## An explanation of the difference between explaining and justifying — or maybe its a justification… you decide by Juli Dixon

The terms “explain” and “justify” are used frequently when discussing expectations for how students should support their solutions in mathematics. But what do we really mean? Is there a difference between explaining and justifying? I think there is a difference. To me, an explanation describes what the student did (frequently this is simply the steps taken to perform a procedure) and a justification includes why what the student did is mathematically okay.  While this description is helpful, it isn’t enough. The purpose of this post is to make sense of that difference through the use of a mathematical task.

The terms “explain” and “justify” are used frequently when discussing expectations for how students should support their solutions in mathematics. But what do we really mean? Is there a difference between explaining and justifying? I think there is a difference. To me, an explanation describes what the student did (frequently this is simply the steps taken to perform a procedure) and a justification includes why what the student did is mathematically okay.  While this description is helpful, it isn’t enough. The purpose of this post is to make sense of that difference through the use of a mathematical task.

Two students are asked to compare 4/7 and 4/5 to determine which fraction is greater.

The students both conclude that 4/5 is greater than 4/7.

One student says that the numerators are the same so the denominators can be compared. Since five is less than seven, 4/5 is greater than 4/7.

Another student says that both fractions describe four parts of a whole. With 4/5 the whole is divided into fewer pieces than with 4/7. Since there are fewer pieces in the same-size whole, the pieces must be bigger. Therefore, 4/5 is greater than 4/7.

In both responses, the students provide a description of their thinking. However, only the second response provides a window into why the student’s thinking is mathematically valid. It is interesting to note that the explanation used the terms numerator and denominator while the justification did not include mathematical vocabulary. Students can provide justifications even if they do not have mastery of academic vocabulary. Students can reason in their own words. Reasoning is important. An explanation is provided without evidence of mathematical reasoning. A justification is based on mathematical reasoning.

In the fraction comparison example, the student’s explanation is fairly complete. You might be more likely to observe a student simply explain that the common numerator strategy was used without going into further detail. The explanation provided earlier includes a deeper description but it still does not include information regarding why the denominators can be compared so it is not a justification.

I believe that we should require students to justify their thinking as they build conceptual understanding. Once students transition from making sense of concepts to applying more efficient procedures, explanations of what they did should suffice, with the expectation that they can justify their thinking when asked. Since concepts should be taught before procedures then it follows that students would provide justifications for solutions to problems before they would provide explanations without justifications. This fits well with my position on academic vocabulary (see my earlier blog post below). Justifications do not rely on academic vocabulary – I would even go as far as to argue that withholding some academic vocabulary is useful in providing a justification. By avoiding the terms numerator and denominator with the fraction comparison problem, the second student needed to explain their meanings.

Justifications are time consuming to produce. Once students are comfortable justifying their thinking about a mathematical idea, the justification is no longer necessary. At this point, a simple explanation should suffice. Explanations are most efficient when they are provided through the use of academic vocabulary as in the example above where the student noted that with common numerators, one can simply compare the denominators.

There is a difference between explanations and justifications. Both are important, what matters is when they are emphasized. This is the case with many aspects of mathematics instruction.

Notice the trends and how they work together –

– We should teach concepts before procedures.

– We should support students to use everyday language before academic language.

– We should expect students to provide justifications before explanations.

How we organize our instruction influences how students come to make sense of mathematics. With this in mind, when do you think it makes sense to give students notes on mathematical ideas? This will be the topic of my next post.

