**Five Ways We Undermine Efforts to Increase Student Achievement
**

**(and what to do about it!)**

Blog Post Part 5 of 5

I am excited to provide this fifth part in a five part series of posts unpacking the Ignite session I provided at the NCTM Annual. You can view the Ignite on the NCTM Annual website (https://www.nctm.org/annual/) beginning at timestamp 21:40.

This fifth potentially (Un)Productive Practice focuses on making the most of teaching rigorous standards. This fifth practice is the most nuanced of all. First, we need to make sense of rigor and how it is currently addressed in schools where efforts are in place to support it. Second, we need to acknowledge that it might not be working as well as we had hoped.

Rigor, as it relates to the shifts associated with recent state standards, is often defined as the need to include conceptual understanding, procedural skill, and application in mathematics teaching and learning. Those who embrace the idea that rigor is important have also come to understand that concepts need to be addressed before procedures. The key here is that students should make sense of the mathematics by exploring “the why” before they develop fluency – I guess this could be thought of as “the what”. Fractions provide a helpful context for unpacking this.

Consider the expression 4 ÷ 1/5. If students are taught the algorithm to *keep *the four (or in this case write it as the fraction 4/1), *change *the division to multiplication, and *flip *the 1/5 so it becomes 5/1 there is very little motivation to make sense of this algorithm as it relates to computing with fractions. In contrast, if students were provided with a word problem that could be modeled with this expression, access to the concept is more obvious.

Consider the following word problem:

I have four sticks of butter. If one mini-pan-sized chocolate chip cookie uses 1/5 stick of butter (I know – that isn’t much butter – but you get my point :), how many cookies can I make?

It would not be difficult for students to draw a representation of this problem. It would likely look something like this:

Students would see that 20 cookies could be made from 4 sticks of butter. This would be an important accomplishment. Students would understand that dividing by a fraction can be interpreted as determining the number of groups the size of that fraction they can make. This experience should certainly be provided prior to introducing the “invert and multiply” algorithm. However, this alone is not sufficient and we are beginning to see this as we examine data related to student achievement. According to anecdotal data, students are still demonstrating that they are not remembering algorithms or they are applying them incorrectly (this includes some students with I have recently worked with). What are we missing?

I think I have the answer! We are neglecting the connection. We need to continue teaching concepts before procedures but we need to do more than that. We have to be intentional about connecting them. We need to make the connection explicit. I will use the cookie problem to illustrate what I mean.

When we explored 4 ÷ 1/5 in context, we saw that we could make 5 cookies for each stick of butter because there are 5 one-fifths in each whole stick. Since there were four sticks of butter we could conclude that there were four groups of five so 20 cookies could be made in all. Another way to model four group of five is 4 x 5. Therefore, 4 ÷ 1/5 was solved by finding 4 x 5.

What if the cookie recipe called for 2/5 stick of butter? Rather than being able to make 20 cookies from four sticks of butter, only 10 cookies could be made. Since each cookie used twice as much better, half of the number of cookies could be made. This can be represented by dividing the original product by two. Therefore, 4 ÷ 2/5 is solved by finding 4 x 5 ÷ 2. This is not exactly the same as how the algorithm is typically described but it is more clearly connected to the concept of dividing by a fraction. And, as students become comfortable connecting their new understanding of dividing fractions to this computation, instruction can be used to link this procedure to the standard algorithm of invert and multiply.

What is important here is that students will be more likely to remember how (and when) to divide fractions because the procedure they use will be connected to what they have come to understand. It is during the lessons focused on making these connections that academic vocabulary should be introduced (see part 4 of this series for a more extensive conversation related to academic vocabulary).

This practice was more difficult to explain than I anticipated. I hope you will send any comments or questions you have my way by tweeting to @thestrokeofluck. Please keep the comments public so we can engage more mathematics educators in our discussion.

As with every post in this series, my goal is to support increases in student achievement for each and every learner. Because of this, I have come to realize that I should have shared SIX potentially (Un)Productive Practices. Because of this realization, I plan to share a sixth post in this five-part series within the next few weeks (would this qualify as a fraction greater than one? I’m sure we wouldn’t call it improper :). Stay tuned…

Please tweet your thoughts, comments, and ideas on this post to @thestrokeofluck

**Five Ways We Undermine Efforts to Increase Student Achievement
**

**(and what to do about it!)**

Blog Post Part 4 of 5

I am excited to provide this fourth part in a five part series of posts unpacking the Ignite session I provided at the NCTM Annual. You can view the Ignite on the NCTM Annual website (https://www.nctm.org/annual/) beginning at timestamp 21:40 but you are probably best off starting from the beginning because the other ignite sessions are excellent.

