# Getting right answer shows ‘deep understanding’

Students display their “deep understanding” of math problems by getting the right answer, argues James V. Shuls, director of education at the Show-Me Institute.

His second-grade daughter’s school is stressing “real world” word problems this year. His daughter subtracted correctly to find how far the snail crawled, but got the lowest rating, “does not meet expectations.”  She didn’t use the prescribed process.

“Interestingly, there have been other problems where she reached the wrong answer, but received a higher score,” writes Shuls.

The teacher apparently wanted students to follow an 8-step process that includes drawing a unit bar for each variable.

It’s clear his daughter didn’t just stumble on the right answer, he argues. She had a process that made sense.

“When a student can correctly identify the type of problem and can solve for the answer using some type of process, they understand the concept.”

“Knowing you’ve done something right and then getting criticized” anyhow is discouraging for students, Shuls writes.

Rather, we should celebrate correct answers and, when necessary, demonstrate more efficient methods or other ways of thinking about problems. This should be done while keeping in mind that what matters most is that our students have a method that works and is transferrable to other problems.

We tell students to “think like scientists” and “act like mathematicians,” Shuls writes. “Do you know what good scientists and mathematicians do? They get the answer correct.”

1. L. C. Burgundy says:

Lord, as an engineer I am thankful I was actually taught arithmetic and not that edu-school busy-work nonsense shown in that picture.

2. lgm says:

Getting the correct answer here takes no understanding. 1/2 of students who are guessing will choose to subtract the numbers in front of them. The other half will choose to add them together. Of those who choose to subtract, some will subtract in the right order and know how to regroup to obtain the right answer. That is not proof that they are doing anything more than the method below grade level readers are forced to use: find two numbers somewhere in the word problem, subtract smaller from larger, label answer with units that are next to the numbers. The fact that the student was unable to present a solution is significant, especially since the instruction has been given in class.

• SC Math Teacher says:

Maybe she knew how to do it and knew that the extraneous stuff was a waste of time. Maybe she knew that “less than” called for subtraction in this instance. I suppose the teacher could have required her to explain herself in a sentence or two, but even that is overkill in my view.

3. Jena says:

Looks as though this was a lesson on model drawing, which can be useful – one would conceivably begin with simple examples. I believe Singapore math uses a lot of model drawing, although I don’t that this is how they would do it. As a teacher there are times when I have very specific directions that need to be followed and following them IS part of the assignment. From what I see in this post it seems as that the girl did not follow the directions for model drawing.

• Deirdre Mundy says:

Model drawing for word problems is a great tool, because it can help kids move from the concrete to the abstract. HOWEVER, once a kid has grasped the abstract, it’s a waste of time and graphite.

Not all kids move from one stage to the next at the same time.This child appears to have moved on. Maybe other kids in the class have not – but it makes no sense to force her to keep using models that she doesn’t need.

(I’ve noticed that Saxon, in particular, tends to ask kids to draw out their answers long after my kids need it. So our rule is ‘draw it if you feel like it.’ Because sometimes, they WANT to get artsy at math, and sometimes they just want to get FINISHED.

4. Crimson Wife says:

We use Singapore Math in our homeschool and drawing a bar model can indeed be very helpful in solving word problems. But it’s a tool, and not always necessary. If my child had written the above in his/her Singapore book, I would have marked it correct.

5. Yeah, not so much. I get that this problem was done correctly and came up with the correct answer, but I can give many examples where the incorrect method coincidentally gave the right answer … such as this one from my math class discussion starters: http://matharguments180.blogspot.com/2014/01/day-2-improper-fractions.html
26/65 is 2/5 if you cancel the sixes.

To say that the correct answer is all that’s needed is simplistic at best. This is always the teacher’s job – to suss out whether students knew what they were doing. The real problem here is the over-emphasis by administration on “problem-solving” skills development. That procedure is inappropriately complex for this situation and for this group of students, but it probably enabled her to click a checkbox on the self-evaluation forms. “Taught problem-solving.”

• cranberry says:

“This is always the teacher’s job – to suss out whether students knew what they were doing.”

However, teachers aren’t telepathic. It isn’t realistic to expect they are.

This is a variation on the “show your work” debate. Is the point of teaching a child math to teach them how to draw models? Or is it to teach math? If a child who drew the models, but arrived at the wrong answers, would receive a higher mark, there’s something wrong here.

I like the Singapore Math method of drawing models to assist students in visualizing word problems, but there’s a danger of elevating a crutch over deep understanding. The question was too simple to need bar models.

If the kid’s an advanced student, he/she could arrive at answers by various methods. All the methods could be useful. Obsessing over methods rather than understanding is not a good idea.

