Math teaching/readiness question

[Kezia]

Over breakfast yesterday, he randomly asked me, “How do you figure the area of a circle?”

Uh, well... that involves pi, I rattled off the formula and briefly explained radius and diameter. Took 60 seconds. 

But should I even teach that just because he is interested? I love that he is interested and asking questions, but I followed that interest with multiplication and taught him my old school way last year way before we considered homeschooling, which he greatly preferred over BA’s methods. He can use BA’s methods but reverts to mine every chance he gets and shows no interest in BA’s methods maybe being easier mentally in some ways. 
 

BTW we use BA as the sole curriculum and are currently on BA 4A, exponents with logic chapter thrown in as variation. We also went further than BA level 3 did by going with level 4 fractions because again, that interest driven math topic took us there. 
 

Do people really do interest driven math? If he asks and shows interest, does that mean he is ready? Can I cause more harm than good by letting his interest direct us in math?

Edited February 13, 2021 by Kezia
Clarification

[wendyroo]

In that situation, I would have gone the other direction. Instead of giving him a formula that he is not really ready to understand and apply, I would have used his question to reaffirm basic principles that he already does know.

"How do you figure the area of a circle?"

The same way you find the area of any other shape - you measure how much space is inside it. With any shape, we could draw little 1 cm by 1 cm boxes inside, count how many we could fit, and know the area. Obviously, with a rectangle it is easy because we can figure out how many rows of boxes we can fit, and how many columns of boxes we can fit, and multiply them. But it is the same process with other shapes. For now, your best bet would be to estimate it by inscribing the circle inside a square and finding the area of the square. You would then be able to estimate the area of the circle. Eventually you will learn how to calculate it exactly, but for now, a well reasoned estimation will probably get you close enough.

[Kezia]

I learned math procedurally. Area of a circle popped a certain formula in my head. 
 

I would need the curriculum to guide me in the teaching if we pursue things like this so he understands it with better depth than I ever cared to know. 

[wathe]

We did an exploration of area of circle  and pi at about that age.  We cut up a lot of different sized circles into wedges, and rearranged in to rectangle-like shapes.  And lo and behold, the length of the long side (pi*r) of the rectangle divided by the length of the short side (r) of the rectangle was always three-and-a-bit, which we called pi.   And the area of this rectangle is very visually r*(r*pi).  It was a fun demonstration

Edited February 14, 2021 by wathe

[*****]

5 hours ago, Kezia said:

Do people really do interest driven math? If he asks and shows interest, does that mean he is ready? Can I cause more harm than good by letting his interest direct us in math?

I can't recall off-hand what BA math is.  But have you ever looked at Mortensen Math?  You can find many videos on youtube, see Crewton Ramone.  It is all about hands-on math.  Your son sounds like a good candidate for the use of this kind of learning. My daughter was a question-asker.  Homeschooling was so fun because of her varied interests. Use it to your advantage!

[Ellie]

Of course, you should have told him. You should tell him anything he asks. It doesn't mean you need to do *only* interest-driven math.

[sweet2ndchance]

You might check out the Sir Cumference book series at the library. Sir Cumference and the Dragon of Pi may be of particular interest right now for your son.

Your son might be interested, he might not, but it never hurts to indulge an educational question from a child. 

Ds8 has been asking a lot about chemistry lately but I don't have the formal study of chemistry planned until next year but that doesn't mean we can't chase rabbit trails now while he's interested.

Edited February 14, 2021 by sweet2ndchance
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[Kezia]

8 hours ago, wathe said:

We did an exploration of area of circle  and pi at about that age.  We cut up a lot of different sized circles into wedges, and rearranged in to rectangle-like shapes.  And lo and behold, the length of the long side (pi*r) of the rectangle divided by the length of the short side (r) of the rectangle was always three-and-a-bit, which we called pi.   And the area of this rectangle is very visually r*(r*pi).  It was a fun demonstration

We will follow this little rabbit trail and do this fun hands on exercise. If I ever learned where pi came from, I have long since forgotten, and I really love seeing him do things like this and come to his own conclusions. 

 

1 hour ago, sweet2ndchance said:

Sir Cumference and the Dragon of Pi may be of particular interest right now for your son.

I will check to see if our library has this as well. 
After the rabbit trail, we will go back to the planned curriculum, likely skipping around a bit within. 
 

Thanks for the great ideas! 