This fourth potentially (Un)Productive Practice focuses on putting the word wall in its proper place. This fourth practice is both potentially unproductive and extremely common. Teachers are expected to lead instruction with academic vocabulary. New vocabulary is introduced and defined at the start of a lesson on a new topic. The words are added to the “word wall” and referred to frequently throughout the lesson.

I just don’t understand. When we introduce new topics in mathematics instruction by starting off with defining the academic vocabulary of the lesson, we are beginning with procedures. We are saying, “here is the new word and here is what it means, now let’s make sense of it within our lesson.” How is that leading with concepts? It isn’t. This is just one reason that leading with academic vocabulary is a potentially unproductive practice.

Explore what I mean by adding the fractions below using a think aloud process before reading the text that follows the fraction problem. Feel free to write down what you do but be sure to talk through the process you use.

Did your self-talk sound something like this?

You probably didn’t need to think too much about this process. It is a procedure you know well from much practice. Now try to add the same three fractions without using the words “numerator,” “denominator,” “top number,” or “bottom number.”

This process was likely less automated. First you had to determine how to describe what you were doing to the fractions. You needed to find a way to describe what happens when you find common denominators. Your self-talk might have sounded something like this:

At this point you probably slipped and said that you would then add the top numbers before you realized you couldn’t say, “top numbers.” You had to pause and think about what you were doing. This is a good thing. You might have continued like this:

How does this second description compare to the first? It is more conceptual in nature. The sense making is more evident. By withholding the academic vocabulary, the process becomes more conceptual than procedural. It also becomes more accessible. We need to take a lesson from research on English language learners and lead with everyday language (Cummins, 2000). The understanding will transfer as academic language is introduced as long as the experiences are connected (I’ll talk about this more in my next blog :).

Access is another important reason for moving the placement of the word wall. If we move the word wall to the end of the lesson, more students are likely to have access to the concepts in the lesson. This is especially the case if the problems students explore are presented in context so that students have the everyday language connected to the context to use to describe the mathematics. Think about the fraction addition problem. If you had been describing pieces of cookie your language might have been more natural. I explored this with fifth-grade students in a small group setting. I describe that lesson in a book I wrote with my colleagues Lisa Brooks and Melissa Carli. The book was just recently released for pre-order through Solution Tree. The video of the lesson is included in the book using a QR code. The link to the book is included here in case you are interested: https://www.solutiontree.com/products/coming-soon/making-sense-of-math-small-groups.html.

In my work with the fifth-grade students, it was clear that the students already knew the terms numerator and denominator. In this case I disallowed the use of those terms while the focus of instruction was developing conceptual understanding of the process of adding fractions with unlike denominators. That is how I handle new concepts where the students already know the supporting vocabulary. I challenge students to describe their processes with everyday language rather than allowing students to hide behind academic vocabulary. Then, as we move towards the procedures that support the concepts we just explored, I bring the academic vocabulary back into the conversation.

Let me be clear in saying that I am not advocating for the avoidance of academic vocabulary. What I am encouraging is the practice of leading with everyday language and then naming what we learn with academic vocabulary. This mirrors leading with concepts and then after concepts are understood, providing access to more efficient procedures (although even this practice needs to be examined – it is the focus of my next post).

If you are concerned that your administrators will look for you to lead with vocabulary, please share this post with them! Have them tweet their feedback to me at @thestrokeofluck so we can all join in the conversation. We all want what’s best for our students; we just need to examine our practices to be sure that they truly make sense in the context of mathematics. Be clear that you are still going to emphasize academic vocabulary. You are just going to be more intentional about when you introduce it so that your focus is on student reasoning.

As with my other posts, the changes in practice I suggest have the goal of ensuring that the students are doing the sense making and the teacher is supporting them to meet the learning goal through the task that is chosen and the questions that are used to support the implementation of that task. As stated in the first post, we can transition our unproductive practices to be productive by keeping the learning goal and student engagement at the foreground of our planning and by critically analyzing our instructional decisions and structures. In the last post I discussed the importance of how we provide scaffolding. In this post, I added another component related to access and equity by leading with everyday language and then naming students’ new understanding with academic vocabulary. I am looking forward to reading about your reactions on twitter and continuing this conversation!

