Comments for Mathematics for Teaching

Comment on What is algebra? by Katerina Kalfopoulou (@kalfokat)

Katerina Kalfopoulou (@kalfokat) — Wed, 22 May 2013 04:07:52 +0000

Of course we do!!! With many different ways I have an example here
http://mathandliterature.blogspot.gr/2010/11/blog-post_25.html

Comment on Math investigation lesson on polygons and algebraic expressions by Violentina J Asuncion

Violentina J Asuncion — Tue, 21 May 2013 22:12:12 +0000

I am thankful that you are passionate to share what you have. Thank you you ma’m. I will see to it that your labor is not in vain i am ready to share it with my students. God bless you always!

Comment on Unpacking mathematics – a geometry example by Erlina Ronda

Erlina Ronda — Fri, 10 May 2013 14:11:05 +0000

I was thinking of putting that scaffold, too as to the shape of the AFCE but the applet shows that E and F were both coming from the midpoint of the diagonals. Also DE and FB are congruent. Maybe it’s enough hint for them to make the conjecture. If the static figure will be used then yes maybe students need additional support.

I have a problem uploading in GeoGebra tube. I will just put a direct link to the applet so readers can open it in their browsers.

The idea of ‘unpacking’ is from one of my readings on math teachers knowledge. I could not anymore recall the title but I’m sure I got the ‘unpacking’ term from one of Deborah Ball’s articles.

Comment on Unpacking mathematics – a geometry example by John Golden

John Golden — Fri, 10 May 2013 13:58:36 +0000

Like the post. I think the idea of unpacking is helpful here. Wondering if the students need some support to conjecture AFCE is a parallelogram first. Either show length of the sides (for an opp sides are congruent approach – goes with the proof method) or the slope of the sides (if you want to emphasize the parallel). Sometimes I think about a process of formalizing the logic: notice, justify, prove. What is true about the diagram? Why is it true? Is it always true?

It might be helpful to have a link to the sketch on GeoGeraTube, or a download link for the file. Sometimes my browser just won’t do the java for embedded GGB. On the Tube there would be the option for mobile-ready version, too.

Comment on My issues with Understanding by Design (UbD) by zip

zip — Fri, 05 Apr 2013 02:23:02 +0000

We don’t need the best curriculum, because even if how best is that curriculum, it might not be the best for many..specially those barangay schools..what we really need is the best teacher who can not just teach but dig where our nation must be planted for it to grow…even if how best is the curriculum, if the teacher will only depend on what is in that materials provided (i’m not saying all, but its a fact that there are really many who are doing it)their concern is to finish the topic without looking at the outcome..it’s all nothing..we need best best best teacher…it is always in the teacher…

Comment on Why do we ‘invert’ the divisor in division of fractions by Jonny pornel

Jonny pornel — Mon, 18 Mar 2013 06:03:14 +0000

Great article! it is really useful.

Comment on Why do we ‘invert’ the divisor in division of fractions by Steve Phelps

Steve Phelps — Sun, 17 Mar 2013 17:30:20 +0000

Good post. The question that (imo) still needs to be answers is this: Why when multiplying fractions a/b x c/d, the answer is the fraction ac / bd…you just “multiply straight across?” Why doesn’t division of fractions work the same way? Why can’t you just “divide straight across?”

In fact, you CAN divide straight across. For example, 3/4 ÷ 5/6 = 3÷5 / 4÷6 = 18÷30 / 20÷30 = 18/20 = 9/10.

Now, multiplying fractions and dividing fractions work the same way.

Comment on Why do we ‘invert’ the divisor in division of fractions by Mark Kronenberg

Mark Kronenberg — Sun, 17 Mar 2013 17:23:48 +0000

How about a simple intuitive approach? If you divide a number by 2, you have ½ as much. If you divide a number by 3, you have ⅓ as much. If you divide a number by x, you have 1/x as much. $10/2 = ½ of $10 = $5. $10/$½ = 10 x 2 = 20 (# of half-dollar coins in $10 is 20. A $10 roll of quarters = 40 quarters: $10/$¼ = $10 x 4quarters/$1 = 40. It’s intuitive. It shows a pattern. It isn’t merely memorizing a rule and bypassing understanding. And it isn’t the complex example above with extra steps that will overwhelm and impel struggling kids to give up after being thrown down a rabbit hole of ineffiency.

Comment on Why do we ‘invert’ the divisor in division of fractions by Jeff Gaynor

Jeff Gaynor — Sun, 17 Mar 2013 16:59:34 +0000

My sixth graders suffer from the “Curse of Knowledge.” Because they learned how to solve this kind of problem in 5th grade (invert and multiply) they do not see the need to understand how or why it works. So I ban this method, just as I don’t allow them to move the decimal point when multiplying decimals.

I eventually lead students to the steps in the article, but even more meaningful is the more concrete understanding that the Connected Math Program offers by supplying stories that involve the amount of cheese needed to make a pizza. I also have students draw illustrations of these problems using number lines and area models (rectangular units are easiest):

(Set 1): For 6 ÷ 1/3: If one small pizza requires 1/3 cup of cheese, how much cheese is needed for 1 pizza? 3 pizzas? 6 pizzas?

(Set 2): For 6 ÷ 2/3: If one large pizza requires 2/3 cup of cheese, how much cheese is needed for 1 pizza? 3 pizzas? 6 pizzas?

How do the answers to the 2nd set of problems relate to those in the first set?

With enough examples, students see that the dividend gets multiplied by the divisor’s denominator (Set 1), and then divided by the divisor’s numerator. They can then generalize to division problems in which the dividend is also a fraction.

Do my students like this method? A few do, but most find it rather painful to have to think things through, when most of their school experience has taught them that finding the correct answer quickly – if magically – is what is rewarded.

Comment on Should we do away with calculation? by Jeff Gaynor

Jeff Gaynor — Mon, 25 Feb 2013 02:05:31 +0000

I taught elementary grades – mostly 3rd and 4th grade – during the time when calculators were introduced into the classroom. I considered this to be a godsend. Until this time the standard textbooks spent 90% of the time on teaching algorithms; i.e., computation. Even ‘story problems’ were at most thinly disguised calculations; there was no time for a deeper understanding of number or mathematical thinking. Calculators allowed students to save time, and allow more for higher level processing.

In time I realized the importance of number sense and mental math, and shortchanging the ability to learn basic facts or understanding the relationships and patterns of numbers (e.g, multiplying by 4 is the same as multiplying by 2, then the product by 2 again.) So yes, students are ‘empowered’ by learning the *basics* (good question – we assume we know what these are!) but why would anyone do long division with a 2 digit divisor if there was a calculator handy. Is it important to know how to do long division? Oh, once I thought it was important to be able to find a square root. Of course it is important to know when division is called for, but long division is rather quaint. On the other hand, we should absolutely be able to arrive at a close estimate, so a degree of number skills and analysis is quite important.

I wince when a teacher says what they teach is important because they’ll need to know it for the work in the next grade. Or that we need to know the properties of numbers because they are used in Algebra. This begs the question. Do we really need algebra? Do we need calculus? (ha – show me one person out of a thousand who does). We are too vested in protecting and overvaluing our turf at times. Much more important is to help students understand the world makes sense; ok, at least the mathematical universe does. Providing the concrete and conceptual foundation of why we do what we do, and why it works, is essential. Tedious or excessive manipulation of numbers following a rote algorithm is obsessive and often distorts the value of math.

I think Wolfram is right in the three reasons to learn math. From time to time it is valuable to step back and take a more overarching view of what we are doing, and why.