The key approach to Montessori is flexibility based on the needs, knowledge and interests of the child. The order of activities is flexible, each activity itself has flexibility, even the use of 'traditional' Montessori activities is itself optional and flexible.
To use one example, oftentimes a child will be introduced to the pink tower, but is able to use it quickly and easily without any or much instruction. The child has passed this sensitive period and doing the activity won't have much benefit. It's best to move on to other activities instead of forcing the child to do it a certain way. When the child is bored with an activity, it can be a sign that it's best to move on.
Children can go through activities very quickly if they know the underlying principle. This isn't a bad thing, but something we have to be aware of as we tailor the activities and environment for the child.
Reading is another example of a skill that needs activities tailored to the specific child and their development. A lot of reading becomes 'intuitive' with practice, that is, it becomes a recognition task based on memory. As such, it's often very important to expose children to many examples repetitively so they can memorize the rules and exceptions.
We adapt the lessons for each child at their particular level, not always going through a firmly set sequence. The sequence is only a guide. The order for the equipment can be modified depending on the interests and abilities of the child.
Keeping this in mind, we can integrate other curricula, techniques and manipulatives into a Montessori approach, everything from programming to adaptations for specific native languages.
Oftentimes children will be interested in a specific part of math, for example: fractions, very large numbers or other number systems. The Montessori equipment can be adapted to be used with these, or other activities can be integrated from other traditions, or invented!
Reading software or text-based computer games can be a good adjunct to instruction for children as they learn to read.
To give another concrete example, many children who become very fluent writing sentences with the movable alphabet are already reading and can have the noun grammar cards introduced to them.
If a child has mastered the beadstair and Seguin boards easily, they can usually move on to the arithmetic operations with the gold bead material.
The environment is prepared and controlled to allow for growth, which means we can't always predict which way a child grows or when this growth will occur. The principle is to guide development from concrete to abstract, following the lead of the interest and ability of the students.