2013 newmans error analysis and comprehension strategies
1. Newman’s Error AnalysisNewman’s Error Analysis
The Australian educator Anne Newman (1977)
suggested five significant prompts to help
determine where errors may occur in student’s
attempts to solve written problems.
From reading to processing: Using Newman’s 5 promptsFrom reading to processing: Using Newman’s 5 prompts
Western Sydney Region 2013
2. Making mistakes
Students can make mistakes in answering numeracy
questions for different reasons.
Newman’s research elaborated the different hurdles
in answering a contextual word problem that can
cause students to stumble. Perhaps the two hurdles
currently best known by teachers are reading and
comprehension.
3. The hurdlesThe hurdles
Newman identified that students may have
difficulty in
• reading the words,
• understanding what they have read,
• transforming what they have read so as to be able to form a
course of action,
• following through on procedures,
• encoding the result of a procedure to answer the question.
READING
COMPREHENSION
TRANSFORMATION
PROCESS SKILLS
ENCODING
4. Research carried out in Australia and Southeast
Asia suggests that about 60% of students’ errors
in responding to written numeracy questions
occur before students reach the process skills
level.
In contrast, most remediation programs focus
solely on the process skills.
Food for thought…
5. 1. Reading
• Can students read the words of the problem?
‘Read the question twice. Circle important information. If you don’t know a word leave it out or
substitute another word.’
Newman’s Prompts
Overview
2. Comprehension
• Can students understand the meaning of what is read?
‘Tell me what the question is asking you to find out.’
3. Transformation
• Can students determine a way to solve the problem?
‘What could you do to get the answer? Predict what the answer will look like. Do you need to draw
a picture to help you understand?’
4. Process Skills
• Can students engage in the mathematical process?
‘Try to answer the question and explain to me what you are thinking. Check your answer with
another strategy.’
5. Encoding
• Can students record and interpret their answer in relation to the problem?
‘Write down your answer. Does it make sense? Have you answered what was being asked?’
6. Newman’s Prompt 1 - Reading
Mr Left had 8 apples. His wife
was given 3 apples. How many
apples did Mr Left have?
Many students responded by doing 8
– 3 = 5 or 8 + 3 = 13 Why?
Some students do not have effective
problem solving strategies and guess
or
focus on a key word, e.g. ‘left’
There are 26 sheep and 10
goats on a ship. How old is the
captain?
76 out of 97 students did one of
the following:
26 + 10 = 36
26 – 10 = 16
26 x 10 = 260
26 ÷ 10 = 2·6
Research Bernard Tola, CLIC DEC
We need to read for meaning.
7. 1 Please read the question to me.
If you don’t know the word leave it out.
( highlight any word you do not know)
The problem
• Many students cannot read the words correctly.
• Students may not understand the meaning of
words.
Teaching strategies
• Students can work with a reading partner.
• A team approach to reading may help students
overcome this difficulty.
• Have mathematics word charts displayed in the
classroom.
8. Newman’s Prompt 2 - Comprehension
• Ensure the students
understand the
vocabulary
• Familiarise students
with the text features
• Teach students how to
identify and use key
mathematical language
• Use known literacy
strategies such as:
literal (here)
“reading ON the lines”
interpretive (hidden)
“reading BETWEEN the lines”
inferential (head)
“reading BEYOND the lines”Henry the octopus has 6 spots
on each of his legs, how many
spots has Henry?
9. • What was the
most/least popular
food?
• How many children
bought fruit?
• How many more children chose a salad
sandwich over a cheese sandwich?
• How many children are represented in
the graph?
• If there are 100 children in the school
how many did not choose a food lunch?
• What season do you
think this survey was
taken in?
• Explain why you
think this.
10. 2. Tell me what the question is asking
you to do.
• Most students can find and underline the
question.
• This does not mean that they understand the
question.
11. Comprehension - Super Six Strategies
Comprehension strategies are the cognitive and metacognitive strategies
readers use to accomplish the goal of comprehension.
Comprehension strategies are interrelated and will rarely be used in
isolation.
The six key strategies are:
•Making connections
•Predicting
•Questioning
•Monitoring
•Visualising
•Summarising
12. Six metacognitive comprehension
strategies
Making Connections
Learners make personal connections from the text with:
• something in their own life (text to self)
• another text (text to text)
• something occurring in the world (text to world).
