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I am designing 10 hour weekly summer math learning opportunities for students taking algebra next school year. I would like to know from algebra teachers what skills they wish their students had mastered prior to taking algebra.

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    $\begingroup$ Only partially related but I have an undergrad student, attempting statistics but earning 20's out of 100 on exams. Obviously an extreme case, but her weaknesses are inequalities, simply copying and/or pronouncing numbers consistently correctly, either into or from the calculator or on her paper, and order of operations to be able to evaluate expressions with a calculator. $\endgroup$
    – nickalh
    Commented May 1 at 10:49
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    $\begingroup$ Especially for non-Americans it would probably be helpful to add what student grade and age you are looking at, and perhaps even the specific subset of what they'll learn next year. "Algebra" is a large field. I suppose that it has a clearly defined meaning within the American curriculum but language barriers and differences between the American and other curricula -- and, actually, between different American curricula because schools differ so much! -- may prevent understanding here. $\endgroup$ Commented May 1 at 13:41

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Automaticity on basic arithmetic.

That is, the ability to carry out basic arithmetic effortlessly without conscious thought.

Automaticity goes beyond capability, proficiency, and even fluency. Simply recalling information or executing a skill isn't enough. It must be instantaneous, flawless, and effortless.

Why is automaticity so important?

Because it frees up mental resources for higher thinking.

Think of a basketball player who is running, dribbling, and strategizing all at the same time. If they had to consciously manage every bounce and every stride, they'd be too overwhelmed to look around and strategize.

The same goes for math, especially algebra. Solving equations feels smooth when basic arithmetic is automatic – it's like moving puzzle pieces around, and you just need to identify how they fit together.

But without automaticity on basic arithmetic, each puzzle piece is a heavy weight. You struggle to move them at all, much less figure out where they're supposed to go.

Case study illustrating the importance of automaticity

Suppose that we have three different students – Otto, Rica, and Finn – whose names are chosen to represent their respective levels of automaticity.

  • Otto has developed full automaticity on multiplication facts and procedures.

  • Rica doesn't know her multiplication facts – she recalculates them from scratch. She is able to carry out multiplication procedures, but she isn't fully comfortable with them and has to proceed slowly, writing every single step down.

  • Finn, likewise, doesn't know his multiplication facts – but he doesn't know his addition facts either, so he uses finger-counting for everything. He is not at all comfortable with multiplication procedures.

These students are each given a lesson on cubes of numbers. After an explanation of what it means to cube a number, and a demonstration with a worked example, they're each given a problem to practice on their own: compute $4^3.$

Let's observe the thought processes (both reasoning and emotions) as each of these students solves the problem.

Otto is so comfortable with his multiplication and addition facts that he solves the problem in 10 seconds in his head. He feels it was easy, is excited to try another, and can't wait for harder problems like cubing negative numbers, decimal numbers, and fractions.

$4^3$ = 4 × 4 × 4. I know 4 × 4 = 16, easy, and then 16 × 4 = ... well that's 10 × 4 = 40 and 6 × 4 = 24, together making 40 + 24 = 64. Done, easy! What's next?

Rica solves the problem in 2 minutes, but her answer is not correct. She takes another 2 minutes to correct the mistake but gets tired and wants to take a break before moving on to the next problem. She's not looking forward to harder problems.

$4^3$ = 4 × 4 × 4. What's 4 × 4? I don't know, let's compute it.

4 × 4 is the same as 4 + 4 + 4 + 4, which is … well, 4 + 4 = 8, plus 4 is 12, plus 4 is 16.

Where was I? Oh right, 4 × 4 = 16 and then 16 × 4 = … ugh, gotta go through that multiplication procedure.

Put the 16 on top, then × 4 on bottom, and now we carry out the procedure. First 4 × 6 = 6 + 6 + 6 + 6, count that up to get 6 + 6 = 12, plus 6 is 18, plus 6 is 22. Write down 2, carry another 2. Then 4 × 1 = 4, add the carried 2, write down 6.

