Setting up the mean

What this sequence teaches

To set up the mean, write every value in brackets and add them together, then write how many values there are underneath. Nothing is worked out: the response is (12) + (7) + (23) + (18) over 4, not 60 ÷ 4 or 15. Because the page never asks for the answer, arithmetic slips can’t hide whether students understand the mean — and every whiteboard shows the set-up itself.

The teaching sequence

Example 1 — 12, 7, 23, 18. The base case: four values, not in order, over 4.

Example 2 — the same list with a 0 added. The smallest possible change — a value that adds nothing to the total — and the response still changes in two places: a (0) on top, and the count goes up to 5. Zero is a value, so it counts.

There are only two examples, deliberately. The testing sequence gives students lists that look nothing like these, so they apply the same steps to new-looking data themselves rather than watching it done.

Running the sequence

• Keep it fast — about twenty seconds for both examples.
• Reveal plays the set-up: each value drops from the list into its brackets in turn while a tally underneath counts up, and the final count stays under the line.
• Replay runs it again.

The rule and the script

The rule, exactly as Show rule displays it: For the following examples: to set up the mean, write each value in brackets and add them all together. Underneath, write how many values there are. Every value counts.

• Example 1 — “I put each value in brackets and added them together. There are 4 values, so 4 goes underneath.”
• Example 2 — “The 0 is a value too, so it goes in brackets and it counts. Now there are 5 values, so 5 goes underneath.”

When Why? is open, the values it names — in the list and in the brackets — and the count turn blue and are underlined. Whether students see the rule and the reasons is your call.

Why brackets every time, but not on the count?

A positive whole number on top of a fraction doesn’t need brackets. But a rule with exceptions makes students decide each time, and that decision goes wrong once values are negative: (29) + (−1.5), not (29) − 1.5. The count has no brackets because it isn’t one of the values: brackets mean “this came from the list”.

Why a fraction bar, not ÷?

12 + 7 + 23 + 18 ÷ 4 divides only the 18. The bar keeps the whole total together.

Order

The page keeps the values in the order they’re listed, so each one can be traced to its bracket. A student who writes them in a different order has still set up the mean correctly.

Where this atom sits

Before it: knowing that the mean shares the total out equally.

After it: working out the value (arithmetic, never asked for here); then the mean from a frequency table, finding a missing value from a given mean, and Σx / n.

About the testing sequence

Five items in a designed order. Items 1, 2 and 4 change one thing from the item before; Items 3 and 5 are resets — new lists where everything changes at once. Resets matter because a student can get a run of one-change items right by editing their last whiteboard rather than using the rule; on a reset, the last whiteboard is no help. Reveal each answer before moving on. There is no scoring — the whiteboards are the data.

What each item is diagnosing

• 12, 7, 0, 23, 18. The zero has moved into the middle. Over 4, or no (0), means the zero only counted when it was tacked on the end. A (0) written at the end of the sum is fine: order doesn’t matter.
• 12, 7, 0, 23, 18, 7. A repeated value. Over 5, or 2(7), means the student counted different values rather than every value.
• 54, 9, 31, 0, 31, 88, 12 (reset). Over 6 usually means a skipped zero or a repeat counted once — ask which. A board that looks like the last answer with bits changed means the student patched their previous whiteboard.
• 54, 9, 31, 0, 0, 88, 12. Two zeros. Over 6 or 5 means zeros are being skipped, or counted once.
• 29, 0, 29, 11 (reset). Over 7 means the count was carried over from the last two items. Over 3 means a zero or a repeat was missed.

Common confusions

Working it out (60/4, 15, or a total on top); counting the plus signs, which gives one too few every time; and writing 12 + 7 + 23 + 18 ÷ 4.

Discussion prompts

• “Point to each value as you count them.”
• “Where is the 0 in your answer?”
• “Did you start from the list, or from your last answer?”

About the expansion sequence

The rule began “For the following examples”. These eight items show how far it reaches, with no new instruction. The order does the teaching: each item adds one new thing, so run them in order. The count stays at 4 until the last item, so all the attention goes to the new kinds of value.

What each step adds

1. A decimal.
2. A negative.
3. A fraction.
4. A letter.
5. The same letter twice.
6. Every value a letter.
7. Labelled values: x₁ to x₄.
8. An unknown number of values: x₁ to x_n, over n.

Where classes stall

• The negative. Students write (29) − (1.5), or leave the minus outside the brackets. This is where brackets every time starts doing real work.
• The fraction. A fraction inside the top of the mean looks wrong to some students. It’s one value, so it gets one pair of brackets. Others count its top and bottom as two values.
• The repeated letter. Watch for 2a. The response never collects anything.
• The labels. Some students write (1) + (2) + (3) + (4). The small numbers name the values; they aren’t the values.
• The last item. Students who try to count the symbols they can see write 4. The last label tells you how many values there are: the list runs up to x_n, so there are n values. This is the formula for the mean, arrived at without being told.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “In item 7, how could you tell there were 4 values without counting?”
• “In item 8, why can’t the answer be over 4?”