# How to Succeed in Math: Answer-Getting vs. Problem-Solving

You want your child to succeed in math because it opens so many doors in the future.

But kids have a short-term perspective. They don’t really care about the future. They care about getting through tonight’s homework and moving on to something more interesting.

When kids face a difficult math problem, their attitude can make all the difference. Not so much their “I hate homework!” attitude, but their mathematical worldview.

Answer-getting asks “What is the answer?”, decides whether it is right, and then goes on to the next question.

Problem-solving asks “Why do you say that?” and listens for the explanation.

Problem-solving is not really interested in “right” or “wrong”—it cares more about “makes sense” or “needs justification.”

### Homeschool Memories

In our quarter-century-plus of homeschooling, my children and I worked our way through a lot of math problems. But often, we didn’t bother to take the calculation all the way to the end.

Why didn’t I care whether my kids found the answer?

Because the thing that intrigued me about math was the web of interrelated ideas we discovered along the way:

• How can we recognize this type of problem?
• What other problems are related to it, and how can they help us understand this one? Or can this problem help us figure out those others?
• What could we do if we had never seen a problem like this one before? How would we reason it out?
• Why does the formula work? Where did it come from, and how is it related to basic principles?
• What is the easiest or most efficient way to manipulative the numbers? Does this help us see more of the patterns and connections within our number system?
• Is there another way to approach the problem? How many different ways can we think of? Which way do we like best, and why?

### What Do You think?

How did you learn math? Did your school experience focus on answer-getting or problem-solving?

How can we help our children learn to think their way through math problems?

CREDITS: “Maths” photo (top) by Robert Couse-Baker. “Math Phobia” photo by Jimmie. Both via Flickr (CC BY 2.0). Phil Daro video by SERP Media (the Strategic Education Research Partnership) via Vimeo.

## 33 thoughts on “How to Succeed in Math: Answer-Getting vs. Problem-Solving”

1. Great advice! I’ve also been impressed with the many online math tutorials that my daughter’s school gives them!

1. There are so many great learning resources online, aren’t there? When I was a child, we’d make trips to the library and bring home stacks of books — but now, I can visit a whole world of libraries through this magic portal on my desk. 🙂

You and your daughter might enjoy some of the resources on my Online Math Adventures page.

1. Kim Beezhold says:

Hi Denise,
The link to the article about The Golden Ratio link, “Golden Sales Pitch”, for your online resources, sends you to Science News, but you have to be a subscriber to read it. I read some about the doubts about it on Wikipedia, though. It’s interesting. I still think the Golden Ratio and it’s connection to the Fibonacci Sequence is very interesting and it is worth studying the ratio.

1. Kim Beezhold says:

Hi Denise,
The link to the article about The Golden Ratio link, “Golden Sales Pitch”, for your online resources, sends you to Science News, but you have to be a subscriber to read it. I read some about the doubts about it on Wikipedia, though. It’s interesting. I still think the Golden Ratio and it’s connection to the Fibonacci Sequence is very interesting and it is worth studying the ratio.
I did a search for the golden ratio on the Science News site and found BOOK REVIEW: The Man of Numbers: Fibonacci’s Arithmetic Revolution by Keith Devlin.
You have to be a subscriber to read the review. Have you read his book?

1. Kim Beezhold says:

Thank you☺

2. James Daniels says:

There is room for both learning maths and getting the right answer.
I learnt to subtract using the borrow and payback method, but as a child I never understood how it worked. When I was 9, someone in the class asked the teacher where the “ten” was borrowed from. I remember thinking, “That’s a good question”. They must have been doing a calculation related to money, because the teacher answered, “From the bank”. I remember thinking, “She’s not answered the question”. I contented myself with knowing that I didn’t need to know where the “ten” had come from, as long as I followed the algorithm (a word I hadn’t yet learnt) I’d get the right answer. Thereafter, I never questioned where the “ten” came from until my daughter was being taught how to subtract and engaged with a cluttered assemblage of “tens” being subtracted from the next column.
One doesn’t always need to know exactly how arithmetic processes work.

1. So true, James. One doesn’t need to know how the subtraction algorithm works to use it, just as one doesn’t need to know how a car works to drive it.

The whole point of an arithmetic algorithm is that it can be done without thinking. Crank it through, follow the steps, and you’ll get the right answer. In that way, algorithms are more like magic rituals than like mathematics.

