

Catherine Johnson said...

yup, I liked the "precious class time" line, and used it as a jumping off point to point out, on the Irvington Forum, that the time costs of supplementing an inferior curriculum must be addressed.

Rory

How'd you do that???

I just tried to put a link into Tex's post & it looked like I was going to mess things up so I didn't.

Catherine Johnson said...

M.J. McDermott confirms every cultural stereotype I have:

--military
--science geek (began college with a physics major)
--Scots Irish (McDermott)

Apparently she made this video on her own. No affiliation with any organization; just went out and did it.

Sound familiar?

Catherine Johnson said...

They don't learn any useful math skills because instruction is missing--there's just activities. They don't learn an odd way of doing division--they don't learn ANY way of doing it.

beautiful

Catherine Johnson said...

Ken

I can't subscribe to the DI list serv - any suggestions?

I think I followed the various steps (sent them an email, etc.)

Is there a website I should start from?

Catherine Johnson said...

My guess is the video will be powerful for parents and perhaps for school boards.

You just never get to see or hear an "authoritative parent" and that's what we have here.

Catherine Johnson said...

I've come to think the entire issue is simply (!) and most effectively a question of "the political."

(Ed tells me "the political" is to be distinguished from "politics.")

Here in Irvington I intend to press two issues:

--Math Trailblazers is rejected by mathematicians, physicists, engineers, economists, scientists, etc.; no curriculum rejected by large numbers of experts in the subject being taught should be selected under any circumstances

--parents must have final say over their children's education just as parents have final say over their children's medical treatment

In short: parents need a vote and a veto.

These two arguments are extremely simple, and I suspect there are few parents who would argue against them.

These arguments have the virtue of simply refusing to engage in dueling statistics.

I don't care what "research shows." I care what actual mathematicians say, and I have legal and moral responsibility for my child's well-being.

If I make a mistake raising my children - I've made plenty - it's my mistake to make, not theirs.

I wouldn't force Saxon Math on a parent who didn't want it.

The school should not force Math TRAILBLAZERS on me if I don't want it.

Instructivist said...

"Apparently she made this video on her own. No affiliation with any organization; just went out and did it."

It's inspiring what a determined individual can accomplish. Great technology like the internet now magnifies such efforts thousand-fold and even million-fold if this video goes "viral".

Going "viral" is a new term I heard only recently. If I understand it correctly, it means exponential distribution. Someone sends a link to a couple of friends who do the same and before you know it, it's seen by hundreds of thousands of people. Blog publicity also helps. Akin to the MLM concept. Only it works here and is not annoying.

Instructivist said...

Catherine,

Did you get an e-mail from DI when you subscribed. You need to confirm by clicking something in the e-mail.

Maybe sending another request and responding to the e-mail will do this time:

majordomo@lists.uoregon.edu

Put subscribe di in the body (IIRC)

Anonymous said...

Paula,

The audio is not working on my computer so I could not listen to the video to hear the phrase used in a particular context.

I can only speak about what a number story is in the curriculum that I use. In Singapore first graders are given illustrations and taught to write equations (number sentence) which tell a "story" about the illustration. (The equation is not the story. The story is the explanation about what the equation represents about some particular thing taking place in the world.) I will link you to one such page which I uploaded for you:

http://bp1.blogger.com/_6Bdjlk_XwVc/RbKH8awgqXI/AAAAAAAAABI/HoOOMK4NkTk/s1600-h/number+stories.JPG

The "number story" would be verbalizing something like, "There are seven children altogether, three of them are girls, the other four are boys."

This sort of exercise is a transition to solving word problems later on. It isn't actually the word problem itself.

Refering to an equation as a number sentence has the intent of focusing on the fact that an equation is a mathemtical asssertion that one makes and as such must express a complete thought. Just like a sentence written in English prose, there is a subject and predicate and the assertion is either true or false. This approach is supposed to train them to express themselves coherently when they show their work in word problems. (Although in my experience it takes a whole lot more than throwing out, "Write a number sentence!")

