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I'm a maths tutor, and my students/tutees are aged $11 - 18.$

Obviously I have limited time with students, usually one hour per week. Moreover, if parents don't see improvement over a year or two they will look for a better tutor which is undesirable for me.

I feel like students not knowing their basics well enough, like being quick at multiplication, addition, subtraction, division, and fractions, holds them back enormously in learning more difficult topics. Their learning speed in these difficult compound topics that requires quick arithmetic is massively reduced and this disrupts their learning of the main goals of the harder topic.

When I was a young kid of 4 - 7 years old (or something like that), my Mum used to play a cassette in the car of the times tables, and ever since, I have had instant recall, and this has been one of the main contributing factors for allowing me to learn the more difficult topics easier than others. [I am not comparing myself to others on this website, as most people here are also probably good/ quick at arithmetic. I am comparing myself to the general population]. This is because I don't get distracted from the learning goals of a topic when calculating $7\times 6$ is required to attain that unrelated, more difficult goal: my brain converts $7\times 6$ to $42$ in under a second, and I can get on with understanding the more difficult topic.

I was wondering if it was a good idea to give students - who are not quick or accurate enough at these basic calculations - tools, and a programme that if they stick to, will guarantee improvement in multiplication, arithmetic, fractions. And then only continue on the compound topics once they have proven their speed and accuracy in various tests? Then their newly-found instant recall will not stand in their way of learning harder topics, and might even give them a confidence boost in learning mathematics.

automatic-algebra has one-minute/$30$-second/$15$-second multiple choice tests on different aspects of arithmetic. But the difficulty and allotted time of the tests don't vary, and so the tests aren't very comprehensive/ are one-dimensional. However, I have tried that website with some students and some students do improve a bit with it and some students didn't improve much with it. So I was thinking of creating a website/webpage that was like that one but where, for example, either the difficulty of the questions increased, or the allotted time decreased, or a combination of both.

I was thinking of changing my tutoring to be a strict, hard-line approach: Either they improve so that:

  • they can multiply any two numbers from their $9\times 9$ tables in $< 5$ seconds.
  • Something similar with addition
  • Then on to fractions - obviously more time is needed for this, but they should still need to do them quickly.

But I don't continue tutoring with them if they can't do the calculations within the time period.

Is this approach too hard-line?

Personally, I think it may be a good idea for almost every student who is not quick at these calculations. I just worry as to whether or not this will work. If they haven't bothered learning their times tables by now, and they are reluctant to do so with my instruction, then what? Do I sweep this under the rug and teach them harder topics anyway? I know that students can learn harder topics, but it just takes much much longer for them to be able to do exam questions quickly in these harder topics (because they don't have the instant recall of arithmetic operations), and this exam performant is very important when it comes to tutoring, as parents are more likely to fire me if the student has done poorly in exams.

Another aspect of this is that if I am too "hard-line" in this approach then the student may not like me (but the parents might)...

What do you think? Has anyone tried this approach in practise?

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    $\begingroup$ The quiz website you describe sounds a lot like mine at Automatic-Algebra.org. $\endgroup$ Jun 18 at 19:22
  • $\begingroup$ Are the students allowed to use calculators in class at some point? $\endgroup$
    – Photon
    Jun 18 at 19:44
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    $\begingroup$ @DanielR.Collins that's the one. Thanks $\endgroup$ Jun 18 at 19:53
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    $\begingroup$ Calculators shouldn't be necessary for basic arithmetic. For working out two-digit by one digit multiplication and harder, calculators are fine, so long as the student has demonstrated ability to calculate these quickly without a calculator on paper first. I allow working out of $7\frac{2}{5} - 17\frac{14}{11}$ on a calculator; $\frac{4}{7} + \frac{8}{3}$ on paper; $\frac{1}{2}+\frac{1}{4}$ I would also allow working out on paper, but would expect students to know that off hand ,although unfortunately the reality is that they often have no idea how to simplify this... $\endgroup$ Jun 18 at 20:05

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It depends greatly on the students.

I have had great success training students with the flashmaster app. With this app you can focus on improving time for basic facts. I have used this with my classroom students, insisting they spend a certain amount of time on it.

