Everyone (yes, EVERYONE) who learns new math struggles at first and then slowly puts the pieces together. The only difference is that the people with more experience learning math know what to expect and how to push through it. This is the key to not only understanding math but really to doing anything tough in life.

Think about how it feels when you start a new topic in math. Can you identify these stages in your process? Do you let yourself get caught up in frustration before you push through?

• Stage 1: Bewilderment – I’m sorry. The fact that no one uses this word anymore made it all the more important to me that I manage to use it in a sentence today. Anyway, back to our topic: This is the stage where you have literally no idea what is going on. All you know is that someone is writing symbols on the board and they might as well be making them up. You try to look in the book and it looks more made up than the stuff you saw on the board! Pushing through this stage means asking tons of questions and truly reading your math book. When I say reading, I mean sitting with a pen and paper and trying the examples right along with the book.
• Stage 2: Stumbling – You know you have hit stage 2 when you can do problems in the book, but only if you have your notes. In other words, you get the first step or two and then get stuck. BUT — All it takes is a look at the notes or an example (and a few minutes of frustration) and you can figure out where to go. A lot of people get stuck here because they haven’t learned that the frustration they’re feeling is no different than “feeling the burn” when working out. Those little trip ups and times when you are trying to find you mistake? Those are exactly the moments your brain is building the connections it will need for the next problem.
• Stage 3: Robotic Understanding – After a while of working problems and stumbling about you will find yourself at this stage. Things are starting to feel much different now. While you may not always know WHY you do certain steps, you can do problems on your own and are getting the right answers most of the time. You’re seeing connections between this and things you learned before even though some of the ideas are still fuzzy. This is a good place to be and a place to be careful! Plenty of people figure they got it at this stage and stop practicing/studying. Don’t fall for that! Keeping the work up now will pay off with…
• Stage 4: True Understanding – Truly understanding a math concept or idea means that you can work problems that use it (even if you get stuff wrong here and there – that’s normal!), see how it is connected to other ideas, and even be creative with it (as in do a problem that is different from anything you have seen and apply this new idea to it). In my personal experience, there are probably 100 levels of “true understanding”. Even with all my experience, there have been times where I suddenly realize something about a topic I learned 10 years ago! If you are taking math class, you won’t necessarily reach this stage until you have had a few new topics that your mind can start to connect together. In my experience, the difference between accepting being in the “robotic understanding” stage and pushing to the “true understanding” stage is what separates the A/B’s from everyone else.

Your goal: Next time you are feeling overwhelmed and frustrated learning math, remember: It is part of the process. You will have to find your own way to push through it, but I am hoping just knowing this much will keep you motivated!

math anxiety, math success, studying tips, test-taking

Are you taking an online math class this spring? I could give you tons of advice but my #1 tip is:

Treat it like a real math class.

People feel silly writing notes from something they are reading online. That’s silly. You take notes in a face to face class right? Don’t let yourself become more passive simply because there isn’t the pressure of the professor looking at you funny while you just sit there doing nothing.

Write.

If there are online videos, take notes. If there are online notes, write your own notes from them and try to work the examples as you go. You should never be looking at any math without a notebook in front of you.

math success, online math classes, studying tips

Alright, so you signed up for discrete math or linear algebra and figured “how different can this be from calculus?” right? It even started out that way – mostly calculations and definitions. Then *BOOM* they hit you with this:

Suppose that A,B, and C are nonempty sets. Prove that and .
This isn’t a calculation. This isn’t a “if you see this – do this” type of situation like some math. While you CAN make a set of rules to write many of the basic proofs you will need, true proof writing is an art and there are many correct ways to go about it. It’s likely your book won’t help you too much on setting up the logic and the right kind of thinking for proof writing. It is completely different from the thinking that you would use to “find the derivative” or “solve this linear equation” for instance!

This is why I recommend “How to Prove It – A Structured Approach” to all of my students. The author, Velleman carefully develops logic and technique for writing proofs while introducing mathematical ideas like sets and relations that you will need to understand anyway. There is an ENTIRE CHAPTER on proof techniques and great practice problems.

I truly can’t recommend this book enough. In fact, this is the book that was used in my “intro to proofs’ class in college – that’s how thorough it is (If only all colleges had such a course!). If you want to do well in your advanced math courses, you will be glad you bought this.

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NOTES: The link above is NOT an affiliate link. I live in Illinois, so it isn’t possible anyway . Also, the image is from a pretty funny blog..

concepts, proofs, resources

I realize that for a lot of people, taking math courses can be stressful especially if you are just used to “getting it” before and now having trouble. Personally, I think there is something we could do with the way we teach math that would help everyone relax and appreciate math more but that’s for another post on another day .