I checked a third grade Singapore book and word problems are referred to as word problems. The "number story" is a very short term gimmick used in the first grade which is quickly dropped.

Catherine Johnson said...

ooooo

going viral

cool

Catherine Johnson said...

instructivist

I didn't get a return email from the listserv

Catherine Johnson said...

I wonder, though, how parents who themselves have difficulty with math would react to the video?

Hi Susan!

(I STILL haven't read THE INCREDIBLES, the article you posted - which is why I haven't commented - I was in Evanston on my whirlwind tour of Alden Estates, Target, & Office Office Depot.)

I'm going to guess that a parent who doesn't feel comfortable with the "traditional" algorithms may feel even more uncomfortable with what he or she sees here.

If it was difficult for you to master one algorithm for each of the 4 algorithms, the thought of your child having to deal with even more algorithms would be daunting, I imagine.

Catherine Johnson said...

I feel - and I do mean "feel" - appalled when I watch this video.

I watch this video and see many new opportunities for my child to experience failure at school.

Of course I see this video in the context of our middle schoo's Algebra in 8th grade / Death March to Harvard / Phase 4 Extravaganza, so I'm not normal.

At least I hope I'm not.

PaulaV said...

Hi Myrtle,

When I watched the video, I immediately thought of the email my third grader's teacher sent...the one about writing multiplication stories. I think I thought the same thing Catherine thought...I "see many new opportunities for my child to experience failure at school."

"In Singapore first graders are given illustrations and taught to write equations (number sentence) which tell a "story" about the illustration."

If my son had started off doing Singapore math in first grade, perhaps he would be better prepared to write a multiplication story.

"This sort of exercise is a transition to solving word problems later on. It isn't actually the word problem itself."

Yes, there should be a "transition" before solving word problems. So far there hasn't been a transition in my son's case. Suddenly, in third grade he will be asked to write a story in math. I see trouble ahead. It is hard enough to get him to write on his daily reading log. I cannot imagine having to tackle writing in math as well.

Thank you for the link and the explaination!

--PaulaV

I am one of those parents who (as you say) did not feel comfortable with the standard math algorithms and this is what I have to say.

My son struggled with math continuously from 3rd through 8th grade. He received "intervention" services in the form of teacher consultation, support classes, etc. for those years at our school district. I cannot imagine what the cost of this was over those several years to taxpayers but one can assume it was significant. In addition to the support he received at school, our evenings at home became a grinding routine of math tutoring. I was never a good math student. The best I could do was stay a little ahead, try to do the worksheets with him and then wait for the bad grades to come home (and they did). Through all this effort and expense, he never did better than a C on any report card and was consistently demoralized and/or angry. At 10:30 PM one night, I loaded myself into the car and ran down to Borders Books in dire frustration because the methodology to accomplish the homework he had that night could not be found ANYWHERE in the textbook. There I purchased a simple guide on how to do basic math. There were no pictures, no stories of "what would happen if you wanted to make a kidney shaped swimming pool .." etc., etc., etc. There were simple instructions for each type of problem followed by exercises.

This was a turning point for us as I realized that we (him AND I) were not inadequate, we were simply NOT BEING TAUGHT.

The video fails to mention this -- the schools are all teaching this "progressive" math program but, as mentioned earlier, the end result is that a large majority of kids ARE NOT LEARNING ANYTHING AT ALL. Yet, the homework is designed around getting ready for the standardized tests and the tests themselves assume they have learned the standard methods so...what gives? Are they supposed to be doing this magical sort of learning whereby you just "get exposure" to math concepts and...voila! You can do it! The progressive methods also vary from the standard ones in a very disadvantegous way -- they require a lot more spatial organization. Take any one of them in the video and think about how much "peripheral" calculation was required - numbers were stored to the right and the left of the central problem and then have to be regrouped in order to present the answer. How on earth is this better? Furthermore, the reformed approach assumes time is not an issue so they've created revisions with no thought to how long it takes to complete a problem. Last I checked, every major test these students will take from APs to SATs, ACT, etc. are time-based, meaning they are graded by accuracy within time limitations. Contrary to the methods shown in the video, TIME IS OF THE ESSENCE.