•What do I already know about the problem?
•Have I done similar problems before?
13. Six metacognitive comprehension strategies
Predicting
Learners use information from graphics, text and experiences to anticipate
what will be read/viewed/heard and to actively adjust comprehension while
reading/viewing/listening.
•What operation is the problem asking me to do?
•Will my answer be bigger or smaller than the numbers in
the problem? Why/ why not?
•What words or visuals do I expect to see in this problem?
•Were my predictions accurate? Why/ why not?
14. Six metacognitive comprehension strategies
Questioning
Learners pose and answer questions that clarify meaning and
promote deeper understanding of the text. Questions can be
generated by the learner, a peer or the teacher.
•What information in the problem helps me know
what to do?
•When you read the problem did it remind you of
anything you know about or have done before?
15. Six metacognitive comprehension strategies
Monitoring
Learners stop and think about the text and know what to do when
meaning is disrupted.
•Is the problem making sense?
•Do I need to re-read/view/listen to the problem?
•What can help me fill in the missing information?
16. Six metacognitive comprehension strategies
Visualising
Learners create a mental image from a text read/viewed/heard.
Visualising brings the text to life, engages the imagination and uses
all of the senses.
•How can I represent this problem visually?
•Can I describe the visual I have made for the
problem?
•How did the visuals help me understand the
problem?
17. Six metacognitive comprehension strategies
Summarising
Learners identify and accumulate the most important ideas and
restate them in their own words.
•What are the main ideas and significant details in
the problem?
•If you were to tell another person how you solved
the problem, what would you tell them?
•What information will help you solve the
problem?
18. Newman’s Prompt 3 - Transformation
To overcome this hurdle children need to:
• Construct a simple visual representation of the problem
eg. teach them to draw a Tape Diagram as a thinking tool
• Have the problem modelled several ways, as good comprehension of
mathematical texts aids transformation
• Write own problems – builds familiarity of features of word problems
• Have distractors in problems so they are forced to explore a range of
problem solving strategies
• Have text altered for different operations and have students identify key
words and clues that help select operations
• Explicitly discuss the visual clues within the tape diagram
19. 3. Tell me how you are going to
find the answer.
• At this point the function of understanding takes over from
the function of reading.
• The link between comprehension and transformation is very
strong and may require the student to go through several
cycles of asking the question and trying to find ways to find
the answer.
The problem
• The density of mathematical language. There is a lot packed
into a small number of words.
Teaching strategies
• Read the first sentence. What does it mean? What do you
know?
• Make mindmaps or charts to assist students develop the
language of mathematics.
20. What’s the problem with word problems?
• What we want students to think about is how the numbers in
a problem relate to each other.
• To achieve this we often encourage students to draw a
diagram.
• Tape diagrams can be used to provide a common framework
in using diagrams as thinking tools.
• A tape diagram offers students a thinking tool to visually
represent a mathematical problem and transform the words
into an appropriate numerical operation.
21. To get to work, I travelled on the train for 14 minutes, then
I caught a bus which took half an hour.
Finally I walked for 9 minutes. How long did it take me to
get to work?
Teaching Transformation – Tape Diagrams
14 minutes Half an hour 9 minutes
53 minutes
22. Using tape diagrams
There were some oranges in a box.
Because we bought 14 more oranges,
there are 21 oranges in the box altogether.
How many oranges were in the box at first?
Number of
oranges at first 14 more oranges
were bought
21 oranges altogether
+ 14 = 21
= 21 – 14
or
23. Newman’s Prompt 4 - Processing
• Count Me In Too Resources
• DENS Books
• Counting On
• Sample Units of Work
• Talking about Patterns and Algebra
• Red Dragonfly Maths
• Fractions Pikelets and Lamingtons
• Programming support website
• Reciprocal Numeracy
These programs and strategies help us teach processes with a
focus on mental computation and problem solving.
24. 4. Show me what to do to get the answer.
Tell me what you are doing as you work.
The problem
• Students can sometimes find or guess the correct answer
without understanding the problem. They can also use
an incorrect or incomplete procedure.
Teaching strategies
• Ask “What do you need to know to be able to work out
the answer?”
• Make lists or diagrams linking words that are associated
with a process.