[Not_a_Number]

14 hours ago, Kezia said:

But should I even teach that just because he is interested? I love that he is interested and asking questions, but I followed that interest with multiplication and taught him my old school way last year way before we considered homeschooling, which he greatly preferred over BA’s methods.

What are “old school ways”? What are BA methods? I don’t remember BA having any special fancy methods, although we didn’t use the program — I’ve just glanced at it.

[Not_a_Number]

2 hours ago, Kezia said:

We will follow this little rabbit trail and do this fun hands on exercise. If I ever learned where pi came from, I have long since forgotten, and I really love seeing him do things like this and come to his own conclusions. 

The definition of pi is just the ratio of the circumference of a circle to the diameter of a circle. It's always the same because all circle are similar figures -- they are all scaled up versions of each other. 

[kirstenhill]

23 minutes ago, Not_a_Number said:

What are “old school ways”? What are BA methods? I don’t remember BA having any special fancy methods, although we didn’t use the program — I’ve just glanced at it.

I'm going to guess that the OP means procedural vs. conceptual, but maybe doesn't think of it using those terms, necessarily.

Also, for multi-digit multiplication and long division (which the OP wouldn't have gotten to in BA yet), BA never gets all the way to the traditional algorithm most of us grew up using.  I taught my two older kids who used BA the traditional algorithm for both as a "shortcut" once they were comfortable with the longer way.  My youngest refuses to hear me out about the traditional algorithm...and I'm just left getting annoyed that big multiplication problems take a lot of writing...and he has dysgraphia so if he writes it is slow.  At least he let me teach him short division, which is definitely my favorite math shortcut. 😁

[Not_a_Number]

29 minutes ago, kirstenhill said:

I'm going to guess that the OP means procedural vs. conceptual, but maybe doesn't think of it using those terms, necessarily.

I'm also guessing that, but I'd like to hear what she has to say 🙂 . 

 

29 minutes ago, kirstenhill said:

Also, for multi-digit multiplication and long division (which the OP wouldn't have gotten to in BA yet), BA never gets all the way to the traditional algorithm most of us grew up using.  I taught my two older kids who used BA the traditional algorithm for both as a "shortcut" once they were comfortable with the longer way.  My youngest refuses to hear me out about the traditional algorithm...and I'm just left getting annoyed that big multiplication problems take a lot of writing...and he has dysgraphia so if he writes it is slow.  At least he let me teach him short division, which is definitely my favorite math shortcut. 😁

Ah, hmmm. I do like the traditional algorithm eventually!! I think I spent a few years with DD8 not using it, then I did teach it, and now she uses it all the time, because it's SO MUCH faster. 

[wathe]

2 hours ago, kirstenhill said:

I'm going to guess that the OP means procedural vs. conceptual, but maybe doesn't think of it using those terms, necessarily.

Also, for multi-digit multiplication and long division (which the OP wouldn't have gotten to in BA yet), BA never gets all the way to the traditional algorithm most of us grew up using.  I taught my two older kids who used BA the traditional algorithm for both as a "shortcut" once they were comfortable with the longer way.  My youngest refuses to hear me out about the traditional algorithm...and I'm just left getting annoyed that big multiplication problems take a lot of writing...and he has dysgraphia so if he writes it is slow.  At least he let me teach him short division, which is definitely my favorite math shortcut. 😁

Mine have both firmly rejected the traditional algorithm.  For a while they insisted on the long, long way of writing every partial product, which for big problems would lead to errors because there were so many zeros they would lose track of place value, or misalign columns, or lose an important digit in all those zeros.  Now they like lattice.  I'm OK with that.  It's more writing than the traditional algorithm, but not a lot, and it's well organized - really good for my disgraphic kid.

[wathe]

Back to the OP, yes, we do both interest driven and curricular math.  We watch youtubes on fun math topics (Vi Hart, Matt Parker).  Read mathy books (Murderous Math Series, Math with Bad Drawings) and go off on tangents whenever questions come up.  Statistics and exponential growth have been much discussed this year, a la pandemic math.  The freedom to ditch the day's plan and go off on a rabbit trail is one of the best things about homeschool IME.

ETA: we went on a major slide rule tangent this year.  So fun.  The slide rule museum has a great how-to and virtual slide rule to practice on, and DIY paper slide rules are pretty cool too.

Edited February 14, 2021 by wathe

[Kezia]

3 hours ago, Not_a_Number said:

What are “old school ways”? What are BA methods? I don’t remember BA having any special fancy methods, although we didn’t use the program — I’ve just glanced at it.

