Teacher clarity refers to a clear understanding of what students will gain from the instructional unit and how the teacher will guide them through the unit to enhance comprehension and support the development of concepts over time. This involves articulating learning intentions, also known as learning targets in my school, which are statements that inform students in advance of what they are expected to learn from the lessons. Teacher clarity is evident when instructors are organized, capable of explaining instructions effectively, and proficient in assessing students' understanding.

In Chapter 2, it is emphasized that teachers should establish learning intentions that build upon prior knowledge, are engaging, encompass both content and mathematical practices, and address language and social objectives. Additionally, success criteria in mathematics are utilized to motivate students and foster the practice of self-assessment.

2. Learning intentions can help students make connections between current learning and previously learned content. Identify the learning intention for a lesson you have recently taught. 

• What previously learned content is connected to this learning intention? Did your students see the connection? 

• If so, how did this impact their engagement in the learning? 

• If not, how might you modify the learning intention and experience to bring more attention to this connection?

• Which way does your questioning sequence lean?

• How can you make focusing questions a stronger presence in your mathematics classroom?

2. Identify two or three mathematics tasks you've asked your students to work on recently. Think about each task in light of its difficulty and complexity (see Figure 3.1 on page 77). 

• In which quadrant does each task fit? 

• Is each the right kind of task given your learning intentions?

2. In what ways are they used for multiple representations as students work on mathematics collaboratively?

3. What strategies can you use to make these tools more available to your students?

4. Consider the strategies you use, and the strategies you've read about and tried from this chapter to solidify surface learning. What are some effective ways to build surface learning, and why is that necessary?

2. How are these tools used to move learning from surface learning to deep learning?

3. Consider the grouping practices in your classroom. What strategies do you use to form mixed-ability groups and to ensure both group and individual accountability? What new ideas from this chapter could you use to ensure more rich and rigorous collaborative work in your mathematics classroom?

2. Are you using them for maximum transfer impact?

3. How could you refine your implementation of these tasks to make learning most visible?

4. Think about the strategies you use to help learners make connections in their learning. What is the mix of near and far transfer opportunities (describe on page 177 & 178) you provide your students? How do you think about introducing/scaffolding these opportunities for learners?

1. Thinking back over the recent units of study you have taught, what forms of differentiation do you use most often?

2. Which forms would your students benefit from using more often?

3. How can you accomplish this?

I think dialogic strategies are most impactful for my students. I didn't know what they were before reading this chapter, but I know the engagement and conversations I have with my students deepen their conceptual understandings.

We do block teams in my classroom each week. The amount of math skills that I see being transferred within this activity is amazing to see. I recently calculate my students are engaging with 17 different math standards while participating on a block team.

When I work with students in the upper elementary grades on problem solving strategies, the strategy I find most effective is bar modeling. Teaching students how to draw a bar model to represent the information in a word problem provides them with a visual to help make sense of the problem. I find taking the time to teach students about different types of bar models and how to break apart a problem and put the information into the model gives students a strategy for solving a variety of types of problems. The students I work with ability to solve problems has increased when they use the bar modeling. I see this on their unit tests, word problem work I do with them and on placement tests. Giving students an efficient way to visualize a problem is key in helping them understand how to solve word problems.

For the second graders I work with, “You’ve done lots of learning about how to subtract numbers. We’ve used unifix cubes, drawn pictures. Now we’re going to see what happens when you subtract a number from a teen number. We’ll spend some time sorting the equations and looking for patterns (for example doubles). 

The second graders I work with understand the concept of subtraction with regrouping and can use base ten blocks to model problems and explain what they are doing as they solve problems. They get stuck when trying to subtract from a teen number. They know their double addition facts but do not see the related subtraction fact. They understand that teen numbers are ten and some more but do not see that when subtracting 13 - 3 you are left with ten, without counting back on their fingers or using a number line. Using post its with equations to create a sorting activity the students identified these two types of subtraction equations. They enjoyed the activity, got very excited and saw the connection to the doubles facts and teens numbers. At the end of the lesson they were able to solve both types of problems independently. I am curious to see if they held onto this skill this coming week.

