Often when Mathematics is taught, the focus is on processing. For instance, we teach students how to multiply two digit numbers, or how to simplify a fraction.
To solve Mathematical Problems though, there is a lot more to it than just having processing skills. Students need to be able to read and understand the problem. They need to be able to think through what they know about Maths and pick out the appropriate strategies to solve a problem. They have to apply these strategies (processing) and finally check that they have answered the question properly.
Due to the fact that there is so much involved in solving problems, when a student gets an answer wrong, it is often difficult to know whether they have a problem with their processing, or whether they made a mistake at some other point.
In 1977, Anne Newman suggested a series of prompts to help teachers to determine where students are making mistakes in problem solving. In this blog, I will outline these prompts and discuss how I use them to diagnose student’s issues with problem solving. I will also point out the benefits of using these prompts to improve your students’ ability to use and understand mathematical language and strategic thinking.
The first prompt is “Please read the question to me. If you don’t know a word, leave it out.” I tend to say, “Read the question out loud. Are there any words you don’t understand?”
I find that if students stumble when reading the question out loud, it generally means that it doesn’t make sense to them.
One other thing I’ve found is that even when students say that they understand all of the words, what they think a word means and what it actually means are sometimes quite different. For this reason, it can be helpful at this stage to ask students what the key words mean.
The next prompt is, “Tell me what the question is asking you to do.” I tend to say, “Put the question into your own words.”
I find that students seem to struggle with this step when you first start using Newman’s prompts. Even students who are quite proficient problem solvers can struggle here. The prompt is, however, a great literacy exercise and students get better at this step the more they do it. This step is also critical, because it can make it really obvious when students have missed the whole point of the question.
The third prompt is, “Tell me how you are going to find the answer.” I tend to also ask, “What do you know that could help you to answer the question?”
As with the second prompt, students often struggle with this when they start using Newman’s prompts. That’s because many children do not pre-think their working and often use trial and error or whatever first pops into their head!
Apart from being a great tool for identifying students’ issues, this prompt is useful because it trains students to pre-plan their solution. It gets them to slow down a little, to use their reasoning. To consider that there are strategies involved in solving problems and that some will work more easily than others. By forcing students to explain and justify the strategy they’ve chosen, we’ve also given them an opportunity to use language to communicate mathematical ideas.
The fourth prompt is, “Show me what to do to get the answer. Talk aloud as you do it so that I can understand how you are thinking.”
Of course, by this stage you will already know whether students have a strategy to solve the problem. This is just the processing step, where we see whether they can actually do the Maths.
At this point, you may see ‘silly’ mistakes. You should also be able to gauge how quick or slow students are at their processing and see their mathematical confidence (or lack of).
As with the previous step, this prompt requires students to explain their thinking and gives them an opportunity to use mathematical language.
The final prompt is, “Now write the answer to your question.” I tend to say, “Re-read the question and check that you’ve answered it correctly”, or “Check that your answer makes sense.”
I don’t find this prompt to be all that necessary as a diagnostic tool, as it is fairly obvious to teachers when students have made a mistake at this point. This step is great for students though, as it forces them to look back and check what the question was asking them to do. Getting students to think about this step at the end of every problem alleviates the ultimate ‘silly’ mistake of using amazing strategies and reasoning but leaving out a small and vital piece of information right at the end!
In summary, Newman realised that when a student makes a mistake in problem solving, it is not necessarily because of flaws in their processing skills. Students could also have made an error in the reading or interpretation of the question, the identification of which strategies to employ, or in the final communication of their answer.
By using Newman’s prompts, you will be able to identify where your students are going wrong in their problem solving, so that you can accurately assess where they need extra assistance. The regular use of the prompts will also give your students a good framework for solving problems and will give them great opportunities to both understand and use mathematical language more effectively.
Prompts used with permission from Anne Newman.