Done. Result is 62. Oh wait, the teacher says that's close but not quite right. Fine, let's try this again.

(Rica repeats the entire procedure above and this time gets a result of 64.)

Great, teacher says that 64 is right. I know there are more problems to do but that one was kind of hard and I'm tired. Teacher, can I take a break and do the next one later?

Finn takes 10 minutes to solve the problem, but his answer is not correct. He tries again for another 10 minutes but makes a different mistake. The teacher has to sit with him for another 10 minutes to carry him through the problem. By the time Finn is done with the problem, it has almost been a full class period. He is totally exhausted and overwhelmed and dreads doing the rest of the homework.

$4^3$ = 4 × 4 × 4. What's 4 × 4? I don't know, let's compute it.

4 × 4 is the same as 4 + 4 + 4 + 4, which is … ugh, gotta count all this up. This is annoying.

Start at 4, then 4 more is 5, 6, 7, 8.

Start at 8, then 4 more is 9, 10, 11, 12.

Start at 12, then 4 more is 12, 13, 14, 15.

Phew, that took a while, but now I have 4 + 4 + 4 + 4 = 15. Why was I doing that, again? Oh right, I was really doing 4 × 4 = 15.

Wait, we're not even done yet. I did 4 × 4 = 15, but that was because I wanted to do 4 × 4 × 4. Okay so now I need to do 15 × 4. Ew, that's going to be even harder. I don't like this. But fine, let's do it.

15 × 4 is the same as 15 + 15 + 15 + 15, and those are big numbers so I need to line it up on paper.

Put 15 at the top, then another 15 below, then another 15, then another 15.

Let's add the right column:

Start at 5, then 5 more is 6, 7, 8, 9, 10.

Start at 10, then 5 more is 10, 11, 12, 13, 14.

Start at 14, then 5 more is 15, 16, 17, 18, 19.

Write down 9, carry the 1, then add down the left column: start at 1, then 1 more is 2, then 3, then 4, then 5. Write down the 5, we have 59.

Answer is 59. Glad that's over. That took forever. Oh wait, the teacher says that's wrong. Noooo... do I have to do this whole thing over again?! This is way too much work.

(Finn repeats the entire procedure above and this time gets a result of 66, which is still incorrect. He is getting very noticeably frustrated and his teacher sits down with him to go through his work. They find and fix several errors together and arrive at the correct result of 64.)

I can't do any more of this today. I'm too tired. I hate math, and my teacher gives me way too much work. And the next problem looks harder, and there are even more on the homework! This is terrible. Class is almost over so I'm just going to zone out until the bell rings.

The big takeaway

This case study demonstrates that the more automaticity a student has on their lower-level skills,

  • the easier they will find it to acquire new higher-level skills,

  • the more quickly and independently they will be able to execute those skills,

  • the better they will feel about the learning process as a whole, and

  • the more excited they will be to continue learning more advanced material.

Students who develop automaticity will feel empowered, while students who do not will feel overwhelmed and defeated.

And by the way, these results compound. If you don't develop automaticity on prerequisite skills, that's going to prevent you from learning and developing automaticity on new skills, and that's going to compound into a massive "learning debt." Lack of automaticity is like a high-interest loan that, over time, compounds into a massive financial debt.

How does one build automaticity?

Practice. Lots of it. Automaticity demands regular practice over time until skills are second nature.

But not just any practice. It's a 3-stage process:

  1. Start with conceptual understanding. (This does not have to go super deep -- the student just has to understand the intuitive / concrete meaning of the symbols they are working with.)

  2. Once a baseline amount of conceptual understanding has been established, move on to practice activities in an untimed setting.

  3. After a student can successfully and consistently execute a skill in an untimed setting, have them practice in a timed setting. (Careful: if you time a student before they can even execute the skills untimed, all you're going to do is stress them out and make them hate math.)

