# How hints help speed up math performance — and what this says about memory

The link below will take you to a short movie (QuickTime Required). You’ll see a series of seven easy addition problems, which will flash by at the rate of one every two seconds. Your job is to solve the problems as quickly as possible (ideally, you should say the answers out loud).

Click to watch movie

Apart from the possibility that you might have a better memory for some math facts than others, were any of the problems easier for you?

If you’re like most people, you probably responded faster when a problem was repeated, as was the case with 3 + 5. And since the order of operations doesn’t matter in addition, you might have also had an easier time with 7 + 2 since you had seen 2 + 7 earlier. You might have even thought you saw 7 + 2 twice — as both Nora and I did when we tried this demo (yes, it happened to me even though I’m the one who made the demo!).

If seeing “2 + 7” earlier helps you solve “7 + 2” later, then maybe some other mathematical operations can be facilitated in similar ways. In fact, there has been a good deal of research on multiplication and division problems, with a bit of a startling result: While seeing “9 × 4” helps you solve “4 × 9” later, seeing “36 ÷ 9” doesn’t seem to help with “36 ÷ 4”. Aren’t these all equivalent sorts of problems?

Timothy Rickard and others have come up with a model that explains the difference in multiplication and division problems: The multiplication problems have identical elements — the same numbers, just in a different order. The division problems actually involve different numbers; only when the answer is known are the numbers the same. But Jamie Campbell has found that division sometimes is speeded in cases like “36 ÷ 9” and “36 ÷ 4”. Campbell argues that because of the close relationship between these problems and 9 × 4, we do have some advantage solving “36 ÷ 9” when we’ve seen “36 ÷ 4”. But when it comes to addition and subtraction, the rules change somewhat.

While 36 only has a few common factors (2, 4, 6, 9, 18), there are many different numbers that can add up to, say, 12 — every number lower than 12 is a possibility. So practicing “12 – 5” isn’t much help when we try to solve “12 – 7” later on. So Campbell and two other researchers decided to present 48 volunteers with addition and subtraction problems to see whether their model of memory for math facts held up.

The researchers presented every possible addition problem involving digits between 2 and 9 (that’s 28 different problems when “ties” such as 5 + 5 are removed) and the corresponding subtraction problems. They carefully monitored when associated addition and subtraction problems (like 2 + 3 and 5 – 2) appeared (sometimes soon after one another [4-6 problems away], and sometimes separated by as many as 22 problems). The speed of response was carefully tracked.

So, does seeing the same problem again make it easier to solve? What about seeing a similar problem? This graph charts the results:

For both addition and subtraction, seeing the identical problem a second time resulted in significant time savings. For both addition and subtraction, seeing the same problem with only the order of the elements changed (2 + 3 and 3 + 2, or 5 – 3 and 5 -2) also resulted in a significant time savings. But when the problem type changed (2 + 3 and 5 – 2, or 5 – 3 and 2 + 3), there was not time savings.

Unlike in multiplication and division, seeing the associated subtraction problem doesn’t help with an addition problem, and vice versa. And while seeing a previous problem with a different order does help in both addition and subtraction, for subtraction the savings for seeing a problem in a different order is just a half to a third of the advantage for seeing the identical problem.

Campbell’s team argues that this demonstrates that memory for addition and subtraction is managed differently than multiplication and division.

Campbell, J.I., Fuchs-Lacelle, S., Phenix, T.L. (2006). Identical elements model of arithmetic memory: Extension to addition and subtraction. Memory & Cognition, 34(3), 633-647.

1. #1 Joel McKellar
June 2, 2008

The pattern definitely worked for me. A little embarrasing, but did anyone else stumble at the 8+6 problem? The answer in the double digits took me by surprise (everything up to there added to less than 10), and I almost didn’t finish it before the next question. I didn’t have the same problem with 9+6, I guess I was now ready for it?

2. #2 tania
June 2, 2008

Was there any commentary on that HUGE increase in the subtraction compared to the addition or is the graph just on a deceptive scale?

3. #3 Joshua Zelinsky
June 2, 2008

We already know that addition/subtraction and multiplication/division are handled differently since we have stroke victims who have only problems with one.

This result seems consistent with the known result that stroke victims who have memory problems are much more likely to have problems with multiplication than addition. They both seem to point to multiplication/division being more memory intensive than addition/subtraction. This also fits in with how we teach both of these: no one needs to memorize their addition tables, but people do need to memorize multiplication tables.

4. #4 Dave Munger
June 3, 2008

Tania,

The large difference is due to the fact that we’re not as quick answering subtraction problems. When the same subtraction problem is repeated, the benefit of having seen it before is much larger than in addition.

The key thing to notice about the graph is the difference between seeing the same subtraction problem twice and the subtraction problem with a different order (12 – 7 versus 12 – 5, for example).

With addition, there’s no difference between seeing 7 + 5 and then 7 + 5 again, or seeing 7 + 5 and then seeing 5 + 7 — in each case, the improvement is equivalent. But with subtraction, seeing the same problem again helps much more than seeing the problem with a different order.

5. #5 Nemo
June 3, 2008

Hm, I noticed a different pattern with myself: when the answer was 2-digit, it took me more time to say the answer.

