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I recently started giving maths lessons and it seems like I am at my wits end. My own background is: I'm a masters student in physics, already did several tutorials for younger students, especially for a course on mathematical methods in physics. Of course, some years ago when going to school myself, I was helping some of my friends with school maths, so teaching school maths is also not entirely new to me.

However the current situation really gives me a headache. The pupil is going to 8th grade, the current topics are simple systems of two linear equations in two variables and the intuitive methods of solving them. The problem which I and the pupil are facing is however another one: She really has a hard time doing simple calculations with numbers, multiplying, dividing, working with fractions. It is not a problem of understanding. Given a big amount of time she can do all the calculation using long division and multiplication.

However she is entirely missing what I would call "intuition for numbers", i.e. she doesn't see if one number is a multiple of another (i.e. she doesn't notice that 8/4 can be simplified to 2), she doesn't know the most useful fraction to decimal fraction conversions (i.e. 1/4=0.25). She knows the 10x10 multiplication table in principle, but it takes a huge amount of time till she recalls the result. Sometimes she even needs some seconds to calculate 2x3=6.

You can imagine that it is very hard teaching new topics in such a situation. A week ago we discussed the methods for solving systems of linear equations and I intentionally gave only examples with small coefficients and such that no fractions appeared at all. The result was that she understood the topic very well and could solve such exercises. However, if in an exam she encounters exercises with higher numbers or examples where fractions will pop up during the calculations, she again won't be able to solve them.

And now I'm really stuck and don't know how to proceed. Make her learning the multiplication table by heart? But she already knows it, sort of. However, doing operations with numbers in her head just doesn't work properly. Returning to standard exercises from 6th and 7th grade? Takes a huge amount of time and doesn't really solve the actual problem since the exercises will be diluted with other concepts which are not of importance now.

Thanks for any hint, Michael

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    $\begingroup$ Blitz answer for 1 to 10: 1 is the identity. Practice counting by 2s (this should be done pre-K!) and 5s & 10s (should be done by early elementary!). The 3s and 4s are often done with songs (see, e.g., here for 3 to 9). The 9s can be done with the standard hand trick (google if unfamiliar); see MESE 5866 for a bit more about finger counting. With commutativity, the only remaining products are: 6x6, 6x7, 6x8, 7x7, 7x8, 8x8. Drill and kill these. $\endgroup$ – Benjamin Dickman Dec 26 '14 at 7:07
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In my experience, I feel this is best learned by just practicing. I understand that it may seem remedial for whatever lessons you are giving (or perhaps it isn't) and that it can take away from time you need to teach other subjects.

I honestly do find that I have certain values (multiplication tables, special fractions) just "memorized", but I still know how to calculate the values if I ever forget them (i.e., doing the long division for fractions). Whenever I would teach about fractions I would always do the long division first and point out the ones that are really important.

Also, sometimes fractions can be easier to remember when they are related to something used in everyday life. In the U.S., a quarter is .25 or 25 cents. Most students know the value of the quarter so it's just teaching them that $1 \over4$ is another way of saying a quarter. 2 quarters is .50 or 50 cents just like $2 \over 4$ is $1 \over2$ and things like that.

Of course every student learns differently and other countries don't have the same denominations of coins as here in the US, but I was just providing that as an example.

Something that a lot of students these days do have are smart phones and I have seen apps that can test quick calculations like these (it's sort of like the flashcards students used to use when I was younger). Or, perhaps the students can just use actual flashcards with a friend or relative (or themselves).

tl;dr:

I believe for the simple fractions and quick multiplication you mentioned, it all comes down to repetition and practice, but there are plenty of resources to get that practice if the student is willing to provide the time for it (10 minutes on the smartphone before bed doing quick math quizzes instead of candy crush could be good!)

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  • $\begingroup$ Thanks for the answer! Yesterday we made a table of frequent fraction to decimal fraction conversions and she calculated all of them using long division. But the backup in form of doing the long division is always there, the only downsides are that it takes a huge amount of time and, more important, that it gives you, what is called passive but not active knowledge, that is, she might be able to recall the conversions if explicitly asked for it but won't get the idea to use them on fitting occasions on her own. $\endgroup$ – Photon Dec 24 '14 at 18:15
  • $\begingroup$ Smartphone apps are a good idea, I don't own a smartphone myself, that's why this possibility slipped through my fingers. However, I showed her this game which trains factorization of bigger numbers into prime factors: kenkenpuzzle.com So far it had no noticeable effect though... $\endgroup$ – Photon Dec 24 '14 at 18:18
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"She knows the 10x10 multiplication table in principle, but it takes a huge amount of time till she recalls the result. Sometimes she even needs some seconds to calculate 2x3=6."

