Originally, it was a question on Quora that prompted this and got me thinking. The person asked if “math geniuses” struggled with math problems. Hmmm well first let’s answer that question.
Struggling is actually relative to the one solving the math problem. So, if you are asking if a math genius struggles with some of the problems that are difficult in basic classes up through calculus, then probably not. If, however, you are asking if such gifted individuals work through problems of their own without always knowing every step beforehand, then yes.
I provided the picture to exemplify my meaning here because it can be difficult to imagine problems that “geniuses” would find challenging, even for me with a math degree. While the first problem is hopefully easy for those reading through this, a student experiencing math for the first time can have a hard time putting together that one quantity plus another quantity gives a third quantity.
As a tutor and teacher, I have worked with students from the ground up, so to speak. It’s actually really cool watching little ones light up as they make connections and discover patterns. Getting there can be a struggle, though, and we all went through it at one point or another. In fact, the same struggle sticks with us all the way through our discovery and practice of mathematics.
There are problems out there that remain unsolved despite these brilliant mathematicians working together to arrive at a solution to benefit all of mankind. I would say that qualifies as a struggle since they haven’t been able to determine an adequate answer. But this is not a bad thing!
One of my personal education mentors asked me a question that has forever changed the way I approach math both as a student and teacher. Does struggling have to be a bad thing? While at first, of course I was opposed to ever having a student I was pouring myself into go through any distress in their attempts mostly due to the already negative association many people have with math, I came to understand that this is how individuals grow.
No matter how many times I explain a concept to someone, it will always seem easy when I do it because I already know how to perform the calculation or derivation. Until that person realizes for his or her self, they cannot repeat the process on a new problem. The struggle leads to discovery. Working through examples and then problems on their own helps foster both independence as well as appreciation for the effort needed to become a capable mathematician in their own right.
Granted, I don’t mean locking someone in a room and leaving them to their own devices. No, no. What I mean is a guided study that is equal parts help and discovery so that every student can feel that they understand the subject matter.
If you’re like me, then this revelation is one that does not come easy and can lead to a good deal of strife in terms of concern not only for your own progress but those you are responsible for. Hence, one of my favorite quotes from one of the greatest minds that has ever lived.
The famous math genius and physicist himself, Albert Einstein said not to let your struggles get you down because he was sure he had a much harder problem that he was trying to solve. Hard to get more authoritative than that!
Hopefully this not only answers your question but also serves as a springboard for any discouraged students looking toward the future for a little hope.
As always, thank you for letting me a part of your journey through math!
As a private math tutor, I am called upon for a number of reasons. Some parents contact me out of concern that their child is falling behind. Others reach out to me in the hopes that I can help get their kid to the top of their class or achieve the scores they need to get accepted to the prestigious school of their choice. Lately, however, I have been approached to simply help students with their homework. It is not necessarily that the student is behind or looking to expand their knowledge and jump ahead. Instead, the parents see the methods being used and come to me in frustration to answer the questions that their children have.
While it is true that this is good for business, and it is nice to make a living doing what I love helping people better understand math…I would feel too guilty charging for something that I can remedy in less than an hour. That is why I am hoping to simplify some of the “crazy stuff” that kids are bringing home these days so that parents aren’t left scratching their heads wondering what is is that a teacher is looking for.
First of all, let’s start with the basics. If you have a kid who’s in Kindergarten through Second Grade, they are most likely working on adding and subtracting numbers to begin with. Now many reading this might scoff and say, “Addition? Subtraction? Why can’t they figure this stuff out?” Well, let me show you a few things.
Let’s look at an example of a fairly basic problem:
I would hope that this looks simple to the majority of us. Now consider this, how do you know that 4 added to 8 results in 12. In the beginning we were given pictures; one group of 8 somethings then a group of 4 somethings and told to count them all up. Of course, we would count 12 somethings in total. The problem with this is that counting is slow and we would like to eventually move into memorizing this as a basic addition fact. How then can we help a young student to internalize this idea that 8 plus 4 equals 12?
This is where some of the Common Core strategies come into play. Now, let me just preface this with a disclaimer. I like the methods for helping students to learn concepts that are having trouble with more “old-school” techniques, but these are not useful for everyone. I believe I made my stance clear on the matter in my other post here (Common Core Rant). We good? Ok, back to delving into some of these things.
