PLAN,IMPLEMENT, EVALUATE QUALITY MATHEMATICS LESSONS
The series of videos on this page will take you through the steps in the process of planning, implementing, and evaluating the assessment lesson, and the first mathematics lesson that embeds further assessment of, for, and as learning (see DIFFERENTIATION, FEEDBACK AND ASSESSMENT RESEARCH).
The first lesson involves the most planning, as we will be deepening our own relational understanding (see QUALITY MATHEMATICS LESSONS RESEARCH) as we plan the lesson, implement the lesson, and evaluate the lesson.
After planning, implementing, and evaluating the first lesson, subsequent lessons are planned, implemented, and evaluated following exactly the same process. Your relational understanding will continue to grow, and you will have greater and greater understanding of every child’s current Level of understanding following each lesson. Planning will take very little time, allowing you to make the main thing the main thing – implementing and evaluating quality mathematics lessons.
STEP 7: IMPLEMENTING LESSON 1 – INVESTIGATION USING THE ASSESSMENT, DIFFERENTIATION, INVESTIGATION, AND INTERVENTION TO ENGAGE CHILDREN IN DIFFERENTIATED INVESTIGATION WHERE EVERY CHILD IS INVESTIGATING INDEPENDENTLY AT THEIR LEADING EDGE
Transcript of all Steps in the Process:
All of the resources at ALPATP have been created by a teacher, school leader and department and independent maths consultant, with the aims of deepening teacher relational understanding, enhancing teacher metalanguage and enriching teaching pedagogy. If you are looking for quick teaching resources, this may not be the place for you. If you want to develop your understanding of maths and quality pedagogy, as you plan, implement, and evaluate differentiated maths lessons embedding assessment of, for, and as learning, then you have arrived at the right place – A Learning Place A Teaching Place!
STEP 1: USING THE TEACHING VIDEOS, EXPLICIT TEACHING PLAN TO DEEPEN TEACHER UNDERSTANDING
Let’s look first at the Teaching Resources that we will use to plan how we will initially assess our children’s current Levels of understanding, how we will use the assessment data to plan, implement and evaluate the Explicit Teaching and the Investigation in first differentiated lesson embedding assessment of, for, and as learning, and how we will use the assessment data from the first lesson to plan, implement and evaluate the Explicit Teaching, the Investigation, and the Problem Solving in the next and subsequent lessons.
TEACHING RESOURCE: TEACHING VIDEOS
The Teaching Videos have been created to sound like we are teaching children because they are designed to demonstrate what the lesson will sound like. The Teaching Videos are actually designed to enrich teacher understanding, metalanguage, and pedagogy of questioning, while planning to teach children.
During Explicit Teaching, use your teacher professional judgment to decide if parts or all of a Teaching Video may be shown to children.
You may decide to show parts or all of a Teaching Video to the whole class during Explicit Teaching because the metalanguage is unfamiliar to you and you want to make sure to include all of the strategic questions. If a Teaching Video is shown to the whole class, pause it after every question to allow children to think, ask a partner the question, explain their understanding to a partner.
You may decide to show parts or all of a Teaching Video to a small group of children or an individual child during Investigation because they have demonstrated and explained their understanding at the highest Level that has been explicitly taught, and you want them to have the capacity to continue to develop their understanding without taking you away from the rest of the class.
You may decide to show parts or all of a Teaching Video to a small group of children or an individual child during Explicit Teaching because they have demonstrated and explained their understanding at a Level that is much higher than the rest of the class – they are an outlier – and explicitly teaching this Level to the whole class will, in your teacher professional judgment, cause unconstructive and negative confusion.
If a Teaching Video is shown to an individual child, ask them to pause it after every question to allow the child to think about the question, and think about how they would explain their response.
When planning lessons, select the Grade and the Concept at Grade Level, and watch the Teaching Video, taking notes about the relational conceptual understanding, the metalanguage and strategic questions, lower levels of the concept that are vital as a base of understanding, and related concepts.
Don’t think about the children in your class yet – you will have plenty of opportunity to consider them soon – just focus on deepening your understanding.
TEACHING RESOURCE: EXPLICIT TEACHING PLAN
Explicit Teaching involves asking strategic questions to engage children in transformative learning, not just informative learning.
The content of the Explicit Teaching Plan and the Teaching Video is exactly the same – the same deep conceptual understanding, the same consistent accurate metalanguage, the same strategic questions.
