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The Best Evidence Synthesis Iteration Effective Pedagogy in Mathematics (BES) Exemplar One says that engaging diverse students in mathematical inquiry can led to greater achievement for these learners.
A goal of this presentation is to demonstrate how as teachers we can engage students in mathematical inquiry to raise their achievement.
To achieve this we will look at pedagogical practices associated with the development of inquiry based practices that equitably support all learners to achieve in the mathematics classroom.
Group Work
Dominance of ability groupings
• We group in classes to allow for students to learn at their own pace and to provide added support to those that are struggling or need a challenge. • Streaming enables advanced students to move ahead and not
become bored with classroom activities.
Research on Ability Groupings• Wilkinsin (2000) states that ability grouping practices create different learning
experiences that seem to “perpetuate, or even exacerbate, inequalities among students” (pg. 462).
• Hunter (2011), Dweck (2014) and Boaler (2015) realise that ability grouping gives students’ labels, both in their own minds as well as in the minds of their teachers. Teachers then associate students' placement with the type of learners they are and therefore create different expectations for different groups of students (Boaler, 2015).
• Marks (2013) discusses the harmful effects of ability grouping in her article, The Blue Table Means You Don’t Have a Clue’. This is a quote from a Year 4 student.
“Mrs Ellery puts us into different groups and she moved me from here to here. This means [the green table] you are good at math’s, this [orange table] means you are half good at math’s, the blue table means you don’t have a clue (Marks, 2013)
The New Zealand Curriculum states that the Curriculum is for all students, no matter their ability. The 2015 February education review series made comparable statements. “There is a strong research base that shows that teaching students in ability groups has few, if any, benefits for learners. On the other hand, there are studies that have shown, that when supposed low-achieving students are placed with their high-achieving peers they are soon performing at much higher levels than previously” (Education Review, 2015)
Boaler (2009) p. 114 England – 88% of children placed into ability groups at age 4 remain in the same grouping until they leave school.
It is difficult to support a child’s development and nurture their potential if they are placed into a low group at a very early age, told that they are achieving at lower levels than others, given less challenging and interesting work, and separated from peers who would stimulate their thinking.
Reflecting on the use of grouping is an important consideration when thinking about teaching
mathematics and raising student achievement. Grouping as a mathematics education pedagogy needs
to be challenged as part of the socio-political justice agenda because it is the predominant “structure that sorts and labels children” in terms of their capacity to
learn (McDonald , 2013,pg 381).
Developing mixed ability grouping strategies
Social and Cognitive Payoffs
Collaboration and risk taking
Trust and respect
Mathematical proficiency
• Students learn to participate actively in small mixed ability groups.• Mutual respect, support, understanding and tolerance are developed
between students. • Competition is replaced by co-operation.• Students have a more positive attitude towards maths as their self-
esteem and motivation improve. • They develop and improve mathematical skills • Equality of opportunity and outcome are promoted.
Things that you do NOT like people to say and do when you are working in a maths group.• • Being told that I am the smart guy, so you do everything• Being told = I am stupid at maths• Being told = You don’t know that!• Being told – You are smart.• Not participating and helping the group.• Not listening when someone is talking.• Not being included.• Not being helped when you are stuck.• Not even trying because I have not done it before.• Having a negative attitude.• Having to tell the group what to do after they should already know,
or doing the task wrong because the group didn’t listen.• Someone telling me the answer before I have had time to figure it
out.• Not asking when you don’t understand and pretending you do.• Being corrected• Finding it hard but saying ITS EASY• Having an attitude that this is the answer and I am not willing to
listen to your ideas.• Not explaining how you got that answer so that I can understand.
Things that you do like people to say and do when you are working in a maths group. • Giving me time to work out the answer for myself• Being praised for what I do• People saying my way is great.• Having the whole group join in – working together• People helping me figure out the answer when I
get stuck• Being told I can do it.• Having people stop and explain it to me.• Being given credit for what I have tried.• Giving me a helping hand when I am stuck –
giving me a clue• Including everyone’s ideas
• I enjoyed working in my group. We all had a chance to share and I liked that I
could help the others to see the patterns I noticed. They could not see them like I do.
• I learnt from Lui how you need to look at both sides of the equal sign. I did not see that but he showed this with the blocks and I could see it.
• Working with different people was good. I got to work with some of my friends. Show them things I know about the equal sign. I like maths more now.
