Re-inventing Mathematics Intervention: Focus on Conceptual Development by Juli Dixon [Part 2 of Re-inventing Intervention]

The first of Six Features for Re-inventing Intervention is to focus on conceptual development. The argument for the six features is laid out in the introduction to this series. I encourage you to start with the first blog in the sequence if you haven’t read it already. In that blog, I shared an example of a 4th grade intervention group where the teacher presents students with the following multiplication example:

Here is what I would likely say:

Two times 6 is 12, put down the 2 and carry the 1. Two times 8 is 16, add the 1 to get 17. Put down the 7 and carry the 1. Two times 4 is 8, plus 1 is 9. Put down a 0. One times 6 is 6. One times 8 is 8. One times 4 is 4. Two plus 0 is 2 and 7 plus 6 is 13. Put down the 3 and carry the 1. One plus 9 is 1, plus 8 is 18. Put down the 8 carry the 1. One plus 4 is 5.

Should I revise the self-talk I am using? My response might seem to contradict my position, but I don’t think I need to change it. The reason I think my self-talk is okay is because I already understand the algorithm. I know that when I say “carry” I really mean that I am regrouping ten ones to make one ten. I am recording the remaining ones in the ones column and the one ten in the tens column to add to the tens. It is more efficient for me to say “carry the one” than all of that.

Are you surprised? I spend my academic life focused on sense making in mathematics, and then when I perform the standard algorithm for multidigit multiplication when nobody is listening, the language I use for myself is basically nonsense. I start out fine when I multiply 2 times 6 to get 12. However, what does it mean to “put down the 2 and carry the 1”? Is it a 1? How do you “carry” a number? This language is confusing at best and inaccurate at worst. If the goal is to focus on sense making and I use this language with students, I am not supporting them to understand the process of multidigit multiplication1.

Efficiency is the issue though. In our rush to “catch our students up” we are jumping to the efficient language too early. This language should evolve from an understanding of the standard algorithm. When students are in an intervention setting and focused on multidigit multiplication, it is likely that they have already demonstrated struggles with this algorithm. Saying it more often and more slowly in the same way students sat through in whole-class instruction is not the answer. However, we also can’t use our own, adult self-talk as teacher talk. Our teacher talk, at least initially, must be grounded in conceptual understanding. Consider the difference as I talk through the algorithm using teacher talk.

I am multiplying 486 by 12. I think of 12 as 2 ones and 1 ten. I start by multiplying 486 by 2. Two groups of 6 ones is 12 ones. That can be regrouped as 1 ten and 2 ones. I record the 2 in the ones place and I record the 1 ten in the tens column to combine with the tens after I multiply them. Two groups of 8 tens is 16 tens. I add the recorded ten from earlier to get 17 tens in all. Seventeen tens can be regrouped as 1 hundred and 7 tens. I record the 7 tens in the tens place, and I record the 1 hundred in the hundreds column to combine with the hundreds after I multiply them. Two groups of 4 hundreds is 8 hundreds. I add the recorded hundred from earlier to get 9 hundreds in all. I record 9 hundreds in the hundreds place.

Now I multiply 486 by ten. Ten times 6 ones is 60 or 6 tens and 0 ones. I record the 6 in the tens place and the 0 in the ones place. Ten times 8 tens is 8 hundreds. I record the 8 in the hundreds place. Ten times 4 hundreds is 4 thousands. I record the 4 in the thousands place. Now I add 971 + 4860 and regroup as necessary. My finally product is 5832.

Using conceptually based language to describe the multidigit multiplication algorithm is a lot! It probably took twice as long, and that’s without describing the process of adding the partial products 971 and 4860. No wonder so many teachers skip this stage of the process. This leads us to reconsider the point of intervention. Is the goal to promote efficiency or understanding? The answer should be both. What is important to consider is the order with which they are addressed. We need to begin with understanding. Focusing on understanding is typically inconsistent with efficiency of language. It is often cumbersome and slow. But it is necessary. Returning to our multiplication example, to support students to make sense of multidigit multiplication, it is often helpful to back up and focus on sense making with single-digit multiplication and use that to lead into multidigit multiplication. For example, we can make sense of 2 x 6 as two groups of six. What about 2 x 86? This can be thought of as two groups of 86. We can break apart the 86 to find two groups of 6 and two groups of 80 then combine them. This extends nicely to 2 x 486. But what about 12 x 486. This is more complicated. Can we break apart both the 12 and the 486? Absolutely. That is what happens with the multidigit multiplication algorithm when we make sense of it. This is supported by using base ten blocks and the area model for multiplication.