Although less than 5 seconds is great, a student who can find the number in 10 or 20 seconds will also have success, so I think your goal is unnecessarily high.

Another approach that I have been successful with is to give the students that I am tutoring a times table. I tell them that they have to fill it in before we can do the work. This is often effective:

  1. filling in the chart is good practice for them - which means it is not a waste of time.
  2. you can see which facts are hard for them - often they don't know the product of 7 and 8 and I tell them 5, 6, 7, 8; 56 is 7 times 8 and they remember it afterwards.
  3. This can be easier than memorizing. For example a student can count by 3's to fill in the 3's and then double the 3's to get the 6's
  4. Once they have the chart, it will be easy to simplify fractions etc.
  5. Student who do this in class, will write times tables on their tests, so they can refer to them.
  6. If the student really resents having to do this - the alternative is to learn the tables and have them at their fingertips.

I test mastery by mixing up all the times facts from 1 to 10 and telling them that they have to complete it in 10 minutes. It's not super fast but it's enough to succeed.

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As I said in a comment, the quiz website you describe sounds a lot like mine at Automatic-Algebra.org.

One chief reason that I made that site was because I saw students in my college remedial algebra classes be completely unable to progress at the point when they had to do polynomial factoring, when they didn't know their multiplication tables from memory. As in, they'd out-loud give up and say this task was simply impossible for them.

Now, was that solution a silver bullet? No, and I doubt there is any. In recent years success rates in those classes have trended ever-downward, and my major university system (and others) are terminating all of those classes, because approximately no one benefits from them. The current dominant theory is that those basic facts should be reviewed in a just-in-time, mixed-practice, corequisite support context -- but I'm skeptical that will really make any difference.

I will observe that there's at least a continuum of facility at picking up new mathematical (and other) facts -- some of us pick them up mostly on first sight, while others fail to retain them even after many tries. So admittedly a purely logical sequencing of needed-fact-A coming sometime before dependent-fact-B, which is sufficient for highly-proficient math learners, is insufficient for math learners with demonstrable weaknesses. Most people aren't like us, the visitors to this site.

And I think I've seen experiments with classroom digital mastery systems, where students had to succeed at early skills before moving on, but the results weren't better than anything else.

So arguably a policy might be that you should give some amount of review, but move at on at some later point, regardless of whether they digested the early facts or not. Even if they did, they'll likely forget them at some later point anyway. You might need to accept needing to always review those facts when they become critical later on. I suspect such students will be fundamentally limited in how far they can go, but I doubt there's any way to fix that for them.

Related questions:

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    $\begingroup$ This rings true at first parse. But I will need to re-read and think more about this. $\endgroup$ Jun 18 at 19:57
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The question: Should I discontinue tutoring engagements with students who refuse to remediate their inability to perform basic calculations to some standard?

My answer: it depends on your goals as a tutor / for your business.

If your goal as a tutor is strictly to make money and you're worried that you won't be able to replace any students that walk away from your services, then you certainly shouldn't terminate any of these arrangements yourself but rather wait for dissatisfied students to leave on their own (assuming you don't think they're going to leave bad reviews on a website, bad-mouth you, cut you out of some tutoring referral program that exists with local schools, etc).

If you aren't concerned about your enrollment or your frustration with students' lack of progress outweighs your need for the money, then you can implement a policy without fear of losing business. Such a policy would likely be in the long term interest of your students and business because:

  1. Many students in the stated age range (11 - 18-year-olds) could benefit from better arithmetic skills in the classroom (if nowhere else) for years to come.
  2. Seeking mastery of these basic facts appears to be pretty common practice, so the perceived benefits for approaching more difficult problems may be real.
  3. If the benefits are truly real, then you can imagine a virtuous cycle of improved student outcomes, increased morale, positive word-of-mouth leading to new business, increased credibility as a tutor, more compliance from future students, etc.