Like the Teaching Videos, the Explicit Teaching Plans have been created as if we are teaching children because they are designed to demonstrate what the lesson will be like, but actually designed to enrich teacher understanding, metalanguage, and pedagogy of questioning, while planning to teach children.
The Explicit Teaching Plan and the Teaching Video contain the complete sequence of teaching and learning for the Grade Level of the concept, and are not designed to be taught all in 1 lesson. Parts of the Explicit Teaching Plan and the Teaching Video may be planned and programmed to be explicitly and sequentially taught over many lessons. Your teacher professional judgment will tell you how much to teach in each lesson. Because you are teaching using questioning, you will recognise when it is time to end the Explicit Teaching by the much reduced number of children who are able to explain – if you ask a question and no one is ready to explain, it’s time to allow the children to investigate the concept at just beyond their current Level of understanding – see STEP 5: IMPLEMENTING LESSON 1
There is an Overview page in the Explicit Teaching Plan that contains the strategic questions and sequence of teaching and learning in the Teaching Plan. The Overview page could be printed out to use as notes during the implementation of the lesson. The Overview page could also be printed out and placed in your program.
STEP 2: PLANNING AND IMPLEMENTING ASSESSMENT LESSON
It is now time to consider the children in our class, and their Levels of understanding. So how do we identify every child’s current Level of understanding? We engage children in assessment that looks exactly the same as their everyday teaching and learning – which means the assessment is authentic and reliable. It also means that the assessment is not taking time from learning and that teaching, learning, and assessment are seamless as it is assessment as learning. It is also assessment of learning as the data is recorded as point-in-time reporting, and assessment for learning as the data is used to differentiate the next and subsequent lessons.
The Assessment for each concept contains models of all Levels of the concept, with suggested Levels to display for each grade. Use your teacher professional judgment to select 2 or 3 Levels that you anticipate will suit your class, to either display or demonstrate at the beginning of the assessment. The Levels serve as models to remind or suggest to children, the recording they may use to show their strategy and thinking. The Assessment also contains suggested instructions to give to the children to allow them to engage in the assessment to demonstrate their highest Level of understanding.
Following the Assessment, you will use the children’s Levels of understanding to plan, implement and evaluate the first lesson.
STEP 3: TEACHING RESOURCE: DIFFERENTIATE
After implementing the Assessment, you will have a good idea of most children’s current Level of understanding of the concept. This will allow you to identify the Levels to explicitly teach in the first lesson. Children for whom the Assessment did not identify their Level of understanding, use your teacher professional judgment to select Levels to explicitly teach that you think may be suitable for each of these children. Using the Assessment data and your teacher professional judgment, select 2 or 3 Levels to explicitly teach in the first lesson to begin to cater for every child’s Level of understanding. The 2 or 3 Levels you select to explicitly teach in the first lesson may or may not include the grade Level of the concept. Don’t worry, you will include this Level in a subsequent lesson.
The Explicit Teaching Plan and Teaching Video are almost exactly the same for one Level as for every other Level – only the size or complexity of numbers changes. There is generally no need to read more than one Level’s Explicit Teaching Plan or watch more than one Level’s Teaching Video. Use your teacher professional judgment to decide which Level’s Explicit Teaching Plan to read or Teaching Video to watch. The Differentiation for each concept contains models of all Levels of the concept – for the Levels that you don’t need to read or watch, you may look at these models.
STEP 4: IMPLEMENTING LESSON 1 – EXPLICIT TEACHING: USING THE TEACHING VIDEOS AND EXPLICIT TEACHING PLANS TO EXPLICITLY TEACH DIFFERENTIATED MATHS IN THE FIRST LESSON
Explicit Teaching involves asking the strategic questions in the Teaching Video and Explicit Teaching Plan. Using teacher and child questioning allows us to identify the Levels of understanding of our children. We ask a question, allow thinking time, then have children ask a partner the question to ensure every child develops metalanguage.
Children have the opportunity to explain their understanding to their partner, and to the class when they are ready. Children who are not yet ready to explain their understanding have at least asked the question, thus developing their metalanguage in preparation for when they are ready to explain. Children who are ready to begin to explain their understanding are explaining to a child within their zone of proximal development, thus developing both their own, and their partner’s understanding and metalanguage. The strategic questions in the Teaching Video and the Explicit Teaching Plan are not wasted on just a few children – every child has the opportunity to ask and answer the questions, developing understanding and metalanguage!
Continue using teacher and child questioning as you explicitly teach each of the 2 or 3 Levels you used your teacher professional judgment to select. As you explicitly teach each Level, record it on a sheet of paper. Display each Level’s sheet of paper as an Anchor Chart. The Anchor Charts serve as models to remind the children of the recording they may use to show their thinking and solution strategy.