The mathematical philosopher Alfred North Whitehead said:

“It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.”

Civilization may advance as Whitehead claims. But for the purpose of education, I think the “why?” is much more important than following the steps. I want my children to learn to think about what they are doing — and to be able to put those thoughts into words and explain why they got that answer.

1. Kim Beezhold says:

It is possible to follow the Standard Algorithm for Subtraction and understand how and why it works.
I found this on http://www.study.com:

Maybe the non-standard algorithms are confusing because someone just made them up, in which case I agree, they need to be set aside. It was astounded by the number of “Tricks”I saw in the Nix the Tricks” booklet. It is worth reading and some of it will help. I have only seen a few of the algorithms I learned back in public school, in the 80’s, Like FOIL, that we’re fully explained what was a better method and why.
You must use some kind of algorithm, right?

2. Kim B. says:

What if you understand why but get the wrong answer? How do you deal with that, as a teacher?

1. Since we are usually working together (as in my Buddy Math post), I just ask my daughter to explain her work. I ask that all the time, even when her answer is correct.

She usually catches the mistake in the course of explaining what she did.

If she doesn’t catch it, I’ll say something like, “Be careful” or “Are you sure?” and that makes her do a double-take. Often, all it takes is a raised eyebrow. She’ll catch me in mental glitches that way, too, when it’s my turn to do a problem.

1. Kimberly Beezhold says:

Yes, it really works for one on one which is really what I want to do with my kids. ☺

3. Kimberly Beezhold says:

Hi Denise,
I looked up Alfred North Whitehead on Wikipedia. He had something to say about education. He was looking for reform in math education way back then, too.
my real point, though, is that he wanted their to be meaningful applications. How and why were both important to him, I believe because of the following I found on his Wikipedia page:

“Whitehead’s philosophy of education might adequately be summarized in his statement that “knowledge does not keep any better than fish.”[70] In other words, bits of disconnected knowledge are meaningless; all knowledge must find some imaginative application to the students’ own lives, or else it becomes so much useless trivia, and the students themselves become good at parroting facts but not thinking for themselves.” ( Note: only part of this is a Whitehead quote within a quote from the Wikipedia page).

We may never find a balance to the learning how to get the right answer and knowing why we are doing it, however, I propose that knowing Why you are solving a problem a certain way requires work and so it must be work a student is motivated to do in the first place. Everyone is different, so how would it be possible to implement understanding why and how in a group(school classroom or co-op group). That is the real question in my opinoin, for which each teacher needs to find their own answer. Or is it the student who needs to figure out what makes him motivated to do math for himself?
“How can we help our children learn to think their way through math problems?”

We can look for as many opportunities to do real life math problems with our children, so that they can see we find ways to use math as a tool in problem solving. The context of reality, gives the concepts a place to make sense. As a homeschool parent, I can start doing a certain math algorithm( let’s say Lattice multiplication) because I want to do it. Then, at least one of my children will want to follow along, pushing their interest, instead of assigning them work to do. We have to DO math to understand it. We have to start somewhere that we might not understand yet and go from there. that is my opinion, since you asked.

4. Kimberly Beezhold says:

My tablet’s autocorrect replaced the word “piquing”with “pushing” for some reason. Sorry I didn’t catch it right away. Just know I meant getting their attention by doing rather than directly teaching each concept to them. We have to model using math to solve problems. The answers ARE important. If it’s wrong, you go back to the point where they got off track and learn what was wrong. If it’s the right answer, then you will see that they understand why because you were following along with them until they take off on their own. If they greatly surpass you, they will be successful if they still find math interesting and/or useful.

1. You make a lot of good points, Kimberly! The difficulty is always to find a balance between the How and the Why that works for the students you are teaching.

And the prerequisite for *that* is for the teacher to understand that there actually is a difference between How and Why — that getting the right answer does not equal understanding. Have you read the book by Liping Ma, Knowing and Teaching Elementary Arithmetic? It’s a fascinating (at least to me) study of what it means to understand math well.

1. Kim Beezhold says:

Yes, I’ve seen some of Ma’s work. I think her research is important. I will read as much of it as I can. I think it will help me understand the why part, as it pertains, for example, what is borrowing in column subtraction and what does it mean.

