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I tutor a 16 year old girl. As far as I can tell, she has average talent and interest in math.

However, her knowledge of math is that of a 10 year old or even below. She knows the basic operations on positive integers below 100, but that's mostly it. She has problems sorting negative numbers. She has insufficient understanding of fractions: she would understand what eg. 3/4 would mean graphically, but does not understand that it's the same number as 6/8, and can not do operations with fractions. She can not solve simple worded exercises like "A new phone costs double as an old one. If the new costs 200$, how much does the old one cost?" that is, she can not translate the words to mathematical operations.

She would say things like 0.9 + 0.1 = 0.10, or that 3/4 and 6/8 are different numbers, or that -1 < -2, which I suppose at some point are normal part of development, and these struggles eventually lead to development of more abstract concepts of numbers (given proper guidance and enough patience, of course).

As far as I can tell, she has average cognitive abilities (intelligence, verbality, memory etc). I suppose that the reason she has fallen so much behind is that she went to an extremely weak elementary school and somehow no one raised a red flag before.

I have been teaching her once a week for half a year online (we're 1000 miles apart).

We made some progress, but it is extremely slow. I realized that she completely misses the necessary concepts, and therefore we need to take it slow and work on hands-on direct experience and not force abstractions, just as someone would go about teaching a smaller child.

The problem is, that most teaching material for her level focuses on much younger kids, so I really struggle to come up with activities that are suited to her age, especially ones that can be done online (unsurprisingly, she isn't very interested in coloring pies). She doesn't hate math and she is neither super interested, nor super disinterested - she is, as far as I can tell, a typical teenager. She isn't particularly lazy either - if I give her homework, she puts in effort to do it.

I understand that motivation is highly individual, so there might not be a general answer.

However, I am curious, if there are any resources or methods that are aimed at teenagers who have, for one reason or another, fallen behind in math.

Note, that our native language isn't English, and she does not speak English much at all. That said, I'm definitely interested in English books and websites etc. it just means I have to adapt them.

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    $\begingroup$ Unfortunately I don't have an answer to your question but I feel the need to say that while she may be below average for her age, you might be surprised just how bad average is (of course it depends on the country). Some of the examples you mentioned are not surprising at all for a 16 year old, and I think coloring pies is exactly what a lot of students need. $\endgroup$ – Thierry May 22 at 21:56
  • $\begingroup$ Good point. I'm not concerned about comparing her to others or where she "should" be. My goal is to build the basics for the long term so she can finish high school. However, I try to avoid too many "childish" activities because those seem to frustrate her, especially if it becomes repetitive and she still doesn't get it. $\endgroup$ – BKE May 22 at 22:04
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    $\begingroup$ There are plenty of unschoolers, who may be way behind on formal math skills. Most of them are fine with understanding equivalent fractions. I'm afraid that school often makes kids htink math is meaningless. $\endgroup$ – Sue VanHattum May 23 at 0:14
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    $\begingroup$ Is the student's native language mentioned somewhere, or did I miss it? Even if the teacher can adapt English-language materials, perhaps material in their native language might also be helpful, so that information might be worth providing in the question. $\endgroup$ – Obie 2.0 May 24 at 23:18
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    $\begingroup$ @user985366 sure I did that already, apps.mathlearningcenter.org/fractions is a nice app where one can overlay them to illustrate this. Maybe it was not clear from the question, but my problem isn't so much about the specifics, because all of these tips works to some degree, but the question is more about how to increase motivation, how to facilitate self learning and coming up with ideas by herself, and how to increase retention. $\endgroup$ – BKE May 25 at 13:11
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One of the skills that math education (or any education, for this purpose) includes is the ability of the student to learn on his own. Judging by your description, your student lacks this skill. I have also found from my experience with teaching teens that these students often also lack retention skills and do not ask meaningful questions during the classtime.

Because of that, consider starting with a more frequent schedule (three to four times a week), then gradually decreasing to weekly once the student learns to learn on their own, to retain previous concepts and come up with questions.

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    $\begingroup$ That is a good point, higher frequency and study "volume" would definitely help. It might even be required to make progress. In our specific case it is quite complicated in practice, will see what I can do. $\endgroup$ – BKE May 23 at 13:07
  • $\begingroup$ There were other great tips and important advice in this thread. I decided to accept this answer because I feel that increasing the frequency of learning would make the most sense and would have the most impact. $\endgroup$ – BKE May 24 at 21:20
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Most people tend to only understand math when they can translate it to the real world, something they can understand. You've given several examples of theoretical concepts that look like magic to someone who can't translate it into something meaningful.