During the Explicit Teaching, if, after asking a question, very few children in the class are yet able to explain their understanding, stop the Explicit Teaching, and allow all children to Investigate at just beyond their current Level of understanding. During the Investigation, pay closer attention to the children who you think are close to developing understanding of the concept at the Level of the last question, prompting, questioning, observing and listening, to allow them to begin to develop their understanding to this Level. In the Explicit Teaching in the next lesson, you may again include this Level of questions, to allow these children to begin to explain their new understanding.
STEP 5: PLANNING LESSON 1 INVESTIGATION
TEACHING RESOURCE: INVESTIGATE
After explicitly teaching your selected Levels of the concept, children are ready to begin investigating at just beyond their current Level of understanding.
How do we get children to Investigate at just beyond their current Level of understanding? Initially you may have to use your teacher professional judgment to assign Levels to children to investigate, until they are ready to confidently identify the Level that is just beyond their current understanding. Older children, who are used to having a teacher or adult assign their Level to them, often take longer than younger children, to develop this capacity.
At the end of the Explicit Teaching, ask the children to point to the Level that they feel ready to Investigate. Explain that this is not a Level they already understand and can explain, it is the first Level that they do not yet understand.
When children settle down to Investigate, ask them all to fold their arms while thinking about the Level they have chosen for themselves. Stand behind each child in turn, and quietly ask them which Level they have picked. Use your teacher professional judgment to decide whether you agree their selected Level is appropriate for them. If you agree, simply ask them to begin Investigating.
If you think a child has chosen a Level that is too easy for them, you may quietly say something like, ‘Ok, try that Level, and then if it is a little easy, try the next Level’. The child may wish to develop their capacity to follow the investigation process on an easier Level before venturing to a higher Level.
If you think a child has chosen a Level that is too challenging for them, you may quietly say something like, ‘I’d like you to show that you understand the first Level before you go onto that Level’.
This strategy contributes to authentic growth mindset (not to be confused with the contrived culture of pretty posters and cute quotes). You may be pleasantly surprised that many children recognise the Level that is just beyond their current understanding quite accurately!
This strategy also releases children to Investigate one child at a time. This makes guiding chldiren through the Investigation process much simpler.
The first Investigation listed is based directly on the Explicit Teaching Anchor Charts and allows children to investigate in a familiar situation. The first investigation is almost exactly the same for one Level as for every other Level – only the size or complexity of numbers changes. There is generally no need to read more than one Level’s Investigation. Use your teacher professional judgment to decide which Level’s Investigation to read.
The other Investigations are not based directly on the Explicit Teaching Anchor Charts and allow children to investigate in an unfamiliar situation. Use your teacher professional judgment to decide which Investigations children will engage in.
Over a few lessons, you may end up with multiple Anchor Charts. Because the Anchor Charts have been created collaboratively with the children, the children understand them, and have ownership of them. As the number of Anchor Charts increases, children have no trouble identifying their Level because the Anchor Charts have been added to one at a time over time. The Anchor Charts can be permanently displayed to create a wall that teaches or a bump it up wall – and serve as models of success criteria.
STEP 6: TEACHING RESOURCE: INTERVENTION
After implementing the Assessment, you will have a good idea of most children’s current Level of understanding of the concept. You may have some children who were not yet ready to demonstrate any understanding of the concept, even at a low Level. This may include children requiring learning support for learning difficulties or children are new to English speaking. These children will investigate an Intervention while other children investigate a Level of the concept.
The difference between a Differentiation and an Intervention is that a Differentiation involves children investigating the class concept but at different Levels, while Intervention involves children investigating a different concept that provides vital understanding to enable them to investigate the class concept, for example counting concepts including counting forwards and backwards, recognising numerals, writing numerals, and counting items – these are essential understandings for all number concepts, including adding, subtracting, multiplying and dividing.
Each Intervention has its own Anchor Chart and Recording Sheet. The Anchor Charts have been created with images to allow children investigating an Intervention to follow the investigation process independently over time. Each Intervention Anchor Chart and Recording Sheet is a different colour to allow children to recognise each Intervention when they are investigating more than one Intervention in a lesson, for example, they may spend 10 minutes investigating counting forwards and backwards, 10 minutes investigating recognising numerals, 10 minutes investigating writing numerals, and 10 minutes investigating counting items within one maths lesson.
