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]]>When you’re studying math on your own, a question you must continually ask yourself is how much time to spend on the problem you’re working on. That question also comes up when you’re taking a traditional class, though your options are more limited in that case because of the fixed class schedule. I wrote about this topic last year in the context of competitive programming problems, but there are some special considerations for math problems.
Before you work on a math problem, consider your time budget. Do you have minutes, hours, or days available to spend on the problem? And are you willing to spend all the time you have available without giving up early and looking at the solution? The answer to those questions will affect how you approach the problem.
At one extreme, assume you have unlimited time available. The unlimited time mindset gives you the freedom to practice all your problem-solving and creative thinking skills. You can re-read your notes or relevant textbook sections. You can fill up as many sheets of paper as you want with ideas. You can do Internet research. The only thing you can’t do is look up the solution.
Days
In reality, you won’t need “unlimited” time for textbook problems. (Research problems are another story). After a few days of work time (perhaps spread out over a longer period of calendar time), you’ll either find a solution or you’ll run out of useful work to do. But the unlimited time mindset encourages you to use all the tools at your disposal. And if you don’t find your own solution, you’re likely to learn something from the textbook’s solution when you read it, since you have spent so much time with the problem.
Hours
If unlimited really means days for textbook problems, then the next option is hours. This option works for:
For the first two cases, someone else has imposed the time limit. If a class requires that you complete a set of 10 problems each week, or a 5-problem take-home exam over the course of a few days, then that’s the time you have. You can’t spend a day per problem, and if you have other things going on, you can’t even spend a few hours per problem. So you have to do what you can in the time available.
For the third case, the time limit is self-imposed. If the goal of your project is not just to spend a particular number of hours studying math, but to learn a particular set of math topics, then you have to keep an eye on how much time each problem takes. In this example, you can still spend hours per problem. So there’s time to sit and think and use a multi-step problem-solving process. But if a problem turns out to be too involved, you might decide to skip it or peek at the solution.
For my project, I’m using more of a days approach than an hours approach, since I don’t have a fixed curriculum. But either approach can work depending on your goals.
Minutes
If you only have minutes to spend per problem, then you’re probably taking a timed exam or contest. Or you’re working on a self-study project where your goal is to burn through a large set of easy problems in order to work on your math fluency.
For an exam, the best approach is to prepare enough in advance that each problem on the exam is just a variation on a problem type you have solved many times before. Then on the exam, you only need a few minutes to work out the details. The same might be true of a contest, though contests like IMO have unique problems that are not amenable to cookbook solutions (or so I have heard).
For the “how much time” question I’m considering here, the minutes option isn’t too interesting, since the decision is largely made for you. You just have to budget your time and do your best to demonstrate what you know.
For the competitive programming version of this article, I looked through many recommendations from people on Quora. Specific time recommendations for how long to work ranged from 1—6 hours to 1—3 days, with one person suggesting a week and one suggestion a month. But such time recommendations from other people aren’t too useful. As I did in that article, I recommend instead taking a two-part approach:
One way that math problems differ from programming problems is the amount of feedback you get as you’re working on the problem. When you’re solving a programming problem on an online judge, you get feedback from your editor (syntax highlighting and autocompletion) as you write your solution, feedback from your computer (compiler errors and program output), feedback from the online judge (a verdict when you submit your solution), and feedback from your peers (when you look at implementations and editorials). This makes it easier to hack away at the problem for hours, using feedback to motivate you to keep working and find a solution on your own.
With many textbook math problems, it’s harder to get feedback as you work. There’s no feedback as you brainstorm a solution with pencil and paper. A computer usually won’t compile and run your solution or give you a verdict. (Online platforms like Khan Academy can do this for some types of problems, but they aren’t great for practicing proofs). So your first and only feedback opportunity is when you read the solution.
Because feedback is limited, it’s especially important with math problems to decide in advance what mindset you’ll use as you work on the problem. If you can, approach problems as if you have unlimited time. This allows you to brainstorm for longer, write longer proofs, and generally extract more learning benefits from your problem-solving sessions.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
(Image credit: prelude2000)
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]]>The post Why are Textbook Proofs so Short? appeared first on Red-Green-Code.
]]>A mathematical proof can be the size of a novella. For example, Andrew Wiles’s published his famous proof of Fermat’s Last Theorem in two journal articles covering 129 pages. But proofs in introductory textbooks like Rosen often contain just a few sentences. It might seem obvious that these proofs are short because they’re easy. But that’s not exactly right. Last week, I wrote a moderately long post covering an “easy” proof about how the div operator behaves when the first argument is negative. Why was it so much longer than the version in the solution guide?
The goal of a proof is to convince your reader that a mathematical statement is true. To do that effectively, you have to make some assumptions about what prior knowledge the reader brings to the proof. Which assumptions you make can significantly change the length of the proof, since they determine how you describe each mathematical fact you use. You could use a fact in several ways:
Proofs in introductory textbooks generally use #4 for facts covered in other parts of the textbook. For example, a proof step might say, “By Theorem 3, we can…” or “We can use Theorem 2 from Chapter 5 to….” When the textbook author needs a fact from another mathematical discipline, they might use #1 or #2 for basic topics, and #3 for more specialized topics that the reader might remember only vaguely.
Here are some examples of how I used each of these options in the div proof:
As anyone who has studied math or science textbooks knows, they require different reading techniques than other books. In How to Read a Math Textbook, I quoted a writer who compared the process to digging something out of the ground. If you want to encourage a DIY mindset in your readers, it may not be worth explaining things too carefully. Just make sure your proof is correct and precise, and let the reader dig out the rest.
I can understand this point of view. If you read through the div proof at regular blog post speed, you might not follow it, even though I spell out the math in extreme detail. There are just too many symbols. So if readers have to read through the proof line by line anyway, what’s the benefit of adding more detail? Again, it comes down to the audience the proof is written for. If you’re writing it for an instructor, then more detail demonstrates that you understand the proof. If you’re writing it for yourself, then writing more detail can improve your understanding of the proof, and ensure that you know how each step works. But if you’re writing a textbook or an answer key, it may be pedagogically best to let the reader fill in some gaps.
The most obvious reason a proof is short is time and space constraints. Textbook authors don’t have unlimited pages. Students don’t have unlimited time to finish assignments. Readers of Wikipedia don’t have unlimited time to read math articles, even though there’s no article size limit and editors may want to keep writing. So the proof eventually must end.
