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I will soon tutor someone I know in math. Because of the bad results on tests, combined with the feedback from the teacher, he has a very bad sense of achievement. Which in turn, has given him a low motivation for the subject. I think the problem is a lack of basic understanding in some very early topics in math, which has given him problems keeping up in class.

How should I go about finding the holes in his math understanding? Finding what it is he is struggling with? When I ask, he gives me very general answers, usually just the current topic. I'm thinking that a test would not be the best way, but I might be wrong. Do you have any other suggestions?

If anybody has experience with a similar situation, any other advice would also be appreciated.

Thanks for the help.

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    $\begingroup$ Also, asking the student to provide a live narrative as he is doing his work, about why he chooses various steps, what he's intending, and so on. A "running commentary" (or other verbose answers) can provide more insight than the usual rather-cryptic, telegraphic style that is popular. $\endgroup$ – paul garrett Feb 16 '17 at 20:46
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Since you didn't mention what level your student is, I'll try to make my answer as general as possible, leaning more toward secondary math students.

In my experience tutoring and teaching secondary level mathematics, I have observed that many students who struggle most have a number of skill domains that are far below grade level, and that these deficiencies hinder progress, in some cases completely. If students have such deficiencies and are having trouble factoring quadratic expressions, for example, discussing quadratic expressions and laying out procedures and completing examples and exercises of quadratic expressions for hours and hours will produce very little, if any, improvement. So spending a lot of time assessing secondary topics or procedural knowledge of their current topics, or even topics from "earlier this year" is not your best choice. One of the biggest mistakes we can make as educators is to assume our students understand something just because it is simple, or remember something just because they have been exposed to it before.

I find that the most effective use of assessment during tutoring is to start with very, very basic maths. You might be surprised what you find. Here are some examples of elementary skills you should assess:

  • Multiplication tables
  • Factoring composite numbers and writing the prime factorization
  • Multiplying and dividing multi-digit numbers manually
  • Multiplying and dividing multi-digit numbers mentally
  • Adding and Subtracting multi-digit numbers manually
  • Adding and Subtracting multi-digit numbers mentally
  • Adding and subtracting small positive and negative numbers mentally
  • Computing squares of integers and roots of perfect squares
  • Solving one-step and two-step linear equations mentally
  • Ordering rational numbers
  • Simplifying rational numbers
  • Identifying Order of operations and simplification mistakes.

To compose a more comprehensive list, find a source of primary education standards in mathematics and jot down learning outcomes from 1st to 6th or 7th grade. Whether your student is a 35 year old struggling in a college calculus course or a 16 year old struggling in high school algebra, the best place to begin looking for holes is primary education.

Thankfully, though these are elementary skills, secondary students are usually ready to discuss numbers as objects, so you can work on strengthening these skills without having to use primary education methods, such as number tiles, but instead focus on the "structure of numbers", which is why factoring numbers is at the top of the list.

Once you determine which of these skills is lacking, it should be easy to determine a course of action. You should not spend the time drilling, but instead teaching the student and their parent drills that they can do daily at home to strengthen these fundamental skills. And give them access to sources of exercises, like the infinite math software or deltamath.com. You should convince the parents that the one hour or so that you spend with them each week is meaningless unless they spend a little bit of time practicing each day.

In addition to elementary skills, you should assess neatness, organization, and study skills. Look at the student's homework. Is it easy to identify the task they were given? Is it well formatted on the page and easy to follow? Are similar problems done in a similar manner each time? Even the most casual assignment should be done with attention to detail. This will allow students to identify their own errors, a skill that you should teach them as well. It's possible that the student has the background to understand the material, but is not able to follow their own work or identify when they have completed the task because they lack organization skills.

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  • $\begingroup$ It is indeed secondary education. Brilliant and detailed answer. I'll make sure to write down a list of early topics in math and find out what of it he understands. Thank you very much for the help! $\endgroup$ – Smebbs Feb 15 '17 at 17:35
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    $\begingroup$ On the subject of drill software for basic skills/assessment, I created Automatic Algebra (automatic-algebra.org) to cover many of the key, mentioned skills -- no account or login required. $\endgroup$ – Daniel R. Collins Feb 17 '17 at 7:30
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    $\begingroup$ @DanielR.Collins I will check it out, thank you! $\endgroup$ – Andrew Feb 19 '17 at 18:02
  • $\begingroup$ @DanielR.Collins Wow, great work. I have some suggestions for options and additional drills. Care to collaborate? $\endgroup$ – Andrew Feb 22 '17 at 21:53
  • $\begingroup$ @Andrew -- I'd love to hear more suggestions. If you use the email link on the "about" page of the site above, I'll get it. $\endgroup$ – Daniel R. Collins Jul 12 '17 at 14:54
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As an experienced elementary school teacher and high school tutor, I would like to suggest some modifications of the list of elementary skills in the accepted answer.