This is absolutely not knowing the multiplication table. As a community college lecturer with lots of remedial courses, I see this a lot; a student will say they know the times tables, but they're really sequentially adding mentally. Frankly, they don't understand that "knowing times tables" means instant recall, total automaticity. Later on they'll be near-helpless when it comes to factoring, fractions, estimating, etc.

So the first thing in this case is that she really needs to sit down and memorize the times tables for real. I tell my students to practice it in a timed environment every day until they never make any mistakes. Here's the website I set up for that; note that the first and fundamental skill is times tables: http://www.automatic-algebra.org/

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    $\begingroup$ Thanks, a very nice website! $\endgroup$ – Photon Oct 1 '15 at 18:39
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Permit me to direct you to read an answer to another question by another user Benjamin Dickman first, notably the second part of the answer that begins with the line "Given the above discussion, I would like to make one additional comment"

I cannot improve on the linked answer but I can connect it to your question. What you call an "intuition for numbers" and I might classify as "number sense", I feel could be improved with an emphasis on delving into the prime factorizations of integers. An earlier answer by @Richard suggests making lattices of these factorizations (though I know them as "trees"). In my experience, many students who struggle with the same things that you say your student is struggling with have more success when working more with prime factorization.

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    $\begingroup$ Factorisation of monic quadratics is something that I have used also. Unlike the integer factorisation lattices (trees usually do not find every possible factor), quadratic factorisation combines addition and multiplication, so students get a feel of the difference between the two operations, which aids number sense. $\endgroup$ – Richard Dec 26 '14 at 5:26
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In fourth grade, our daily math class included a three minute timed exercise. Each student had a page of approximately 50 or 100 math problems. All of the math problems were of one kind -- either adding 2-digit numbers, or subtracting 2-digit numbers, or multiplying numbers between 0 and 12, or dividing numbers between 0 and 144 by numbers between 1 and 12.

I think you should get a book of these worksheets for your student, and have her do one 3-minute exercise (on one worksheet) per day. The worksheet itself won't improve her skills, but it will:

  • give her a chance to practice
  • let her see her improvement
  • give her a goal of being able to do arithmetic quickly.
  • learn to spot what she knows how to do, and do those problems first.

You can also teach her some valid shortcuts and estimating techniques. I would teach one new shortcut per lesson, and have her see how much that shortcut helps with her 3-minute timed exercise.

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  • $\begingroup$ Thanks, I will try that! I think, I will write a little bash script generating TeX code for such sheets using a random number generator. :) However, 3 minutes seems to be a little tough for 50 problems, this gives you less than four seconds for each including reading. I would estimate about 15 minutes or more for such an exercise in her current speed. Probably rather start with less exercises in the beginning... $\endgroup$ – Photon Dec 25 '14 at 8:24
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    $\begingroup$ @Photon As far as I understand this proposition, the point of having 50 question is precisely that it is not (or barely) doable. The student will not feel ashamed not to have done everything, and will be happy to do more each week. $\endgroup$ – Benoît Kloeckner Sep 28 '15 at 13:15
  • $\begingroup$ Thanks, I get the idea! $\endgroup$ – Photon Oct 1 '15 at 18:39
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I have 2 ideas for you.

First, I have helped a student get number intuition by getting him to building factorisation lattices. For instance, 12 can be represented by a 2D, 2 node by 3 node grid/lattice:

            12

      6            4

3           2

      1

They get really pretty when you get 3D lattices (3 unique factors, like 60). It can help to draw the connecting lines in different colours, but it is essential that each factor has its own axis/direction. More than 3 unique factors is almost impossible to draw nicely, but is not needed until 210 (2*3*5*7). Having multiplicities of factors is not a problem.

It gives students a good feel for numbers and how they interrelate, apart from good practice with division/multiplication.


HOWEVER... (idea 2)

My personal experience is that mental arithmetic is overrated in importance for future academic achievement. If a student does not have good mental arithmetic by Year 8/9, most teachers I know would give them a calculator.

But I do not like calculators!

Calculators often do not help visualisation, or allow for easy experimentation. But I have had amazing results available with a tablet using iPython with a web interface. Other Computer Algebra Systems may work just as well, but I have not tried them with students.