If you’re studying math on your own, most of these constraints disappear, and it’s mainly a question of how long to work on a problem before looking at the answer. When considering that question, the principles that apply to programming problem practice also apply to math proof practice: how many problems you want to cover, the benefits of struggling with a problem, how hard the problem is, whether you have the right background, and whether you’re still getting something out of the process. But although you have to stop eventually, for the purpose of learning proofs, I think longer is usually better.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
(Image credit: Insomnia Cured Here)
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]]>Last week, I suggested a process for getting better at writing proofs. To illustrate that process, here’s an example of how to use it to prove a theorem from Rosen.
Prove: If $a$ and $d$ are positive integers, then $(-a) \textbf{ div } d = -a \textbf{ div } d$ if and only if $d \mid a.$
This proof is an exercise from a section in Rosen called “Divisibility and Modular Arithmetic.” It’s an odd-numbered exercise, so I’ll be able to compare my proof with the one from the solution guide.
For this step, we’ll think about the proof statement and collect any information that might be useful.
When I first read the proof statement, I found the equation $(-a) \textbf{ div } d = -a \textbf{ div } d$ to be ambiguous. It probably has a precise meaning when taking into account order of operations rules, but I prefer to be more explicit, writing it as $(-a) \textbf{ div } d = -(a \textbf{ div } d)$. I’ll assume that this is equivalent to the original equation.
The $\textbf{div}$ operator relates to a theorem known as the Division Algorithm (Rosen 4.1.3 Theorem 2). This theorem states that given integers $a$ and $d$, with $d>0$, there are unique integers $q$ and $r$, with $0 \leq r<d$, such that $a = dq + r$. In this equality, $q$ is called the quotient, and it is defined as $q = a \textbf{ div } d$
$d \mid a$ means $d$ divides $a$ (with no remainder). It follows that $a/d$ is an integer.
With the notation defined, the next step is to try some examples.
Informally, we can think of $a \textbf{ div } d$ as meaning “divide $a$ by $d$ and discard the remainder.” If $a=4$ and $d=2$, standard division says $a/d=4/2=2r0$ ($2$ with a remainder of $0$), so $a \textbf{ div } d=2$, the same result provided by the $/$ operator. But if $a=4$ and $d=3$ then $a/d=4/3=1r1$, so $a \textbf{ div } d=1$. The remainder is ignored.
By definition in this context, the remainder must be nonnegative. We have to be careful about this rule when dividing negative numbers. For example, if $a=-4$ and $d=2$, $a/d = -4/2 = -2r0$. For zero remainders, this works as expected. But if $a=-4$ and $d=3$ then $a/d=-4/3=-2r2$, not $-1$ with a remainder of $-1$. We have to adjust the quotient to get a nonnegative remainder. I find it useful to write this as $-4/3=(-6/3)+(2/3)$. The $-6/3$ term is the quotient, which evaluates to $-2$, and the $2/3$ term is the remainder, which is usually written as $2$, without the denominator.
Next, let’s check that the theorem works for a couple of examples:
Example 1:
Let $a=4$ and $d=2$. Since $2 \mid 4$, the theorem says:
$$(-a) \textbf{ div } d = -(a \textbf{ div } d)$$ $$(-4) \textbf{ div } 2 = -(4 \textbf{ div } 2)$$ $$-2 = -(2)$$ $$-2 = -2$$
So the first example works.
Example 2:
Let $a=4$ and $d=3$. Since $3 \not| ~4$, the theorem says:
$$(-a) \textbf{ div } d \neq -(a \textbf{ div } d)$$ $$(-4) \textbf{ div } 3 \neq -(4 \textbf{ div } 3)$$ $$-2 \neq -(1)$$ $$-2 \neq -1$$
So the second example works as well.
These examples suggest why this theorem is true. It has to do with how the $\textbf{div}$ operator works with negative numbers. When $d \mid a$, there is no remainder, so $(-a) \textbf{ div } d$ is the same as $-a/d$, which is an integer according to elementary division rules. So we can just evaluate $a/d$ and then negate it.
But when $d \not| ~a$, evaluating $(-a) \textbf{ div } d$ takes a bit more work. It’s not the same as $a/d$ because of the rule that a remainder must be positive.
Let’s make these $\textbf{div}$ rules more precise. It can be shown that $a \textbf{ div } d = \lfloor{a/d}\rfloor$ (see Rosen 4.1.2, Definition 2, Remark). We can write this equivalence a few different ways, which will be useful in the proof:
$$a \textbf{ div } d = \lfloor{a/d}\rfloor \tag{1}$$ $$(-a) \textbf{ div } d = \lfloor{-a/d}\rfloor \tag{2}$$ $$-(a \textbf{ div } d) = -\lfloor{a/d}\rfloor \tag{3}$$
Let’s consider some ideas for the proof. Since we’re proving an “if and only if” statement, a standard approach is to prove each direction separately. So we will have a Part 1 proof that if $(-a) \textbf{ div } d = -(a \textbf{ div } d)$ then $d \mid a$, and a Part 2 proof that if $d \mid a$ then $(-a) \textbf{ div } d = -(a \textbf{ div } d)$.
We’ll use these tools:
The equalities $(1)$, $(2)$, and $(3)$, which relate the $\textbf{div}$ operator and the floor function.
Properties of the floor function.
The Division Algorithm, which guarantees a unique quotient and remainder when two integers are divided.
For Part 1, we start with a $\textbf{div}$ equation. Using $(1)$, $(2)$, and $(3)$, we can convert it to an equation in which the floor function operates on fractions. Then we can use the Division Algorithm to express each fraction as a quotient and a remainder. Divisibility is related to remainders, so we can use the information about remainders to show that $d$ divides $a$.
For Part 2, we start with the assumption that $d$ divides $a$, which means $a/d$ is an integer. This means the floor functions have no effect on $a/d$, so when we use $(1)$, $(2)$, and $(3)$ to introduce floor functions, they just disappear. This makes it straightforward to get to the desired $\textbf{div}$ equation.
We’ll see that the way $\textbf{div}$ works with a negative argument is related to how the floor function handles a negative argument.
We now have all the information we need to prove the “if and only if” statement in two parts:
Part 1: If $(-a) \textbf{ div } d = -(a \textbf{ div } d)$ then $d \mid a$.
Let $a$ and $d$ be integers such that $(-a) \textbf{ div } d = -(a \textbf{ div } d)$. Using $(2)$ and $(3)$, we can convert this equality to floor notation and write $\lfloor{-a/d}\rfloor = -\lfloor{a/d}\rfloor$.
By the Division Algorithm, we know that given our positive $a$ and $d$, we can find unique integers $q$ and $r$ such that $a=dq+r$. Dividing by $d$, we get $a/d=q+r/d$. Multiplying that equation by $-1$ gives us $-a/d=-q-r/d$. We now have $a/d$ and $-a/d$ expressed as an integer quotient and a fractional remainder between 0 (inclusive) and 1 (exclusive).