Student reflection
Group Roles Facilitator
Gets the team off to quick startMakes sure everyone understands the information on the task
card.Organizes the team so they can complete the taskKeeps track of time Substitutes for absent roles
“Who knows how to start?”“I can’t get it yet… can someone help?”“We need to keep moving so we can…”“Let’s find a way to work this out.”
Resource Manager
Makes sure the team is using all resources well, especially people.Calls the teacher over for a team questionCollects supplies for the teamCares for and returns suppliesOrganizes clean up
“I think we need more information here.”“I’ll call the teacher over”“We need to clean up. Can you… while I…?”“Do we all have the same question?”
Recorder / Reporter
Gives update statements on team’s progressMakes sure each member of the team records the dataOrganizes and introduces report
“We need to keep moving so we can…”“I’ll introduce the report, then…”“Did everyone get that in your notes?”
Reflection Leader
Helps the group reflect on their work during the task and at the end.Asks questions about the group’s activity:
“What strategies have we used?”“What worked?”“What isn’t working/didn’t work?”
1. Teachers need to be aware of the social process of math discourse and develop a shared perspective so that everyone in the group participates.
Focus on how students participate with each other. Do they actively engage in listening, discussing and make sense of what others
are saying? Do they understand the reasoning of other perspectives? Can they develop a collective view – does everyone in the group understand? Develop ways that allow them to disagree and challenge politely as well
justifying their position so they have ownership of their learning.
How to establish groups
2. Develop roles for the members in the group.3. Be aware of the different status (Hunter, 2007) that
students have. Position students so they have a voice and the confidence to use it.
4. Praise effort, not ability (Dweck, 2014)5. Use authentic open-ended tasks. These support the
notion that there are multiple ways that students can develop and support each other in the construction of explanatory reasoning and justification (Hunter, 2007, pg. 6).
6. Create prompts that students can use to ask questions
How to establish groups - cont
The effect of status
When students work in small groups the differences in status (not ability or motivation) shapes who talks, who others listen to, and who’s ideas direct what decisions are made. It is better to consider students as having low status instead of low kids, low achievers, struggling students because this means teachers need to look for more effective ways to open up the maths for all students
- All students participate.
- The responsibility for learning rests with the group.
- The responsibility for learning rests with the individual.
Talk Moves For Students
Talk MovesWe will use ‘talk moves’ to help us
share our ideas with the class
You are expected…• to explain and justify• to repeat what someone else has
said.• to agree or disagree• to question others
• We will use ‘talk moves’ to help us share our ideas with the class
• You are expected…• to explain and justify• to repeat what someone else has said.• to agree or disagree• to question others
Participation patterns support collaboration
scaffolding of questioningTeacher: If you don’t understand, what questions do you need? Sandra: I don’t understand, could you please repeat it? Teacher: If someone didn’t understand it though and the same thing was said to them…student responsibility to explain and re-representTeacher: Explain it in a different way, an easier way, or a clearer way. How did you work that out? Can you show us how you did it and what you used?
Small group collaboration
expectation of collective sense-making. Teacher: I want you to explain to the people in your group how you think you are going to go about working it out. Then I want you to ask if they understand what you are on about and let them ask you questions. Remember in the end you all need to be able to explain how your group did it so think of questions you might be asked and try out how you will answer them.
Student perception of math talkExpect inconsistency in responses•Not all students recognise the value of talking in the
maths classroom.• Some see it as useful because it exposes them to
different ways of thinking• Some find it frustrating because they are not sure how
to access the thinking •Varies across cultures and social groups
How can children’s talk support learning?
1. In presenting ideas students need to clarify and organise their thoughts.
2. Facilitates personal and collective sense making.3. Supports building connections between representations and
multiple strategies.4. Use others as a resource of ideas to challenge and broaden
understanding.5. Help students learn mathematical language.6. Sense of authority moves from teacher to discipline7. Support development of mathematical identity.8. Provides a resource for teachers – build on their thinking.9. Allows students to see mathematics as created by communities
of learners.
Math tasks
Tables and Seating Problem
a) At least 79 parents said they are coming to a meeting in our
hall tonight. They will sit at large tables that seat 5 people each. How many tables do we need? Are there any parents left standing?
b) At least 373 parents said they are coming to a meeting in our hall tonight. They will sit at large tables that seat 5 people each. How many tables do we need? Are there any parents left standing?
c) At least 1264 parents said they are coming to a meeting in our hall tonight. They will sit at large tables that seat 8 people each. How many tables do we need? Are there any parents left standing?