2. One math task my students recently worked on was multiplication math word problem cards. Students were grouped based on ability. The cards for the group needing support had a hint at the bottom, telling them how to develop an equation. The cards for the on grade level group had just the word problem, no hint. The cards for the group needing enrichment had word problems that involved multiple steps instead of just one. The goal of the lesson was for students to have additional practice multiplying two digit by two digit numbers. The group below grade level would fall into the fluency quadrant with low complexity and low difficulty, and is the right kind of task given the learning intention is not developing equations from word problems. The group on grade level would fall into the stamina quadrant, with high difficulty and low complexity, and is the right kind of task given the learning intention. The enrichment group would fall into the strategic thinking quadrant, with low difficulty and high complexity. This is the right task based on the learning intention because these students can already fluently multiply two digit by two digit numbers.

This past week I gave a group of fourth graders a division task. The task was based on a short story problem and asked the students to build a division model using manipulatives. We had used the same story problem and worked together to build models using different numbers. In this task I asked them to work in a group of three to build the model. This task fit in between the lower-level and higher-level demands, We had worked on previous problems using a procedure to build a model (lower level), however the task asked the students to build a model which increases conceptual understanding. What happened was that one student was able to complete the task independently while the other two observed. I then asked the student who was successful to work independently a related question, while I worked with the other two students. The level of difficulty was appropriate for the successful student. While I assumed the level of difficulty was appropriate for all three, I was wrong. Two of the students needed more questioning and prompting to relate the division equation to a model. They still don’t have a clear understanding of what each number means in a division problem. To be successful they needed smaller numbers and teacher guidance. My types of questioning changed while working with the two students. It became more specific, such as “What does this number mean? Where is this number represented in our model?” With the first student I asked her, “How do you know your model matches the equation?” 

2. While I do not teach math I do support with interventions during our flex period. As of late my group and I were working on adding decimals up to the hundredths place. This was something the students knew how to do but needed more practice remembering how to line up the decimal point. The students were given five addition/ subtraction of decimals problems. I believe this task would fall into Low difficulty/complexity, as the students were just practicing the step of lining up the decimal points. Another task was using hundreds blocks to show adding and subtracting decimals. Students had to represent the problem using the blocks. I believe this would be a low difficulty/ high complexity task. This is because they needed to grasp the understanding of the tiles and use them properly.

When I did teach math I would approach my students with strategic thinking which is  low difficulty and high complexity. Fluency is low difficulty and low complexity. These tasks are used to help students solve complex math problems more efficiently. The fluency will help with understanding and not just memorization. It also builds confidence and transferable to other subjects which makes them a better learner in all subjects, not just math. 

I just did a lesson on polygons on Friday.  This chapter made me think about that lesson and how I ran it.  My kindergarten students did not know what a polygon was.  I know this because I started my lesson by asking them.  They had some knowledge that they were shapes but that was all they knew. I believe they made this connection because I started the lesson by asking them what we are currently, specifically focusing on in math. Their response was, 2D and 3D shapes. I had some examples of polygons and non polygons and my student's tasks were to sort them onto a piece of chart paper. I called one student at a time and asked them if a particular shape was a polygon or not.  They shared what they thought and why, if they did not know, they asked a peer to help them. I was purposeful in who I called and when, as I wanted to give some students some more time and examples to see before they came up and did their shape. There was some really good dialogue going on between all the students, but this was still surface-level learning as they were working on the new vocabulary that I had given them through direct instruction and have them try to make sense of it. It did involve thinking about each shape.  One shape really stumped a student as it had straight lines and angles, and it is closed.  I love the picture of her deep in thought, trying to figure out if it is in fact a polygon or not. She finally announced that it “is not a polygon because it has curved lines”. There is a picture of her deep in thought and another of her choice in where to put her shape.
This quote from this week's chapter really resonated with me ”when students explain their thinking verbally, in a way that other students can understand, all students are better able to consider the ways that other people think and adopt some of these practices themselves. By listening to others think, the student is guided through the same thought processes that someone else used, as if an apprentice”. After this lesson, my students had an opportunity to create their own polygons on a giant geoboard. This was fun and they were all highly engaged, they also used pattern blocks to make polygons.  The hands-on element using manipulatives and other materials and the collaborative engagement during these activities will cement the content of my earlier direct instruction to a deeper level of understanding. There are pictures above of the activities and the awesome polygon 4 of my students made together.

