Math Teaching Tips: Digging Deeper
Over the years of homeschooling, I have become better and better (I think) at getting out more of math work than the kids just getting the correct answer. I’ve learned techniques to help us dig deeper and check if there is true understanding or whether there was just rote memorization of a process that resulted in a correct answer but may not be of benefit if the process is forgotten in future or cannot be applied/usedeasily if the problem does not fit a certain format.
Below are some techniques or “checks” that I use when I give the kids mental math problems, but I also use for written work.
1. Does my student understand what he is looking for?
Make sure that your student understands what he is being asked to find. Sometimes the kids may get a wrong answer and then we find out that they didn’t hear the problem correctly or mixed up the numbers. In other cases, they just may not be sure what they need to find. Have student repeat/or restate what they need to find.
2. Is the answer reasonable?
After your student comes up with an answer, ask them if the answer is reasonable. Do this before they give you their answer. If they need help or say yes, but it really isn’t, I try to guide them along by telling them to estimate the answer or predict what the answer should look like.
Example 1: Student is asked to estimate 25 x 37. To check for reasonableness of an answer, I would suggest that student start with 25 x 10, since 10 is a rounded number. 25 x 10 = 250, so then student could compare his answer with that. Then, I would suggest that my student multiply 25 x 40 (but remember that 25 x 10 was 250, so 25×40 would be 4x as large since 40 is 4x as large as 10).
One problem we had recently was 4 ¼ divided by some number = 5. I asked, what should the answer look like? Well, usually when you divide by a number, your answer is smaller than what you started with. But I pointed out the fact, in this case, the answer is larger. So, in this case the number we are dividing by would have to be a fraction (less than one) I had to guide my student with this, but I tried to stress this fact or generalization so it could be applied in the future, in sha Allah.
Try to be consistent with having student check for reasonableness to help student build the habit of learning to automatically do this on their own, in sha Allah.
3. How did you get your answer?
Have student explain the process they used to get their answer. Do this regardless of whether the problem is incorrect or correct. Sometimes after my student gives their response, I try to help them revise it to make it more concise.
I may also, then tell them toexplain (as if they were explaining to someone)how to do a similar problem (in general, regardless of the numbers). This requires that they use math vocabulary that can be applied to any problem (e.g. if you want to tell someone how to multiply ¾ x ¼ you might say, multiply the 3 x 1 and then 4 x 4, but in general (for any problem), you might say, multiply the numerators and then multiply the denominators. Thus now you have an explanation for any such problem.
If student is stuck on a problem, I have them tell me what might their first step be, then what, then what……
4. Give a related problem to help student see patterns.
After my student has worked out a problem, I might give a related problem to see if they can work out a pattern that can be applied.
For example, we had a problem of 5 percent of 80. Once they got the answer, I asked them what 20 percent would be, what 40 percent is. I try to get them to see that 20 percent is 4 times 5 percent and that they could just multiply their initial answer by 4 and then to get 40 percent, multiply the initial answer by 8. I also teach them to identify what 10 percent of a number is and then use that to find what is asked if it is a multiple or factor of 10 (5 percent, 20 percent, 60 percent, etc. I think that doing this consistently builds up their ability to find easy percents like this faster.
5. Make the problem simpler.
One of my famous lines is to “make the problem simpler” before they get into calculations. So, for example, if they are dividing 40/20, I tell them to take off the zeroes and then they have 4/2. Whenever I have them make the problem simpler, I try to make sure that they understand what is going on (that taking off the zeroes is like dividing both by 10). In another problem, they might have 1000/25. In this, I suggest a “tricky” shortcut and they have to be sure to correct for this. So I say, take off one zero and now I have 100/25 which is 4 and they have to remember to put the zero back on on top and thus adding a 0 to end of their answer of 4 to get 40.
If the problem was 7 x 1/7, I would remind student to use the multiplication shortcut by cancelling out the 7s as they could think of the problem as 7/1 x 1/7 so we have 7 x 1/(1 x 7) but they can rearrange it so that 7 x 1/ (7 x1) and the 7s cancel out to make 1 and they are left with 1 x 1 = 1.
6. I’m stuck!
When doing written work, when one of the kids doesn’t understand what to do, he writes “don’t understand.” I have started to ask him to write specifically what it is he doesn’t understand or how he is stuck. In this way, I can get him toattempt to go past the “looking at the problem and drawing a blank stage and just moving on.”
7. Perimeter and Area problems
In our review work (I use NC’s Week by Week Essentials regularly) they are often asked to find perimeter or area. I noticed that they would get the two mixed up (i.e. find perimeter when they needed to find area). In other cases, they would simply find the area of say a rectangle pool but what was asked was the area of a walkway around the pool. So, I told them, for any perimeter or area problem to ALWAYS do the following:
- Draw a picture
- Write a formula
and I kept drilling this simple routine into their heads each time and Alhamdulillah, now, they draw a picture and write a formula pretty automatically and their work of this type is more accurate now.
8. Make a plan
In general, I now see how helpful it was back in school when my math teachers would have us write out the following for each problem:
- Find: (What I am asked to find)
- Given: (What I know)
Sometimes, I have the kids write up a little paragraph on the steps of how they solve a general problem (such as how to determine if a number is odd or even). For my kids that are not as skilled in expressing themselves well in writing, I might provide a frame:
How To Tell If a Number is Odd or Even
Numbers can be ____________________ or _________________. Here is how to tell if a number is _____________ or _______________. A number is _____________ if __________________________. For example, _______ is __________ because _______________________________. A number is _____________ if ____________________________. The number ______________ is ____________ because ___________________________. Now, you know how to tell if any number is ___________________ or ___________________.