2. Kim Beezhold says:

Hi, I put a hold on this book. In the meantime, I found a very interesting review of the book on the MAA website:
https://www.maa.org/press/maa-reviews/knowing-and-teaching-elementary-mathematics-teachers-understanding-of-fundamental-mathematics-in

These 2 points were found to be important to the author of the review:

“1. Chinese teachers are specialists, and teach only mathematics, but American teachers, being generalists, are required to teach a range of different subjects.

2. In China, mathematics teachers are allocated a good amount of time for preparation of lessons, and they operate within a national curriculum that offers clear guidelines and relevant teaching materials. Conversely, U.S. teachers spend the whole day in front of a class. Working in relative isolation, they tend to rely too heavily upon commercially produced textbooks.”

I kind of observed #1 when I was in college and found out that the elementary school teacher majors only had one class for mathematics to prepare them for teaching math. Last summer, a substitute teacher, who was teaching high school math, lamented the fact that many of his students did not know basic math facts. High school math teachers, at least in Illinois, study mathematics as their major, and then student teach, with a mathematics professor and an education professor overseeing them. Elementary school teachers do not study math, and yet they are responsible for laying the foundations of math skills and understanding. Generalization does not seem to be working.
I don’t really know what to say about the author’s point #2, except that I’ve seen good and bad presentations of elementary math topics online and think parents should take back there control of their children’s education since teaching groups of children does not seem to be an efficient way to learn.
I guess I didn’t get to the part about what it means to understand math well. I will look for that specifically when I read the book.
Any thoughts?
🙂

1. I think you’re right that generalization is not working well for elementary math. I’ve been excited to see the growing number of math specialist teachers around the internet and the interest in learning math that many teachers show on social media. Things are improving, but the system is a behemoth that doesn’t turn easily.

Liping Ma’s book revolves around four questions that represent problems one might face in an elementary or middle school classroom. When you read the book, you’ll get the most from it if you stop and answer each question before reading what the Chinese and American teachers said. The best way is to close the book and actually write out your answer in a few paragraphs before reading on.

1. Kimberly Beezhold says:

I have the book now. I will try writing down the 4 questions.
Thank you.

5. Kim Beezhold says:

Yes, I’ve seen some of Ma’s work. I think her research is important. I will read as much of it as I can. I think it will help me understand the why part, as it pertains, for example, what is borrowing in column subtraction and what does it mean.

1. Kim Bee says:

I actually didn’t like this book. Not really research that involves statistics and control groups, or very scientific approach. I guess education can be a social science which is not hard science.
I just wanted to say what I’m thinking because there’s 2 sides to every story. Thank you for listening.😺

1. Kim Bee says:

I did get some good things from LipingMa that you and others interpreted, so I think I still learned from it.

6. Kimberly Beezhold says:

Ok. I looked and found the four questions. I figured out how I would repond to the first three. I wrote it out like you said.
The perimeter/ area relationship is going to take me a little bit longer, I think, because it’s more complicated than the other three.
I discovered a method called partial differences for subtracting.
I drew a picture for the fractions and solved the problem without using the reciprocal.
I tried out Lattice Multiplication, and it eliminated the messy lining up digits and remembering to put zeros to hold places. I did have to carry, though.
It’s a start🙂.
Thanks for the idea to write out my own reponses.

7. Kimberly Beezhold says:

Unfortunately, I have to return this book to the library because pages are going to fall out! It looks like a bad binding of a paperback that was laminated.

How far did you get with the relationship between perimeter and area?
I was having a hard time. Would ADEPT help, from Better Explained?

1. For the rectangles, experimentation offers the best way to explore. Get some graph paper, draw a variety of rectangles, and measure them. How do the area and perimeter compare? Can you draw different rectangles with the same area, or with the same perimeter? What patterns do you notice?

1. Like any human creation, the Common Core standards have good and bad points. I like the emphasis on mathematical thinking and problem-solving. I don’t agree with some of the order of topics. I hate the focus on testing — especially the new, confusing style of “show your work” standardized tests.

As for the quote you linked to, Oak Norton took Daro’s comment way-far-away out of context to put his own political spin on it. I think the politicization of math education is the absolute worst thing to come out of the whole Common Core argument.

1. Kim B says:

You seem to have a good balanced view of all these math educators. I agree with you 😺

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