I have been reading The Wonder Weeks, which documents your baby's mental leaps based on cognitive science (ie. notable changes in brain wave activity and growth of head circumference during specific weeks in their development).  Their suggestions for activities to do with your baby to promote that mental leap are super compatible with Montessori. 
Here are the notes I've made for the first six months, and the corresponding Montessori materials.  I will update this to include links as I go back and document what we used over the first months!
THE WONDER WEEKS 
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Montessori Materials 
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Newborn - 5 weeks (1 month) 
World of strange sensations 
Cannot distinguish between himself and the outer world, whether his discomfort comes from his body or outside it 
Sight: can see about a foot away (to mother’s face), beyond that vision is blurred 
React best to strong contrasts: black and white holds their attention the longest, red and white etc, is also effective 
Can track moving objects as long as the movement is slow and deliberate 
Sound: likes sounds that resemble the thump of the heart, veins and other internal organs familiar from the womb 
Smell: can recognize mother’s smell if presented with her clothing 
Feel: can sense temperature changes but cannot regulate her temperature; being touched is the best amusement 
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The black and white Munari mobile to stimulate vision 
The topponcino, which the mother sleeps with before birth, to comfort baby with mother’s smell 
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Week 8 (2 months) 
World of Patterns 
Ay 6-8 weeks, baby’s head circumference increases dramatically and there are changes in their brain waves  
Baby no longer experiences himself and his world as one universe 
Discovers his hands or feet, will voluntarily grasp a toy instead relying on reflex 
Looks at luminous and shiny things 
Makes sounds, sings, interrupts when others are talking 
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Simple grasping toys: grasping beads, bell rattle, puzzle ball 
Montessori mobiles: the shiny octahedron mobile, gobbi mobile (our baby batted at this one) 
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Week 12 (3 months) 
World of Smooth Transitions 
Can recognize subtle and gradual changes 
Better control of bodily movement 
Vision improves and will soon be as good as an adult’s 
Studies and plays with your hands 
Can shift between loud and soft sounds 
Produces sounds that resembles vowels 
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Montessori mobiles: gobbi mobile in gradations of colour, dancers mobile (this one moves very subtly) 
Bell rattle 
Box of textured fabrics 
*maybe a good time to introduce baby sign language 
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Week 19 (4.5 months) 
World of Events 
String together patterns and transitions to become events eg. he can reach out for a toy, grasp it, shake it, turn it around to inspect it, put it in his mouth 
Speak in babbling sentences 
Recognize short flowing sounds  
Recognize familiar tunes 
Can understand simple, familiar sentences 
Enjoys mirror games - recognizes mom and her reflection 
Dramatic changes in brain waves and increase in head circumference 
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Bell rattle 
Puzzle ball 
Sing musical scale 
Movement mirror 
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