To do my part to help, I’ve written about how you can do well in calculus, and even had a guest post filled with tips for NOT FREAKING OUT when you hit your first really tough math class.

Based on how much people are reading the calculus articles, I’m betting a lot of you guys are engineering or science majors. If so, you definitely should check out this article over on College Info Geek: Tips for Engineering Students. In fact, if you are taking any math class whatsoever, I would follow these tips – regardless of your major!

My favorite sentence in the whole article “Take an active interesting in learning.” ::nods::

math success, studying tips

Everyone who has studied math has felt overwhelmed, confused, and downright frustrated at one time or another. The real stuff starts to happen when you decide how you are going to handle it. Are you going to give up and try to avoid it all as much as possible or learn to adjust a bit (kinda like working out hard – hey everyone gets sore sometimes but that doesn’t mean you stop!).

Today, Laura Laing was kind enough to drop by and offer some advice. And she knows what she is talking about too! She is the author of the book and blog “Math for Grownups” which is all about being REAL in math. In other words, answering questions like “how do you actually use this stuff in life?”, and “why is it this way?” in a completely laid back, nice to read way. Definitely worth checking out! On to her tips…

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By the time you hit high school or college math, you might have a similar realization as Dorothy’s in The Wizard of Oz: “Toto, I’ve a feeling we’re not in Kansas anymore.” Even if you’ve done well in math before, you may find that the problems look a lot different and the concepts are more difficult to understand.

That’s because math is much, much bigger than most people might think. Mathematicians usually end up specializing—in statistics, abstract algebra or calculus, for example. But until you get to that point, you will need to learn a little bit of everything. And that can get pretty darned overwhelming.

So how do you handle feeling overwhelmed in a subject area that you once felt comfortable with? Or how do you get through a class when the math just isn’t clicking?

Breathe deeply
It never pays to panic. If you feel yourself getting overwhelmed—or worse, thinking that you are too dumb to understanding the material—practice some deep breathing and look for ways to reframe your thinking. Be your own internal cheerleader, or find others who can boost your confidence.

Take small bites
By definition, feeling overwhelmed means there’s too much going on. Instead of looking at the big picture, try tackling smaller pieces. With homework, this can mean really focusing on each problem individually and taking breaks between problem sets. In lectures, focus on asking specific and concrete questions. Make notes to ask your instructor for more information, if something doesn’t make sense.

Look for connections
We all have some sense of the math around us. And with years of math education behind you, you definitely have some math skills that you can build on. The trick is to draw on this experience, even in math classes where all of the material seems brand new. Look for touch points in your classes—information that is familiar or that you can connect to skills you already have.

Get some help
No, you don’t need a psychiatrist, but it may be a good idea to ask for assistance from your instructor or find a tutor. This is not a sign of weakness. Everyone needs help sometimes—even mathematicians. In fact, most of history’s greatest mathematicians collaborated at least occasionally. Now is not the time to let your pride get in the way of success.

Write about math
One of the best ways to check your understanding is to restate the concepts in your own words. You don’t need to keep a math journal, but it can be helpful to summarize your class notes—especially if the concepts are challenging and your understanding of them is fleeting.

It can be upsetting to walk into a class and feel lost. But with a few strategies, you can get past feeling overwhelmed and right to the task at hand: learning the math.

math success, resources, tutoring

Lectures might not be the most exciting thing to sit through (well unless you are in MY class, yeah couldn’t resist saying it, of course I do more than lecture but lets get back on track…) but let’s face it: most math classes are strictly lecture oriented. Therefore, if your plan is to do well, you MUST figure out the way to get information this way even if your learning style is a little different.

Hmmm...This may look awesome, but if your notes look like this you could have a little trouble following them.