But most of these (still somewhat elementary) concepts can easily be shown using real world examples.

She has insufficient understanding of fractions: she would understand what e.g. 3/4 would mean graphically, but does not understand that it's the same number as 6/8

Take a pizza, cut it into 4 pieces. Ask her how many slices she wants (let's say she says 2). Then continue cutting the pizza in 8 slices, and give her two of those slices. Wait for her to complain, or if she doesn't, ask her if she received as much pizza as she asked her for. Try to get her to explain to you that you didn't give her what she asked for.

The core idea to convey is that 2 slices when there were only 4 slices is exactly as much pizza as 4 slices are when there are 8 slices. Therefore, 2/4 = 4/8.

While a pizza may be impractical for a tutoring session, the same applies to a small cake or a chocolate bar.

and can not do operations with fractions

My elementary school teacher used water and measuring jugs (each containing a liter, but each jug was marked in 4 parts, or 5 parts, or ...) to first prove to us that e.g. 2/4 + 1/5 = 7/10. At this point, it was no longer a wild magical claim, but it was provably the case.

At that point, it became easier to try and understand (using unintuitive mathematical tricks) how we could've figured out that this was the case. You can do this in steps:

  • First, using two jugs that are divided into 4 parts, prove that 2/4 + 2/4 = 4/4. She should be able to predict that when she sees it done with the jugs.
  • Then, using a 4-part-jug and a 2-part-jug, prove to her that 2/4 = 1/2. Pour the liquid from one jug to the other to prove the point.
  • Then ask her what 2/4 + 1/2 is going to be. Try to avoid asking her for a fraction, instead try to get her to state whether it's going to fill the jug or not.

To this day, I still use those jugs when doing fractions in my head. It really stuck with me because it's a visual and tangible representation of something that is otherwise just some arbitrary number magic.

Similarly, you can use the pizza/cake/chocolate bar to come up with similar real life examples.

She can not solve simple worded exercises like "A new phone costs double as an old one. If the new costs 200$, how much does the old one cost?"

Try to come up with simpler and simpler worded exercises, to the point of balantly easy maths. Try to figure out the point where she starts getting stuck.

For example:

  • A new phone costs double as an old one. If the new one costs \$200, how much does the old one cost?
  • A new phone costs \$10 more than an old one. If the new one costs \$200, how much does the old one cost?
  • A new phone costs \$10 more than an old one. If the old costs \$200, how much does the new one cost? (this is slightly easier as the two sentences map more clearly to each other)
  • You intended to buy a \$150 phone, but you ended up buying the fancier \$200 phone. How much did you overspend?
  • You're going to buy a \$100 phone, but you want to also buy the \$20 phone case that goes with it. How much money should you bring to the shop?

If she really struggles with visualizing this, she may be more comfortable talking about an actual purchase she made instead of a theoretical one. Ask her about something she's bought or is wanting to buy. You can get her to do the same puzzle solving depending on the situation)

  • If her parents paid for some of it, but not all, ask her how much it cost, how much her parents paid, and how much she paid for herself.
  • If she's still saving up to buy it, ask her how much it costs and how much she has already saved. Then ask her how much she still needs to save.

Whatever you end up talking about, then follow up with a theoretical example that exactly mirrors the real life situation she already understands. This will very much telegraph to her that these two cases (the one she knows and the one she doesn't) are effectively the same but with different numbers/words.

Doing this repeatedly will eventually teach her to translate an unknown situation into a known one.

Stick to what they know

It's a cliché example, but in The Office there is a character called Kevin Malone, whose main characteristic is being allround dumb. It's established at one point that he can't count money, but he can count food. If ask him to count pies instead of dollars (or salads), the question becomes answerable to him.

Here's the scene in question

It's a really stupid joke, but it actually gets at the heart of how people do math: visualization. People are intuitively able to visualize that which they understand, and no one is able to visualize math without having been taught how to translate the abstract into the real (and equal), or vice versa. That is your job as a math educator.

Your task is to find the visualization that works for your student. That's going to be a highly personal process. Relate it to her hobbies and past experiences.

Ask her things the already (subconsciously) understands, no matter how trivial, and then present more theoretical examples that are blatantly the same situation as the one she already understands.

