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.
