Adds instead of multiplying
Students need to re-read the problem. If they can then see that they should be multiplying rather than adding ask them to repeat the questions. If students do not understand that the problem requires multiplication, or they are not a multiplicative thinker, then they need to do some of the early part of Book 6: Teaching multiplication and division to help them move from additive to multiplicative thinking. They could list the numbers out, circling each 5th number,
e.g., 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

This could lead onto skip counting. They could then draw a diagram of what the garden looks like for part a) if each child planted a separate row. This can then lead to a discussion that the answer is 5 + 5 + ... + 5, i.e. the repeated addition model for multiplication.

Counts all the objects that they draw in a diagram
These students are still at a counting stage. If they display a rectangular array, then they have this physical model of multiplication which could move them onto skip counting or repeated addition as indicated above.

Skip counting or repeated addition error
Counts one group of objects short or over. These students need to keep track of the number of additions or skips they have done. This could be done by writing down the number of additions or skips they have done. For simple problems such as these, students could tally the number of skips on their fingers, but some annotation is better.

Performs an inappropriate multiplication
Students need to re-read the problem paying close attention to all the words used. They may have ignored the word "each" reading the problem as "Four children planted 8 carrots", or "Three children planted 12 cabbages". Discuss with them how important the word "each" is, as it changes the whole sense of the question. Mathematics often requires precise close reading skills.

Insufficient or incomplete working
Students who showed insufficient or no working may not have experienced the need for working or not believe it is important to show working if they know the answer. Developing the ability to make a mathematical statement (such as how they worked it out) is a valid part of the mathematics learning. It can also provide significant formative information about their learning needs.

Students who showed their working as repeating or summarising information (e.g., 6 x 5) from the question may need to discuss what constitutes working (or an argument), or even that working is their rationale for their answer. Students could share their working and identify if it is sufficiently explicit to be a justification for the answer they have given. As the use of multiple strategies is an important aspect of contemporary mathematics pedagogy (as well as a key indicator of an advanced stage in the Number Framework), value needs to be placed on encouraging students to communicate their strategies, both orally (see whole class discussion), and in writing.

Strategies

• Students who used the more sophisticated strategies of partitioning or doubling (that are higher up the Number Framework) have correspondingly higher mean abilities.
• Students who gave an expression (e.g., 6 x 5 or 5 x 6) had mean abilities close to those using partitioning of doubling strategies, except for part c). This indicates that although these students are using basic facts rather than strategies, especially for parts a) and b), they may well be able to suggest appropriate strategies.
• Students using repeated addition, skip counting, or diagrams had similar mean ability. The exception was those using repeat addition in part c) had a higher mean ability.
• The success rates for different strategies varied between 80-90%, regardless of their sophistication.

Click on the link Analysis of strategies [pdf] for more details of strategy use.