Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

Americans pronounce dance as DANS (/dæns/). In "dance", the "a" vowel before M or N raises and fronts toward [eə] — the tongue pulls up and forward, breaking the vowel into a tense glide as it anticipates the nasal. This is called the Cat-Vowel Before M/N, and it's why Americans sound more relaxed than the textbook. It comes out as DANS. You'll hear it in sentences like "The choreography in the dance numbers was spectacular".
Record yourself saying "dance" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 4 sounds. Explore each sound's mouth shape and how it's made.
Touch the tip of your tongue to the roof of your mouth just behind your teeth. Add vocal cord vibration as you release.

The tongue relaxes down in the back and the corners of the lips relax before the consonant. This adds a schwa-like 'uh' relaxation after the /æ/. Think of it as 'relaxing out of the vowel' — it is no longer a pure /æ/ sound.

Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "dance", the "a" vowel before M or N raises and fronts toward [eə] — the tongue pulls up and forward, breaking the vowel into a tense glide as it anticipates the nasal. The "/æ/" vowel raises and fronts before M or N — tongue pulls up and forward, producing a tense [eə] glide (between /e/ and /ə/). Not a pure /æ/.