Thursday night’s #hsmath (homeschool math chat) on Twitter was a peaceful the-more-tools-the-better lovefest, except for my continued ~~vocal~~ typed support of ease of access to the much maligned calculator, and my disparagement of using flashcards to help students memorize basic math facts.

Think I’m a heretic? Well, you probably aren’t wrong, but…

**1. Flashcards, by definition, involve presentation of math facts devoid of context. ** Yes, I hate worksheets full of ‘problems,’ as well. Math is far more than arithmetic, and making memorization of basic math facts the primary point of entry into the field is just plain boring. Kids wind up thinking that they hate math (or worse, that they are ‘stupid’), when what they really hate is being forced to *memorize*, especially in bulk! Studies of memory show that the mind needs hooks in established memory to which it can anchor new material; the more hooks, the more likely the new material will be retained, and the more pathways will exist to facilitate retrieval of the new material.

ETA: If you want to learn more about memory and the teaching of mathematics, check out Republic of Math’s post on the subject!

**2. Flashcards make it too easy to sneak a peek at the answer.** Don’t get me wrong – I love flashcards for quizzing myself on structural formulae and other content that one can’t read at a glance, but the times tables? I don’t know about you, but I never had the will power to really commit to an answer before I peaked. Maybe they’d work better if someone else was holding the cards, but they could just as easily be holding a written out table, in that case.

**3. Flash cards reinforce two fallacious ideas, one for students and one for instructors: that knowledge of basic math facts IS mathematics, and that one must know one’s math facts before one can move on into other mathematical study.** Much like studying a map without understanding the history, culture or ecology of the places ON that map is sure to turn students off of geography, studying math facts in isolation is sure to turn students off of mathematics. Keeping ‘the good stuff’ of mathematics; the exploration and answering of actual questions, until after one has shown sufficient rote memorization skills is bass-ackwards. Children are exploring machines, and, given the opportunity to explore mathematical questions, will either eventually pick up math facts on their own, see the value of putting the work into memorizing them, or work out other ways of accessing the answers they need when they need them to answer questions they care about.

**4. Flashcards focus on knowing ‘correct answers’, and, usually, on knowing them as rapidly as possible.** I get that instant recall of basic math facts is useful, but I really don’t see that the stress of timed drills is worth the result, esp when there are other ways of working out the answers. (Don’t remember 8 x 4? Double 4 x 4 instead!) Flashcards are usually answered verbally, adding an element of embarrassment to a potentially already stressful timed drill. I can’t think of many instances (outside of math class) in which it is critical to have a completely accurate answer (as opposed to a reasonable estimate) but in which a calculator can’t be utilized. (Spend the time learning to recognize and discern what a reasonable answer is, instead!)

**5. Students are in control of working out the answer when given a calculator.** Yes, they can easily punch in the keystrokes needed to get that answer, but doing so, and thereby finding that answer, is in THEIR hands, not those of an all-knowing educator. Subtle, but important distinction, as anyone who has ever fought a kid for control of a computer keyboard knows. Students using a calculator regularly will also quickly learn that they have to be careful when inputting their equations, or else they get crazy answers. They thereby learn to evaluate the reasonableness of their answers with such regular exposure, and will input equations again to be sure of the accuracy of their keystrokes if an answer doesn’t make sense to them. Calculator use thus promotes the development of an interior locus of control, critical thinking skills, and a healthy understanding of the limits of technology.

**6. By inputting equations into calculators themselves, students are repeating the entirety of the math sentence in the process of generating their answers.** Doing this repeatedly amounts to the same thing as rote drill, helping students learn their basic math facts while removing the sources of stress inherent in flashcard quizzes, and may eventually render the calculators obsolete, but *in manner that supports the student as long as they desire such support.* Calculators are also useful for kinesthetic and spatial learners, as they can learn to associate the position of keys with the answers generated, providing other modalities in which to learn math facts other than in strictly visual and auditory methods.

**7. Calculators can be used to find answers when needed in context of something the student cares about, and can allow access to thinking about mathematical problems.** Flashcards, unless they have succeeded in securing rote memory, are useless in actual problem solving situations, where as calculators are tools that can be used in situ. Even if rote mastery of math facts does not occur over time, students using calculators are actively engaged in the solving of real problems, and are thus doing real math, expanding their understanding of the world, and their confidence in the abilities of their own minds. No one would say that Stephen Hawking’s using a computer to speak means that he can’t tackle the mysteries of the universe, so I really don’t understand why a student’s ‘reliance’ on a calculator should be seen as a bar preventing them from entering the world of higher mathematics.