Many heated arguments in math education ultimately amount to people focusing on their personal favorite part of this pipeline and deeming the rest of the pipeline unnecessary.

But what about creativity?

Some people think that because automaticity requires repeated practice, it turns students into mindless robots, and to leverage the power of human creativity, one needs to break free from that robotic mindset. But this is a false dichotomy.

In reality, automaticity is a necessary component of creativity. The whole purpose of automaticity is to reduce the amount of bandwidth that the brain must allocate to robotic tasks, thereby freeing up cognitive resources to engage in higher-level thinking. If a student does not develop automaticity, then they will have to consciously think about every low-level action that they perform, which will exhaust their cognitive capacity and leave no room for high-level creative thinking.

As a concrete example, consider what is typically considered one of the most creative activities: writing. Effective writing requires a frictionless pipeline from ideas in one's mind to words on paper. If a writer had to consciously think about spelling, grammar, word definitions, transitions between sentences, when to make a new paragraph, etc., they would become bogged down in low-level robotic tasks and would have no mental bandwidth to think about high-level creative details like vivid imagery, logical cohesiveness, and emotions evoked by various phrases and ideas.

Further reading: I've written extensively on this. See Chapter 14: Developing Automaticity and Chapter 19. The Testing Effect (Retrieval Practice) in the working draft here for more info and plenty of scientific citations to back it up.


Addendum: defense against misinterpretation

  • This answer in no way suggests to put fluency before conceptual understanding. You should know what multiplication means before you memorize multiplication facts.

    But for the vast majority of students, especially those who don't enjoy thinking about math outside of class (which is virtually everyone), fluency doesn't naturally happen after conceptual understanding, and you can't create fluency by beating conceptual understanding to death.

  • This answer in no way advocates for giving up on students who haven't developed automaticity on basic skills.

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    $\begingroup$ From someone who has always struggled with math, you've finally given a word for the my struggles: automaticity. It's been 35 years, and I still haven't developed this skill, to be frank. And yes, without this skill, I've found solving most algebra problems to be nigh impossible unless I can write out each step. It has always baffled me how people can just skip steps and know the answer. This is how they can. Note to self: read chapter 14... $\endgroup$ Commented Apr 30 at 21:25
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    $\begingroup$ The "Where was I?" in Rica's recounting is, IMO, the core of the issue. Automaticity allows you to keep your attention on the main problem while doing the nitty-gritty calculations. I have heard stories from intro to calculus professors that of all the things they can test at the beginning of the semester, the one thing that best predicts how well a student does in the course is how quickly they do algebra problems, pointing at automaticity. $\endgroup$
    – Arthur
    Commented Apr 30 at 21:34
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    $\begingroup$ I couldn't agree more with the necessity of this for any hope at real comfort in mathematics. I teach chemistry and am continually surprised by the number of students I teach who are in precalc or higher who can't figure out how on Earth I can do something like 1.428 / 7 in my head. And the amount of basic algebra I wind up teaching because the problems involve decimals and variables other than x, y, and z ... well, it's a lot. Keep fighting the good fight, math teachers. $\endgroup$ Commented Apr 30 at 21:50
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    $\begingroup$ I came here to post this but not as thoroughly. Research into cognition and learning that chunking happens when smaller pieces of knowledge are memorized. So even science agrees that facility with the fundamentals enables better learning of more advanced topics. Otto also could have realized $4^3=2^6$ and counted up 2, 4, 8, 16, 32, 64. $\endgroup$ Commented May 1 at 5:14
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    $\begingroup$ @N.Virgo this answer in no way suggests to put fluency before understanding. Just that for the vast majority of people, fluency doesn't naturally happen after understanding, and you can't create fluency by beating understanding to death. $\endgroup$ Commented May 2 at 14:10
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For me, the top items are automaticity with:

  • Times tables
  • Negative numbers
  • Order of operations

I have a timed drill site on these skills at Automatic-Algebra.org.