One effective way I have built surface learning is in the teaching of multiplication facts. I often use a number talk to show how multiplication facts build off of one another. For example if you know your double facts (x2) you can double the double to know x4. Then double double the double for x8. Number talks have also been helpful in helping students recognize patterns. I find writing equations in word form is very helpful in teaching students how a basic fact like 2 x4 can help you solve 2 x 40. This moves students past the misconception that they are simply adding a zero. 

During independent practice, Students have access to  manipulatives to use while solving problems and used for Differentiated Instruction. 

Using fraction circles, students can build different fractions and visually compare them to understand equivalence and ordering. Using manipulatives in math instruction, teachers can provide students with a more interactive, engaging, and effective learning experience, helping them develop a deeper understanding of mathematical concepts and make connections. 

To make manipulatives more available to any students during that unit of instruction only the manipulatives that work best during that time will be accessible to all students in the classroom. Parent involvement and resource sharing is also valuable to students. 

I believe building surface learning is crucial as it provides the foundational knowledge. Students still need Practice, repetition, visual aids, representations, scaffolding and questioning. It helps provide better retention, confidence and deeper understanding. 

To get my students to a deeper level of learning, we play with the tools, by playing with the tools we get a lot of ah-ha moments as concepts begin to make sense.  Seeing four blue balls fall into a numbered tray, next to three green balls and seeing they equal 7 is one thing.  When we take those balls out and do it again with the three green balls first followed by the four blue balls, the students can see that even when we reverse the numbers, the sum is still the same.  Students then play with the balls to see what other ways they can make 7. I find doing math in this hands-on visual way supports a deeper and sometimes faster understanding of different concepts. 

I mix my student groupings up all the time, depending on the task at independent stations. If the task involves reading, like in word problems for example, I will ensure each group has a strong reader to help support this aspect of the group, otherwise, the groupings are quite random.  I switch up groups each day based on work that needs to be completed.  I also think about who has not really worked together much and try to put them together.  Sometimes, I place close friends in the same group and sometimes I don’t.  I have a couple of really quiet ESOL students this year so I tend to separate them so they are engaging in conversations with strong peer role models. Even if they aren’t always verbally engaging, they are hearing the rich conversations between their peers. I will support their engagement within the group by asking them questions specifically so they can gain confidence in speaking within a group setting. My students have assigned seats for tabletop, and then they have assigned groups for math and literacy groups.  When my students come into the room on a Monday morning, they have to find their new tabletop spot.  Mixing the tables like this supports relationships between all my students as they get to play with every other student in the class.  There are plenty of opportunities throughout the day for them to also choose who they want to sit and work or play with. If I need to work with a specific small group of students or an individual student on a specific task or concept, I pull them from whatever group they are in and they come to me at a different table. These homogeneous groupings are also ever-changing as they are focusing on a specific skill.  A child can be in one group for one math skill and a completely different group for another math skill.  This type of grouping also prevents students from seeing themselves as a particular leveled learner which is so important for their own sense of self-confidence as a learner.  During math, I tend to rotate through the ath stations.  I can engage with any group during this time and support discourse and accountable talk. I can go where I see a specific need and address misconceptions or struggles as they occur in real time. 

Accountable talk and manipulatives are crucial tools needed to transfer surface learning to deep learning. These tools are used to move surface learning to deep learning by memorizing formulas and definitions to understand the concepts and connecting them to different areas of math. In chapter 5 it states, “Meaningful mathematical discourse happens when students work on engaging tasks in thoughtfully selected small groups, so they can push each other to be accountable for deep learning about the mathematics at hand. Whole class discourse provides opportunities for students to share and debate both misconceptions and solution strategies.”

