Self-regulated learning in the mathematics class
Researchers: Charles Darr and Jonathan Fisher
Context: proportional reasoning (Year 7)
Two researchers taught a series of lessons on proportional reasoning to a class of Year 7 students over a 4-week period. Classroom interactions were videotaped and five students interviewed before and after the unit of work. Two instructional components were identified as promoting the kinds of attitudes and behaviours that support SRL. They were:
· using thinking models to represent proportional problems
· reflecting on learning in written journals.
Using thinking models to represent proportional relationships
Thinking models help students to form a representation of a problem situation. They can involve concrete objects, or be more abstract. Thinking models used included double number lines, geometrical shapes, cuisenaire rods, and decimal pipes.
The double number line
One of the most successful models was the double number line. The double number line allows the elements in a proportional relationship to be modelled on a two-sided scale.
The double-number-line was introduced to students through a series of hands-on activities involving two-litre milk containers. As a thinking tool it:
· elicited thinking and supported discussion
· helped students to illustrate proportional relationships
· allowed students to concentrate on observing and controlling the problem solving situation rather than holding all the details in their minds
· helped students to become familiar with strategies that could be used to solve similar problems. /
Journalling
Writing journal entries allowed students to reflect on the experiences, ideas, and feelings involved in their mathematics learning. Journalling was used on six different occasions and was often initiated with a prompt from the teacher. One example was:
"Write some instructions to a Year 5 explaining how to find two thirds of 60."
We found that using journalling:
· allowed teachers to identify emerging thinking
· began learning conversations between teachers and students
· allowed teachers to give feedback
· helped focus lesson planning
· sometimes resulted in students sharing their journal entries with others, allowing them to observe other ways of thinking and explaining concepts.
Providing opportunities for students to observe and reflect on their own thinking is critical if they are to develop the ability to self-regulate in mathematics. Integrating thinking models and reflective journalling into the normal flow of mathematics instruction can help students become aware of their own ability to organise thinking and learn mathematics.
To read more, see this PDF document.
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Journalling in mathematics
Alex Neill (2005)
What is journalling?
The benefits of journalling
Teaching tips for journalling
Writing prompts for journalling
References
What is Journalling?
Journalling involves students writing about their learning in mathematics. What they write can be based on a prompt given by the teacher – for instance, on a topic currently being covered in class. Alternatively, it can be more self-directed, with students choosing from a menu of prompts, or simply free-writing about their thoughts, feelings and ideas. Teachers will often read students’ journal entries and respond with comments or questions. Sometimes this can evolve into an ongoing written conversation between student and teacher. Journalling can happen before, during, or after a session of mathematics learning.
The Benefits of Journalling
Journalling has benefits for students, teachers and the mathematics programme as a whole. These include: promoting understanding, promoting a sense of involvement with mathematics, promoting better informed teaching and promoting teacher student relationships.
Promoting Understanding
Journalling promotes understanding by building on the powerful links between writing and learning. According to one researcher:
... writing can engage all students actively in the deliberate structuring of meaning: it allows learners to go at their own pace; and it provides unique feedback, since writers can immediately read the product of their own thinking on paper" (Emig paraphrased in Borasi and Rose,p.384).
Journalling in mathematics allows students to:
· establish connections between mathematical ideas. For instance, by asking students to explore and write about various strategies for solving similar problems;
· focus on what they do and do not know;
· develop more precise ways of communicating;
· clarify, organise, and refine their thinking;
Promoting a Sense of Involvement with Mathematics
Researchers report that journalling can help students to deal with the struggles that are often involved in learning and doing mathematics. It can also allow students to examine the nature of mathematical knowledge and what it means to think mathematically.
Journalling promotes a sense of involvement with mathematics when it encourages students to:
· summarise goals, strategies, and reactions to mathematics;
· record their accomplishments and openly vent frustrations;
· consider the relevance of mathematics in their lives.
Promoting Better Informed Teaching
When teachers read their students’ journal entries they are able to gain a new perspective on how individual students are coping with their mathematics learning. This information can be used to develop next learning steps.
Journalling promotes better-informed teaching by providing teachers with:
· formative data about students' understanding and dispositions, including students’ emerging ideas about mathematical concepts;
· a basis for learning conversations (written and oral);
· an opportunity for feedback and feed-forward;
· information that assists in short and long-term planning.
Promoting Teacher Student Relationships
Journalling can also help teachers to make deeper connections with their students. It does this by:
· enabling students to actively participate in a dialogue with teachers though a private, non-threatening means;
· allowing students to share ideas, opinions and feelings about mathematics, the mathematics programme and their own learning styles;
· providing a permanent record for the teacher and student to examine.
Teaching tips for Journalling
Careful management is needed to maximize the potential of journalling. The following tips provide useful advice for using journalling as a learning technique in the classroom.
1. / Classroom organisation is important. Teachers need to have a manageable system, especially if they are going to take part in ongoing journal conversations with students. This might mean responding to one group of students at a time, rather than the whole class in order to give deeper levels of feedback.2. / Daily journal writing is not a necessity. Journalling should be used strategically to complement the classroom programme.
3. / Journalling is a form of transactional writing. Students will need time and scaffolding to develop as journal writers. Teachers can assist students by modeling examples of journal entries.
4. / Like any classroom innovation, the success of journalling depends on the quality of student teacher relationships and the belief and commitment that both parties have to the journalling process. In particular, students need to know the benefits of writing in a journal.
5. / Provide feedback on what is important. Commenting on writing conventions such as spelling and paragraphing, is not as important as providing feedback on mathematical thinking.