The goal is to get her to realize that these seemingly different situations are the same. The more blatant the similarity, the faster she'll figure it out.

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    $\begingroup$ Thanks. I know and have used all of these examples eg. pizza, jugs etc. but still it is very useful to document here. These kind of work, but she gets very quickly bored with it and retention is low. I think one can't assume that "real world" example will motivate, especially an adolescent. I understand motivation is highly individual, and maybe I just haven't figured her out yet. But maybe there are some tips for adolescent girls that work. $\endgroup$ – BKE May 24 at 6:39
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    $\begingroup$ I have a hazy memory of my own maths teacher, back when I was a teenager, describing some research where they took 1950s British textbooks, full of problems about "boy stuff" (cricket scores, warplane engines) and translated the same problems into typical girl hobbies (cooking, dressmaking etc), and found immediately that the female students were more motivated and suddenly scored equal or better than the male students in tests. As this answer suggests, given that you're going to have to adapt/customize anything you find anyway, try to make the subject matter close to her own interests. $\endgroup$ – Daniel Hume May 24 at 10:02
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    $\begingroup$ @BKE: What would help is (innocently) trying to con the student. Some people really respond well when there is a direct benefit to figuring something out. For the pizza example, if you take the other six pieces for yourself, it can trigger a "that's not fair!" reaction which gets them to evaluate why it's not fair - which indirectly teaches the concept you're trying to teach. $\endgroup$ – Flater May 24 at 11:15
  • $\begingroup$ @Flater that's another good way. She indeed has a strong sense of justice which is something I can build upon. $\endgroup$ – BKE May 24 at 21:24
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A warning: I have read some of the books I mention here but I have not tried them on actual students.

I recommend getting your student some of Danica McKellar's books. Your student might want to start with "Math Doesn't Suck" which is targeted at ages 9-12 and covers fractions, decimals, and percentages. If this is too difficult, then try "Do Not Open This Math Book" (ages 6-9, addition and subtraction) then "The Times Machine!" (ages 8-10, multiplication and division). After she understands arithmetic, let her try "Kiss My Math" (ages 11-13, pre-algebra), then "Hot X: Algebra Exposed" (ages 12-14, Algebra 1), then "Girls Get Curves: Geometry Takes Shape" (ages 14-16, geometry).

To get some idea why I think Danica's books are quite different from the others, see my blog post here. She writes in a very clear way, and her books are actually perfect for parents trying to understand how to help their children with their homework.

My PhD dissertation was about conceptual and procedural ways of teaching mathematics, and I was curious about which of the two approaches Danica favored. I was actually very pleasantly surprised to see that she managed to explain things using a very good combination of both.

From my blog post about "Math Doesn't Suck":

I think the inspirational messages spread throughout the book are the most important thing about it. Here’s the last part of the final paragraph of the final section (“Great Expectations”) of Chapter 5:

Every homework problem you think you can’t do—but then through determination, you solve—every time you exercise your brain and your beauty, inside and out, you’re becoming the young woman you aspire to be. I’m here to tell you from personal experience that you can be a glamour girl and a smart young woman—who can certainly do math.

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  • $\begingroup$ Oops, I posted this answer before I saw your comment that your student's native language is not English. Perhaps someone can read the book with her, translating as necessary. $\endgroup$ – Joel Reyes Noche May 23 at 8:52
  • $\begingroup$ No worries, I checked them out and I also think those books are quite good. I can use or adapt at least a few ideas I am sure. $\endgroup$ – BKE May 23 at 13:22
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Beast Academy is for younger kids, but she might still like it. The format is "guide books" that are like graphic novels, where the beast students are taught by beast teachers, along with practice books. You can see examples on their website. It has levels 2 to 5. But it was designed for gifted kids, and it offers plenty of interesting challenge, whatever level you're at, along with excellent conceptual instruction.

She could look over the website, and if she likes it, she could sign up for their online version, which would allow her to move through the levels at whatever pace works for her.

I will be using it with an adult student this summer, who just failed my calculus class. I told him he has lots of holes from what I can tell and filling in the holes will help him succeed when he retakes calculus.

If you suggest this to her, and it does work for her, I'd like to know. I'm trying to get clarity on how helpful it is for adults (and teens).

