Over the last decade and a half, I've been fortunate to train under a variety of martial arts instructors - some well-known, others completely unknown. I've been especially fortunate since all of my instructors have been completely open to discussing their methods and techniques in a critical fashion (some instructors find it disrespectful to question how their technique works or why it may not work etc).
I learnt early on that while the views of effective individual techniques vary from person to person, all of my instructors agree on three basic physical facets that are required to progress in the martial arts. In fact, the vast majority of my instructors all used the same analogy to describe these three facets - The Triangle Hypothesis.

In their words, each corner of The Triangle represents one of these facets. There is Technique, Power, and Speed (bear in mind, that these are the basics, so aspects such as 'Flexibility' would fall under Technique, whilst 'Explosive Force' would fall under Power etc.).
Each facet is interconnected - Improving any one corner has the potential to improve any other corner (for example, if you improve your technique - power and speed are likely to increase as a result).
In order to grow as a martial artist, you have to make your triangle bigger. That is, you have to improve your Technique, Power, and Speed. Improving any one aspect will help you, but in the long-run, it may leave you unbalanced.
So why does everyone say that Technique is the most important of these three aspects? The reason is this: Technique is the base of your martial arts training. Speed and Power will help you progress tremendously, but in the long-term - speed and power will decline (more so for some than others) and there is only Technique that remains (included here, is Knowledge and Experience).
As a result of my own experience, I've humbly chosen to add 'C' to the centre of my Triangle. C stands for Cardio/Conditioning - I feel that this is crucial (also starts with a ‘C’....alliteration...nice.) because as you tire, your technique deteriorates, it becomes harder to generate power, and your body begins to slow down.
How you develop your own Triangle is up to you. No doubt, your instructors and training partners will have an impact on how you grow but ultimately, it is you who decides. Consider this whilst examining your goals in the martial arts. What kind of triangle is best suited to you and your goals?

(Interestingly enough, the Gracie family logo is also a Triangle - it may be a homage to the Triangle Choke, but I suspect it probably has something to do with what I've written above - correct me if I'm wrong though!)
Half Guard is a very common position in BJJ, especially for beginners who have not yet developed a solid Closed Guard and as a result are often passed. The position itself is simply this: 'when you trap one of your opponent's legs between your own'.

Traditionally, the Half Guard was considered an inferior position in which you are one step away from having your entire guard passed. However, views on the position have changed considerably as more and more practitioners have shown the offensive potential associated with the position.
The Lockdown
The 'Lockdown' is a Half Guard position made popular in recent years by Eddie Bravo. Typically, the position leads to various sweeping options though it does offer some submission possibilities as well. It is also one of the tightest Half Guard techniques possible in my opinion - great for preventing your opponent from passing into Side Control or the Mount.
The technique itself is simple:
1) Step over and trap your opponent’s leg with your outside leg.
2) Hook your outside leg's foot into the pit of your inside leg's knee.
3) Hook your inside leg's foot under your opponent's leg.
Key Points to Remember:
- You can stretch out the Lockdown to help break your opponent's posture.
- Keep your opponent's posture broken down using over/under hooks (double under hooks to best utilise the Lockdown’s sweeps), otherwise you are open to an Indian Death Lock counter as well as a ground n pound when striking is permitted.
- Squeeze your knees together to keep the position tight.
Common Mistakes:
- There's no need to constantly stretch out the lockdown. True, it frustrates your opponent but it tires you out needlessly (since you're not gaining any ground) and makes the 'Whip Up' to your side a lot harder (Try holding and dialling your mobile phone using only one hand whilst keeping your fingers stretched out to see what I mean).
- Don’t start camping - Keep your game active. If you feel pinned to the ground, try grinding your knuckles into your opponent’s ribs to make some space (but only just enough for you to turn on to your side and begin working sweeps/submissions!).
Half Guard is a position I'm very familiar with - it's a rare occurrence when I don't transition through Half Guard during a roll (I find it easier to escape to Half Guard when mounted than to sweep my opponent; it's also a lot more comfortable than trying to lock my full guard against someone with a big waist!). Though getting double under hooks came to me quickly, performing a successful Whip Up was a whole other story - I think I must have stalled using the Lockdown for a month before the Whip Up started working for me.
Whenever the Whip Up isn't working for me, I usually look towards establishing a butterfly hook or, (if there's enough space) transitioning back into full guard.
Don’t fall into the Lockdown campsite method – Keep active, there's good reason Half Guard was considered an inferior position for so long! Even if you feel pinned - do something... anything! Try switching from Lockdown to its calf-crushing (even though it’s traditionally an ankle lock) Judo cousin 'Ashi Kannuki' (just mirror the position of your legs - inside leg traps opponents leg etc).
Another problem that can occur is when someone diagonally mounts you from the half guard (as seen in the photo example). In this situation you can attempt a submission via Arm Triangle or, you can release the Lockdown, and revert to some basic BJJ bridging escapes (I find the latter to have a higher success rate).
Next Step: The Whip Up
Four new members joined our BJJ class last week. It was interesting to observe how each of them approached the art differently especially, how each approached the rolling segment of the class.
A & B rushed forward into each other with little recollection of the techniques and positions presented to them just a short while before; C spent the majority of his time trying to keep as much distance as he possibly could; and D, vigilantly worked towards closing the gap...and then pushing away his partner to reopen it.
Each of the new members demonstrated a different strength - A&B showed aggression and endurance. C, constantly moving, demonstrated speed and a high work rate. And D showed perseverance whilst trying to play a slower game.
BJJ is probably one of the most complex and intricate martial arts in the mainstream. It's no surprise that beginners to the art may have little to no idea what to do when they roll - it can be a little overwhelming! It takes a lot of training and a fair amount of time before one's hard work begins to blossom into successful submissions, escapes and transitions (not necessarily in that order!).
A valuable thing one can do straight away is to start thinking BJJ - I'm not suggesting one turn into a human technique encyclopaedia...I'm referring to developing an overall understanding of the BJJ game. Not only will this provide one with direction during their rolls, it'll help understanding in the application of different drills.
Stephan Kesting's e-book 'A Roadmap for Brazilian Jiu Jitsu' is a great place to start. Having stumbled across this some weeks back, it's a resource I wish I had read years ago! The e-book is free to download and distribute; I'll also add a link to Stephan's 'Beginning Brazilian Jiu Jitsu' site (which has a free e-course as well) to my link bar. Alternatively, you can download the book here (it's only 3MB):
http://rapidshare.com/files/169132969/BJJ_Roadmap_1.2.pdf
'A Roadmap for Brazilian Jiu Jitsu' goes over all the basic positions in BJJ and then takes things one step further by explaining the hierarchy of positions. Stephan has also included transition flowcharts, photos and instructions/ideas of what to think about whilst rolling. For experienced BJJ practitioners, this e-book can act as a little reminder to what your juniors are thinking and how they're feeling - everyone needs a dose of empathy once in a while!