## Small Group Instruction by Juli Dixon [Part 6 from the (Un)Productive Practices Series]

Five Ways We Undermine Efforts to Increase Student Achievement
(and what to do about it!)
by Juli Dixon

Blog Post Part 6 of 5

Yes – you read it right, this is post six of a five part series focused on the five potentially (Un)Productive Practices and what to do about them. It is a spin off from an Ignite session I provided at the NCTM annual meeting in April 2018. The Ignite can be found at the NCTM Annual website here: https://www.nctm.org/Conferences-and-Professional-Development/Annual-Meeting-and-Exposition/Past-and-Future/2018-Washington-DC/ beginning at timestamp 21:40.  However, I did not address this sixth practice during my session. You can only do so much in five minutes with 20 slides…

This sixth potentially unproductive practice has to do with small group instruction. Before I share why I feel the need to include this post, I want you to think about small group mathematics instruction you have observed recently, or possibly even one you facilitated. What came to your mind? You likely pictured a kidney-shaped table with four students all facing the teacher.  The students were probably like-ability. Were the questions posed by the teacher high level or low? I would guess that they were low.

During the professional development I provide, when I ask teachers to imagine small group instruction in mathematics the images they share are similar to what I provided here. This is a problem. What is described here is not best practice, especially when the goal is developing conceptual understanding of mathematics.

What is most concerning is that teachers and coaches are sharing with me more and more that they, or the teachers they support, are being directed to use small group instruction in every lesson every day. This is a problem. We know that the learning goal should influence the structure of the lesson because flexible instructional structures promote mathematical sense making. Yet small group instruction, in the way I just described is being promoted as a blanket requirement.

This is where the practice becomes unproductive. This structure may be valuable for practicing procedures, but it is not appropriate for developing concepts.

So what should take its place? At times, whole-class discussion is a better option, as is the use of concurrent small groups (students simultaneously working in small groups on the same task with the teacher pushing in to the groups as opposed to pulling small groups to the kidney-shaped table). Sometimes, the pulled small group is the best option, even when focusing on conceptual development. However, the small group needs to be re-imagined if your vision was like what I described earlier.

Students should be grouped moderately heterogeneously – not with outliers but with some differences across learners to enhance the discourse. Tasks should be worthwhile and the questions that support them should engage students in reasoning. The time should be used to collect evidence of student learning as well as their gaps in understanding. Those of you who know my work and that of my colleagues with DNA Mathematics know that what I am describing here is applying the TQE Process to the small group. I recorded a short video discussion on this topic. If you are interested in hearing more you can find it toward the bottom of the page at this location: https://www.solutiontree.com/products/product-topics/dixon-nolan-adams-mathematics-resources/making-sense-of-math-small-groups.html.

If you want to explore the topic of small group instruction in mathematics more deeply according to what I discussed here you might find the book that my colleagues Lisa Brooks and Melissa Carli and I just wrote valuable. It will be released July 13, 2018 but it is available for preorder at the site I shared in the last paragraph.

I think this is it, the last part in a five part series on potentially (Un)Productive Practices. However, I think I like blogging. I’ve also been encouraged by your responses. I will continue as long as people find what I share to be of value. I am taking requests for topics. The first request I received was to focus on what to do when students don’t share what we need them to share during instruction. I will blog about that topic here next.

Please tweet your thoughts, comments, ideas on this post, or suggestions for future posts to @thestrokeofluck

## Neglecting Opportunities to Connect Concepts and Procedures by Juli Dixon [Part 5 from the (Un)Productive Practices Series]

Five Ways We Undermine Efforts to Increase Student Achievement
(and what to do about it!)
by Juli Dixon

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/Conferences-and-Professional-Development/Annual-Meeting-and-Exposition/Past-and-Future/2018-Washington-DC/) 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

## Leading Instruction by Introducing Academic Vocabulary by Juli Dixon [Part 4 from the (Un)Productive Practices Series]

Five Ways We Undermine Efforts to Increase Student Achievement
(and what to do about it!)
by Juli Dixon

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/Conferences-and-Professional-Development/Annual-Meeting-and-Exposition/Past-and-Future/2018-Washington-DC/) 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.

## Providing Scaffolding Just in Case by Juli Dixon [Part 3 from the (Un)Productive Practices Series]

Five Ways We Undermine Efforts to Increase Student Achievement
(and what to do about it!)
by Juli Dixon

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/Conferences-and-Professional-Development/Annual-Meeting-and-Exposition/Past-and-Future/2018-Washington-DC/) 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.