Long division is perhaps as difficult for a 4th or 5th grader as Calculus is for a high school senior. The multiple steps are so complex that they need to be drilled until they often lose all meaning for a child. Cute mnemonics and hours of drill might result in a procedural competency for many students (unfortunately not for all), but they do not build number sense, estimation skills, mental math or mastery in solving word problems.
We’ve all heard of the benefits of the curricular approach “CONCRETE > PICTORIAL> ABSTRACT” (C>P>A) — what does that look like in our classrooms, and what does it look like for long division?
Truth #1: Children learn best using the C>P>A approach
We’ve seen this internationally, (the highest ranking countries use it) and we’ve seen it in our own classrooms. We use it because it works. Continue reading Teaching Long Division So It Makes Sense
If you were at the CMC conference in Asilomar today, I (Kathleen) promised to post the Power Point slides from the talk “Making Sense of Word Problems”.
Feel free to use or edit as you like. (I fixed the typos!)
1. Two More Fun Warm-ups Reviewing Word Problems
We have seen our 8th graders struggle with algebra problems that involve “the number of bills” and the “value of those bills”. They can write x + y = 27 if there are 27 five and 10 dollar bills altogether, but stumble over the value equation: 10x + 5y = 210 when told that the 27 bills add to a value of $210.
So we decided to try to start such distinctions earlier – 5th and 6th grade. Here are 2 Power-Point Warm-Ups that help students begin to make this journey. As always, use manipualtives (we used Cuisenaire rods and Monopoly money, but any blocks will do) and give them time.
Word Problem- Money Continue reading More Word Problems
REVIEWING MADE FUN!
We found on our second quiz that many students were still struggling with word problems. (Correct student quiz answers here.) So we adapted some of Steve Wyborney’s “Esti-mysteries” to continue reviewing word problems, and students seem to enjoy these ‘mysteries’ and look forward to them.
Try these engaging warm-ups:
Important: Take TIME when you show these. We try to slow down the process as much as possible (without totally ruining the tension!), in order to allow more students to spend the time they need thinking through the problems. Number 3 can be used as a “Number Talk’ to see how many ways students can see the problem 7 x 13. Continue reading ‘Mystery’ Warm-Ups, plus: Starting our Unit on Patterns
We found Lego-type blocks at the Dollar Shop! Continue reading Fractions with Legos
Word Problems – YAY!
Our intensive use of Cuisenaire rods came to fruition now as we attempt to transfer fraction visualization to word problems. Here are the 2 lessons we spent on this so far:
CW Frac to WP1 and CW Frac to WP2
We used a format called “Builders and Scribes”. Continue reading Fractions: Transferring From Blocks to Word Problems
According to this article by Jo Boaler — professor of mathematics education at Stanford and co-founder of www.youcubed.org — math memorizers scored poorly on the international PISA test, and the U.S. has more memorizers than most other countries in the world. The highest achieving students internationally were those who thought of math as a set of connected, big ideas.
Here’s what we see:
1. A visual approach to fractions gives students better number sense, and better access to word problems.
When we require drawing, every problem becomes a word problem. In the problem below, all students recognized that 1/2 is 6 out of 12, visually. This is a “12-peak Toblerone”, so a total of 17 twelfths (by simply counting!) . Then this student imagined moving one 12th from the top row to make the 2nd row equal to one, leaving 5/12 on top. This shows number sense! Our students can do fraction addition and subtraction mentally. More importantly, visualization helps facilitate the transfer to word problems, as below. Continue reading Why Withhold the Fraction Algorithm?
First of all, we’re glad we did this new unit. It uses Cuisenaire rods extensively.
However, it was hard! There were pitfalls and things we would do differently next time, but aren’t there always?
So, here goes. Continue reading Egyptian Fractions… This Learning Pyramid was STEEP!
Raise your hand if you remember hating homework as a child … Raise your hand if you have children and hate it when they have homework … Raise your hand if, as a teacher, you have ever received homework submitted with teardrops on it 🙁 If you haven’t raised your hand yet, you were born under a lucky star, or you have a faulty memory.
On the plus side, homework
- offers a chance for students to independently consolidate skills they learned in a group setting
- builds skills of responsibility and time management
Continue reading Homework – How Much? How Hard?
Enjoy our four consecutive days of investigation lessons, plus one BIG question: “Why the investigations? Are they worth the time invested?”
Continue reading Four Investigations and a Question