The simple book from Borders starting making all the difference. With this realization, I took him over to a Kumon Center. They gave him a simple assessment test. I treated this warily thinking this was just part of the "sell." He was in 8th grade at this point and his assessment was 3rd grade level. I cried that day -- not because of his deficiencies but because it was all so clear to me -- he couldn't do math because HE DIDN'T KNOW HOW TO - as stupid as this sounds, this was a revolutionary concept in our family at that time and certainly not one ever expressed to me by an educator.

I have often thought of the current style of math instruction as a system in which we want to create skillful experts but for some reason, we don't want to let them know what tools they can use to hone their skills. We're enchanted with the idea that children will somehow derive the higher meaning in numbers and we shouldn't clutter up their mind with repetitive exercises. Tell that to the piano student, the dance student, the tennis player, the golf player, the French student and so on and so on and so on....frankly, I just don't know how on earth we got to where we are. I am so sorry for the lengthy post but heck, I feel better!

Tex said...

So many excellent comments that concisely communicate the issues!

“We're enchanted with the idea that children will somehow derive the higher meaning in numbers and we shouldn't clutter up their mind with repetitive exercises. Tell that to the piano student, the dance student, the tennis player, the golf player, the French student and so on and so on and so on....frankly, I just don't know how on earth we got to where we are.”

I think about how this can resonate with many parents whose kids are not so involved in math.

Tex said...

This all makes me think of the great “Taught my dog to whistle” image.

http://www.kitchentablemath.net/twiki/bin/view/Kitchen/TaughtMyDogToWhistle

“I said I taught him, but I didn’t say he learned.”

Instructivist said...

"Take any one of them in the video and think about how much "peripheral" calculation was required - numbers were stored to the right and the left of the central problem and then have to be regrouped in order to present the answer.

This is a superb point!

It also occurred to me, while watching the TERC method, that this cumbersome method makes heavy demands on the pathetically limited working memory people have. Remember the magical number seven, give and take?

Inevitably, such "peripheral" calculations provide a rich opportunity for error. Can you imagine a kid, possibly struggling with organizational issues, having to keep track of all these ancillary calculations?

Catherine Johnson said...

correction

Rory says M.J. McDermott created the video in connection with the Where's the Math folks.

I'm trying to think where I got the idea she did it on her own....it's possible I read that on Linda Moran's TERC listserv.

I'll check later on.

Catherine Johnson said...

our evenings at home became a grinding routine of math tutoring

I for one am looking forward to a grinding Sunday of math tutoring.

Christopher is down to a D- or thereabouts in Ms. K's class & has to make up a test tomorrow, so today promises to be a horror.

On one of her recent tests, possibly the one Christopher is making up, she included a problem requiring the kids to:

WRITE AN EQUATION WITH VARIABLES ON BOTH SIDES IN ORDER TO FIND THE MEASURES OF A COMPLICATED SET OF ADJACENT, VERTICAL, SUPPLEMENTARY, AND COMPLEMENTARY ANGLES

The Math dad emailed Ms. K pointing out that she had never taught the kids how to set up or solve an equation with variables on both sides.

She replied that she had taught it last year.

This is the new mantra; the math chair told us the same thing about the test Christopher got his 60 on. The reason Ms. K was justified in giving the kids a small, complicated, hard-to-read drawing of a submarine to find the volume of was that "she taught it last year."

She did indeed teach how to find the volume of a cylinder last year, or, rather, Ed taught the subject. The class (Ed) spent perhaps 5 days on the subject and iirc she never returned the final test they took which I presume Christopher and many of his friends bombed.

Not a problem for the math chair.

"She taught it last year."

Math dad knows every single topic Ms. K has taught inside out; he knows when she taught it, too.