Comments:These problems are great for the big idea: Distributive Property
• What is 8 + 5? How can you use 8 + 2 to help you solve 8 + 5?• How can you use 3 × 7 to solve 6 × 7?• A friend is having trouble with some of his 6 times
facts. What strategy might you teach him?• Ella solved 6 + 8 by changing it in her mind to 4 + 10.
What did she do? Is this a good strategy? Tell why or why not. What strategy do you use to solve 6 + 8?
50/5 + 10 = 20 write a story to explain what happens here?
3 X 5 = 30/2 - true or false why?
Create your own stories using the 2,3,5 times tables.
9 x 0 = 0 Why?
Fibonacci Project 1,1,2,3,5,8,13,21,..
05/01/2023
Tyler: Why is a circle 360 degrees?
• The Sumerians watched the Sun, Moon, and the five visible planets (Mercury, Venus, Mars, Jupiter, and Saturn), primarily for omens. They noticed the circular track of the Sun's annual path across the sky and knew that it took about 360 days to complete one year's circuit. Consequently, they divided the circular path into 360 degrees to track each day's passage of the Sun's whole journey. This probably happened about 2400 BC.•
Sent from my iPad•
Questioning
Encouraging student talk and reflection
Summary•How do you give self and peer assessments?
• Setting a learning goal clarifies for students what they need to master (e.g., My goal is to understand the difference between mean and median and know when they should be used).
• Students assess peers' as well as their own progress towards the learning goal. For example, students complete assignments individually and then swap assignments, grade one another, and provide feedback.
• Students take more responsibility and are more aware of their learning.
•In a study, students who engaged in self and peer assessments did better than students who engaged in discussions.
• Students given the opportunity to do peer and self assessments outperformed students in a control group in three assessments, with low-achievers benefiting most. Low achievers behaved more like high-achievers, studying more effectively.
•For more information on self and peer assessments, visit nclrc.org.
Plan to collect evidence throughout the year which shows:
*students solving problems and modelling situations *what the students can do independently and most of the time *evidence from across strands/the curriculum
Students should be encouraged to identify their own best efforts where possible.
Planning
Big Ideas
Problem
Possible misconceptions
Likely solution strategies
Equipment
The Bag of Marbles Task
Strategy Who and What Order
FractionDetermine the fraction of each bag that is blue marbles (x is ¼; y is 1/3; z is 1/5). Decide which of the three fractions is larger (1/3). Select the bag with the largest fraction of blue marbles (bag y).
PercentDetermine the fraction of each bag that is blue marbles (x is 25/100; y is 20/60; z is 25/125). Change each fraction to a percent (x is 25%; y is 33 1/3%; z is 20%). Select the bag with the largest percent of blue marbles (bag y).
Ratio (Unit Rate)Determine the part to part ratio that compares red to blue marbles for each bag (x is 3:1; y is 2:1; z is 4:1). Determine which bag has the fewest red marbles for every 1 blue marble (bag y)
Ratio (Scaling Up)Scale up each bag so that the number of blue marbles in each bag is the same (e.g., x is 300 R & 100 B; y is 200 R & 100 B; z is 400 R & 100 B). Select the bag that has the fewest red marbles for 100 blue marbles (bag y).
AdditiveDetermine the difference between the number of red and blue marbles in each bag (x is 50; y is 20; z is 75). Select the bag that has smallest difference (bag y).
Other
The Five Practices Model
The five practices are:
1. anticipating student responses to challenging mathematical tasks;
2. monitoring students’ work on and engagement with the tasks;
3. selecting particular students to present their mathematical work;
4. sequencing the student responses that will be displayed in a specific order and
5. connecting different students’ responses and connecting the responses to key
mathematical ideas.
Problem: Fruit juice consists of twocups of concentrate for every three cups of water. If there are 240 campers and each camper has ½ cup of juice,how much concentrate and how much water will be required?
Big Mathematical Idea :Finding the Highest Common Factor can help us to solve Ratio, Proportion and Percentage problems.
Anticipated Strategies - least to most sophisticated
Names of children Stage Standard Equipment /diagrams to move students to the next level
Transitioning to mixed ability groupings within an inquiry classroom is a process that requires time and reflection.
Teachers must be patient, make mistakes and learn from them.
Learning outcomes to expect from this change.
Substantial progress in terms of academic achievement and student agency.