Please tweet your thoughts, comments, and ideas on this post to @thestrokeofluck

__Reference:__

Cummins, J. (2000). *Language, power and pedagogy: Bilingual children in the crossfire.*Clevedon, England: Multilingual Matters.

**Five Ways We Undermine Efforts to Increase Student Achievement
**

**(and what to do about it!)**

Blog Post Part 3 of 5

As promised, I am providing the third in a five part series of posts unpacking the Ignite session I provided at the NCTM Annual a little over a week ago. You can view the Ignite on the NCTM Annual website (https://www.nctm.org/annual/) beginning at timestamp 21:40. While you are there, you should check out all of the Ignite sessions – they are worth the time!

I am especially excited to discuss this third potentially (Un)Productive Practice because of my personal focus on supporting students who struggle. You can learn more about how I developed first-hand knowledge related to students who struggle by reading about my family’s story at www.astrokeofluck.net. This will also clarify the genesis of my twitter handle (@thestrokeofluck).

Now back to the topic of this post – scaffolding. In education, scaffolding describes supports provided to students to assist them in meeting a learning goal. So what does scaffolding have to do with supporting students who struggle? It has everything to do with supporting students who struggle when the answer to the question, ”How do you provide differentiation?” is “By scaffolding.” This becomes an issue when the scaffolding is provided “just in case” students might need it rather than “just in time” when students *demonstrate *the need.

Just-in-case scaffolding creates issues of both access *and *equity. When scaffolding is provided before students have the opportunity to make sense of a challenging task without the extra help, students are inhibited from developing productive perseverance. All too often, so much support is provided through the initial scaffolding that the cognitive demand of the task is significantly decreased (Boston & Wilhelm, 2015). If this sort of scaffolding is provided for students who struggle, then these same students are denied access to cognitively demanding tasks. When access is denied, equity becomes an issue.

So how do we provide differentiation that is equitable? We can still provide differentiation through scaffolding. The key is to provide the scaffolding just in time rather than just in case students need it. Just-in-time scaffolding helps to develop productive perseverance by allowing students to engage in demanding tasks and then assisting them to maintain the engagement when they struggle by providing support through teacher questioning.

This is where I wish I could show a classroom video to model what I mean by providing scaffolding just in time. We will need to imagine the classroom instead. Consider a class of students in grade 7 who are tasked with solving the following problem:

*Alex used up all of the money she had saved from her lawn care job to buy a skateboard. She then borrowed $17 from her mother to go to the movies with her friends. After she went to the movies she bought a soccer ball for $15. She borrowed that money from her mother as well. Alex keeps track of her money in her notebook. **What should she write in her notebook to indicate her balance now? Explain your reasoning.*

What are your thoughts as you read the problem? Are you thinking that the wording might be confusing for your readers who struggle? Are you thinking about issues students often have when computing with integers? Those are all legitimate thoughts. It is what we do in response to those thoughts that either supports or inhibits students’ achievement. It is all about the scaffolding.

Often, when I observe teachers using tasks like this with learners who struggle, I see scaffolding in the form of teachers unpacking the word problems to the point that the task becomes an exercise. It might sound something like this:

*Teacher: Would the $17 be positive or negative?*

*Students: Negative.*

*Teacher: What about the $15?*

*Students: Negative.*

*Teacher: What do we do when we add negative numbers?*

The cognitive demand of the task is greatly diminished by this “scaffolding.” The students are no longer left to make sense of the context and determine the operation to be performed. The teacher is providing supports for students in anticipation of a struggle. This is providing just-in-case scaffolding.

In contrast, imagine a classroom where the teacher provides space for students to do the sense making. It might sound something like this:

*Teacher: (reads the problem to the students) Class, spend 2 minutes thinking about this problem and then share your ideas with your partner.*

*(Teacher waits 2 minutes before circulating the room to listen to discussions of students)*

*Student: (addressing teacher) We don’t know how to solve it.*

*Teacher: What is the problem asking?*

*Student: What Alex should write in her notebook to show how much money she has.*

*Teacher: What do you know?*

*Student: We know that she spent all her money and then borrowed money from her mom.*

*Teacher: Now work with your partner to see how you might represent the money she borrowed. (Teacher walks away).*

What you probably notice is that the teacher provides processing time for students to do the sense making. When students indicate a struggle, the teacher provides just enough information so that students can re-engage with the task without lowering the cognitive demand of the task. Which classroom image most closely represents the opportunities you or the teachers you support are affording students? How would you describe your classroom?