Substituting these results into the floor functions, we get $\lfloor{-a/d}\rfloor =\lfloor{-q-r/d}\rfloor$ and $-\lfloor{a/d}\rfloor=-\lfloor{q+r/d}\rfloor$.
For a positive argument, the floor function simply discards any fractional component. So $-\lfloor{q+r/d}\rfloor=-q$.
For the negative argument, $-q-r/d$, the expression $\lfloor{-q-r/d}\rfloor$ evaluates to either $-q$ (if $r=0$) or $-q-1$ (if $r>0$).
(To see why this is, think of the floor function as moving a real number to the left on the number line to the next lowest integer. So for a positive real number like $3.14$, the result is just $3$, with the $0.14$ stripped off. But for a negative real number like $-3.14$, the result is $-4$, the next lower integer.)
But our initial assumption is $(-a) \textbf{ div } d = -(a \textbf{ div } d)$, which means that $-\lfloor{q+r/d}\rfloor$ and $\lfloor{-q-r/d}\rfloor$ are equal. Since $-q \ne -q-1$, it must be true that $r=0$, which is the definition of $d \mid a$, as required.
Part 2: If $d \mid a$ then $(-a) \textbf{ div } d = -(a \textbf{ div } d)$.
Let $d$ and $a$ be integers such that $d \mid a$. By definition, that means $a/d=q$ for some integer $q$. Multiplying by $-1$ gives us $-a/d=-q$. Taking the floor of both sides: $\lfloor{-a/d}\rfloor=\lfloor{-q}\rfloor$. But this is just $-q$, since taking the floor of an integer doesn’t change it. By $(2)$, $(-a) \textbf{ div } d = \lfloor{-a/d}\rfloor$, so we have $(-a) \textbf{ div } d = -q$.
Consider the equation $a/d=q$ again. Taking the floor of both sides gives us $\lfloor a/d \rfloor=\lfloor q \rfloor=q$. Multiplying by $-1$: $-\lfloor a/d \rfloor=-q$. By $(3)$, $-(a \textbf{ div } d) = -\lfloor{a/d}\rfloor$, so we have $-(a \textbf{ div } d) = -q$ as well. Therefore, $-q=(-a) \textbf{ div } d = -a \textbf{ div } d$, as required.
The combination of Part 1 and Part 2 proves the theorem.
The Rosen solution guide takes a somewhat different approach to this proof. Instead of proving the “if” and the “only if” parts of the theorem, they prove Part 2 and its inverse. By the rules of logic, this is equivalent to proving Part 1 and Part 2. Here is that proof, which I have expanded with additional detail.
Solution Guide Part A: If $d \mid a$ then $(-a) \textbf{ div } d = -(a \textbf{ div } d)$.
Let $d$ and $a$ be integers such that $d \mid a$. By definition, that means that $a/d=m$ for some integer $m$, and therefore $a=md$. Multiplying by $-1$, we get $-a=(-m)d$, which means
$$-a/d=-m \tag{4}$$
Since $d \mid a$, we know that $a/d$ is an integer, so $a/d=\lfloor a/d \rfloor$, which is equivalent to $a \textbf{ div } d$ according to $(1)$. Multiplying by $-1$ gives us $-a/d=-(a \textbf{ div } d)=-m$ according to $(4)$.
Since $d \mid a$, we also know that $-a/d$ is an integer, so $-a/d=\lfloor -a/d \rfloor$, which is equivalent to $(-a) \textbf{ div } d$ according to $(2)$. So we have $-a/d=\lfloor -a/d \rfloor=(-a) \textbf{ div } d=-m$, according to $(3)$.
Since both expressions are equal to $-m$, we have shown that $-m = (-a) \textbf{ div } d = -(a \textbf{ div } d)$, as required.
Solution Guide Part B: If $d \not\mid a$ then $(-a) \textbf{ div } d \neq -(a \textbf{ div } d)$.
Let $d$ and $a$ be integers such that $d \not\mid a$. This means we will have a nonzero remainder when dividing $a$ by $d$. Expressed in Division Algorithm notation: There exist integers $q$ and $r$ such that $a=qd+r$, $0<r<d$, and $q=a \textbf{ div } d$.
Multiplying the Division Algorithm equality by $-1$ gives us $-a = (-q)d-r$. Let’s manipulate this as follows:
$$-a = (-q)d-r$$ $$-a=-qd-d+d-r$$ $$-a=(-q-1)d+(d-r) \tag{5}$$
Since $0<r<d$, we know that $0<d-r<d$. This means we can interpret $(5)$ as a Division Algorithm equation with dividend $-a$, divisor $d$, quotient $(-q-1)$ and remainder $(d-r)$. By the definition of $\textbf{div}$, this means $(-a) \textbf{ div }d=(-q-1)$, the quotient.
But the definition of $\textbf{div}$ also tells us that $q=a \textbf{ div } d$, which means $-q=-(a \textbf{ div } d)$. Since $(-q-1) \neq -q$, we have shown that assuming $d \not\mid a$ leads to the conclusion that $(-a) \textbf{ div } d \neq -(a \textbf{ div } d)$, as required.
Since we have proven statements of the form “if $A$ then $B$” and “if not $A$ then not $B$” (the inverse), this suffices to prove “$A$ iff $B$”.
In last week’s post, I mentioned Cal Newport’s approach of reproducing proofs for his Discrete Math class using a Quiz and Recall system. A prerequisite for using this system is having something to recall. The proofs shown above are probably too detailed to learn and recall. But writing proofs out in extreme detail forces you to understand every step. This is good preparation before you settle on a more compact version to use for study.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
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]]>The post How to Practice Writing Proofs appeared first on Red-Green-Code.
]]>Proof-writing skills are important for all college-level math. But there’s a special relationship between proofs and discrete math. In the “Goals of a Discrete Mathematics Course” section in the preface to his textbook, Rosen puts Mathematical Reasoning first in the list. He writes:
Students must understand mathematical reasoning in order to read, comprehend, and construct mathematical arguments [proofs]. This text starts with a discussion of mathematical logic, which serves as the foundation for the subsequent discussions of methods of proof. Both the science and the art of constructing proofs are addressed.
Rosen Chapter 1 is “The Foundations: Logic and Proofs,” and that chapter ends with sections on “Introduction to Proofs” and “Proof Methods and Strategy.” A textbook that specifically covers proof techniques, Daniel J. Velleman’s How to Prove It, begins with chapters on these same topics, and includes chapters on logic and on mathematical induction which Rosen also covers. So it’s not surprising that many of the exercises in Rosen ask for proofs. Here is a process I use to get the most out of these exercises.