Students are able to use mathematical language to support
their explanations and to clarify their understandings of others’ explanations.
Mixed ability groupings will have benefited students in
social and behavioural areas. We will see improvements in student self-concept, social interaction, time on task, and positive feelings toward peers and maths.
An increase in the number of students achieving ‘at’ or
‘above' the National Standards as well as a decrease in the attainment gap between ethnicity and gender.
A raised achievement standard for Maori and Pacific
students who were over represented in the ‘below’ and ‘well below’ National Standards groups.
Teacher planning that includes deliberate, relevant and
authentic learning contexts based on student interests.
There are significant changes in teacher knowledge and pedagogy in using effective mathematical practices that promote students thinking.
A collaborative, school-based, professional learning
process that is ongoing.
Teachers that are effective and culturally responsive with good content and pedagogical knowledge, and have the willingness to inquire into doing things differently.
Talk Moves for Teachers
Teacher has a critical role in orchestrating productive talk – Talk Moves 1. Revoicing by both teacher and students
2. Teacher-initiated requests for a student to repeat another students’ response.
3. Teachers’ elicitation of a student’s reasoning (do you dis/agree, why do you think that?)
4. Teachers’ request for students to add on
5. Revise your thinking
6. Turn-and-talk
7. Wait time
Revoicing• Often used in the early stages of discussion.• Can be useful to:• Clarify a muddled/unclear response (check with student if this is what they
meant)• Help students clarify their thinking and improve their understanding • Make sure everyone heard• Sometime done at the end of more than one students contribution (a kind of
summing up move).
Important positioning/power factor in revoicing
“Are you saying that…?” “so, you are saying…” ; “so let me see if I’ve got your thinking right…?”• Opens up a slot for the student to chime in, to agree with or disagree
with the formulation of the student’s meaning that the teacher has put forward. • It is the student’s idea that is being formulated and made public, not the
authoritative knowledge of the teacher. • Teacher and student are positioned, momentarily on equal footing, in
co-constructing the jointly explicating an idea.
Repeating: Asking students to restate someone else’s reasoning• Restating of another student’s contribution marks the contribution as being especially
important and worth emphasising.
• Signals to the student that his or her ideas are being valued
• Provides a second chance for other students to catch up on something really important
• Sends a message that they better be prepared/listen as they may be asked to restate idea
• Makes everyone aware that the discussion is a discussion among the whole class and not just teacher-one student.
• Note: only ask a student to restate when the original ideas are clear and comprehensible.
Reasoning: Asking students to apply their own reasoning to someone else’s reasoning
• Supports habits of reasoning about why their mathematical claims or suggestions are valid.
• Press students to explain why they agree or disagree. • Importance of convincing others.• Sometimes students agree but ways of reasoning differ.• Sometimes disagree, and need to find out whose reasoning is correct.
Agree/disagree starters
• I think 4 x 8 and 8 x 4 are/are not the same because….
• Mrs J gave her students the equation 7 + 8 = _ + 5 and asked them to tell what number should go in the blank to make the equation true. Kane said that a 15 should go in the blank and Keyon said that a 10 should go in the blank. Who do you agree with , and why?
• Casey said that a square is a rectangle. Do you agree with Casey? Why or why not.
• Barlow, A., & McCrory, M. (2011). Strategies for promoting disagreements Teaching Children Mathematics, 17(9), 530-539.
Adding On (Say more, Teacher press): Prompting students for further participation
• Prompting a wider range of students to contribute adds more ideas to the discussion. • Enables students to carefully consider the ideas, to think about
what they understand, and to put it into their own words. • Prompt can be open to all, or specific to student; or range from
general (can you say more) to specific idea (e.g., Why did you chose 2?, other examples are? What do you mean by…?).
ReviseAllows students to revise their thinking as they have new insights• Has anyone’s thinking changes?• Would you like to revise your thinking?
Student: • “I thought …..but now I think because….”• “I’d like to revise my thinking…”
Turn-and-talks
What are the benefits to students? • Allows students to clarify and share ideas• Encourages students to orient themselves to each other’s thinking
What are the benefits to you? • Circulate and listen to partner talks, use this information to chose
whom to call on.
Using wait time
• Giving students time to compose their responses signals the value of deliberative thinking.
• Recognises that deep thinking takes time.
• Creates an environment that respects and rewards both taking time to respond oneself and being patient as other take the time to formulate their thoughts.