Some of the skills on the list have no bearing on the ability to do high school math such as multiplying and dividing multi-digit numbers mentally. Furthermore many students have a calculator that they are allowed to use in their high school math classes and it would be good to verify if that is the case here. Even if they can use a calculator, knowing there multiplication and division tables through 10 by 10 is essential knowledge.

One of the most important skills that teachers complain about is their students lack of understanding of fractions. Therefore operations with fractions (addition, subtraction, multiplication, division, simplification) should certainly be on the list.

I certainly agree that neatness counts and have dealt with too many students who couldn't read what they wrote in the middle steps of a problem and messed up because they thought the 6 was a 0 etc.

Finally, I would like to suggest that you can work on the current material and see what gaps there are and then go back. I have had much success with this method since students tend to be more motivated when they know how the basics connect with what they are currently learning.

Good luck

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  • $\begingroup$ "Multiplying multi-digit numbers manually" uses the first-outside-inside-last (FOIL) algorithm for multiplying polynomials (with the same numbers of terms as the numbers have digits). "Dividing multi-digit numbers manually" is a special case of using partial fractions to divide polynomials. Both skills require thoroughly following an algorithm through several steps. Multiple kinds of sanity checks can be applied to the answers. $\endgroup$ – Jasper Feb 19 '17 at 2:14
  • $\begingroup$ @Jasper I didn't disagree with manually, I disagreed with multiplying and dividing multi-digit numbers mentally. I agree that there is value in multiplying and dividing mentally... but why would you need to multiply multi-digit numbers mentally. I myself cannot multiply 56 by 43 without a pencil and paper and still feel quite prepared for high school math $\endgroup$ – Amy B Feb 19 '17 at 17:07
  • $\begingroup$ @Andrew You seemed to have missed my point which was to help the OP figure out to best prepare the student. Many of the students I prepare are thrust into classrooms where they use calculators all the time That being the case it would seem inefficient of the OP's time that his students need to learn mental math of multi digit numbers in order to succeed in their current class. $\endgroup$ – Amy B Feb 19 '17 at 19:58
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    $\begingroup$ @Andrew - I don't overuse calculators in my class & don't inspire my students to be lazy. My students calculation skills far surpass their peers. I believe that students should know how to use a calculator & when to use it. I introduce calculators for 1 chapter (statistics) & allow its use on a test. Many students have no idea how to use a calculator before I introduce it. Some think it's cheating. Others assume they will no longer have to think. They're quite surprised to find out that neither is true. $\endgroup$ – Amy B Feb 19 '17 at 20:05
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    $\begingroup$ @Andrew I picked an example to reflect what you said -mentally multiplying multi-digit numbers. That's very different than products of powers of 10 such as 50 times 4000 which all students should know. $\endgroup$ – Amy B Feb 22 '17 at 8:56
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This is an addition to Andrew's answer which is to long for a comment. :-)

Your question suggests that your student shows a low level of efficiency expectancy which results in a low motivation (as you describe it). It's important that your student thinks he is able of showing good results in order to be motivated. So I would recommend you to enhance his self esteem in mathematics and his belief in his ability to successfully solve mathematical problems.

This is mainly influenced in the way he attributes past actions and their (positive or negative) outcome. For example successes in mathematics need to be attributed internal and stable ("I got a good grade because I am good in mathematics") while negative outcomes shall be attributed external and/or unstable ("I got a bad grade because the exam was to hard" / "I got a bad grade because I didn't learn enough but I can change this the next time").

Ambrose et al. (2010) recommend the following strategies for enhancing the students' efficiency expectancy:

  • Identify and create an appropriate level of challenge
  • Provide early success opportunities
  • Articulate your expectations
  • Provide constructive feedback
  • Be fair
  • Educate students how to attribute success and failure
  • Describe effective study strategies

I hope this will help you. I guess it will be a long way to change your student's perspective on his mathematics aptitude but it's worth it. I also just gave an answer on how to enhance students' motivation which you might find helpful. Besides I recommend you the article by Ambrose et al. (2010) where the above strategies are detailedly described.

References

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