BA uses the area model, distributive property, and the multiplication algorithm, none of which is the way I was taught. For ex. 18x3, I would multiply 8x3 and carry the two tens and think (3x1) + 2. I would also use zero as a place holder when multiplying multi-digit numbers by another another multi-digit number. 

BA is not using special or fancy methods, just not the way I was taught. Upon further investigation, BA frowns upon the way I was taught much of upper elementary math concepts. 

[Not_a_Number]

14 minutes ago, Kezia said:

BA uses the area model, distributive property, and the multiplication algorithm, none of which is the way I was taught. For ex. 18x3, I would multiply 8x3 and carry the two tens and think (3x1) + 2. I would also use zero as a place holder when multiplying multi-digit numbers by another another multi-digit number. 

So, to be clear, the way you were taught is a logical consequence of the stuff BA teaches 🙂 . It's too bad they never do the algorithms, though -- I think they are an excellent outcome of their methods and they follow easily once you understand the methods well. 

 

14 minutes ago, Kezia said:

BA is not using special or fancy methods, just not the way I was taught. Upon further investigation, BA frowns upon the way I was taught much of upper elementary math concepts. 

I don't always agree with their methods, but I do agree with them on that. I think it's important to start with the definitions and work with them before learning algorithms. 

[Not_a_Number]

41 minutes ago, wathe said:

Mine have both firmly rejected the traditional algorithm.  For a while they insisted on the long, long way of writing every partial product, which for big problems would lead to errors because there were so many zeros they would lose track of place value, or misalign columns, or lose an important digit in all those zeros.  Now they like lattice.  I'm OK with that.  It's more writing than the traditional algorithm, but not a lot, and it's well organized - really good for my disgraphic kid.

So, just a tip: let them do whatever they want, then try it with them again. My kiddo wanted to play and play and play with multiplication, until she had explored it enough for it to be boring old news. Then she was ready for the traditional algorithm, which really IS the fastest way if you need to multiply quickly without a calculator. 

I'm not saying it's necessary or anything... it's just worth knowing if the resistance decreases 🙂 . 

Edited February 14, 2021 by Not_a_Number

[Kezia]

3 hours ago, kirstenhill said:

I'm going to guess that the OP means procedural vs. conceptual, but maybe doesn't think of it using those terms, necessarily.

Yep I learned all math procedurally or at least that is how I remember it. If it was ever taught to me conceptually, I do not remember it. I remember formulas and algorithms. That is why I would rattle off a formula with regards to finding area of a circle. I admit that I lack a deep understanding of the concepts. My passion was language related, not math and numbers. I never wondered or asked about math concepts. 
BA advocates kids finding these things on their own through exploration of the concepts. I printed some extra materials from BA, where they specifically say do and encourage this, not that. The “that” they are referring to is my multiplication method and applying just formulas I remember. They want the kids to come to those conclusions from working through the problems to gain a much better understanding. We have gotten through the BA multi digit multiplication section, I did teach it to my son, but he likes the original way I taught so much better. I never learned short division either but it looks much easier than my long division and BA teaches it. 
 

 

[wathe]

7 minutes ago, Not_a_Number said:

So, just a tip: let them do whatever they want, then try it with them again. My kiddo wanted to play and play and play with multiplication, until she had explored it enough for it to be boring old news. Then she was ready for the traditional algorithm, which really IS the fastest way if you need to multiply quickly without a calculator. 

I'm not saying it's necessary or anything... it's just worth knowing if the resistance decreases 🙂 . 

Oh, for sure.  I model the traditional algorithm when I need to do a quick calculation all the time.  So far, no dice. 

 

 

[wathe]

And really, why learn an algorithm when one can whip out one's pocket slide rule? 

JK, sort of.  Eldest really does carry a pocket slide rule these days.

[Not_a_Number]

12 minutes ago, Kezia said:

 We have gotten through the BA multi digit multiplication section, I did teach it to my son, but he likes the original way I taught so much better.

Well, of course he does. It's much EASIER. You know what's a lot harder, though? Remembering it when you have 10 other algorithms to remember, and you have no clue where any of them came from so no way to check whether your memory is right or not. That's the kind of thing that really limits you in higher math, when the formulas multiply to a terrifying list you can't possibly memorize. 

The point of going through the logic isn't just to get to the Holy Grail of the traditional algorithm. It's also to model for kids how to think about the mathematics and how to work with definitions to figure things out. 

I wouldn't let him use this algorithm if he can't explain it. Leave it as encouragement for later, and make sure he can do things using the definition before you move on from that.