This year I coached a year 8 student with poor mental arithmetic and very little algebra through a Physics course that used numerical integration techniques and vectors. How did he do it? He programmed it all in Python. Mental arithmetic never came into it. Algebra was never an issue, as he found computer manipulation of variables much more intuitive than algebraic manipulation. Now he is experimenting with fluid dynamics - beyond what I studied at university.

His times tables are still atrocious, his algebra is improving (through practice with Physics equations and DragonBox), but he is doing university level physics and applying it to complex problems.

The use of technology presents problems in exams, but appropriate guided use can give students confidence and experience that will eventually translate into maths intuition. It also provides the practical technology/maths skills that are of use outside a maths classroom, something traditional maths curricula struggle with.

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  • $\begingroup$ Thanks for your ideas! The lattice idea sounds nice (that's about what you need if you are playing the Kenken game I mentioned in one of the other comments). I will try that! Unfortunately, I don't expect my pupil to easily get into programming. However the question, whether spending much time and effort into mental calculations makes sense, is a valid one. On one hand, from next year on the students will be allowed to use calculators in the exams, on the other hand this year they are not and the current aim is to pass this year. Also, being able to do mental calculations will give confidence. $\endgroup$ – Photon Dec 25 '14 at 8:49
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Permit me to recommend this book as a source of fun topic ideas:

Benjamin, Arthur, and Michael Shermer. Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks. Random House LLC, 2008.

And see Art Benjamin's YouTube video:


      ArtBenjamin


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    $\begingroup$ Thanks, those videos look really funny! However, I have the impression that they are of much use if one is already able to do the multiplication in the conventional way to shorten up the calculation. But I wonder whether they can substitute the standard way of calculation, since they are only applicable in special cases. $\endgroup$ – Photon Dec 24 '14 at 18:29
  • $\begingroup$ This one actually looks very nice: youtube.com/watch?v=v1Ih3-mDPUk :) $\endgroup$ – Photon Dec 24 '14 at 18:46
  • $\begingroup$ @Photon: Also the amazing Art Benjamin! Thanks for that link. $\endgroup$ – Joseph O'Rourke Dec 24 '14 at 19:05
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I'm not a teacher. I have an idea on how to do it but it should be taken with a grain of salt. If you tell her that she needs to learn another method because gets the result faster, she might not listen or understand. She might be like "How can it be done faster." However, it is true that there is another reason to learn the other method. That's for the sake of knowing how to do it by that method and showing the work. I think it's better if as a teacher, you tell her that reason and then give her a simple task of solving a problem such as $\frac{3}{4} + \frac{5}{6}$ by writing $\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}$ and writing it that way on the test. Maybe she's so used to computing the value of a fraction by using long division that it's not obvious to her that $\frac{3}{4} = \frac{9}{12}$ until after she computes both of them by long division. Maybe she's really smart and realized that the value of a fraction can be defined using long division and that until you compute the value of both fractions using long division, you don't have a proof that they're equal according to that definition. It's just like questioning the axiom of choice is smart. Unfortunately, if that's the case, thinking for herself won't get her the marks. I would love it if I could get the way the world works changed legally so that students are taught using a student centered approach and the student is not pushed to agree that it's obvious that $\frac{3}{4} = \frac{9}{12}$. Hopefully, if you teach her that $\frac{1}{4} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12}$, she will understand why that's true. Then it will be easy for her to see why $\frac{3}{4} = \frac{9}{12}$. If she still can't understand why $\frac{1}{4} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12}$, and she's not willing to perform calculations on fractions the way she was taught when she doesn't understand why it works, I think it's not worth pushing her to do it and she should have a right to make her own choice to not answer test questions whose answer requires that type of calculation and not get the marks.

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  • $\begingroup$ My question is quite old and thus I don't teach this pupil any more. But I think, her problem was not that she questioned reducing fractions as a concept. For reducing a fraction it is necessary to find a common factor in both nominator and denominator and this was the challenge she faced. $\endgroup$ – Photon Oct 10 at 12:38
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    $\begingroup$ @Photon I think you're right. It's too late. This answer can no longer solve your problem and get a check mark but that's okey. That's the way Stack Exchange works. If you have another student like her, this answer might solve your problem. It's better if there isn't another student like her but if there happens to be one, then maybe things can be improved by using the idea in this answer and if it works, putting a check mark beside it. Then it can then be used to research how to fix an already existing problem of students not understanding things properly. $\endgroup$ – Timothy Oct 10 at 18:47

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