I have a feeling it might be the pencil. Don't skimp on pencils, the best are Ticonderoga Dixon, #2. (might be Dixon Ticonderoga). it has the familiar green stripe, writing and end cap-eraser part.
Also, I don't know if they make these anymore, as I tried to find it online, but a good, hand crank pencil sharpener has always worked best for me. Mine is a Berol, with 6 size settings and I bet it's 30-35 years old. Still works great! Sending a link (or whatever you call it) to a sharpener that I've used before, but don't own. It's the hand crank with 5 or 6 size settings.
Hope this helps. I'm pretty persnickety about my pencils. I'm a teacher and I often bring pencils from school home to sharpen on my old faithful Berol hand crank, as opposed to the battery-op at school. I haven't been able to find another like the my other, but now that I found that one online, I think I'll give it a try myself.
www.reuels.com/reuels/Lead_Pencil_Sharpeners_Staplers_Pap...
KS Pencil Sharpener Boston Model
$19.99 $14.99 On Sale!
HT1031
P.S. Get good, not cheap, cheap, cheap pencils. You can pay a quarter or 50 cents for glue, Crayolas, spirals, markers, etc. but I always buy Ticonderosa or another good brand (Stafford) of pencils.
Good luck!
Mary