Elementary Problem Solving: The Early Years

Posted on by Denise Gaskins

[Rescued from my old blog. To read the entire series, click here: Elementary Problem Solving Series. Photo by Studio 757 via Flickr (CC BY 2.0).]

You can begin to teach your children algebraic thinking in preschool, if you treat algebra as a problem-solving game. Young children are masters at solving problems, at figuring things out. They constantly explore their world, piecing together the mystery of how things work. For preschool children, mathematical concepts are just part of life’s daily adventure. Their minds grapple with understanding the three-ness of three blocks or three fingers or one raisin plus two more raisins make three.

Wise homeschooling parents put those creative minds to work. They build a foundation for algebra with games that require the same problem-solving skills children need for abstract math: the ability to visualize a situation and to apply common sense.

Continue reading Elementary Problem Solving: The Early Years

Confession: I Am Not Good at Math

Posted on by Denise Gaskins

I want to tell you a story. Everyone likes a story, right? But at the heart of my story lies a confession that I am afraid will shock many readers. People assume that because I teach math, blog about math, give advice about math on internet forums, and present workshops about teaching math — because I do all this, I must be good at math.

Apply logic to that statement. The conclusion simply isn’t valid. …

Update: This post has moved.

Click here to read the new, expanded version

Project Follow Through Story Looks Interesting

Posted on by Denise Gaskins

Project Follow Through was an almost-30-year study that compared the effect of different teaching methods on over 20,000 students nationwide. I have started reading The Outrage of Project Follow Through: 5 Million Failed Kids Later [site no longer exists, but try this book: Project Follow Through: A Case Study of Contingencies Influencing Instructional Practices of the Educational Establishment], which explains the research and its results in layman’s terms. So far, I have enjoyed the book, which is being released chapter-by-chapter every Monday. The introductory chapter will be available only for the remainder of this week, however, so if you are curious, you had better act now. I recommend downloading the pdf file to read at leisure: Right-click on the link for each chapter, then choose the “Save” option.

[Hat tip: D-Ed Reckoning, who is running a series of articles (part 1 here) highlighting his favorite parts of the book.]

Elementary Teacher Education

Posted on by Denise Gaskins

Unfortunately, this is all too believable:

Received an email from a parent.

Not one of our students, but rather the parent of a high school student who plans to attend this university. The parent is looking for advice on how to get the kid out of math. Seems that the kid has already taken the bare minimum number of units of high school math needed for graduation and has stopped taking math. The parent is wondering if the kid can take some sort of test (before forgetting any more math) to fulfill the university’s math requirement.

Guess what career the kid is planning on? School teacher.

From Rudbeckia Hirta at Learning Curves.

Mathematics and Imagination

Posted on by Denise Gaskins

Comments by W. W. Sawyer, in his wonderful, little book, Mathematician’s Delight:

Earlier we considered the argument, ‘Twice two must be four, because we cannot imagine it otherwise.’ This argument brings out clearly the connexion between reason and imagination: reason is in fact neither more nor less than an experiment carried out in the imagination.

Continue reading Mathematics and Imagination

Number Bonds = Better Understanding

Posted on by Denise Gaskins

[Rescued from my old blog.]

number bondsA number bond is a mental picture of the relationship between a number and the parts that combine to make it. The concept of number bonds is very basic, an important foundation for understanding how numbers work. A whole thing is made up of parts. If you know the parts, you can put them together (add) to find the whole. If you know the whole and one of the parts, you take away the part you know (subtract) to find the other part.

Number bonds let children see the inverse relationship between addition and subtraction. Subtraction is not a totally different thing from addition; they are mirror images. To subtract means to figure out how much more you would have to add to get the whole thing.

Continue reading Number Bonds = Better Understanding

It’s Elementary (School), My Dear Watson

Posted on by Denise Gaskins

[Rescued from my old blog.]

From Time magazine, June 18, 1956:

“[M]athematics has the dubious honor of being the least popular subject in the curriculum… Future teachers pass through the elementary schools learning to detest mathematics… They return to the elementary school to teach a new generation to detest it.”

Quoted by George Polya in How to Solve It. I finally got my very own copy of this excellent book, so I can quit pestering the librarian to let me order it from library loan again…

Blogger Rudbeckia Hirta teaches math to pre-service teachers, and it seems that not much has changed since 1956. Hirta says the test answers shown were representative of her class — for instance, 25% of her students missed the juice problem. Too bad these students never read Polya’s book, in which he discusses a four-step method for solving problems. Step four is to look back and ask yourself whether the answer makes sense. Good advice!

Continue reading It’s Elementary (School), My Dear Watson

Our Tax Dollars at Work

Posted on by Denise Gaskins

Well, the new year has come, and it’s time to start gathering up receipts and thinking about tax forms.

Would you like to know that our tax dollars are doing good in the world? The National Science Foundation has spent many millions developing and promoting “reform” math textbooks, with encouragement from the U.S. Department of Education. Surely our public schools will now rise out of the doldrums and surge ahead in mathematical achievement, right?

Try for yourself this problem from one of the more famous/infamous of the reform math textbooks:

Can you find the slope and y-intercept of this equation?

10 = x – 2.5

And then check out this editorial[editorial has disappeared] at edspresso.com. You’ll be amazed at the answer!

Update: Checking on back-links, I discovered that this page had gone AWOL, so I’ll give you the “answer” from the teacher’s manual. The “slope” is 1 and the “y-intercept” is -2.5, according to Connected Math. Unfortunately, this equation actually describes a vertical line (undefined slope) at x=12.5 (never touches the y-axis).

Doesn’t bode well for “CMP helps students and teachers develop understanding of important mathematical concepts…”

The “Aha!” Factor

Posted on by Denise Gaskins

[Rescued from my old blog.]

For young children, mathematical concepts are part of life’s daily adventure. A toddler’s mind grapples with understanding the threeness of three blocks or three fingers or one raisin plus two more raisins make three.

Most children enter school with a natural feel for mathematical ideas. They can count out forks and knives for the table, matching sets of silverware with the resident set of people. They know how to split up the last bit of birthday cake and make sure they get their fair share, even if they have to cut halves or thirds. They enjoy drawing circles and triangles, and they delight in scooping up volumes in the sandbox or bathtub.

Continue reading The “Aha!” Factor

Math Quotes II: The Ultimate Goal of Mathematics

Posted on by Denise Gaskins

[Rescued from my old blog.]

I thought you might enjoy the quote I’m going to use on the blackboard in math class:

The clearer the teacher makes it, the worse it is for you. You must work things out for yourself and make the ideas your own.

—William F. Osgood, quoted in
Out of the Mouths of Mathematicians
by Rosemary Schmalz

Continue reading Math Quotes II: The Ultimate Goal of Mathematics