As with my other posts, the changes in practice I suggest have the goal of ensuring that the students are doing the sense making and the teacher is supporting them to meet the learning goal through the task that is chosen and the questions that are used to support the implementation of that tasks. As stated in the first post, we can transition our unproductive practices to be productive by keeping the learning goal and student engagement at the foreground of our planning and by critically analyzing our instructional decisions and structures. In this post, I added an additional intention to access and equity by withholding scaffolding until it is necessary. I am looking forward to continuing this conversation!

Please tweet your thoughts, comments, and ideas on this post to @thestrokeofluck

__References:__

Boston, M. D., & Wilhelm, A. G. (2015). Middle school mathematics instruction in instructionally-focused urban districts. *Urban Education, 52*, 829–861.

**Five Ways We Undermine Efforts to Increase Student Achievement
**

**(and what to do about it!)**

Blog Post Part 2 of 5

Wow – I am blown away by the response to my posted Ignite video! You can view it on the NCTM Annual website (https://www.nctm.org/annual/) beginning at timestamp 21:40. I am also thrilled with the conversations shared on twitter (@thestrokeofluck). I didn’t know what to expect as I am relatively new to blogging, so please keep the feedback coming so I can support you to the extent I am able. I will also share the second installment related to these posts regarding administrators’ six spheres of influence regarding teaching and learning mathematics in a blog hosted by HMH within the next week or so. The first post in that series is posted here: https://www.hmhco.com/blog/an-administrators-6-spheres-of-influence-in-mathematics-teaching-and-learning.

Let’s dig in to the second of Five (Un)Productive Practices.

During the Ignite presentation I said that I wanted to talk about gradual release and then gradually get rid of it. I must admit that what I shared was a bit stronger than I had planned. I guess I got caught up in the momentum of the Ignite session. I was probably still reeling from the outstanding presentations provided by my fellow Igniters. It was very stressful to watch such excellent and entertaining five-minute presentations knowing that I was going to share some ideas that were intended to make people feel uncomfortable – and I only had five minutes to do it in!

So what did I intend to say? I intended to say we need to be critical about our over-use of the classroom structure referred to more commonly as “I do, we do, you do.” This teaching technique is being overused during mathematics instruction and people are realizing it! This is a good thing, but it is not enough. In an effort to hold on to a structure that is no longer appropriate to support all aspects of teaching for rigor, people are responding by saying, “you can enter gradual release at any phase.” I can’t be the only one that finds this to be nonsense…

The entire idea of graduate release of responsibility is to begin with the teacher in control of the sense making by modeling a problem or idea (see Pearson and Gallagher (1983) for the original coining of this term in an article on reading comprehension). The teacher then moves to a more facilitated role by supporting students to engage in the task along with the teacher, in essence, replaying what the teacher has shared. Finally, the teacher relinquishes control so that students can demonstrate their understanding.

Changing the order of gradual release, or entering it at a phase where control is already relinquished is no longer gradual release! You can’t gradually release something you didn’t have at the start! OK, enough exclamation points – but this really gets to me.

With all of that said, I do believe that there is a time and place for gradual release. It is absolutely appropriate for teaching procedures. If your goal is to teach a lesson on long division or polynomial division, I strongly encourage you to model the process first, then provide guided practice as your students use the procedure with you, and finally, allow space for your students to practice the algorithm independently. The implementation of gradual release, without modification, is appropriate for procedural lessons. What about lessons more conceptual in nature? This is where gradual release needs to be replaced, not revised.

My colleagues and I in DNA Mathematics (#DNAmath) developed a lesson delivery structure that is appropriate for use with conceptual lessons. I will introduce it here. However, if you want a more comprehensive exploration, I invite you to check out the *Making Sense of Mathematics for Teaching *series (https://www.solutiontree.com/products/product-topics/dixon-nolan-adams-mathematics-resources.html).

The *Layers of Facilitation *describe a lesson structure that is more student-centered than Gradual Release of Responsibility. The goal of this structure is to privilege classroom discourse while maintaining a focus on the learning goal for the lesson. The teacher implements a task through the use of whole-class discussion. The teacher supports students to engage in the task through questioning. Full-class discussion around a problem is followed by students all working on the next problem (or set of problems) in concurrent small groups with the teacher pushing into groups to provide support through questioning and to collected evidence from student responses. Finally, individual accountability is supported as students work on problems on their own to provide evidence of where they are relative to the learning goal for the lesson.

The key here, again, is that the students are doing the sense making and the teacher is supporting them to meet the learning goal through the task that is chosen and the questions that are used to support the implementation of that tasks. As stated in the first post, we can transition our unproductive practices to be productive by keeping the learning goal and student engagement at the foreground of our planning and by critically analyzing our instructional decisions and structures. I am looking forward to continuing this conversation!