To refine the implementation of task-based learning strategies and make learning more visible, we can incorporate clear learning goals and success criteria, use of technology and digital tools and scaffolding support and differentiation would be best. The teacher I work with uses Gradual progression, real world connections, scaffolding support and diverse assessments. So for example problem solving within a unit would be used and homework would be given on the same unit for practice. 

I can accomplish this and any teacher can by using clear learning objectives, real world relevance and last but not least scaffolding and support.

With the students who are struggling in math I still focus on the use of different strategies,  however I modify the size of the numbers and may adjust the complexity of the problem. I have found that students struggling with understanding multiplying two-digit by two-digit numbers for example, do best with methods that are more conceptual. For example, the standard algorithm is confusing. They forget what to multiply first, to put a zero as a place holder. However when using the area model the students understand when they are multiplying tens to tens, and ones to ones. It adds a visual component that helps them make sense of what they are doing. Most importantly I give the students the choice on how they want to solve the problem. If they always use repeated addition, I provide feedback to the students that it is a strategy that works but there are other more efficient strategies. I know that I need to find ways to show the student that multiplication can be easier. I first have to determine why they aren’t using multiplication. It’s typically because they don’t know their multiplication facts. 

All of my students could benefit from more self-regulatory feedback. I can accomplish this by being more precise in stating learning objectives in child friendly language and asking students to pause and reflect on what they are doing and how effective it is. I can ask them to think about their successes for that day or session and what they may do differently or need to work on next time.

The near transfer in this task is the prior learning.  Each student needs to know what a number represents and they need to be able to consistently count up to 10 objects. The far transfer happens when students understand fluency to five and fluency to ten. Students can take this knowledge and transfer it to a real-life situation.  You are at the store and your mom says you can pick some treats for the week. There are seven days in the week so you can pick seven treats.  What will you pick and how many of each will you pick? 

Another problem-based task I so it making shapes out of shapes.  My students pick a card with certain pattern blocks on it and they have to use exactly the same blocks to create a specific shape.  There is one where they have to use multiple shapes to create a triangle.  This one is challenging for many of them. There are multiple ways to put the shapes together and they are so proud of themselves and each other when they finally accomplish the task.  We also do a 2D and 3D shape hunt around the school.  This helps students notice that shapes are everywhere, how they are all connected, and why different shapes are used for different reasons. They then transfer this knowledge to their own block-building time.  They have a deeper understanding of how certain shapes and sizes of blocks work better for certain aspects of their building.

I think my students would benefit from me devoting about ten minutes in each day to building fluent retrieval of basic arithmetic facts. To accomplish this, I could redesign my math rotations to ensure one station is dedicated to fluency practice.  If I ensure my other stations are focused on the current content being taught, and practicing previously taught concepts, I can have students do fluency practice and fluency games at another station.

With the students who are struggling in math I still focus on the use of different strategies,  however I modify the size of the numbers and may adjust the complexity of the problem. I have found that students struggling with understanding multiplying two-digit by two-digit numbers for example, do best with methods that are more conceptual. For example, the standard algorithm is confusing. They forget what to multiply first, to put a zero as a place holder. However when using the area model the students understand when they are multiplying tens to tens, and ones to ones. It adds a visual component that helps them make sense of what they are doing. Most importantly I give the students the choice on how they want to solve the problem. If they always use repeated addition, I provide feedback to the students that it is a strategy that works but there are other more efficient strategies. I know that I need to find ways to show the student that multiplication can be easier. I first have to determine why they aren’t using multiplication. It’s typically because they don’t know their multiplication facts. 

All of my students could benefit from more self-regulatory feedback. I can accomplish this by being more precise in stating learning objectives in child friendly language and asking students to pause and reflect on what they are doing and how effective it is. I can ask them to think about their successes for that day or session and what they may do differently or need to work on next time.