This article isn’t exactly about how to prove theorems. (See the Velleman book for that). Instead, it’s about how to practice proving theorems. I ran into this distinction several years ago when I wrote an article called How to Attack a Programming Puzzle and got feedback that essentially said: how can I practice a problem I don’t know how to solve? It’s certainly worth studying problem-solving techniques. However, a good practice process is as useful as any problem-solving heuristic. By practicing many types of problems at the right level of difficulty, you naturally pick up problem-solving skills.
Here’s a six-step process for improving your proof-writing skills.
Step 1: Find a proof to practice
You can find the best practice proofs in the main text of a textbook that’s written at your level. If you use a good textbook, these proofs will have good explanations. You might also find explanations of the same proofs online, but as with any online source you have to be careful about quality. A textbook from a major publisher will have gone through multiple reviews from experts, and you can find reviews from students online to see what they think of it.
Another source of practice proofs is odd-numbered textbook exercises, which often come with answers. However, answers to exercises are usually less detailed than discussions in the main text. A good solution manual can provide more detail.
Finally, you could use exercises that you don’t have answers for, but that can be problematic if you don’t have someone to give you feedback on your work.
If you’re taking a class, the professor does Step 1 for you, and you also have a built-in source of feedback.
Step 2: Brainstorm
For any moderately difficult proof, it’s unlikely that you’ll be able to start with the premise and proceed step by step until you reach the conclusion. So you need some way to come up with ideas on how to approach the proof.
Some options:
Step 3: Write a draft
Once your brainstorming seems to be leading somewhere useful, try writing a first draft of the proof. In this version, you should have a good idea of why each step leads to the next. When you read through the complete draft, it should seem like a reasonable argument. If you get stuck, return to Step 2 and repeat as necessary.
Step 4: Fill in the details
You can write a proof at varying levels of detail depending on the intended audience. If you’re writing for a mathematical journal, you might skip proof steps that you assume the expert audience can fill in. For a college class, I like to err on the side of providing too much detail in my answers. Once in a graduate Theory of Computation class, I was presenting a proof about Turing machines and the professor asked me to skip a few slides because I was going over details that the class already knew. I don’t know if they really did, and I still think it’s better to explain things too much than two little. But you have to keep your audience in mind.
If you’re preparing a problem set to turn in for a class, this step is an opportunity to read through your draft proof and provide more explanation for steps that might not seem as obvious as when you first wrote them. You might also convert your written proof into $\LaTeX$ if you have time. Seeing it nicely typeset might encourage you to perfect your argument. And just the act of writing it again can make errors more obvious.
If you’re practicing on your own, there’s less incentive to polish your proof. But this is still a good time to verify that you’re convinced by your own argument. Once you move on to the next step and look at someone else’s solution, especially one written by an expert, there’s a temptation to accept that as the proof. But mathematicians can prove theorems in multiple ways, so yours can be different and still be correct. The time when you’re still working on it is a unique chance to use your unique experience to invent your own proof, even if you later discover that someone else found it hundreds or thousands of years earlier.
Step 5: Read someone else’s version
The best way to learn from your practice is to have an expert evaluate your work and suggest areas for improvement. But if that’s not an option, it’s still worthwhile to see how an expert (like a textbook author) proved the theorem that you just finished proving.
Especially if the proof you’re reading is a concise answer at the back of a textbook, it’s important to read it the same way you read critical sections of the textbook: go line by line, fill in any gaps, and generally write your own version of it. When you’re done, you should be able to follow the author’s proof as clearly as you follow the final version of your own proof. If the author took the same approach as you, this might be as simple as comparing your proof with their proof and ensuring that yours has at least as much detail as theirs does. If their proof is very different, then you may have to do some research to understand their argument, at which point you can decide which version you like best.
Step 6: Practice it later
As with any other math skill, if you want to really learn a proof, you have to practice it more than once. In his classic blog post on studying discrete math as a student, Cal Newport describes creating a canonical version of each proof covered in his class, and quizzing himself on these “proof guides” to make sure he could prove all of them on demand.
Completing the prior steps gives you one or more detailed proofs that you can use as source material for learning a mathematical topic in detail. Completing this last step gives you more return on your learning investment by solidifying the proof technique in your mind, to be deployed in the future when you need it.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
(Image credit: Euclid)
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]]>If you search for advice on how to read a math textbook, you’ll find plenty to choose from. I’ll link to some of my favorites at the end of this article. But first, here’s an overview of my experience reading math books, and what techniques I found to be useful.
An article in The Harvard Crimson warns that to survive Harvard’s infamous Math 55 class, you need to be willing to write your own textbook. That sounds dramatic, but I think it’s the right approach for learning math at any level. That’s because no math textbook is exactly right for you. You can avoid reading books that are much too easy, or that require background you don’t have. But the ones that remain still make assumptions about what you know, and those assumptions aren’t always right.
The solution: write your own textbook. As you read, take notes that spell out the important ideas from the text in exactly the way that makes sense to you. For some ideas, this will mean writing multiple pages in excruciating detail, perhaps drawing on other sources besides the main textbook. For other ideas that you already know well, you might just write a formula that you can use for reference.
If you follow this approach rigorously, you end up with a customized document that provides exactly what you need for future study or application. For every explanation in the original textbook, your customized textbook fills in the details to make it obvious (to you) why each step follows from the previous one. If a section of the original textbook covers a topic you have no use for, your customized textbook can omit it. When you look something up in your customized textbook, you get an answer that’s at exactly the right level of detail. Or if it isn’t, you can keep updating your textbook until it is.
To write a section of your custom textbook, you first have to understand the topic you’re writing about. That’s where the reading process comes in. There are more and less elaborate processes for reading a math textbook. I’ll focus on a simple one involving two reading speeds: moderate speed, and line-by-line speed. When you start a new section in the textbook, begin at moderate speed. The author probably starts with a few paragraphs introducing and motivating the section (though more advanced texts may skip this part). When you reach a definition, theorem, example, or any nontrivial block of mathematics, switch to line-by-line speed.
When reading line by line, it can help to cover up subsequent lines (in a physical book) or adjust the window size (in the electronic version). This helps you focus on the current line. More importantly, it allows you to think about a claim the author is making before you read more details about it. For a theorem, this means you can try to prove it yourself before reading the author’s proof. For an example, you can work it out on your own before looking at how the author solves it.
Reading a textbook section in this way is like getting access to bonus problems that come with detailed solutions. Then when you get to the exercises at the end of a section, you’ll be warmed up to tackle them. And your custom textbook will give you the information you need to get started on the problems without having to refer back to the original textbook.