* * *

**In short, one can think of the student first entering the world of mathematics much like someone first learning to ride a two-wheeled bicycle.** Using flashcards to demand that a child master basic math facts before proceeding is akin to putting the child on said two-wheeler and giving them a good shove; some may fly, their feet sticking the pedals and making the most of the boost you gave them, but most will crash and or fall, perhaps over and over again. Some of these will come to fear getting back on the bike, and some will never do so again, tension grabbing at them as they enter math classes for the rest of their education, having internalized that they are neither supported nor safe in those environments, and likely falling, and failing, again and again, thus reinforcing these conclusions. The student given free access to a calculator, on the other hand, has been given training wheels. Should the often-felt stigma around use of such training wheels be removed, that student can ride to their heart’s content, exploring the entirety of the mathematical world, knowing that they are unlikely to fall, and *therefore be unlikely to fear.* They may eventually decide (or come to realize) that they don’t need or desire the training wheels anymore, in which case they can easily remove them. In the meantime, they have felt safe and supported, and have not only explored the mathematical world, but have likely come to *enjoy* it.

Love this! My visual-spatial 2E kids sooo need context for math facts… cooking, building, sewing, etc. They need ‘reasons’, not ‘rote’. Hands-on iPhone & Android Apps help too. They love to push those buttons!

Getting an answer correct is so much more rewarding when you can really perceive the results, as well. This week my girls and I are going to measure our garden walk, work out what size pavers we should get, go buy them, and put them in! I’ll let you know how it goes! 🙂

I’m not sure what skill teachers are trying to promote by rapid answers to flashcard questions.

Apart from the question of what aspect of memory this activity is promoting, I liked your point 3 about conflating mathematics with this sort of relatively mindless activity.

That, coupled with the anxiety this generates, means it’s something I would never do.

What is the point of making kids nervous in math class?

What is the point of rapid fire answers?

Thanks for highlighting this issue.

Hi Gary! Thanks for coming by and commenting!

I’m curious as to what you mean by “apart from the question of what aspect of memory this activity is promoting” re my point #3. Could you please elaborate?

Yes, I really don’t understand why so many people are baffled by why we have a epidemic of math phobia in this country, when flashcards, rote memory, and fear are the lessons so many were taught in math class!

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Indeed if flashcards are used as a testing device, all you have said holds true.

Sounds to me like you’ve seen flashcards used incorrectly and calculators used incorrectly.

In my experience, I’ve seen the opposite: calculators used incorrectly and flashcards used correctly. So my thoughts on the matter are quite reversed.

We (all people who coach math) need to get our acts together and use the tools. All of them. And learn to use them correctly.

Bon

Egad! I wrote that one sentence wrong. It should read “Sounds to me like you’ve seen flashcards used incorrectly and calculators used

correctly.”I’ve seen kids CHOOSE to make themselves math flashcards, and then use them *on their own terms* to learn basic math facts, which I have no problem with. My issue is with the pressure for speed, accuracy, and know-it-yesterday-ness of so much of what I’ve seen with flashcards. As a tool used by a student to learn something THEY have decided to learn, they can be a great tool. I just hate seeing kids bludgeoned with them, and ending up hating what they then think of as being math!

Indeed, Siggi – I’m all about using them “right.” Which means letting the kiddos use them as they see fit.

Again, we’re on the same page, just a different paragraph (to make a metaphor worse than it was ever meant to be).

😀

(missing you)

3. Flash cards reinforce two fallacious ideas, one for students and one for instructors: that knowledge of basic math facts IS mathematics, and that one must know one’s math facts before one can move on into other mathematical study.

This is so true. The third grader I’m tutoring because he’s bored with math in school, we pretty much did some fun problems where he didn’t need to know how to multiply (since he was just learning that) and then skipped to 3D geometry. We did a temporary shift to a little experimentation (toy bungee jumping) then back to 3D geometry. We play with these toys: Zome and Jovo which is a cheaper version of Polydron that also has another knock-off brand, Geofix.

Great to have you here, Rosina! 🙂

Zome is a great tool, but I haven’t heard of Jovo before – must go check it out! I love math toys and tools that don’t require reading or writing – just thinking!!!

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To discuss this properly, I think we should get below the “flashcards vs calculators” level, and ask ourselves this: how useful is it to “know by heart” certain mathematical facts, such as that 144 is a square number, or that 7 x 8 gives us 56? If it is not very useful, then flashcards, or any other method that is designed to result in allowing children to know these facts by heart, are not very useful.

But if it is useful to know things like this “by heart”, then those of us who use flashcards need to be shown that some other method will lead to the same results.

So … is it useful, or not? Or, to be more precise, in the grand scheme of acquiring mathematical competence, just how useful is it?

This is a great question, Douglas; thanks for asking it here!

I’d have to answer that basic math facts ARE useful things to know over the course of one’s mathematical journey, but I don’t understand why our mainstream culture demands that we constrain our youth into rote memorization of these facts when there is SO much more math with which to enthrall young minds! I’ve seen it over and over again; a child isn’t good at rote memory tasks, starts thinking that these are all math IS (since working on them is all they do in school), and then they start thinking that math is A) boring and B) something that he or she is ‘no good at.’ Tragic!

Calculators let us PLAY with numbers, on OUR terms, exploring math as we go. Some folks will internalize (ie memorize) specific facts as they go, and others won’t; if some of those latter group CHOOSE to use flashcards as a way to help them internalize those facts *when they have decided that it would be beneficial for them to do so for their own reasons*, then power to them.

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