5: DUOLOG MATHS: METHOD
Section Contents:
Just as Paired Reading is a set of generalised tutoring behaviours for reading which can be applied to any book (Topping, 2001b), there was a need for a set of generalised tutoring behaviours which could be applied to any mathematical problem. Mathematics is a very wide ranging area, and this proved considerably more difficult for mathematics than for reading.
However, after a study of successful tutoring behaviours by professional teachers, a set of (much simpler and complementary) tutoring behaviours suitable for use by non-professionals was designed. This system was termed "Duolog Maths". A duologue is a dialogue between two people, and the title was selected as the preferred name by a large group of teachers in the USA - hence the American spelling.
A flowchart indicating the coaching behaviours identified in the study of teacher behaviour will be found below. The behaviours are arranged here in a logical and strategic order. The behaviours indicated by an asterisk (*) were those considered appropriate for inclusion in a system designed for use by non-professionals. (The definition of the others need not concern us here.)
The resultant flowchart for the Duolog Maths system (see below) includes only these behaviours, again arranged in a strategic and sequential order. It is intended that this flowchart be used in training tutoring pairs and then a copy be given to each pair in order to remind them of the interactive flow required of them. The following page then gives more detail concerning the definition of each tutoring behaviour. It is intended that this be photocopied onto the back of the flowchart, for reference by tutoring pairs.
A listing of "More Tips for Tutors" is included below. Teachers will need to use this with considerable care, given the dangers of information overload for the participants. However, this forms a useful list for reference by professionals, who might select single items from this list for discussion with participants in the light of particular problems which seem to be emerging from the monitoring of the interaction between the pairs.
Duolog Maths - Tips for Tutors
- Listen - give your tutee time to struggle to explain what their difficulty is. Do not just jump in to fix what you assume their difficulty is.
- Read - your tutee might be having trouble reading a word problem. If so, read it for them and check their understanding
- Question - ask helpful and intelligent questions which give clues, to stimulate and guide student thinking, and challenge their misconceptions. Examples: "what kind of problem is this?"; "what are we trying to find out here?"; "can you state the problem in different words or a different way?"; "what important information do we already have?"; "can we break the problem into parts or steps?"; "how did you arrive at that?"; "does that make sense?"; "where was the last place you knew you were right?", "where do you think you might have gone wrong?", "what kind of mistake do you think you might have made?". Do not say "that's wrong!" - ask another question to give a clue. Ask "why?". Try to avoid: closed questions which require only a yes or no answer; questions which just rely on memory; questions which contain the answer; the question "did you understand that?". Try to avoid answering your own questions. Avoid indicating the "difficulty" of any step.
- Pause for Think-Aloud - give your tutee some thinking time, before expecting an answer. Encourage them to tell you what they are thinking all the time. Then you will find out where and how they are going wrong. Remember tutors need time to think, also! If you are not sure, say so. You are not supposed to know everything.
- Make It Real - try to make the problem seem real and related to the life of your tutee. Ask the tutee try to imagine what the problem would look like in real life. Encourage them to use fingers, counters, cubes, sticks or any other objects to show the reality of the problem. Or have them draw dots, a picture, a list, table, diagram, graph or map. Useful charts include a number line, a multiplication matrix, and a place value chart. With your tutee's permission, mark their written working out with lines, arrows, colours, or numbering to help them. Have the tutee think of what they have learned before or problems they have solved before, relevant to the current problem. Work through a similar but simpler problem. How can this kind of problem be related to people, places, events and experiences in the home/community life of the tutee? Or those of someone they know or have seen on television? Make up a similar problem using the student's own name. Try to use everyday language
- Check - check that your tutee eventually gets the right answer. But remember there is probably more than one "right" way to solve the problem. Only if all else fails show your tutee how you would do it (while you think aloud).
- Praise & Encourage - give your tutee praise and encouragement very often, even for a very small success with a single step in solving a problem. Keep their confidence high.