To ensure each child investigating an Intervention is investigating at just beyond their current Level of understanding, we need accurate assessment data. This will need to be implemented one-on-one, but will not take long.
INTERVENTION 1 ASSESSMENT (No colour as the child does not see the Recording Sheet): We need to identify the number that each child can consistently and accurately count forwards to and count backwards from. Simply ask each child to count forwards, naming higher and higher numbers, until you identify the child’s highest number. Repeat for counting backwards. Record these numbers on the Intervention Anchor Chart and Recording Sheet.
INTERVENTION 2 ASSESSMENT (Blue): We need to identify the numerals that each child recognises accurately and consistently without counting, when asked, ‘What number is this?’ Record these numerals on the Intervention Anchor Chart and Recording Sheet.
INTERVENTION 3 ASSESSMENT (Red): We need to identify the numerals that each child writes legibly. Record these numerals on the Intervention Anchor Chart and Recording Sheet.
INTERVENTION 4 ASSESSMENT (Green): We need to identify the number of items each child counts accurately and consistently. Record this number on the Intervention Anchor Chart and Recording Sheet.
Prepare an Intervention bag for each Intervention for each child investigating that Intervention. For example, Child A may be ready to Investigate numbers from 1 to 5. Child A may be able to count forwards and backwards to and from 5, and recognise numerals to 5, but needs Interventions for writing numerals 2, 3 and 5 and for counting 3, 4 and 5 items, while Child B may be ready to Investigate numbers from 5 to 10. Child B may need Interventions for counting from 5 to 10 forwards and backwards, reading numbers 8 and 10, writing numerals 5, 8 and 10, and for counting more than 6 items.
For Child A we will prepare a bag for INTERVENTION 3 (Red) with the lowest numeral the child cannot yet write – 2 and Intervention 3 Anchor Chart and Recording Sheet in it; and a bag for INTERVENTION 4 (Green) with the lowest numbers which the child can count items with plus 1 further number – 1, 2 and 3, some counters and Intervention 4 Anchor Chart and Recording Sheet in it.
For Child B we will prepare a bag for INTERVENTION 1 with Intervention 1 Anchor Chart and Recording Sheet in it; a bag for INTERVENTION 2 (Blue) with the numerals up to and including the lowest numeral the child cannot yet recognise – 1 to 8 and Intervention 2 Anchor Chart and Recording Sheet in it; a bag for INTERVENTION 3 (Red) with the lowest numeral the child cannot yet write – 5 and Intervention 3 Anchor Chart and Recording Sheet in it; and a bag for INTERVENTION 4 (Green) with the lowest numerals the child can count with plus 1 further numeral – 1 to 6, some counters and Intervention 4 Anchor Chart and Recording Sheet in it.
These Interventions may be investigated with an Intervention teacher or other adult. The Intervention teacher / adult may explicitly teach each child their Intervention/s. They could then place the Intervention materials, and the Intervention Anchor Chart and Recording Sheet – after recording the session’s progress – into a bag for the child to take back to class. In class time, when other children are independently investigating the class concept at different Levels, the child investigates the Intervention independently (see Intervention).
The Interventions may be investigated in class without an Intervention adult explicitly teaching them the Intervention, just as other children investigate different Levels of the class concept. Initially the child is guided through the Intervention investigation until they are able to follow the process independently (see Intervention).
STEP 7: IMPLEMENTING LESSON 1 – INVESTIGATION
USING THE ASSESSMENT, DIFFERENTIATION, INVESTIGATION, AND INTERVENTION TO ENGAGE CHILDREN IN DIFFERENTIATED INVESTIGATION WHERE EVERY CHILD IS INVESTIGATING INDEPENDENTLY AT THEIR LEADING EDGE
So what does differentiated Investigation of the class concept or an Intervention look like?
Children sit in mixed-abilities – selecting their own place to sit is great! Children may choose to sit with a child who is investigating the same Level as they are, or a child who is investigating a higher Level, a child who is investigating a lower Level, or a child who is investigating an Intervention. At times you may ask a child to sit with a particular child as they investigate independently, for example, a child who may be experiencing productive struggle at a Level that the other child has developed a little deeper understanding of.
Initially you will guide the children through the Investigation until they can follow the Investigation process independently. During this time, children are simply learning how they will investigate – they may not be learning any maths yet.
Once children can follow the Investigation process, they will begin to Investigate independently. This is when they will begin to both develop and demonstrate their Level of understanding. Children are not actually learning when we are teaching. Children are learning when they investigate a concept at just beyond their current Level of understanding.