6. #6 Jonathan Vos Post
June 3, 2008

This is important research!

What is easy and what is hard in Mathematics? That is important to Math teachers — and to the many students who flunk out or drop out of high school, with Math usually being the class that ends their education.

I’ve been an Adjunct Professor of Mathematics at a private university, and have spent a year teaching Math in malfunctioning middle schools and high schools in Pasadena, Calaifornia.

There are well-studied procedures for teaching the Mathematically unprepared student, which I have employed. Some states have also legislated with respect to “Dyscalculia” or “Mathematics Disorder.” I
have become very experienced in this specialized literature and its applicability. I was formally considered for a federal Title V Grant earmarked for Remedial Mathematics at Woodbury University in Summer 2005. The Federal government also has studied and funded programs relating to Dyscalculia.

What strategies/methods/formats could be used to address these needs?

Have I personally utilized any of the techniques that I mention?

There are specific established methodologies which differ by the
learning style of the student (i.e. Verbal, Visual, Tactile, Kinaesthetic). The bottom line: roughly 1/3 of students with
Dyscalculia have a neurological, medical, or psychiatric situation beyond the ability of classroom instructors as such.

The good news: 2/3 of students with Dyscalculia suffer from bad teaching in earlier education, and/or ineffective strategies for parsing, manipulating, acquiring, and mastering Mathematical subject materials, and/or a social tolerance for Dyscalculia (i.e. parents, peers, even teachers in other subjects saying “don’t worry, I have trouble myself with Math.”) This is addressed in the hit CBS-TV show NUMB3RS, whose Math Advisor is my former professor, Dr. Gary Lorden, at Caltech, whose letter of recommendation is on file at Charter College of Education.

I am back in graduate school 35 years after I first started at graduate school (M.S. and Ph.D. — all but degree — in Computer Science). That’s because the school system, pressured by the horrendous unfunded mandates of No Child Left Behind, considers me to be unqualified to teach Math, until I complete my Teaching Certificate, which requires 6 quarters at the college of education.

The public school system of America has resegregated into decent schools for the middle and upper classes and in the suburbs, and intolerable urban schools for the African American and Hispanic students, who are treated like prisoners, by inadequate teachers, with minimal parental involvement, in decaying infrastructure. This resegregation by race and income is shocking. And, once again, a key question is: what is easy and what is hard in Mathematics?

Campbell’s team’s team is to be praised. Let’s hope that it leads to solutions to the hard problems — not solving equations, but solving the destruction of America’s public schools.

7. #7 Tony Jeremiah
June 3, 2008

..Jamie Campbell has found that division sometimes is speeded in cases like “36 ÷ 9” and “36 ÷ 4”. Campbell argues that because of the close relationship between these problems and 9 × 4, we do have some advantage solving “36 ÷ 9” when we’ve seen “36 ÷ 4”.

I wonder if this is heavily influenced by learning algebra. I can see how this would work given that one can explicitly express how multiplication and division are interrelated in algebraic form. In the case of the examples here, it would be:

36x = 9; x = ? and 9x = 36; x = ?

I’m guessing the advantage disappears for pre-algebra students.

Unlike in multiplication and division, seeing the associated subtraction problem doesn’t help with an addition problem, and vice versa. And while seeing a previous problem with a different order does help in both addition and subtraction, for subtraction the savings for seeing a problem in a different order is just a half to a third of the advantage for seeing the identical problem.

Campbell’s team argues that this demonstrates that memory for addition and subtraction is managed differently than multiplication and division.

My guess is this still has to do with learning algebra and/or algebraic knowledge as represented in memory. In this instance, the interrelatedness between addition and subtraction expressed in algebraic form is clearly not as straightforward as multiplication and division (in which you only need to divide both sides by a particular number to solve for X):

2 + 3 = X; X = ?

that’s not too hard.

But then when it comes to the mental gymnastics of interrelating the numbers:

2 = X – 3; X = ?; 3 = X – 2; X = ?

things quickly become complicated as there are more ways to solve for X (probably explains why the advantage is cut by 1/2 or 1/3). Also, I’m guessing the SNARC effect may be involved, particularly as it concerns the relative advantage of addition over subtraction, and especially if participants are giving manual responses with their right hand.

8. #8 ecoli
June 5, 2008

“The pattern definitely worked for me. A little embarrasing, but did anyone else stumble at the 8+6 problem? The answer in the double digits took me by surprise (everything up to there added to less than 10), and I almost didn’t finish it before the next question. I didn’t have the same problem with 9+6, I guess I was now ready for it?”

Same for me… but I don’t think it was because I was ‘ready for it.’ I think, since nine is closer to 10, adding it is simplier (since we are trained to think in base ten, it’s easiest to add 10 to anything). Adding 9 is easy, because it’s just adding 10 and subtracting 1. Adding 8 is like adding 10 but subtracting 2 or adding 5 and adding 3. This takes a bit longer to do in the old noodle.

9. #9 Amiya Sarkar
June 7, 2008

Arguably, addition (and subtraction) took less time because it seems to have an evolutionary basis, which may be conserved across species.

Multiplication or division, in comparison, are rather synthetic; can be deduced indirectly from the rules of addition (or subtraction).