25. Newman’s Prompt 5 - Encoding
• Does it make sense?
• Does it answer what the question is asking?
• Think of ways of recording the answer.
• Have you used the correct units?
26. 5. Now write down the answer to
the question
The problem
• Students often forget the question, especially
in measurement questions and do not
answer all of the question or answer using
an incorrect unit.
Teaching strategies
• Have students reread the question and
compare it with their answers.
27. The 5 questions - RECAP
1. Read the question to me.
2. What is the question asking you to
find out?
3. What method did (could) you use?
4. Try doing it and as you are doing it tell
me what you are thinking.
5. Now write down your answer.
These can be used to diagnose where the problem lies
28. NEWMAN’S ERROR ANALYSIS
STAGE 2 ASSESSMENTS
Kate has 96 pencils. Each box
of pencils holds 10 pencils.
How many full boxes of
pencils can Kate make?
If pencils come in boxes of 15,
how many pencils are in 20
boxes?
1 READING
2 COMPREHENSION
3 TRANSFORMATION
4 PROCESS
5 ENCODING
6 No Problems
29. The first international cricket
team to tour England was an
Aboriginal team. The team won
14 matches, drew 19 matches
and lost 14 matches.
How many matches were
played?
Natalie paddled 402 km of the
Murray River in her canoe over
6 days. She paddled the same
distance each day.
How far did Natalie paddle
each day?
STAGE 3 ASSESSMENTS
33. Newman’s in the classroomNewman’s in the classroom
Where do you think James had problems?
Reading
Comprehension
Transformation
Process Skills
Encoding
35. Newman’s in the classroomNewman’s in the classroom
Where do you think Ricky-Ann had problems?
Reading
Comprehension
Transformation
Process Skills
Encoding
36. Newman’s method in the classroomNewman’s method in the classroom
In using Newman’s error analysis in the
classroom, it is recommended that you always
go at least one step past where the first error
occurs.
37. Want to find out more about NEA?
• http://www.curriculumsupport.education.nsw.gov.au/prim
• http://www.curriculumsupport.education.nsw.gov.au/seco
38. Enter your class data
into the NEWMANS
worksheet.
The NEWMANS
reporting worksheet
will automatically be
updated
Reading Comprehension and Transformation are significant hurdles, 66% of children breaking down at these points
What is the difference between cognitive strategies and metacognitive strategies? Cognitive strategies are mental processes involved in achieving something. For example, making a cake. Metacognitive strategies are the mental processes that help us think about and check how we are going in completing the task. For example, ‘Is there something that I have left out?’ Cognitive and metacognitive strategies may overlap depending on the purpose/goal. For example, as the cognitive strategies involved in making a cake proceed (following the steps in order), the metacognitive strategies assess and monitor the progress (to check that a step has not been missed). How does this relate to comprehension? Cognitive strategies assist in understanding what is being read. For example, predicting Metacognition is particularly relevant to comprehension. Metacognitive strategies allow individuals to monitor and assess their ongoing performance in understanding what is being read. For example, as a text is being read, the reader might think: I don’t understand this. I might need to re-read this part.
Learners make a personal connection from the text with : Something in their own life ( text to self) Another problem (text to text) Something occurring in the world ( text to world)
Learners use information from graphics, text and experience to anticipate what will be rad/viewed/heard and to actively adjust comprehension while reading/viewing/listening. Teachers/students might say: What do I/think the operation might be? Why? I estimate the answer to be around…. Why? My predictions was right because…? My prediction was incorrect because ..? Teaching idea Before and after chart: Students predictions before and during the problem. As they work through the problem they either confirm or reject their predictions.
Learners pose and answer questions that clarify meaning and promote deeper understand of the text (problem). Questions can be generated by the learner, a peer or a teacher. Students/ teachers might say: What in the text helped me/you know that? When you read/viewed the question did it remind you of anything you already know? What am I /you trying to find? Is all the information in the problem we need to be able to answer the question?
Learners stop and think about the problem and know what to do when meaning is disrupted – e.g. can’t read the persons name – Tujla Students/ teachers might say: Is this making sense? What have I/you learned? Do I need to re-read/ use a known name etc? What does this word mean? Teaching idea Coding – I understand, I don’t understand/I think I have worked it out
Learners create a mental image from a text. Visualising brings the text to life, engages in the imagination and uses all of the senses. Teachers/ Students may say: What is the picture I/you have in my/your head as I/you read the text? Can I/you describe the picture or image you made while you read/heard that part? Teaching idea Sketch to stretch: as student reads problem students sketch their visualisation. In groups they share their sketches and discuss reasons for their interpretation.