He told her, "Maybe you're thinking of last year's 7th grade class. You probably taught it to them."

Catherine Johnson said...

The great thing about Math Dad is that he's even more ferocious than I am.

Ed talked to a parent who knows him; the parent kept saying, "He has a strong personality. He has a very strong personality."

I like that in a man.

Catherine Johnson said...

The progressive methods also vary from the standard ones in a very disadvantegous way -- they require a lot more spatial organization.

THAT'S IT!

Every time I see one of these problems I feel overloaded and bewildered, but I could never put my finger on the reason. I assumed my problem was just that the algorithm is unfamiliar.

But you're right: the demands on spatial organization are enormous.

Catherine Johnson said...

Catherine Johnson said...

When Christopher and I took the KUMON placement test he placed into 2nd grade and I placed into 4th.

So you shouldn't feel too bad.

Catherine Johnson said...

I gotta get back to KUMON.

I told Christopher the other day that I think he's going to be going back to KUMON and he pitched a fit.

I'm going to stay away from the KUMON topic today.

Instructivist said...

I still can't fathom how Google searches, even though I am pleased with the result.

A blog search for "math" has me in second place after linda moran, out of nearly two million candidates:

1,983,256 posts matching math - showing 1 through 10

Meteorologist critiques TERC and Everyday Math
3 hours ago by lindamoran - References
As a parent and math teacher, after ten months of studying the "Math Wars" upside and down, inside and out, I have to admit, I've come out pro-reform math. But the wrinkles yet to be ironed out of this revolutionary new approach to math ...

Teens and Tweens - http://www.lindamoran.net/blog_teen/

Fuzzy math video
17 hours ago by Instructivist - References
Math video.

Instructivist - http://instructivist.blogspot.com/index.html

In her latest post, linda moran, after much scrutiny, comes out on the side of the fuzzies, even though serious wrinkles still need to be worked out, she admits.

I wondering if the "wrinkles" couldn't be so fundamental that they are the program itself, its essence. The analogy with Communism comes up again.

SteveH said...

"... but because it was all so clear to me -- he couldn't do math because HE DIDN'T KNOW HOW TO - "

This is not about which algorithm is best to teach. It's about competence, both curricular and teaching. It's nice to have students take responsibility for their learning. They just shouldn't have to take responsibility for the teacher's teaching.

Me said...

Thanks so much for sharing your experiences. You are saying that at least by the time their child has been having trouble with the crazy TERC-like math, the video would make it very clear to any parent what's going wrong.

I'd been thinking about the case of parents watching the video before their child had been exposed to the program and perhaps not realizing the seriousness of the situation before it was too late to do anything.

Catherine Johnson said...

What we're experiencing, which I'm going to have to start writing about, is a horrifically bad "traditional" course that I think is horrifically bad because of the constructivist ideology all of our new young teachers have absorbed in ed school.

Ms. K teaches everything "from the top" - i.e. all skills and concepts are "embedded" in a whole mess of other skills and concepts.

This month the kids were asked to set up and solve their FIRST equations with variables on both sides:

* ON A TEST

* IN A COMPLICATED VERTICAL, COMPLEMENTARY & SUPPLEMENTARY LINE DRAWING

They had spent no classtime learning how to set up these equations; they had spent no classtime to my knowledge learning how to solve these equations; they'd had a few days at most on vertical, complementary, and supplementary angles.

All of this stuff - all of it - was "taught" as one big mush.

I've come to believe that the constructivist ideology of ed schools is horrifically damaging to older kids for these reasons:

1. new teachers have NO idea how to break a concept down into its component parts

2. new teachers have NO idea that they SHOULD break a concept or skill down into its component parts

3. new teachers have NO idea that "understanding" is distinct from "being able to do it on a test"

4. new teachers have NO idea that learning for all living creatures happens through a process of spaced repetition which, in math, translates to distributed practice

The constructivst ideology of young teachers explains why the math chair thinks it's fine to tell us that "if students need distributed practice parents can find worksheets online."