Please tweet your thoughts, comments, and ideas on this post to @thestrokeofluck

__References:__

Pearson, P. D. & Gallagher, M. C. (1983). The instruction of reading comprehension. *Contemporary Educational Psychology, 8*, 317-344.

**Five Ways We Undermine Efforts to Increase Student Achievement
**

**(and what to do about it!)**

Blog Post Part 1 of 5

Thank you for the amazing response to the ignite session I provided at the 2018 Annual Meeting of the National Council of Teachers of Mathematics (#NCTMAnnual2018) [to view the Ignite session, go to https://www.nctm.org/annual/ and click on the Ignite video – Juli’s talk begins at 21:40]. Based on the feedback I received I decided to write a series of posts to provide a more in-depth exploration into these unproductive practices and what to do about them. What follows is my first in a series of 5 blog posts I will share here over the next few weeks. Please submit comments, questions, and ideas so that I can work them into my responses for my subsequent posts. I am hoping that we can use these posts as a catalyst for some important dialogue here and on twitter (@thestrokeofluck).

While we are focusing on these unproductive practices here, I am also sharing related posts regarding administrators’ six spheres of influence regarding teaching and learning mathematics in a blog hosted by HMH (https://www.hmhco.com/blog/an-administrators-6-spheres-of-influence-in-mathematics-teaching-and-learning)

I imagine that those of you who were not at the ignite might be curious about these unproductive practices by now. Here is the slide I shared at ignite:

These teaching practices are commonplace and often required by administrators and/or districts. Many of them may have been generated from our colleagues in English language arts (ELA) and might work very well in their content areas, however, upon reflection, you will see that they are often unproductive when applied during mathematics instruction. My goals in this series of blogs are to help you to see why these practices are unproductive and also to assist you in generating a plan for what to do about it.

This post will focus on the first unproductive practice of posting the lesson objective (or essential question) for conceptual lessons. Posting the lesson objective for conceptual lessons at the start of the lesson has the potential to undermine students’ efforts to engage in sense making. What does that mean and what can we do about it?

We need to start by acknowledging that the learning goal should determine the tasks that we use and the questions we choose to support those tasks. If a lesson is conceptual in nature, like making sense of division or understanding what factoring accomplishes, then the ways students interact with the content in those lessons should necessarily be different than with procedural lessons like those focused on long division or polynomial division. The issue here is that if students are told what it is they are supposed to “discover” at the start of the lesson then the students have been robbed of the discovery process, even if that process is highly facilitated. There should be some aspect of discovery in conceptual lessons. This is not necessarily the case with procedural lessons. If a lesson is procedural, it is appropriate and even desirable to post the lesson objective at the start of the lesson (Wiliam, 2011). If lessons are conceptual, the teacher guides students to uncover the lesson objective through guided facilitation with the tasks the teacher chooses and the questions teachers use to support those tasks.

When I share this with teachers I get some push back because of the requirements that teachers *must *post the essential question or lesson objective a the start of the lesson. My gut tells me to respond by saying that we need to work together to change the rules! My pragmatic side reminds me that we don’t have time for this. We must adjust now and change takes too long. While we work together to change the rules to support student achievement in mathematics, we must also work together to survive and flourish within the system we find ourselves. Here is my response to that need.

Using the Four Queries for the Essential Question is a good start. As with any lesson, our first task is to make sense of the learning goal. If the lesson is conceptually based *and*there is a requirement to post the essential question or lesson objective, then a useful practice is to “zoom out” on the essential question. By this I mean to word the question or objective in a way that provides students a general idea of the goal of the lesson while protecting the inquiry that should be inherent in a conceptually based lesson. For example, if the lesson if focused on contrasting sharing into groups and making groups to divide, the posted question could be, “How can I divide in different ways?” By the end of the lesson, the student should be able to describe the difference between sharing and grouping to divide. The key here is that the student does the sense making by the end of the lesson rather than being told by the teacher what to think at the start of the lesson. We can transition our unproductive practices to be productive by keeping the learning goal and student engagement at the foreground of our planning and by critically analyzing our instructional decisions and structures. I am looking forward to continuing this conversation!

Please tweet your thoughts, comments, and ideas on this post to @thestrokeofluck

__References:__

Wiliam, D. (2011). *Embedded Formative Assessment*. Bloomington, IN: Solution Tree Press.