Once students understand what is happening when we multiply the partial products (the 486 by 2 ones then the 1 ten in the task included here), the focus can shift to connecting the process to a more efficient solution path. The next blog in this series will address the need to be explicit when connecting conceptual development to procedural fluency. Stay tuned! In the meantime, please share your feedback on twitter @thestrokeofluck!


1 My colleagues and I discuss the conceptual underpinnings of multidigit multiplication in depth in chapter 2 of Making Sense of Mathematics for Teaching Grades 3-5 (Dixon, Nolan, Adams, Tobias, & Barmoha, 2016).

Re-inventing Mathematics Intervention: Making Time for Understanding by Juli Dixon [Part 1 of Re-inventing Intervention]

Unfinished Learning in the Spotlight

Recent conversations regarding unfinished learning in mathematics have brought topics related to intervention supports into the spotlight. This attention is long overdue. What follows is a culmination of many years focusing on this topic. My thoughts are informed by best practices in mathematics teaching and learning as well as insights gained from my experiences with my daughter and my work with students, teachers, and administrators in schools. 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). This story goes back many years – my daughter, Alex, is now 25 and works as a public-school paraprofessional in a pre-kindergarten class. I am a proud mama! I have been developing a concept of productive intervention since Alex was 12.

As we examine mathematics intervention, I think we will see that it needs to be reinvented. Imagine a scene where an interventionist is working with a small group of students in grade 4. The students are ability grouped; all having demonstrated struggle with multiplying multidigit numbers. The teacher shares the following multiplication problem and helps students to step through the process of the standard multiplication algorithm.

The teacher says to multiply the 2 by each digit in 486 and then to put down a zero as a place holder and then to multiply the 1 by every digit in 486. Students follow along and then the teacher has students practice with two more problems. The teacher provides immediate feedback if students make errors like forgetting to put down the zero while solving the problems. There is very little student talk other than to provide short answers to the teacher. Students are mostly successful following the steps, but when they encounter this in class again, not too much later, they struggle.

To recap, the intervention is highly structured and highly procedural, and is only minimally supported by student discourse and conceptual understanding. These sorts of scenarios are described in detail in Making Sense of Mathematics for Teaching the Small Group (Dixon, Brooks, & Carli, 2019) and suggestions for changing the tasks, structures, and general experiences in small group instruction are provided. What I am discussing here is more specific. The focus here is on intervention groups and can be identified with many different names. What I am talking about here is pulling small groups to support those who are significantly far behind. The argument I will attempt to make is that, with best intentions, we are trying to do too much and ultimately accomplishing too little.

Intervention Reimagined

The first step in transforming intervention is to embrace the idea that students who struggle in mathematics likely do so because there are foundational gaps in their conceptual understanding of prerequisite content knowledge. Those gaps widen as more content is addressed. The urge is often to “pre-teach” upcoming content so that when the learners who struggle encounter the content, they are more familiar, and ultimately more successful, in class. Returning to the fourth-grade example, we can imagine that this intervention might occur in preparation for a lesson on the same topic so that students may be more likely to experience success with the topic in class. The flaw in this plan is that the pre-teaching is often necessarily procedural because the conceptual understanding is elusive without the foundational knowledge the students lack. Confounding the issue is that many people who are tasked with providing the intervention lack access to professional development on best practices in mathematics teaching and learning.

What does this mean? It means that we are using precious instructional time with close proximity to a teacher (in a small group setting) in a manner that will likely not lead to desired outcomes. What do we do about it? We need to re-invent intervention.

