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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
- 30. SNAP 2006 36 % and 14 % ATSI correct
- 31. Teaching Transformation with Newman’s Prompts – another example using a Tape Diagram
- 32. JamesJames
- 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
- 39. Lunch Time…
Notas do Editor
- 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.