Because every child is Investigating at their leading edge, no child needs an adult sitting with them for longer than about a minute. If you need to sit with a child for longer than about a minute, the Level they are Investigating is too high for them, and not within their zone of proximal development.
Because you are differentiating, and providing Intervention, even the children with the lowest Level of understanding do not need an adult sitting with them, because they, like every other child, are investigating at just beyond their current Level of understanding. If you have children who are not used to investigating independently, for example, children who have been trained to think that they only learn when an adult is sitting with them, it may take some time to re-train them to learn independently. If the child is only able to demonstrate understanding with an adult sitting with them, they do not really have understanding.
Because no child needs an adult sitting with them, the teacher is available to every child. You may roam, observe, question and listen to every child. You are free to provide just enough support to children who are experiencing productive struggle. You are free to guide children to move between Levels – up when a child demonstrates and explains understanding of their current Level, and down when a child demonstrates they have not yet developed understanding of a prior Level that is fundamental to their current Level. Over a little time, children will recognise when they are ready to investigate at a higher Level, or when they need to return to a lower Level, and move between Levels themselves. This is true child-led learning.
For children who take some time to learn the investigation process involved in either the class concept and an Intervention, to allow them to investigate independently, including children who are used to having an adult sitting with them while they learn, children who have experienced trauma, or children who are new to English, your first goal is to develop their independence.
Do this for children investigating a Level of the class concept by ensuring their Level of understanding has been explicitly taught and is displayed on an Anchor Chart (you may print an individual Anchor Chat for these children to have in front of them – see Differentiation), or, for children who are investigating an Intervention, by ensuring they have the Intervention Anchor Chart and Recording Sheet in front of them and the materials they need.
As you roam, stop by each child who is still developing their independence, and give them their first instruction from their Anchor Chart, then walk on. You walk on to give the child the opportunity to develop their confidence and capacity to follow an instruction independently.
Continue roaming, stopping briefly at other children as you notice they need a little support, observing, prompting and questioning. Because the children are sitting in mixed-abilities, as you walk from one child who is still developing their independence to the next, you will roam past every other child. This includes children who are experiencing productive struggle – children who have just moved onto a higher Level; children who are struggling with an aspect of their current Level; children investigating a related concept to apply it to their current Level – see Integration; children investigating a lower Level to apply it to their current Level; as well as other children investigating an Intervention. You are available to provide just enough support to every child, every lesson.
Return to each child who is developing their independence. If the child has followed the first instruction, praise them briefly, give them the second instruction and walk away. If they have not yet followed the first instruction, demonstrate the first instruction, replace the materials you used in the demonstration, then walk on. Repeat.
Some children will develop independence sooner than others. Some children, whose confidence is low, or whose belief that they only learn with an adult, may take a long time, with many emotions rising to the surface. It is worth persevering – when these children who have never learnt independently before, finally learn that they can learn independently, this is a huge amount of learning! They will then be ready to learn maths independently at their leading edge.
Of course you may use your teacher professional judgment to decide to bring a child or a group of children to a place in the room to revise a Level that they are experiencing difficulty with or to extend them to a higher Level of understanding that has not yet been explicitly taught – but it is important that they then investigate independently, because it is during independent investigation that deep relational understanding is developed.
An alternative to the teacher reteaching children who are experiencing difficulty investigating a particular Level is to allow them to sit with a child who has just developed their own understanding of the Level. This helps both children – the child who has just developed their understanding has the opportunity to explain this new understanding to a child who is ready to learn it; and the child who is experiencing difficulty has the opportunity to learn with another child within their zone of proximal development.
Children pause regularly during investigation to share their developing understanding with their partner for the lesson. To learn, talk is necessary, and talking to others who are also investigating the same concept, deepens understanding, and enhances the metalanguage children need to explain. As Einstein said, ‘If we can’t explain it simply, we don’t understand it.’
Because children are investigating at their leading edge independently, every lesson is an opportunity for assessment of, for and as learning using both in-class observation, questioning, and listening, as well as child recordings.
STEP 8: EVALUATING LESSON 1
At the end of the first and every lesson, collect the children’s books and place them in piles of demonstrated understanding. To make this easier, you may collect one Level at a time, or ask children to place their books in individual piles according to the Level they investigated.
Spend a few minutes looking at the recordings in each book, to identify each child’s current Level of understanding, and to uncover any misconceptions or challenges. This will allow you to plan the next lesson, including the Levels to explicitly teach, and identifying the children you plan to ensure you observe and guide during Investigation.