If you're going forward with this, next steps would be to iteratively refine your approach (basically some of the steps you already outlined):

  1. Make a clear list of outcomes that are actually needed for success (cursory review online suggests that 5 seconds for a basic addition/multiplication fact is pretty lenient). Having a realistic/achievable goal is good for morale, ensures you're not chasing diminishing returns, and improves the likelihood of success on the following point...
  2. Create a program, if followed, which actually produces desired outcomes outlined in point 1 (May require student trials.) This is important because if the program doesn't work, that hurts your credibility with compliant students. They need to believe both on this subtask and the whole initiative that everyone is aligned on what the goal is and that you, the tutor, know what it takes for them to achieve said goal.
  3. Craft your elevator pitch when you broach the topic with each student (some may be more receptive to analogies with sports, others music, etc) but the basic idea that these are fundamentals that you need automaticity with to elevate your performance should be easy enough. This builds momentum for an otherwise boring (borderline embarrassing depending on age?) task.
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In my opinion, it depends on the age of the student and if they will typically be allowed access to a calculator in their math classes and on tests.

Focusing on arithmetic fact fluency (times tables, adding within 10 or 20) requires that the student have reasonable memorization skills and a solid sense of numeracy. Some students do not have these skills: for example, students with processing disorders, students with dyscalculia or other math-related learning differences, and neurodivergent students may struggle more to acquire these skills. For these students especially, it may be unrealistic to ask them to achieve strong working fluency with arithmetic facts. For some, it is just not possible; for others, it would take a tremendous investment of time and effort that would be better used to focus on (for example) algebra skills. In this way, there is a real accessibility concern associated with demanding arithmetic fact fluency from all students. Of course, if the student's course requires that they have strong recall of their facts without use of a calculator, then this is a worthwhile topic for you to focus on as a tutor; however, for students from age 11-18 (middle school & high school), they are typically transitioning out of courses with a computational focus (and where calculators are not allowed) into courses with an abstract algebraic problem-solving focus (where scientific or graphing calculators are commonly allowed).

As a tutor, your student is paying you to help them achieve success in their coursework. Of course, this doesn't mean only focusing on homework as it is common that a student needs significant remediation before they are able to engage with their current course level. However, the job of a tutor is to help your student develop effective strategies to achieve success in their current and upcoming courses. For students in elementary school, this does mean that an emphasis on computational fluency and knowledge of arithmetic facts is appropriate, since arithmetic knowledge is a "high-mileage" skill that will serve them well in many problem contexts, especially since students at that level can rarely use a calculator in class or on tests. However, for a student in algebra 1 or beyond who is able to use a calculator in class and on tests, they may likely need more support navigating the shift from courses focused on rote computation to courses focused on algebraic skills. In my opinion, you would be doing those students a disservice by skirting the main issue which is their need to develop fluency with more abstract problem-solving techniques applicable to high school and college math classes. Many students who struggled with computational math classes early in their education can succeed and excel in more abstract and algebra-based courses later in their education; similarly, a student can breeze through their computational math classes yet struggle with the transition to algebra work (this is a very common context for students to begin working with tutors). So these two skill sets and course contexts, although certainly not disjoint, don't overlap as much as you might think.

For those topics which do require knowledge of arithmetic facts (eg. factoring quadratics), I typically have students rely on calculators to identify factor pairs. I do sometimes teach divisibility rules if the student is ready for it. Many students, even those who struggled to achieve arithmetic fact fluency in their earlier years, can observe the patterns in factor pairs and develop a stronger sense of numeracy through practice with a calculator. There are some students who need a calculator for any numerical computation (even adding within 10). These students should not be shamed for their lack of fact knowledge, nor should their efforts in developing algebra fluency be derailed in factor of a computational approach that they will rarely need to use in the future. Rather, when working at the level you indicated (age 11-18 and beyond), a tutor should help the student find strategies that they can use in solving diverse types of problems, especially problems which require strong knowledge of algebra techniques. In this context, calculators are an accessibility tool that broadens access to the field of mathematics and increases the odds of student success at high levels.

Ultimately, although fact fluency is a cornerstone for many computation-based classes in elementary and early middle school, the prevalence of calculators in algebra 1 courses and beyond typically makes it unnecessary for students to have a strong command of arithmetic facts. It is unlikely to be an effective use of instructional/tutor-contact time to focus on fact fluency for students whose presenting problem concerns the development of algebraic problem-solving skills. I recommend checking with each student if they are able to use a calculator on tests/quizzes and in class, and if they are allowed, then teach them how to use it!

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