- Summarise & Generalise - have your tutee summarise the key strategies and steps in solving the problem. Point out any errors or gaps, then summarise the key strategies yourself. Talk about how these might be applied to another similar problem (generalised).
Two additional more detailed prompt Sheets are provided, one for the "Questions" behaviour and one for the "Make It Real" behaviour. These would not be issued at initial training, but could be used subsequently as necessary at the teacher's discretion. They could also be printed back-to-back as required.
DUOLOG MATHS Questions Prompt Sheet
Remember:
- It's OK for helpers to say "I don't know" or "I'm not sure"
- Give clues or prompts, in questions
- Do not do it for your tutee
- Do not just give your tutee the answer
Example Questions
- What kind of problem is this?
- What are we trying to find out here?
- Can you state the problem in a different way/words?
- What important information do we already have?
- Can we break the problem into parts or steps?
- How did you arrive at that?
- Does that make sense?
- Where was the last place you knew you were right?
- Where do you think you might have gone wrong?
- What kind of mistake do you think you might have made?
DUOLOG MATHS Make It Real Prompt Sheet
Remember:
- It's OK for helpers to say "I don't know" or "I'm not sure"
- Give clues or prompts, in questions
- Do not do it for your tutee
- Do not just give your tutee the answer
Ways to Make It Real
- Imagine what the problem would look like in real life.
- Use fingers/counters/cubes/sticks to show the problem.
- Draw - dots, a picture, a list, table, diagram, graph or map.
- Use charts - number line, multiplication matrix, place value chart.
- Mark written working with lines/arrows/colours/numbering.
- What relevant has the tutee learned before/what problems solved before?
- Work through a similar but simpler problem.
- Relate to people, places, events in the life of the tutee.
- Relate problem to someone/something they know or have seen on television.
- Make up a similar problem using the student's own name.
DUOLOG MATHS More Tips for Tutors
Your aim is to help your tutee:
- see the use and value of maths
- believe they can be good at maths
- believe effort and practice makes a difference
- by showing interest in maths yourself
- have some fun with maths!
Remember:
- Very few people are brilliant or hopeless at maths - most of use are somewhere in the middle, and it's a bit of an effort for all of us.
- With successful practice, maths becomes less of an effort - but then we go on to harder problems, so it still seems an effort!
- Doing maths together can make it seem less of an effort.
- Doing maths together helps real understanding.
- There is a big difference between doing sums, doing problems, and doing maths in real life.
- Doing maths together helps to relate maths to the real world.
How did your tutee go wrong? Was it because:
- They just did not see what the aim or goal was?
- They lacked confidence and begin to panic?
- They could not read accurately?
- They did not pay attention to the information?
- They could not select the important information?
- They could not put the relevant information in the right order?
- They could not categorise the relevant information?
- They could not estimate or guess to roughly check ?
- They could not see what the solution would look like?
- They did not know a maths fact?
- They did not understand a maths idea or operation?
- They did not understand how two maths ideas linked together?
- They did not understand how to use maths facts and ideas?
- They could not select which maths facts and ideas to use?
- They could not select in which order to use maths facts and ideas?
- They could not transfer maths from one context to another?
- They were careless in counting or calculation?
- They failed to check on themselves?
- They made mistakes in copying or writing down?
When your tutee goes wrong, you might want to:
- Signal that an error has been made
- Indicate where an error has been made
- Indicate the type of error that has been made
If all else fails:
- Show your tutee by thinking-aloud through your way
- Remind them that this is only your way
- Then talk about it
- Then have your tutee try it their way
- Check again later that they can still do it by themselves
About
Evaluation of the Duolog procedure at the time of writing has been with parents rather than peer tutors. This was a controlled study in Glasgow with P4 to P7 pupils who were weaker mathematicians. Pre-post curriculum-based mathematics tests showed the experimental group made significantly greater gains than the control group (Topping, Kearney, McGee & Pugh, 2001 - see section 7 of this manual).