To differentiate instruction means to provide a different learning experience for every individual student in the class. Perhaps there is a student who is just learning English in your class. And perhaps that student sits next to another who wants to have an in-depth discussion about Shakespeare. Should these two students prove difficult to teach at once, a normal person might consider what the root problem is -- that they shouldn't be in the same class. But the wise education bureaucrat knows that any problem here must be the teacher's -- he must not have differentiated his instruction enough.Not to worry, it's entirely possible to teach "algebra" (however a non-math person wants to define "algebra") to someone who doesn't understand fractions.

Hat tip to Joanne for the link above.

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Don't worry -- this phenomenon will disappear in the next 3-5 years. As soon as the grant money runs out.

That's just the problem, it doesn't require grant money, it saves it! Ed School types say this crap to try to sound important, and school districts pick up on it because it saves them money by being able to just dump kids in any class, prepared or not, and make their learning solely the responsibility of the teacher.

Can you elaborate on teaching algebra to someone who doesn't understand fractions? Or were the quotes supposed to indicate irony?

-Mark Roulo

So THAT's where it came from. Last year my district when through this whole big rigamarole over "differentiated instruction". Just the paperwork was horrendous, but moreso was the website where we had to periodically report strategies, notes home and other actions taken to "encourage success" in the student. Bless her heart, the data coach was a nice lady but I honestly dreaded seeing her because it was always more work for me than it was worth in comparison to the results. What is sad is that most good teacher do adapt material for students that are struggling, but somehow putting it into another program which probably cost my district a few millions, it just all has a sour taste.

My comment was the very definition of sarcasm. Seriously, go look it up!

Thank you, Darren.

I've been working on the belief that mastery of rational numbers is the gateway to polynomial algebra, which is the gateway to calculus.

I will proceed as panned :-)

-Mark Roulo

Run that by me in a bit more detail?

The claim is a two-parter:

1) Many of the failures in Calculus (possibly the majority) can be traced to poor algebra skills.

2) Many/most of the failures in Algebra can be traced to poor mastery of rationals/fractions/mixed-numbers.

In support of (1), there is the Euler quote from here (http://www.math.uconn.edu/~troby/allquotes.txt):

"Most of the difficulties students have learning calculus can be traced to weak algebra skills." -- Leonhard Euler

The quote might even be a legitimate quote! In any event, I have seen this position many times in discussion threads on teaching calculus and why calculus seems to be so difficult to so many. There *ARE* other issues that make calculus difficult, but there does seem to be some consensus that for many students who crash and burn in calculus poor algebra is the primary cause.

I don't have an equivalent quote for (2), but the general sense seems to be the same: Much of algebra crash-and-burn can be laid at the foot of poor understanding of using and manipulating rational numbers.

-Mark Roulo

I find problems with both rational expressions and negative numbers.

Thanks for the heads up on negative numbers, Darren.

Interestingly, I had decided that the next few weeks (starting yesterday...) are the time for my son to get very good/comfortable with negative numbers :-) He already knows what they are, but hasn't had much experience in manipulating them formally.

I'm not anticipating too many troubles as adding/subtracting go as a unit and build very naturally off of adding/subtracting positive numbers.

Multiplying/dividing will probably take longer as I'm planning on introducing the unit circle as a framework here, rather than just "cancelling negative pairs." The thought is to lay the groundwork for roots greater than 2 that solve out to complex numbers.

We'll see how it goes. I never did teach him "invert and multiply" for dividing fractions as I consider it a Very Bad Idea(tm). Dividing fractions took longer for him to learn because of this. Multiplying negative numbers may take longer, too!

-Mark Roulo

There's nothing wrong with "Invert and Multiply" if you show, using concrete examples, how and why it works. Example:

10 divided by 5: How many $5 bills would you get for $10?

10 divided by 2: How many $2 bills would you get for $10?

10 divided by 1: How many $1 bills would you get for $10?

10 divided by 1/2: How many half-dollars would you get for $10?

etc.

As for teaching multiplying negatives, here was what was shown to me to make it "concrete". Imagine a pool being filled with water at 1 gallon per minute. The pool currently (time t=0) has 100 gallons in it. How many gallons will it have in 3 min? 2 min? 1 min? Now? 1 minute ago (which is -1)? etc.

What I do not like about "invert and multiply" is that even if the students are shown why/how it works, I expect that it rapidly becomes a magic "trick." I fear that this contributes to the sense that math is a bunch of unconnected/arbitrary tricks rather than an internally consistent logical system.

Multiplying a negative by a positive is not the problem. The problem will be multiplying a negative by a negative. Or raising a negative to the 7th power.

Interestingly, I often don't need to make things "concrete" other than to tie them in to word problems. My son is perfectly happy working with math in the abstract. I think that this is very odd as he is NOT a "math brain" :-)

-Mark Roulo

For negative times a negative, make the pool draining. How much water will be in it in 3 min, 2 min, 1 min, now, 1 minute ago, etc.

Another way to teach division of fractions is to get a common denominator, then "divide straight across" like we multiply. The answer's denominator will always be 1, since we had a common denominator when we divided, so the answer will be the quotient of the 2 numerators. Try it out, it's fun :)

Thanks for the negative × negative with water suggestion. I'll try to work it in.

I think I'm going to skip the division of fractions via common denominator approach for now. My son does division like this:

http://www.mistybeach.com/mbra/topics/math/dividing_fractions_technique.html

It works, he seems to understand the pieces, and he remembers it. Right now I don't want to create any unnecessary confusion.

-Thanks,

Mark Roulo

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