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    $\begingroup$ Thanks! I forgot to mention that the native language isn't English, moreover she doesn't speak English much at all. I will check it out though and see what I can adapt for her. $\endgroup$ – BKE May 23 at 8:26
  • $\begingroup$ What is the language? $\endgroup$ – Sue VanHattum May 23 at 20:01
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    $\begingroup$ Hungarian. Not too widely spoken, except, well, Hungarians :) $\endgroup$ – BKE May 23 at 20:45
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Based on my own experience as a learner (and occasionally 'teacher'), I can think of a couple of other possibilities:

  • She may objectively be able to understand maths and even intelligent, but lacking the confidence to believe that she can. I remember it from myself, on occasion, but also from children I have helped a few times. If that is the case, then she needs a lot of encouragement, and guidance along the lines of 'what do you think?' - the sort of questions that provoke discussion and opinion.

  • Personally, one thing I found incredible demotivating was the lack of deeper insight. In my elementary education it was mostly about the mechanics of how to count, add, subtract, etc, and what I realised later was that I crave deep understanding: I derived enormous pleasure from learning what a natural number is (a representative of a class of isomorphic sets), from understanding the reason why multiplication of multidigit number work the way it does (~ a number such as $5348$ is actually the sum $8 \times 10^0+4 \times 10^1+5 \times10^2+5 \times 10^3$ - if you use the sum-rule, you get to the right method).

Modern school education is still far too focused on rote-learning of rules, just get students up to a certain level of function, rather than encouraging interest and independent opinion.

I hope this helps or perhaps inspires you in some way.

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Age should not be confused with ability. Cognitive understanding of mathematical principles is indicative of higher-order thinking which in turn is ability and skills based. The fact that this student is 16 is irrelevant in terms of ability or skills.

Regarding ability, ability is defined by an understanding of key foundational concepts and the development if skills is in turn dependent on this. The skill needed to solve word sums is dependent on an ability to understand the core fundamentals of plus, minus, add and divide. Shortcomings in ability is addressed by skill. Just because you struggle to understand mathematics naturally should not be an inhibitor to learning the skill.

The same logic applies to things like emotional intelligence. Just because you struggle to communicate naturally does not mean that you will always struggle, it simply means that you need to acquire a skill that others naturally posses. Obviously this is a generalised statement and there are exceptions. Dr Andre Vermeulen at Neuro-Link is an expert with regards to this and some of the material on their website might aid you in developing a learning plan which encourages the development of Neuro-agility.

There are a number of possible contextual explanations to what you are experiencing with this student however have you considered the possibility of a repressed problem? As an example, a person who is colour blind accepts their reality as being THE reality. They do not know that they are colour blind as they think that is the norm and they will not say it either. They only discover that they are colour blind through others around them or through testing or in most cases, by chance. Other examples of this include something as basic as eye-sight. If you struggle to see and you have accepted it as being the reality chances are you will not address the problem because to you it is not a problem. Perhaps you should advise your student to have their eyes tested if possible?

I also see that one of the answers here makes reference to self-learning. Indeed that is an incredibly important point and it is a shortcoming in the majority of education approaches in the world. Most education systems promote learning through the idea of content delivery. I.E. a teacher stands in front of a class and gives info that the class must absorb and this is accepted as being "learning". The students are seldomly really encouraged to self-learn and textbooks are written so as to be able to replace the teacher. The problem with this is that people are not taught to learn for themselves, you listen in class, if you do not understand you read the textbook and just apply the steps laid out therein. I.E. learning is taken to mean understanding.

Arguably the best math program in the world which encourages self-learning is Kumon. In fact if you read through the bio on their website, Toru Kumon the creator of the Kumon method, developed Kumon to be a program based around self-learning. I would strongly suggest you advise this student to join Kumon in addition to you tutoring her. This approach might yield the best results.

The last thing I want to touch on is attitude. Yes it is the cliche explanation for the root of many problems however its effect on the development of ability and skill is vastly underestimated. I have seen this happen in a number of cases, people start off at school with the idea of "Math is difficult and I struggle with it" and never are they taught otherwise. By the time they finish school they have convinced themselves "I will never be able to become and engineer/accountant/scientists et cetera because I can't do math". In a couple of years they went from saying "I struggle" to "I can't" purely because of attitude. The most powerful outlook on the understanding of math is to say "I don't understand it YET".

If you spend your entire life trying to understand something, you have not failed, you have had an entire life of learning and more than that you can't ask for.

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