Difficulties encountered by first-year teachers of science and mathematics; were investigated in this study conducted in May 1981 in Illinois schools as part of a broader study involving teachers in other subject areas as well. A questionnaire was designed and mailed to 138 science and 104 mathematics teachers. Responses were received from 46 science and 32 mathematics teachers. For mathematics teachers, the highest percentage reporting considerable or extreme difficulty occurred in the teaching of the more advanced subjects: algebra II or trigonometry, plane geometry, and advanced placement mathematics or calculus. Problem solving was identified as the activity of greatest difficulty. Among science teachers, the highest percentages of difficulty were with chemistry and physics. Activities of greatest difficulty involved organizing and supervising laboratory activities and teaching computer skills, scientific method, and science problems in relation to mathematical preparation. Links with the community also gave both groups difficulty, and use of computers and calculators was relatively low. Implications for teacher education programs are then briefly discussed. (MNS)