If a child is having trouble quickly remembering that 8 and 4 make 12, then we have to come up with another way to come up with the answer that is more efficient than drawing objects and counting. The first step would be to recognize which number is larger. In this case, 8 is bigger than 4, so we would begin with 8 and count up by 4: 9, 10, 11, 12. So 12 is our answer. This is still slow, but it is slightly quicker than counting up to 12 with two groups. The reason this step is important is for problems like this:
Though this is the same problem, initially, students don’t realize this. Showing young children that a group of 4 and a group of 8 is the same thing as a group of 8 and a group of 4 is another building block that helps with fact families and more difficult problems later on. For a student to realize that he or she can start with the larger number will also save time as counting up 8 from 4 will take longer than counting up 4 from 8. This idea of addition being Commutative is a concept that we get to show students even though they may already have memorized 8+4=12, which is one reason that a teacher may spend extra time on some of these facts.
After establishing that we can count on from the bigger number, we can extend this further. Knowing something called Ten Facts, we can help students to break problems down into easier pieces. For instance, we know:
Since we are adding 4, we need to add two more to get to our answer 12. Ok, there is a lot going on here. Our number system is based on tens, so being able to recognize how to get to ten, and later to multiples of ten, can help improve students’ efficiency later on in life. So recognizing all the tens facts,
will help with knowing how to make it to ten for the first step in addition problems involving answers over ten. Again, notice that both 8+2 and 2+8 are listed as facts. This is a chance to reinforce the idea that addition can be performed in any order so that with extra practice, students are more likely to internalize the concept. The second important skill necessary here is to know that 2+2=4. With the tens facts, a child can see how to get from the larger number up to ten, but that won’t do any good if he or she doesn’t know how much left to add.
Being able to decompose numbers into parts or groups is not only a way for children to perform the rest of this problem and others like it, it also helps to lead into subtraction later on. Knowing doubles like 2+2=4, or being able to see facts like 1+3, 2+2, and 0+4 all equal 4 will help a student to figure out how much more must be added after figuring out the distance from 8 to ten. All these methods are explained in school, and the student decides which is easiest or which he or she remembers the quickest to know that 2 more must still be added.
This brings us to the final step of adding 2 to 10. Being able to add single-digit numbers to ten is useful because it is one of the more quickly learned skills (which builds confidence in students who are struggling) and can carry forward to adding to multiples of ten like 20, 30, 40, and so on. One of my favorite ways to show students how to add numbers is to say them close together. For example with 3 and ten, if you say them quickly, you get “threeten,” which sounds a lot like thirteen. The same with “fourten” and others. With 11 and 12, the sound doesn’t play out so well, but after practicing the others, the student usually pics up the pattern (and finding a pattern is the best we can hope for our students as a teacher). Whatever the path, the student will find a way to finish up to see that 10 +2 =12. So let’s review:
hmmm… I know:
so if I added 2 to 8 already, I need to add 2 more:
Phew, that’s a lot of steps, but look at all the good things we learned to get there. In the end, memorization is the goal. However, simply memorizing isn’t always the best. As I stated in Common Core Rant, students who have a deeper understanding of more basic concepts are better prepared to extend that knowledge when working on more difficult problems later on. Yes, this takes more work, but it could be the only way that your child is capable of getting to the answer if he or she just isn’t remembering the fact initially.
Ultimately, if you prefer that your child learns in a different way, that’s what tutors such as myself are around for, to supplement the education that they are receiving. However, my goal here is to show the method to the madness and help explain away some of the confusion that many parents have come to me with. Hopefully, next time your young student comes home with this “crazy new math,” you’ll have an idea of what is going on and what the teacher is trying to convey through these longer processes. Again, not every student learns the same way, which is my problem with Common Core, since teachers are required to teach the entire class the same methods even though it may not be the most effective. The techniques are useful, however, and we should appreciate the thought that has gone into helping all children better understand math.
I will continue on with some of the subtraction techniques in my next post, and from there I will go on to cover multiplication and division. Thanks for reading, and be sure to like or subscribe as this will most likely become a series I am writing!