I am not a very good cook. I have tried making stews or casseroles. I put meat, vegetables and potatoes in a casserole dish in the oven, a slow cooker, or on the hob in a low heat. Sometimes the potatoes dissolve, and give me a delicious thick sauce. Very nice.  Quite often, though, I'm left with tasteless white lumps in water. Not nice at all. I can't work out what I did differently. Maybe the variety of potato,  or the other ingredients. Any suggestions? 
The effect you are looking for is not congruent with typical preferences. Most cooks and eaters, on finding that the potatoes have disintegrated in their stew, would call them "overcooked". So you cannot rely on recipes to produce this result - most are geared towards having potatoes remain in distinct cubes. There are a few factors you can tweak to get closer to your preferred kind of stew.  Acid. Make sure that your stew doesn't contain any acid, or the potatoes will stay firm. For this purpose, you should count not only acid seasoning such as vinegar, but also any form of tomato, and any milk, cream, cultured dairy and fresh cheese. Semi-hard and hard cheeses (cheddar, gouda or firmer) are OK. For taste, you can add sourness after the cooking time is over. Potato variety. Potatoes come in three broad categories - "firm", "mealy" and "mixed" or "all-purpose". Use the mealy ones. Cooking time You have to cook for a really long time until the potatoes fall apart. If your recipe suggests a cooking time, start checking at that time and after that every half hour until you see the result, for maybe up to 5-6 hours - if it hasn't happened by then, I don't think it will happen later. The temperature shouldn't have any direct effect on the disintegration part, but for purely practical purposes, such long cooking is done with low temperatures, as low as you can get while still having your food gently simmering. Size As Greybeard said, if you cut them up smaller, they will fall apart quicker. Heat source Once heated, the oven is great for low and slow food - but it does take a long time until the food is heated enough to start cooking. I have stopped using raw potatoes in potato-based casseroles, since it frequently takes 3+ hours to get them to cook through (in the "distinct pieces" sense). The hob is a better place, but more difficult to manage over the long time. As you mentioned a slow cooker, it will be easiest to learn how to do it consistently there. If there are oven recipes you really like, give them even more extra time, or parboil the potatoes before assembling the casserole. I would peel, then boil - skin-on potatoes have always seemed to soak up less water to me, and you can make use of the "cut smaller" effect too.  On a side note, if you like this kind of sauce, you might consider ranging out into recipes that use more conventional thickeners for stew. It will be a different taste, but will allow you more flexibility, both with shorter cooking times, and with the ability to add sour ingredients. 