6. / Students should look back and reflect upon their own journal writing experience. This may be done orally, or in writing. There is evidence that such reflection makes the students more aware of themselves as learners. Click on Reflecting on reflective journalling for a further discussion.
Writing Prompts for Journals
Journalling tasks fall into two distinct groups, self-directed, and teacher directed.
Self-directed journalling
Free-choice journalling is where students choose to write about whatever they wish. This can be informed by a checklist of possible starters. Such lists can be developed within the class. One particular list is referred to on ARB resource NM1226. The items on this list are not meant to be exhaustive. The teacher may encourage students to self-select different aspects from this checklist over time so that they reflect more widely on their learning.
Possible prompts for self-directed journalling
1. / What I learnt today in mathematics- What mathematics did I learn?
- Describe how a method works.
- What are my opinions on the mathematical ideas I learnt today?
2. / How I felt learning mathematics today
- I was feeling …
- What did I enjoy or find good?
- What didn't I like?
- My feelings about mathematics
- My favourite thing in maths is?
3. / Things that I need help more on
- I find … hard
- I need to practice …
- I need to remember …
- I need to improve at …
4. / Things that help me learn, or about my learning
- Where did I learn new ideas from?
- How do I learn best?
- How is my learning progressing?
5. / Things that were useful or interesting
6. / Things I found easy or I already knew
7. / Did I have an "AH HA" moment
8. / Any other ideas you would like to write about.
Teacher-directed journalling
Sometimes the teacher can choose the prompt or topic for journal writing. This might be a particular aspect taken from the journalling checklist above, or instead center on various aspects of the mathematics that the students are currently encountering. Some general suggestions follow, along with particular examples.
Possible journaling prompts regarding mathematical content
1. / How it works: Students explain how a strategy works.For example: Write instructions to a friend on how to do front-end estimation. (Go to the ARB resource NM1217 for more information about this example.)
2. / How-to’s: Students explain in writing how to do something.
For example: Someone tells you that two thirds of the 80 cars in the parking lot are green. Is it possible to work out how many cars there are in the parking lot altogether?
How do you find 25% of a number? (Note the student is asked for a general explanation, that is how to find 25% of any number rather than 25% of a specific number.)
3. / Definitions: Students write their own definitions of terms.
For example: What is a percentage?
When are two lines parallel?
What is a triangle?
4. / Trouble shooting: Students explain errors they or their teachers have identified.
For example: Joseph was counting up in 0.1s. He went, "0.5, 0.6, 0.7, 0.8, 0.9, 0.10." What mistake did Joseph make?
5. / Arguments: Students develop an argument to support a point of view.
For example: Think of two ways to find ¾ of a number. Which one do you think is the best and why?
6. / Problem Posing: Students construct (and model answers for) their own problems.
For example: Write a problem, which involves finding the area of a shape.
7. / Pre-writing: Students write before a maths session, looking forward to their learning.
For example: What can you do to day that will help you learn in mathematics.
References
Borasi, R., & Rose, B. (1989). Journal writing and mathematical instruction. Educational studies in mathematics 20(4): 347-365.
Free journalling – Teacher information
All strands - Any objective - All levels - Communicating mathematical ideasDescription
Students write about aspects of a lesson that they have just had. A list of possible starters is used to provide topics for the students to write about.
Keywords
Journalling; estimation.
Task administration
• / After conducting a lesson get the students to spend five to ten minutes writing. You may get the students to write in their maths books, or to use a book dedicated to journal entries.• / Show the students a copy of the possible starters. Emphasise that what they write about is their choice, and does not have to come from the list.
• / Encourage the students to focus on one or two of the starters.
• / Explain that they can communicate ideas to you through this writing.
• / Read the students’ journal entries. Note areas of strengths and weaknesses both for individual students and across the class. Try and give the students feedback, either in written form in their journals, or orally in a one-to-one conference or a group or class setting.
• / Get the students to repeat this exercise from time to time, either in their mathematics book, a journal or any other suitable place.
Teaching and learning
This task is designed to be conducted from time to time as part of a journalling programme that is run alongside your mathematics programme. For further information on journalling, click on any of these:
• / Journalling in mathematics• / Reflecting on reflective journalling
• / Self-regulated learning in the mathematics class
Examples of student responsesThe following responses are based on daily journals kept by nine Year 8 students during a unit on computational estimation. The series of lessons introduced students to a broad range of different estimation strategies. The items on the journaling starter list was compiled largely from these students’ own ideas of what they thought they may write about in their journals. For more information about estimation, click on Computational estimation information.
The percentages quoted show how many of each type of response was written. For example, comments about "The mathematics I learnt today" accounted for 46% of the self-selected journal entries.
The mathematics I learnt today (46%)
"I learnt a few new strategies and easier ways to solve problems."
"I learnt more about estimation and rounding."
"I learnt why we add zeros when we multiply by 10, 100, 1000 … "
"Today I didn't really learn anything new because I already knew [it]."
• / Descriptions of how strategies works (12%)
"I learnt an easy way to do 100 ´ 10. I switched one zero so it looks like 10 ´ 1000= 10 000"
"You can get an accurate but quick answer by rounding just one number e.g.
8682 – 6124 is about 8682 – 6000 = 2682."
Opinions about mathematical ideas (4%)
"I prefer to use front end (rather than rounding) on big numbers or on lots of numbers."
"I like front-end because it's quick and easy and you can do it in your head."
"I don't really like the front-end way of estimating."
How I felt today (17%)
Aspects that were good or that I enjoyed (8%)