• Try to look at the book beforehand – I know that in reality, it is hard to find the time to look at things before class. You don’t have to read and understand the whole chapter though! Here, a quick read through to make mental notes about anything that seems particularly confusing can help you focus on those things during the lecture (and will help you ask better questions as well)
• Don’t write down EVERYTHING! – Definitions are in your book as well as proofs and “rules”. Instead of writing them down again, use the time in class to really try and understand them and ask questions. If they’re raelly confusing, make yourself a quick note to go and research it more.
• Focus on examples and the “why” – Hopefully your teacher or professor does some examples in class that are different from the text. These are studying GOLD. Every worked problem you have in your hands is a problem you can practice later on your own (with a full solution to see what you got wrong if you mess it up). This is where you should be writing the most careful notes. Also pay close attention to the “why”. Why do you use this method and when. This can be tough to follow in a math book so its worth writing down pointers for yourself.
• If you think you are “getting it” then test yourself – Suppose the professor is doing two examples and you really “got” the first one. Ok, try to solve the second one before he does! This way you can see if you really are understanding things and if you get stuck, you are in the perfect position to ask a good question.
• Try using different color pens or pencils – I know this sounds crazy, but I even do this when I use the board in my classes. It is so much easier to read notes (or the board) when the problems are black, the solutions are blue, and confusing parts or things you want to ask questions are in red.
• Rewrite your notes as part of studying – Now if you write slow, my previous tip may not work for you. But you could take your notes like normal and then rewrite them later in this color coded format. In fact this would be a really nice way to catch anything confusing and help solidify the information in your head. I had a friend in grad school that rewrote all her notes as part of studying and it worked wonderfully for her!
• Please ask questions. PLEASE! – The professor really wants you to ask a question. Seriously. Questions help us transition into new ideas and see how well our students are understanding things. The rest of the class wants you to ask a question (well except for that guy in the back who doesn’t care and wants out early, but he shouldn’t be there anyway). They are nervous and probably have the same question its just that someone needs to break the ice. I know everyone says it all the time but its true, so ask questions already!

I’m sure there are tons more tips out there so what are yours? What have you done in math classes that has helped you? Or if you teach, what do you wish your students would do?

math success, note-taking, studying tips

If you think about it, we have evolved to want to try and use the least energy possible. When you have to hunt or gather for every single bit of food – well, this just makes sense. My experience has been that this is no different when it comes to trying to learn something new. If you give your brain a crutch, it will use it all in the name of saving energy.

In order to understand math, you have to practice the same way you would if you were learning a language or even a musical instrument, right? … and the easiest way to practice for most people is to use the problems in the textbook, right? My argument is that when you combine these with the brain’s urge to be lazy sometimes you can actually undercut the learning process!

All of the problems in the textbook are in order – a big issue! If you know that chapter 4 is all about using a certain formula then you brain won’t try to really read the problems for more than that – it will shift into “let’s apply that formula” mode and it will feel like you are learning a lot while, in reality, you are just getting really good at that one technique. Not only that, you are only getting good at that one technique when you know its the technique to use. A computer can do that!

Most math problems have many ways to do them and the real skill is reading carefully to know which method to use. This is the human / creative element to math! Knowing what section or chapter a problem came from even subconciously can take this element out of it and its the most important element to deeper understanding.

So, how can you get around this and make your brain work harder? You have to mix the problems up! A great way to do this is to make flashcards out of random problems from each section, then mix them up and do your practice problems that way. This is especially good for final exams or standardized tests which are typically not “in order” at all.

Some tips:

• Make sure that you put a page number on the back with an answer so that if you get really stuck you can go back and read the section
• Try to get as much of a variety of problems as possible – the goal is to get your brain used to seeing problems that are totally different to eachother back to back
• Still use your book for the routine stuff! – You still need to practice the skill of simply applying a technique. At first, go ahead and work the basic problems and then use this when you need to start learning to pick methods better.
flashcards, math success, Memorization Tricks

Yesterday, I ran across a great article on Math for Grown Ups about different ways of attacking math problems. While this article talks about day to day math, I was wondering how many people really think about this for more advanced topics like algebra or calculus.

I have a habit of giving my students a problem and asking them to find *at least* a couple of ways to find an answer. (In fact, I did one of these as a calculus problem of the week recently). A lot of time, I find that the students are worried they will get confused if they know more than one way – they are simply too focused on just getting an answer! Talking about this always leads to a great discussion which I tend to start off with these two questions:

How do you know what method makes the most sense to you until you have seen several different methods?

How can you be sure you really understand something until you can look at it from a different perspective?

If you have found yourself able to “go through the motions” of math without much understanding then this is a great exercise to try. In fact, this is something that anyone who is trying to understand something better should try! Stop focusing on just getting the answer and start trying to see how things work together. There are patterns hidden all throughout math waiting to be noticed. It’s these patterns that can help you get a deeper understanding and get better at finding fast or efficient ways to “problem solve”.