The previous sections provide a general approach to reading math, with some specific advice. But there’s a lot more advice out there. Here are some resources I found useful.
The canonical question for this topic on Mathematics Stack Exchange is How to read a book in mathematics? The answers present a few perspectives, and some links to other resources, including How to Read Mathematics by Shai Simonson and Fernando Gouvea. This essay is worth reading in its entirety, but here is one key concept: To fully understand a mathematical idea, you need to “make the idea your own” by “following the idea back to its origin, and rediscovering it for yourself” and “mak[ing] the idea fit in with your own perspective and experiences.” The authors quote the book Emblems of Mind by Edward Rothstein, who makes the same point this way: “Reading mathematics … involves a return to the thinking that went into the writing.” Writing your own textbook formalizes this process by providing the structure for you to explain the topic in the ideal way for your background.
A couple more articles, and key points:
How to Read a Math Textbook by Macalester College
How to Read a Math Book by Stan Brown
Brown quotes from a 2003 post by Chan-Ho Suh on the Usenet group sci.math. Here’s how the post starts:
Let me make a distinction between two kinds of understanding a math text. The first is in being able to follow, reconstruct, etc. a proof (proofs make up the bulk of most math texts). The second lies in rewiring one’s brain so that it’s obvious why the theorem is true, and indeed, there is no way the theorem could be false in the context of all your knowledge.
Ideally the first type of understanding leads to the second type. As you learn how scholars have proven theorems in an area, you gain a better understanding of that area, until eventually you can rely on your intuition to lead you in the right direction.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
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]]>In their first 13 or so years of school, students cover a standardized math curriculum. Last week, I covered how Khan Academy approaches that curriculum. Notably absent from that list are many topics in discrete mathematics. But what is discrete mathematics, anyway? I’ll answer that in two ways: with a definition, and with a curriculum.
Unpacking the term itself, discrete mathematics refers to the mathematical study of discrete (distinct) objects, as opposed to connected ones. For example, integers are discrete objects because there are no integers between the integer $n$ and the next integer, $n+1$. In contrast, given any two real numbers, you can always find another real number between them. So the real numbers form a continuous line, while the integers consist of discrete points.
The textbook I’m using this year, Rosen’s Discrete Mathematics and its Applications, defines discrete mathematics in a “To the Student” section. Rosen writes that discrete mathematics deals with countable sets (which the integers are, but the reals are not). He also points out that computers see data discretely, which makes discrete math especially applicable to computer science.
The lead paragraph of the Wikipedia article on discrete mathematics claims:
[T]here is no exact definition of the term “discrete mathematics.” Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
I’ll use the opposite approach and discuss what is included. Specifically, which topics Rosen includes in his introductory textbook.
The latest edition (8th Edition, Copyright 2019) of Rosen has the following thirteen chapters.
Chapter 1: The Foundations: Logic and Proofs
Proofs become more important the farther you go in math. Rosen covers the theory of mathematical proof from the ground up, by describing propositional logic, a system for making and evaluating formal arguments.
Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrics
K-12 math classes cover sets, functions, sequences, sums, and matrices in varying levels of detail. So this chapter is mainly a review of previous math courses. But it also formalizes notation that will appear in subsequent chapters, and covers a few advanced topics like the cardinality of infinite sets.
Chapter 3: Algorithms
This is a short chapter that introduces a few of the topics explained in much more depth in an algorithms textbook like CLRS or Sedgewick: the definition of an algorithm, examples of well-known algorithms, and how to evaluate the time and space complexity of algorithms.
Chapter 4: Number Theory and Cryptography
Rosen wrote another textbook called Elementary Number Theory and its Applications. This chapter introduces the key aspects of number theory he covers in that book, and concludes with an introduction to cryptography.
Chapter 5: Induction and Recursion
According to the introduction to this chapter, “Understanding how to read and construct proofs by mathematical induction is a key goal of learning discrete mathematics.” The chapter begins the process of accomplishing that goal, and also demonstrates how to use induction to prove that algorithms correctly solve the problems they are designed to solve.
Chapter 6: Counting
Counting problems are popular as puzzles and interview questions. For example, you can generate anagrams using counting techniques. Fermi problems, which used to be popular in interviews, require a type of counting. This chapter covers the basics of counting and enumeration, permutations and combinations, and binomial coefficients. These topics appear in precalculus and other K-12 classes, so this is another chapter that works as a review and formalization of previous math experience.
Chapter 7: Discrete Probability
Discrete probability concerns phenomena modeled by discrete random variables, those that take one of a countable number of values. For example, you could model dice rolls using discrete probability theory, since each roll of a die produces an integer result (e.g., an integer from 1 to 6). Actual dice have a finite number of faces, but discrete probability can also handle results from countably infinite sets. In contrast, continuous probability theory deals with random variables that can take values in a continuous range (e.g., any real number).
Topics in discrete probability often come up in primary school. This chapter expands on those topics, and includes sections on Bayes’ Theorem and the expected value and variance of a random variable.
Chapter 8: Advanced Counting Techniques
This chapter expands on the counting techniques from Chapter 6. It has several sections dealing with recurrence relations, and also covers generating functions and the inclusion-exclusion principle for counting the number of elements in the union of $n$ sets.
Chapter 9: Relations
A relation is simple to describe. It’s an association between a member of set $A$ and a member of set $B$, or between members of $n$ sets. For the first type, a binary relation, this chapter suggest the example of a person and their phone number. For the second type, a $n$-ary relation, it uses the scenario of airline flights (airline, flight number, origin, destination, departure time, and arrival time).
From this simple starting point, the chapter considers properties and applications of binary and $n$-ary relations; how to represent relations using matrices and digraphs; equivalence relations; and partial orderings.
Chapter 10: Graphs
Programming contests and algorithms classes commonly use graphs. This chapter considers graphs from a mathematical perspective, without getting into implementation details. So although there’s a section on representing graphs using lists or matrices, translating those representations into code is left to the reader. And although the chapter includes famous graph algorithms (like Dijkstra’s and Floyd’s), they appear in pseudocode, not a programming language.
Chapter 11: Trees
Like the previous chapter, this one takes a mathematical approach and includes several pseudocode algorithms. For this chapter, the topic is the special type of graph called a tree, a connected graph with no simple circuits.
Chapter 12: Boolean Algebra
This chapter covers concepts relevant to designing digital circuits.
Chapter 13: Modeling Computation
The book ends with this chapter on grammars, finite-state machines, and Turing machines, basic concepts in theory of computation.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
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]]>Khan Academy’s math program is designed to help children and young adults learn and practice a particular set of math skills. These skills map to school curricula like the US Common Core, so that when students learn something on the site, it translates to success in the classroom. But Khan Academy can also help adults review the fundamental skills necessary to learn more advanced math. The idea is to fill in “swiss cheese gaps” in knowledge that often accumulate when learning math in school. These gaps can slow down further learning, since math success in later courses depends on knowing the skills taught in previous courses.
