A “granny summary” oft he article Dror Dotan and Naama Friedmann
Reducing interference improves the memorization of multiplication facts in case of Hypersensitivity to interference
Why is memorizing the multiplication table so hard? A nice answer to this question involves a simple example. Try memorizing the following addresses:
Charlie David works on Albert Bruno Avenue
Charlie George works on Bruno Albert Avenue
George Ernie works on Charlie Ernie Avenue
Now, without looking, please tell me: where does Charlie George work?
Difficult, isn’t it? How can we remember such meaningless facts? The child trying to memorize the multiplication table is facing the same kind of difficulty. The example above is taken from the book “The Number Sense” by Stanislas Dehaene (which is very recommended for those interested in how we understand numbers), and the sentences are not random at all: Dehaene took facts from the multiplication table, and turned every digit to a name with a matching first letter (Albert = 1, Bruno = 2, George = 3 etc.). Thus, the 3 sentences above correspond with 3 multiplication facts (the one I asked you about: 3×7=21). The main difference between this example and the multiplication table is that 3rd grade children, unlike you, are not required to memorize 3 facts – they are required to memorize 42 multiplication facts!
To demonstrate the difficulty of memorizing the multiplication table, Dehaene used sentences (and not shapes, for example). This choice is not arbitrary: we remember the multiplication table verbally, in fact – even in the language we study it (try thinking how immigrants multiply). As a result, our ability to memorize the multiplication table is subject to the limitations of our verbal memory.
One of the interesting phenomena about verbal memory is that remembering similar words is difficult. Suppose, for example, you started studying French, and in today’s lesson you learn some similar words – e.g., mieux (better), maillot (shirt), moi (I) and mais (but). Chances are you will find this list difficult to memorize. The reason is that similar words interfere with each other. They have similar representations in our memory, so the brain struggles to distinguish between them. This difficulty, distinguishing between similar words, becomes a nightmare when we try to memorize the multiplication table. Here we have 10 words (10 digits) repeated constantly, in 42 different multiplication facts. No wonder it’s difficult. If anything, it’s surprising that anyone succeeds in this crazy task! (you can relax though – most people don’t. Most of us know most of the multiplication table, but almost all of us have a few “black holes” in the middle. Honestly, do remember what is 7×8? How about 9×6?)
Everyone finds it hard to memorize similar words, but some people find it especially hard. These people can learn a list of dissimilar words like anyone else, but they find it almost impossible to memorize a list of similar words. It’s as if their brain can’t deal with the interference caused by the similarity between the words. This phenomenon is called “hyper-sensitivity to interference”.
What happens when these people need to memorize the multiplication table? It’s a disaster. They just can’t do it. Several studies showed this, and we too encountered a woman with this problem. When I met Ofra (fake name) she was 40 years old, and she still maintained the bitter memory of constant failure when trying to memorize the multiplication table in school. She tried private teachers, songs, everything, but nothing helped. Indeed, Ofra had hyper-sensitivity to interference. How did we know? Because her inability to memorize similar items was not limited to the multiplication table; it was observed also in verbal memory tasks that were not related to numbers.
OK, we said, so we know why she can’t learn the multiplication table. Let’s teach it to her anyway!!
How can we do that? The idea was very simple: Ofra can’t learn the multiplication table due to her hyper-sensitivity to interference, so if we get rid of the interference, she should be able to study properly. And how can we get rid of the interference? Simple: in the multiplication table, interference is caused by similar multiplication facts (e.g., 8×8=64 and 6×8=48). If we teach her only dissimilar facts (e.g., 8×8=64 and 3×7=21), she should be able to learn easily. And how can we still teach similar facts, e.g., 8×8=64 and 8×6=48? We just make sure to teach them in separate lessons, with sufficient delay between them. Say, in two different weeks.
So the plan went as follows: we identified 16 multiplication problems that Ofra didn’t know, and divided them into 4 sets with 4 facts in each. Each week Ofra learned just one set. The learning method was plain memorization: I said a fact, she repeated it, then I tested her and corrected her mistakes, and we repeated this 3 times for each fact. All this took was 5 minutes, and goodbye. We repeated this 3 times a week – i.e., 15 minutes in total to learn each set. Critically, three sets included dissimilar facts, and one control set had similar facts. We expected Ofra to learn the dissimilar-fact sets, but still have trouble with the control set.
The method worked – in fact, even better than expected. Unsurprisingly, Ofra struggled with the control set. She just couldn’t memorize it. But as for the rest of the facts – not only did she learn them, but she did it easily. In one of the sets, she learned the facts perfectly after hearing them only twice! Let me repeat that: twice. Namely, in that specific week, by the end of our 5-minute phone conversation, she has perfectly learned the 4 facts – something she never succeeded in 12 schooling years – simply because we used correct method. She also remembered the facts very well two months after the study has ended!
So we finish with an optimistic message: learning the multiplication table is hard, but possible. Furthermore, we can see how, by detecting the specific learning disorder that disrupts learning the multiplication table, we can help these individuals overcome the difficulty.
What about people without hyper-sensitivity to interference? Does our method work for them too? The answer is yes, but this is a topic for another “granny summary”.
One last thing: how do children learn multiplication in schools? The typical method to memorize the multiplication table is by columns: first all of the products of 2, then of 3, and so on. This means that in each given lesson, the children memorize similar multiplication facts. In terms of similarity, this method is almost the worst one possible! But before we change how we teach multiplication in school, wait: the columns method is probably bad in terms of similarity, but it has other important advantages. For instance, the inner logic of the multiplication table, and its relation with addition, is more transparent this way. Still, dissimilarity-based memorization may be better in some situations, at least outside of the classroom. For example, in computer games that teach the multiplication table, or when you try memorizing something verbal that isn’t the multiplication table, like Spanish.
Interested in more details? The full article is here.