Learners identify and accumulate the most important ideas and restate them in their own words. Teachers/students may say: What thing will help me/you summarise this text – list, mind map, note-taking, annotations etc.? What are the main ideas and significant details from the reading/viewing/listen? Be careful not to use the term key words (See Below) If you were to tell another person about the problem in a few sentences what would you tell them? Teaching idea Have students discuss the difference of the following: Prepositions can change meaning Increase by 7 Increase from 7 Increase to 7 The order of the words can change the meaning 60 is half of what number? Half of 60 is what number? Using key words can be detrimental as they are not always used to convey the same meaning, ‘and’ is often associated with addition, ‘left’ with subtraction. Some words can be classified to indicate a ‘gain’- bought, found, took, got or to indicate ‘loss’- lost, gave, left, sent The order in which information is presented in language is often at odds with the order in which it is processed in mathematics Thus, often the seemingly insignificant words such as ‘to’, ‘by’ ‘of’ are vitally important in creating meaning. In mathematics every word is important and this is why a KEY WORD APPROACH to reading in mathematics is often counterproductive.
Often children can read and comprehend what the question is asking but have no idea what to do Knee to knee , think pair share and other strategies advocated in literacy can be used during discussion. Transformation: How can we build student ability to transform mathematical texts into mathematical processes? Good comprehension of the text vocabulary language of maths understanding of text features of mathematics texts As stated before, we need to build in these students: A range of effective problem solving strategies The ability to choose from these strategies Build towards more abstract strategies Create classrooms where learning to read maths problems occurs frequently and where solving problems is the focus of mathematics lessons
Approaches to problem solving can include any or a combination of many strategies such as act it out, make a table or list, guess and check, look for a pattern, work backwards, use logical reasoning, draw a diagram, brainstorm, simplify the problem, etc. Tape diagrams are an example of the draw a diagram strategy.
This is how Caera represented the problem using a tape diagram.
This slide has been animated to show how each part of the problem is represented in the tape diagram. On each click of the mouse button (or forward arrow) the next part of the problem will be displayed, then after a 1 second delay the tape diagram will be automatically updated. This can be modeled in the classroom with prepared components of the tape diagram.
Encoding: Discuss possible ways of recording the answer
It is not mandatory to do both questions, however, you can if you want to. Use your own judgement to choose the most appropriate one for your students where you will garner the most useful information
Take the role as a student ; Follow the reading prompt – tick the box Complete the comprehension prompts – write the question in the box Transform the problem into a tape diagram using one bit of information at a time Complete the processing in the processing box and check using inverse operations Encode the answer down the bottom Don’t forget to update the spreadsheet each time you test
What could students do in the classroom to solve this problem? Suggestions Make a drawing Draw a table
This was Rita’s response to this item. Not unusual! When I discussed the problem with Rita, this is how she solved the problem using a tape diagram. Knowing about tape diagrams enabled me to have a common ground with the student upon which to discuss this problem.
Did JAMES have difficulty reading the task? NO Did she have difficulty understanding it? NO Transforming it into a mathematical operation or sequence of operations? YES Was she able to carry out these operations accurately? Was she able to express the solution in an acceptable written form? After the interview, I felt quite sure that if Eugenia had been asked to work out the algorithm 402 6, she would have been able to do this, but she was not able to choose and use division in a problem solving context.
Did Eugenia have difficulty reading the task? NO Did she have difficulty understanding it? NO Transforming it into a mathematical operation or sequence of operations? YES Was she able to carry out these operations accurately? Was she able to express the solution in an acceptable written form? After the interview, I felt quite sure that if Eugenia had been asked to work out the algorithm 402 6, she would have been able to do this, but she was not able to choose and use division in a problem solving context.
To find out more about using NEA in the classroom, requires another workshop. You can begin by looking in the Curriculum Support website, where strategies are provided to assist students who are identified using NEA as having difficulties with Reading, Comprehension or Transformation. You can also view there a student who works successfully through the five steps of NEA.