This week, I’ll go over the topics available on Khan Academy. Next week, I’ll look at how they relate to discrete math topics.
Khan Academy is in the middle of a mastery system upgrade. The previous World of Math system coexists with a new mastery system based on courses. Fortunately, the two systems share mastery status for each skill. So if you use one system for practice, you get credit in the other system and don’t have to review topics you already know.
I find the World of Math system simpler to navigate: you can just keep clicking the “Start mastery challenge” button, and problems keep appearing. You don’t have to worry about which subjects the problems come from. But this flat structure makes it harder to see how problems fit into the overall math curriculum. Since I’m writing about curriculum this week, I’ll use the new system’s course-based structure.
Khan Academy currently offers 57 math courses, but many topics are covered by multiple courses. For example, the Differential Calculus, AP®︎ Calculus AB, and Calculus 1 courses each work as a first course in Calculus. The World of Math mission draws from many of the 57 courses, but it picks a subset of the exercises to avoid duplication.
Here’s how I would organize these 57 courses into 15 topic areas by consolidating courses that cover similar topics:
Early math (through 2nd grade)
This is what kids learn in their preschool years and the first three years of primary school. Topics include counting, addition and subtraction, place value, measurement and data, and basic geometry.
4 courses: Early Math, Kindergarten, 1st grade, and 2nd grade.
Primary school (3rd-5th grades)
These courses review and expand on the early math topics, and add multiplication and division, fractions and decimals, rounding, calculating area, word problems, negative numbers, factors, unit conversion, and algebraic thinking.
11 courses: 3rd grade, 3rd grade (Eureka Math/EngageNY), 3rd grade foundations (Eureka Math/EngageNY), Arithmetic, Arithmetic (all content), 4th grade, 4th grade (Eureka Math/EngageNY), 4th grade foundations (Eureka Math/EngageNY), 5th grade, 5th grade (Eureka Math/EngageNY), and 5th grade foundations (Eureka Math/EngageNY).
Later primary school (6th-7th grades)
These courses add more advanced topics to prepare students for later studies: ratios and proportional relationships, rates, percentages, properties of numbers, variables, expressions, equations, inequalities, probability and statistics, ratios, rational numbers, and calculating volume.
8 courses: 6th grade, 6th grade (Illustrative Mathematics), 6th grade (Eureka Math/EngageNY), 6th grade foundations (Eureka Math/EngageNY), 7th grade (Eureka Math/EngageNY), 7th grade, 7th grade (Illustrative Mathematics), and 7th grade foundations (Eureka Math/EngageNY).
Pre-algebra and geometry (8th grade)
The 8th grade courses combine pre-algebra and basic geometry topics, to prepare students for full courses on those subjects in high school:
6 courses: 8th grade, 8th grade (Illustrative Mathematics), 8th grade (Eureka Math/EngageNY), 8th grade foundations (Eureka Math/EngageNY), Pre-algebra, and Basic geometry.
Algebra
A first course in high-school algebra, covering algebraic expressions, solving linear equations and inequalities, graphing lines, slope, systems of equations, expressions with exponents, quadratics and polynomials, functions, linear word problems, sequences, systems of equations, absolute value, piecewise functions, rational exponents, radicals, exponential growth/decay, polynomials, factoring, exponential and logarithmic functions, quadratic functions, complex numbers, conic sections, vectors, and matrices.
4 courses: Algebra basics, Algebra I, Algebra (all content), and Algebra I (Eureka Math/EngageNY).
Geometry
A first course in high-school geometry, covering lines, angles, shapes, triangles, quadrilaterals, circles, the coordinate plane, area and perimeter, volume and surface area, the Pythagorean theorem, transformations, congruence, similarity, right triangles, trigonometry, analytic geometry, solid geometry, constructions, and proofs.
3 courses: High school geometry, Geometry (Eureka Math/EngageNY), and Geometry (all content).
Algebra II
A second course in high-school algebra, covering functions, complex numbers, polynomial arithmetic, radical relationships, rational relationships, exponential growth and decay, exponentials and logarithms, trigonometry, advanced equations and functions, series, conic sections.
2 courses: Algebra II and Algebra II (Eureka Math/EngageNY).
Trigonometry
Trigonometry for right triangles and general triangles; the unit circle definition of sin, cos, and tan; graphs of trigonometric functions; trigonometric equations and identities.
1 course: Trigonometry.
Precalculus
Everything you need to know before you study calculus, assuming you mastered the previous courses.
Topics: trigonometry, conic sections, vectors, matrices, complex numbers, probability, statistics, combinatorics, series, rational and exponential functions.
2 courses: Precalculus and Precalculus (Eureka Math/EngageNY).
Also, these three courses cover the high school curriculum through precalculus: Mathematics I, Mathematics II, and Mathematics III.
Calculus
The traditional pinnacle of high school math, which the previous courses lead up to. Includes courses to prepare for the two AP®︎ Calculus tests.
Topics: Limits, continuity, derivatives, techniques for differentiation, applications of derivatives, analyzing functions, parametric equations, polar coordinates, vector-valued functions, integrals, differential equations, applications of integrals, infinite sequences and series.
6 courses: Differential Calculus, Integral Calculus, Calculus 1, Calculus 2, AP®︎ Calculus AB, and AP®︎ Calculus BC.
Multivariable calculus
Post-high school calculus.
Derivatives of multivariable functions, applications of multivariable derivatives, integrating multivariable functions, Green’s, Stokes’, and the divergence theorems.
1 course: Multivariable calculus.
Differential equations
First order differential equations, second order linear equations, Laplace transform.
1 course: Differential equations.
Linear algebra
Vectors and spaces, matrix transformations, alternate coordinate systems (bases).
1 course: Linear algebra.
Probability and statistics
Khan Academy statistics coverage is thorough, and includes a course to prepare for the AP®︎ Statistics exam.
Topics: Advanced regression (inference and transforming); analysis of variance (ANOVA); analyzing categorical data; chi-square tests for categorical data; confidence intervals; counting, permutations, and combinations; displaying and comparing quantitative data; displaying and describing quantitative data; exploring bivariate numerical data; inference comparing two groups or populations; modeling data distributions; probability; random variables; sampling distributions; significance tests (hypothesis testing); study design; summarizing quantitative data; two-sample inference for the difference between groups.
3 courses: Statistics and probability, High school statistics, and AP®︎ Statistics.