Try finding another way to do the problem, or change the problem a little and see how the answer changes – basically I say, let yourself explore the topic. Use the textbook problems as a starting point (thats really what they are) and go from there – all you need is paper! Remember, math isn’t about just finding the answer, but instead about problem solving skills.

concepts, math success, studying tips

Students ask me all the time: “Should I get a tutor!?” and my answer is almost always a resounding “YES! OF COURSE!”. The only part that bothers me about this question is that it is usually asked towards the end of the semester by students who are in danger of failing. If I were to recommend when to hire a tutor, it would be to do it at the start of a class before your grades suffer! I guess the natural question is – why do I think that getting a tutor is so great?

• Forced Practice on a Fixed Schedule – The only real way to learn math is through practice and that doesn’t mean doing 100 problems the night before the test. Think of it like something athletic – you have to build up and be consistent. Instead of 100 problems before the test, it should be 100 problems over the few weeks before the test. If you know that you are meeting with a tutor at 3pm every Friday then at the very least you will be practicing your math at that time every week. More likely, to get the most for your money, you will be practicing earlier in the week so that you have lots of questions for your tutor! It’s just another way to keep yourself on track.
• Hearing it Explained Another Way – In the classes I teach, I have students explain ideas to eachother all the time. Why? Well, over and over again, I have seen students who don’t understand the way I explained it hear the same idea explained from a friend and then suddenly understand! Even if it is exactly the same process, everyone has a different way of explaining things and hearing these other ways can help you develop your own understanding.
• A Tutor Can Focus on Your “Style” of Mistakes – I have been doing math practically full time for the past 15 years I can still get all caught up in a problem and miss a negative sign if I’m not careful. I know this and have learned to always do a quick double check of my signs. We all have mistakes we tend towards and since these can be very individual, they usually won’t get addressed in a class. A tutor however, can notice that you always forget to divide by 2a in the quadratic formula and help you figure out how to handle that. You may have a habit that it holding you back that you don’t even know about but working with a tutor can help identify and address it.
• Reviewing that Stuff You’re “Supposed to Know” – Who remembers everything they learned in every class ever? Well, if you do you are lucky as most of us don’t. Unfortunately in math, there can be little topics here and there that come up again in future classes in fact this is pretty much 100% guaranteed. Double unfortunately (ok go with me on that phrase lol), there may not even be time in class to remind you of these tricks or skills. This is where having a tutor is incredibly helpful as they can help you go back and practice this “old stuff” so that it doesn’t hold you back with the new stuff.

If you do decide to hire a tutor, you should really look into all of the options first. Your school may already have a free tutoring program that you can take advantage of. Do your research and remember – tutoring is an investment just like paying for your tuition or books – so take it seriously and try your best to get the most out of it that you can.

math success, tutoring

Whether it is a calculus class or a first algebra class, I have seen students at every level and every natural ability make these mistakes. Why would so many fall into these traps? Well, it really comes down to working too quickly and applying rules without stopping to think about whether or not they apply. If you can avoid these mistakes, you will find yourself well ahead of the curve. In fact, these are the kinds of things people writing multiple choice tests think about.

1. Distributing a Power
   It seems as though everyone wants , and who wouldn’t? Every math exercise in the world becomes much easier if you use this! Unfortunately, in general, . Its true in a trivial case – when a or b equals zero (plug zero in for either one to check). Otherwise, this is a no go and if you aren’t sure, try it out with some nonzero numbers: What if a=1 and b=1?

   while . Clearly we get two different things. Now this isn’t a mathematical proof since I used two specific numbers but the fact that it does not work here should give you pause. In fact, you would have to FOIL the left hand side to find its equivalent expression. In other words, .

2. Forgetting to Completely Distribute
   What if you needed to simplify . Many students would write . Can you see what is wrong?

   The negative in front of the parentheses is basically a negative 1 multiplying all of the terms within those parentheses. That means it must be distributed to EVERYTHING, not just the first term! . This works regardless of what is in front of the parentheses. For instance, .

   Most often, it seems that people understand this but make this mistake when they are going too fast. This is why I advocate writing down every single step when you do any math problem. (or just double check yourself anytime you have had to distribute)

3. Applying the Zero-Product Rule to Everything
   The zero product rule is what allows you to solve an equation like . By realizing that the only way two things can multiply to give us zero is if one or both are zero, we can say and . This is all based on the fact that there is a zero on the right hand side!

   If instead, we had we couldn’t immediately take this approach. This does not imply that and . Again, you can only apply this if you manage to get a product of terms equal to zero. (to solve the new equation, you could foil the left hand side and then bring the two over – going from there)

There are plenty of other mistakes that I have seen but these are by far the most common. In the end, it comes down to applying properties in situations where they do not apply. Learn to ask yourself if the rule you are applying makes sense in a given situation and you will be well on your way to avoiding these!

algebra, math success, test-taking