For my incoming remedial algebra students, I recommend that they practice these skills every day on their own, until they can confidently solve all of them, in the indicated time limits, without any errors. Full disclosure: Likely few of them execute on this. Some few express disbelief that it's humanly possible for anyone to ever accomplish this. For the OP's case with a special summer session, perhaps this is something they could work into their daily preparation.

Operations on fractions would come in at fourth if I had to add another important thing. However, assuming we expect simplified forms, they always involve multiple steps such that we can't ever really call them "automatic" operations in the same way.

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+1 all the above. I would add operations with fractions. No "strategies" for multiplication tables. Just know them cold, forward and backwards.

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  • $\begingroup$ Fractions were always my biggest hurdle with struggling students. They could do basic multiplication etc, but tell them to add or multiply two fractions and they immediately get tied up. Do they need a common denominator? Can I cross multiply? Do I add the denoms together? 1/4 + 1/4 = 2/8? 1/8? 2/4? 4/4? I've seen all of those answers. Let alone something like 1/2 * 3/8 $\endgroup$
    – Robin
    Commented May 1 at 15:42
  • $\begingroup$ Failure to reduce fractions before adding and least (or relatively small) common denominators bother me. But I've never taught Alg. I, so I don't know if it's important. In calc., the denominators can grow exponentially if you don't manage your fractions well. Sometimes a problem becomes unwieldy for a student. Doesn't happen often, though. It's just kind of sad to see $1/8+1/16=24/128$, followed by successive divisions by $2$. $\endgroup$
    – user1815
    Commented May 1 at 17:44
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    $\begingroup$ @user1815 I would say that the understanding of numbers that is required to find common denominators is extremely useful. $\endgroup$
    – Questor
    Commented May 1 at 20:50
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I always tell my students that two of the main things that will mess them up are fractions and signs. Mastery of those is fundamental.

Although since it was mentioned, the idea of automaticity is certainly important and will make any math class go smoother.

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Two things pop into my mind:

  • Elementary set theory: the meaning of sets, subsets, unions and intersections, including Venn-diagrams, must be something they play with.
  • Calculating by heart: nothing as nerve-wrecking as a class where you say "The solution is a quarter of 56" or "... a third of 48" and instead of saying $14$ or $16$, half of the class is just looking numb in front of them and the other half is speeding for their calculator.
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    $\begingroup$ Everything else on this page, I very much agree with. Especially "automaticity". However, set theory? That wouldn't even be in my top 10, maybe not my top 20 or 30. $\endgroup$
    – nickalh
    Commented May 1 at 3:55
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    $\begingroup$ @nickalh I suppose the idea is that if you solve, say, a system of two linear equations what you are actually doing is find the intersection between the two sets of solutions. You do that by treating the equations as statements about the relation between the two variables and ANDing them. Understanding what you are actually doing instead of only going through the motions is probably helpful. $\endgroup$ Commented May 1 at 13:35
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    $\begingroup$ @nickalh: I'm afraid that set theory is that much cemented in your brain that you don't even realise it's there: the notions of natural, integer, rational, irrational and real numbers (often referred to as $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R} \setminus \mathbb{Q}$, $\mathbb{R}$ and their relationships ($\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ and $\mathbb{R} \setminus \mathbb{Q} \subset \mathbb{R}$) are that common knowledge that people don't even see the relation with set theory anymore. $\endgroup$
    – Dominique
    Commented May 2 at 6:16
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Everything that doesn't involve variable; order of operations and operations themselves, types of numbers (odd, even, square, cubic, integers, prime etc), fractions (operations, GCD, LCM), inequalities and number line (not involving variable), roots/radicals, Geometry (perimeter, area, volume etc), unit conversions (meter, km, g, kg, L, mL etc), exponents and the properties of exponents of the same base, basic words problems.

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