Competitive and recreational math
This section includes recreational math topics, plus courses that cover the two tests used to determine qualification for the United States of America Mathematical Olympiad (USAMO): 1) the American Mathematics Competition (AMC) and 2) the American Invitational Mathematics Examination (AIME) (AIME).
1 course: Math for fun and glory.
Khan Academy’s math curriculum is a work in progress. New skills periodically show up in the World of Math and in the course system. But there’s already plenty available to ensure that your fundamental math skills are well-polished.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
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]]>The process of learning math involves mastering thousands of small skills. Khan Academy has exercises that help you practice the first 1500 or so of these skills. But as I discussed last week, the Khan Academy mastery system only gets you to an initial level of mastery. The topic for this week: how to continue using Khan Academy to increase your skill mastery after you have officially “mastered” the skill.
Khan Academy provides a way to prepare for studying a specialized math topic like discrete math by first mastering more fundamental math skills. Knowing the fundamentals frees up your mind when you’re learning advanced skills. For example, some discrete math topics use properties of exponents. It’s easier to focus on learning those topics if you know exponent properties as completely as you know your multiplication tables. The goal is unconscious competence: you want to be good enough at a skill that you can use it automatically without thinking about it.
A potential trap with practicing fundamental problems is trying to learn each skill equally well. Although it’s good to work on the basics, it’s not efficient to practice skills you’re already sufficiently competent at. Your target competence level will differ for each skill based on your goals. For discrete math, it’s valuable to be intimately familiar with properties of exponents. But you probably don’t need to be as good at solving integrals using trig substitution.
Even for skills where you want high competence, you must be selective about which problems to use for practice. You can’t keep getting better forever by solving just one problem type. Although Khan Academy can generate unlimited problems for each skill, the problems follow a few predictable pattern. That means they eventually become too familiar, despite the variations that the problem engine generates. Once you reach that point, it’s time to find another source of practice problems. One option is to look for a more advanced skill that implicitly uses the skill you’re trying to get better at. Or you could find a source that approaches the skill differently than Khan Academy does, perhaps by using an abstract proof-based approach.
One other way to combat excessive familiarity with a problem type is just to wait awhile. After a few weeks or months, most problem types will be more difficult than they were when you were practicing them regularly. The Khan Academy mastery system knows about this technique, but once you achieve 100% mastery it’s best to keep your own records of target skills, since the built-in system doesn’t work as well at that point.
Last week, I proposed a list of four mastery levels to track your competence at skills that Khan Academy says you have mastered. Here are some examples of skills that might appear at each level.
Trivial
A trivial problem requires no effort in any problem-solving step: reading and understanding the problem statement, solving the problem, and checking the answer. You can do it all in your head or you can write the answer as you’re solving the problem.
Early math problems, like basic arithmetic and geometry, are trivial for most adults. But for someone with an aptitude for or interest in math, many other skills can also be trivial. For example, the negative exponents skill requires you to know that $a^{-n} = 1 / {a^n}$. Once you know and remember that identity, any problem in that Khan Academy skill will meets the criteria for a trivial rating.
Even calculus problems can be trivial. For someone who has never studied calculus, the problem “Given $f(x) = 5x^2+3x+2$, find $f'(x)$ ” might be incomprehensible. But even an average calculus student shouldn’t need more than one line on a paper to solve it.
If you truly find a skill to be trivial when you come back to it after a few weeks or months, there’s no need to continue practicing it. There’s no room for improvement, so there’s no point in spending time on it. Find another skill that needs more work.
Easy
If you rate a problem easy, you must know how to solve it without notes or references, and you must be able to perform each step of the solution promptly, without pausing to think about how the solution process works or derive a formula from first principles.
In theory, you could practice every Khan Academy World of Math skill until you could rate every problem as easy. Khan Academy’s problem writers design problems so you can solve them with well-defined steps. The problems require little creativity once you learn the process. But it may not be worth your time to do this for every skill.
An example of an easy skill might be interpret change in exponential models: changing units. Problem in that skill start with a short word problem like:
A virus is infecting computers on a network. The relationship between the elapsed time, $t$, in days, since the virus was released, and the total number of computers infected, $C(t)$, is modeled by $C(t) = 310 \cdot (1.31)^t$. Every week, the number of infected computers grows by a factor of ____ (round to two decimal places).
The calculation part of this problem only takes one step. But before you can do that calculation, you must read the problem carefully to be sure what units it’s asking about. And the numbers are usually chosen so you need a calculator to get the answer. So although there’s not much writing involved in this problem, you do have to concentrate to get the answer. Those characteristics move it out of the trivial category and into easy territory.
Moderate
A moderate problem is one that is familiar to you, but which you can’t solve from memory. If you had to solve a moderate problem on an exam, you wouldn’t get full credit unless you made a lucky guess. In the Khan Academy environment, you would need to check your notes or consult a reference to solve a moderate problem.
The equation of a hyperbola from features skill is a typical source of moderate problems. Although the process to solve these problems isn’t complicated, it requires knowing a few formulas, or deriving them. And unlike properties of exponents, these formulas (which relate the vertices and foci of a hyperbola) aren’t ones you would regularly practice in other contexts.
Hard
Hard problems are problems you haven’t practiced enough yet. Even for hard problems, my assumption is that you have solved at least a few similar problems, since in this article I’m only considering skills where you’re rated as mastered on Khan Academy. But some problems types are complicated enough that they require substantial practice spread out over weeks or months to get to the point where you can solve them from memory.
When I was going through the World of Math, the hardest problem type I found was volumes with cross sections: triangles and semicircles. It requires visualizing a moving graph and evaluating integrals to get the volume of the resulting three-dimensional shape.
For World of Math skills, it’s usually worth getting past hard and achieving moderate or easy level. These skills provide an overview of math fundamentals and a good background for future studies.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
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]]>Before learning discrete math, it’s useful to know how much other math you remember. Last week, I discussed how Khan Academy is uniquely useful for evaluating and improving your skills in math topics through Calculus. But eventually you might reach Mastery level in all Khan Academy math skills. Does that mean it’s time to move on? Not quite.
If you complete enough mastery challenges, you will master all the Khan Academy World of Math skills (except maybe one due to a bug). Once you reach this point, mastery challenges work a bit differently. In a normal mastery challenge, you get five or six problems in different skill areas, at a variety of mastery levels. Once you’re at 100% mastery, you get just one problem (or occasionally two), and it’s always at Mastery level. When you submit an answer for that one problem, the challenge is over.
You can always click the Start button again to get another challenge. The system will keep giving you challenges as long as you want. If you get a challenge question wrong, your rating drops back to Level Two. But if you then start another challenge, the system will present you with a problem from that Level Two skill, and if you get it right, you’ll be back up to Mastered in that skill. So it’s not too hard to stay at 100% mastery until Khan Academy releases its next batch of new skills.
