Some Difficult Math Concepts for 3rd to 7th Graders

Some Math concepts are intrinsically difficult for children. This post is about what I think are the commonly found issues in understanding.

Find the first 4 multiples of 24. Ans: 1,2,3,4. The child found factors (correctly) instead of finding multiples. I don’t see this as a big issue. The child knows the concept. This seems like a K (Knowledge) issue. Meaning the child mistook factors for multiples. I tend to look at lack of knowledge as a lesser problem, since the correction of the same is easy. 

But the following issues are more serious since the child doesn’t seem to be a grip on the topic and seems to have solved the problem mechanically without understanding. They seem like C and A (Comprehension and Application) issues. 

1. Which number multiplied by 36 is 324? A child multiplied the two numbers instead of dividing.
2. In many problems, children don’t do a check to see if the answer seems correct and whether it is in the ball park.
1. Add 300, 400, 1500, 2500: Child left justified the numbers and got answer: 11000.
2. Divide 872 by 8: Ans: 19 (instead of 109) – while we teach estimation, children don’t seem to use this when the topic is not estimation.
3. A child stands on top of a roof, 45m above the ground, and throws a ball vertically up. From the highest point, the ball travels 63m to the ground. How high did the boy throw the ball from the roof? And: 63m. (instead of 18m)
4. While doing decimal multiplication or division, many children put the decimal in the wrong place. I am not talking of a problem that involves a division like 0.004589 / 0.1233 – this is difficult to conceptualize. I am talking about problems like 12.3 *5.6  or 14.6 / 3.1 which are easier for a child to understand and process. 
5. 13 * x = 234. What number HAS to be in the units or ones place in x? Or even a similar problem such as 13 + y = 234. What number HAS to be in the units or ones place in y? The equivalent division or subtraction problem is expected to be even more difficult for a child.
6. Understanding operations with negative numbers 13 – (-8) ? 
7. A similar operation in algebra : Expand (2x+y)/3 – 5(6-2y)/7 is usually very difficult for a child in class 7.
8. I stand facing west. While doing a yoga, first I turn 45 deg anticlockwise, then 135 deg clockwise, then 90 deg clockwise. Which direction am I facing now?

My belief is that the natural ability to solve these kind of problems without error comes from fluid intelligence. With age and experience they can come from crystallized intelligence, though the process will be delayed. C and A are far more difficult to inculcate. (Soft skills are equally difficult to learn and very often good soft skills are seen as far more important than C and A or even good grades - Read this).

These issues (of not being able to solve math problems correctly) are probably found as early as in class 1. When asked which number when added to 7 gives 15, some children tend to make a mistake. They add the two numbers. 

The children that do these kind of problems correctly probably bring an intensity or discipline into problem solving that other children lack, else their understanding of complex questions (for their age) is very high. What happens when that discipline is lacking or the ability to understand complex statements is inadequate? They are NOT mentally challenged. They are "average" children who may posses extraordinary talents in other fields - read this.

Towards improving the ability of average children in solving such questions correctly, the questions that I now have towards people who play the role of teachers are:

• How do we track mistakes of these sort in the responses to homework assignments?
• Do we track at all?
• If we do, what do teachers do after correcting the assignment? Is there a feedback loop into the next week's lesson plan or in some way to correct the mistake? What is the exit criteria - meaning how do we decide that the concept is reasonably understood by students in the class?
• If there is such a feedback loop, what kind of lesson / method of teaching helps those children who have difficulty understanding these topics?

Few months back, a friend of mine recently asked me if I knew of anyway to explain to children why negative times negative is positive. I thought about it for a minute and replied that I didn't.

Recently I thought of a scenario to address the issue raised by my friend. Let's say, the temperature in Delhi is 34 deg C and that in Oslo is -5C. What is the difference in temperature between the two cities? If you plot the two values in a number line, Oslo will be to the left of zero while Delhi's will be to the right of zero. The difference intuitively is the sum of distances between Delhi to zero and zero to Oslo which is 34+5=39C. In mathematical terms we are subtracting Oslo's temperature from Delhi's: 34-(-5) which, as we did earlier, was an addition of 34 and +5. In effect, -(-5) became +5. 

I wonder if this explanation would be useful.

Note: Grade and Class are used interchangeably. A child, in India, studies from class 1 to 12 and then goes to college. Usually a child is about 5-6 years old when he is admitted to class 1 and about 18 when he finishes class 12.

Additional reading:

1. http://vbala99.blogspot.com/2014/06/misc-articles-on-personality.html