She doesn't think distributed practice matters.

She thinks conceptual understanding matters, and she thinks once you've put material on the board conceptual understanding has occurred.

The way to assess whether conceptual understanding has occurred is to give students an extremely difficult, timed test with multiple problems all of which require the student to use multiple concepts and skills to solve.

If a student fails the test, that is taken to be a sign that the student needs "extra help," which means that the student comes in before school and has the concepts explained to him again by the teacher, one-on-one.

Everything is taught via "explanation"; if explanation fails, more explanation is assumed to be the remedy.

If more explanation fails, the student is assumed to be incapable of learning the material and is moved to an easier course in which less material is "covered,"

This gets back to the question of whether there are any "real" constructivist classrooms in the country.

If you looked at Ms. K's class you would call it traditional. The textbook is traditional (harks back to the first New Math, in fact); the lecture format is traditional.

But it is constructivist through-and-through.

SteveH said...

"But it is constructivist through-and-through."

We shouldn't be talking about pedagogy or opinion when the problem is about basic competence. Talking about constructivism gives them way too much credence.

SteveH said...

Looking at the video comments again, it seems as though the fuzzies have called out the troops.

LynnG said...

There are a couple of points being lost when I look at the fuzzy comments -- first, after 15 years of fuzzy math, there is absolutely no measure anywhere that demonstrates that it works. The U.S.,(and States and local school districts) have not shown any improvement when fuzzy math is used. How long are we supposed to wait to see results?

Second, while I find the algorithms in TERC and EM annoying, someone on KTM once made the insightful comment that the primary effect of the alternate algorithms allows students to avoid ever learning quick recall of basic facts. While the standard algorithm for long division forces a kid to really use those basic facts, partial products and the TERC clusters do not. This would be alright if math weren't so incredibly cumulative.

But once you decide to work with fractions, it becomes unmanageable to add, subtract, multiply and divide fractions if you don't have an extremely solid understanding of the basic facts.

EM gets around this problem by having no secure goal for fractions in 5th grade. The hardest fraction addition problem we have had so far in 5th grade had denominators that were really "friendly." You won't know what your kid has missed in EM and TERC until you get to middle school or later.

Catherine Johnson said...

TRAILBLAZERS doesn't teach fraction division in 5th grade, which is the final year of the series.

It also teaches forgiving division instead of long division. (I hear that our school is supplementing like mad, but TRAILBLAZERS itself does not teach long division.)

Catherine Johnson said...

hmmmm....I wonder who SteveH23 might be?

She's wrong. Trust me. Don't check out Saxon or Singapore Math. Don't wonder why so many Kumon centers are opening up. Nychold just doesn't understand. Trust me. It's good for little Suzie to write a story about her favorite number. Teachers are experts in math. There is no need for drill-and-kill. Your child will learn to discover everything. Trust me. If your child ends up in the remedial math track in high school, then they are just not good in math. Trust me.

Anonymous said...

Hey, that was me, Lynn!

I was thinking that again when the forgiving division came up. You could just avoid those pesky ones that you can never remember. It'll all work out.

Anonymous said...

I thought the same thing, Catherine. That seems to be our Steve....

SteveH said...

hmmmm....I wonder who SteveH23 might be?

Hee, Hee, Hee. I suppose I should do some work today.

SteveH said...

It's a good writing exercise. You are limited to 500 characters.

LynnG said...

I don't know if EM teaches fraction division in 5th grade. I've looked at everything through the end of the first 5th grade math journal and the most they do is fraction addition and subtraction using clock faces (again). I don't think they get to fraction division at all in 5th grade, but I'm not certain, because we haven't gotten through the year yet.

On the other hand, SingaporeMath obsesses over fractions in 5A. If the workbook and textbook aren't sufficient, the home instructor's guide gives lots of suggestions on supplementing the student that doesn't master all operations on fractions by mid-5th grade.

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