• Mark Roulo says:

“This is always the teacher’s job – to suss out whether students knew what they were doing.”

However, teachers aren’t telepathic. It isn’t realistic to expect they are.

For math, I would expect that a handful of problems would reveal whether the kid knew what to do or was getting lucky. Canceling the 6s won’t work when reducing 6/16, for example. Isn’t part of the art of problem making to bring these sort of misconceptions to the surface (without having to read the child’s mind)?

• cranberry says:

Mark Ruolo, as a parent I remember my children drawing pictures as part of Everyday Math homework. “seventeen birds sat on a wire. Four flew off. How many were left? (required: draw a picture)

Which of course required my kids to draw two pictures, totalling 30 birds. Just imagine how many subtractions problems they could have solved in the time involved to draw 30 birds! That was just one problem. Writing was invented because it is an efficient method to convey meaning.

Now, there are some practices which are important to learn. Eventually children get to more complicated, multi-step problems. It’s a good idea to get in the habit of writing down the equations and processes involved, so that you can double check your work. At that level, though, they aren’t drawing bar models they don’t need for each step of the problem. (If it’s a geometry problem, they should be drawing figures, of course.)

• Any process which reduces the learning of arithmetic to counting (as the drawing effectively does) should be counted wrong.  Students should be learning to convert word phrases into arithmetic expressions (implicitly or explicitly) and solve them, not do what amounts to using one’s fingers.

• lulu says:

And, while Singapore math would let you do that if you needed it, they quickly move to a big box labeled ’17’, or maybe a box labeled ’10’ and a box labeled ‘7’ and then a smaller box with the difference labeled ‘4’, which then says to subtract 4. And then they move to just writing the problem. They show them all, and you do whatever you need to get the work done.

6. Bill says:

8 steps to solve a two digit subtraction problem.

Perhaps this is why so many students in K-12 are math challenged these days, they don’t actually KNOW how to do math (what they’re being asked to do is explain it), and most employers and colleges would prefer to have someone who could actually perform the math operation.

IMO, she got the right answer, and stated how far the snail crawled.

Sound like the director of the show-me institute could use some basic math instruction, but he probably wouldn’t understand it using his outline.

Sigh

• cranberry says:

The director of the show-me institute is the father of the 2nd grader.

The 8 step outline stems from the teacher, not the father.

7. “However, teachers aren’t telepathic. It isn’t realistic to expect they are. This is a variation on the ‘show your work’ debate.”

‘Show your work’ is useful when
> where there is cheating and copying answers with no thinking,
> where there are variations on a question,
> but most importantly where there are multi-step questions that cannot be solved in one string of calculations.

Teachers can be psychic about the errors kids make if they have some of the steps in front of them. Try these on for size: http://matharguments180.blogspot.com/2014/02/day-14-error-analysis-can-you-be.html

If you just have the answer, you don’t get a chance to correct the process, you can only say “yes” or “no”. For simple problems, it’s okay to do without some of the intervening steps. Where you draw that line depends on the students, the material, and the teacher’s knowledge of both.

The original picture? Correct and we move on. Something more complex … I want the student to slow down and KNOW what steps he is taking instead of monkey-push-button.

• cranberry says:

Curmudgeon, I think we agree more than we disagree. The problem cited is too simple to require a drawing. To mark the child down for not drawing a picture places a higher priority on compliance with unnecessary steps than on understanding.

My children did use Singapore Math at one time. I don’t recall the workbooks requiring them to draw models. The models were used as a method of understanding the concepts. (Pulls out workbook.) There is a great deal of straightforward arithmetic practice without bar models.

A good example of American curriculum writers misapplying something which works. “Singapore Math” “Oh, right, let’s draw bar models.” There is lots and lots of practice in our “real” Singapore Mathematics 3A workbook of addition and multiplication without models. Indeed, practice to automaticity.

When it comes to simple arithmetical operations, we should aim for monkey-push-button.

8. Jerry Doctor says:

“The important thing is to understand what you’re doing rather than to get the right answer.” – Tom Lehrer, 1965

• SC Math Teacher says:

Not if you’re building a bridge, Jerry. Or a space shuttle. Or…well, you get the idea.

Both are important, of course, but once you understand what you’re doing and have mastered it, constantly explaining yourself or following steps such as the ones illustrated here become a waste of time. (Cranberry above seems to echo the same position.)

• Deirdre Mundy says:

Jerry, and Tom Lehrer, agree with you, SC. The linked video is a satiric song, showing the pitfalls of that approach!

• SC Math Teacher says:

And shame on me for not following through. My apologies to Jerry…and all who rightly expect commenters to be thorough.