How does the mastery challenge system select which skill to use for your single post-Mastery problem? Even though all your skills are now at the same level, the system doesn’t present problems randomly. Skills you have practiced recently are less likely to come up than those you haven’t seen in a while. But that seems to be the only rule the skill selection system uses. Even though you have mastered Khan Academy Calculus, you’ll still get problems in basic arithmetic. As a result, the mastery challenge system isn’t as useful as it once was after you have mastered all the skills. It needs some manual intervention.
Even before 100% mastery, the mastery challenge system sometimes presents you with problems from skills you have already mastered. That’s because mastery is not a permanent state. You can forget skills if you don’t continue to practice them. But you don’t need to practice every mastered skill at the same frequency. For example, once you master your multiplication tables in childhood, you never need to practice them again. Partly this is because you practice them in other contexts, but it’s also because they’re not that complicated, so you can commit them to long-term memory. But the Khan Academy mastery system still keeps giving you multiplication problems.
A rating of mastered for a Khan Academy skill just means you got several problems in that skill correct with no intervening incorrect submissions. It says nothing about other aspects of mastery:
The Khan Academy practice and mastery system does try to target these aspects of mastery. For example, problem writers try to write problems that discourage guessing and require an understanding of the skill being tested. But only you know for sure how well you understand a concept, whether you guessed the answer, and whether you used notes.
Despite these limitations, I think Khan Academy World of Math is still useful post-Mastery. It’s an effective way to get unlimited auto-graded practice problems on a topic, and the instructional content is still available to explain concepts you’re not perfectly clear on.
Since the built-in mastery levels aren’t useful once everything is marked as Mastered, my recommendation is to construct your own rating system for post-Mastery skills. Here’s how that process would work. First, select a skill, either manually or by starting a mastery challenge. Solve a problem for that skill. Then pick the description that applies best to your experience solving the problem:
The goal with these ratings is to evaluate your true level of competence for skills you have previously learned to a Khan Academy level of mastery. To get an accurate rating, you shouldn’t have solved any problems of the same type in the past week or so (ideally in the past few months). The Khan Academy mastery challenge system takes care of that timing automatically. The wait time ensures that you have the skill in long-term memory.
There are 1498 skills, so if you evaluate five unique skills per day, you could finish in 10 months. At that point, you would have a list of skills to work on. My suggestion would be to ignore the skills rated Trivial and Easy. You know those well enough. The goal is to turn Hard skills into Moderate ones by studying topics you’re missing, and Moderate skills into Easy ones by brushing up on details and practicing to gain fluency.
There will naturally be a range of skills in the Easy range. Although by definition you can solve them all without referring to notes, some problems will always take longer to solve than others. The other consideration is how well you want to know a topic area for future work. Although every Khan Academy math skill is part of a basic math education, you may reach the point where it’s better to move on to skills that the platform doesn’t cover.
Next week, I’ll cover specific examples of problems in each skill category.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
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]]>Math builds on itself, so trying to learn a math topic without first mastering its prerequisites is just asking for trouble. A good place to learn prerequisites is Khan Academy. Although it’s missing many advanced math topics, it covers the basics well. And the way it organizes practice problems ensures that you regularly get tested on the concepts you need the most practice with.
Back in the day, Khan Academy was just a collection of YouTube videos. It still has plenty of those, and they’re useful for learning new concepts or refreshing your memory on old ones. It’s convenient that they’re integrated into the Khan Academy user interface, so it just takes a click to find a video related to the topic you’re studying. But Khan Academy is far from the only place to find free math videos.
Where Khan Academy distinguishes itself is in its practice problems. The way to learn math is to solve math problems, and Khan Academy provides unlimited problems on each topic it covers. If you’re having trouble with a topic, you can keep drilling practice problems until you have the concept down.
For students who are just learning math, Khan Academy provides a guided tour from $1 + 1 = 2$ through calculus and differential equations. But even for people who have already learned all the math topics Khan Academy covers, its practice problems can be a useful tool. They can pinpoint areas of weakness and let you strengthen them in preparation for future math studies or some other goal. It’s much more effective than leafing through a textbook chapter and saying, “I probably know that.”
Khan Academy uses the term mission to describe the user interface they created to guide students through math exercises and videos. The top-level mission, called The World of Math, contains all the other missions.
Missions contain skills, math topics you can practice and master. Skills are part of units, which are part of courses. For example, Limits of composite functions is a skill in the Limits and continuity unit, which is part of the AP®︎ Calculus AB course.
When you’re practicing a mission, you don’t have to worry about units and courses. You can just navigate your way through the skills. Last year, Khan Academy launched a new course and unit mastery system that may eventually replace the mission-based mastery system. For my purposes, I’m focusing on the mission approach. It’s simpler to navigate and I think it’s more appropriate for experienced students doing targeted learning: dipping into specific skills as needed, rather than progressing through courses in order.
The World of Math mission currently has 1498 skills. Khan Academy periodically adds more. For each skill, you get one of five ratings based on your performance on problems related to that skill: Not Started, Practiced, Level One, Level Two, and Mastered.
Using the World of Math mission, you can solve problems for these skills in two ways: 1) Practice, and 2) Mastery Challenge. Your can start a practice or challenge session from your World of Math dashboard.
Your dashboard will usually have a set of practice suggestions in the skills up next for you section. If you don’t like the suggestions, you can add to that section by clicking a square in the skill breakdown grid. You can then click the Practice button to practice a set of problems (five or so) for that skill. If you get enough problems correct, you’ll increase your skill level to Practiced.
In a browser console, you can also run JQuery code like $('.progress-cell')[658].click()
, which lets you “click” a skill by number. Skill 658 is limits of composite functions, so running that code will add that skill to your practice list.
Also from your dashboard, you can click the Start button in the Mastery Challenge section. This is how you start a mastery challenge to access the higher skill levels. In a mastery challenge, as in a practice session, you get a few problems to solve. But in a mastery challenge, each problem is from a different skill. If you get a problem right, you increase your skill level in that skill. So with one mastery challenge, you can increase your level in several skills.
When I use Khan Academy, I usually use mastery challenges. Since a mastery challenge uses problems chosen by the system with no input from me, I can’t just pick familiar topics. It forces me to prove I still know the skills I previously practiced or mastered.
Here’s some trivia based on my experience with mastery challenges:
With some discipline, you can master all the World of Math skills, as some people have. (It’s a lot easier than finishing the Project Euler problems). Next week, I’ll discuss ideas for continuing to use the World of Math mission after reaching 100% mastery.
I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.
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