The reimagining of intervention must involve a multifaceted approach. What I will share over the next several blogs are six features required for reimagining intervention. These features, when taken together, support the re-invention of intervention.

Six Features for Re-inventing Intervention

Productive intervention must:

1. Focus on conceptual development,

2. Provide a clear connection between concepts and procedures,

3. Prioritize a strategic selection of content,

4. Support discourse through engaging tasks and targeted questioning,

5. Elicit and linger on common errors, and

6. Include professional development focused on content knowledge for teaching for all interventionists.

Key Words are Evil by Juli Dixon

People have told me they know I am giving a talk someplace in North America when a tweet comes across their feed indicating that #KeyWordsAreEvil. Why am I so adamant about spreading this message? For so many reasons, three of which I will attempt to clarify here. What follows is my list of the three most important reasons to avoid teaching key words to students and why these reasons made it to the list. But first, I should define what I mean when I use the term “key words.”

When I pontificate that “key words are evil” I am referring to everyday words and phrases such as, “altogether” and “how many more” that are taught to students as indicators of mathematical operations. I am not talking about mathematical vocabulary terms like “sum,”  “product,” or “quotient.”

Here are two examples of word problems that are often used to highlight key words:

Jessica has 8 key chains. Calvin has 9 key chains. How many key chains do they have all together?

Jessica has 8 key chains. Alex has 15 key chains. How many more key chains does Alex have than Jessica?

Focusing on “all together” and adding the two numerals in the first problem results in students who are rewarded with the correct answer.  Similarly, students who search for key words and find “how many more” and then subtract the lesser number from the greater number in the second problem are also rewarded.  However, what happens when students encounter problems like this:

Jessica has 8 key chains. How many more key chains does she need to have 13 key chains all together?

Students are at least as likely to focus on “all together” and add 8 and 13 as they are to identify “how many more” as the key words and subtract. When I share these examples, supporters of key word instruction are often quick to indicate that this is not an issue for them. They say that they use strategies with students to ensure that their students focus on the “right” key words. This is still problematic. I hope that my three reasons for classifying key words as evil will shed light on why teaching key words is so problematic.

Three Reasons for Classifying Key Words as Evil

  1. Teaching key words undermines our efforts to create students who are problem solvers.
  2. Teaching key words sends the message that mathematics doesn’t truly exist in the real world.
  3. Teaching key words is an equity and access issue.

Teaching key words undermines our efforts to create students who are problem solvers.

A goal of mathematics instruction is to develop students who can reason quantitatively. This should include making sense of problems and determining pathways to reach solutions to those problems. Teaching key words negates this focus by, in essence, instructing students to ignore the aspects of the problems that make them problematic and just providing tricks for students to follow to reach answers.

Teaching key words sends the message that mathematics doesn’t truly exist in the real world.

This second reason for avoiding key word instruction follows from the first. By teaching students that there is no need to make sense of problems that involve mathematics but rather to just find the key words and do what they imply, we are teaching students that mathematics is just a set of arbitrary rules to follow, not a lens through which to view the world. When we use mathematics to examine and make sense of our world we are reinforcing the value of becoming problem solvers.

Teaching key words is an equity and access issue.

Students who struggle with reading and/or mathematics are more likely to be taught to use key words to solve mathematics problems than their peers. The justification for this practice is obvious. They are struggling and key word instruction helps them to get more answers correct. However, my position is that this is not a good enough justification. This practice does more harm than good. When students are singled out for key word instruction their access to problem solving and mathematical reasoning is restricted. When this access is limited, there is less opportunity to develop perseverance. This is an issue of equity and it must be stopped.

What is your responsibility in eliminating key word instruction? First, stop! That is, if you are still using it. Next, help your peers to stop! Share this blog. Tweet about it! It is time to have collaborative and courageous conversations. If you still have a key word poster on your wall – or one that is even remotely close to being a key word poster – take it down! Rip it up! Better yet, take a video of yourself ripping it up and send it out in a tweet tagging me @thestrokeofluck. It is time for this change in instructional practice to trend J

#KeyWordsAreEvil

Reflecting on Questioning by Ed Nolan

How do you use questions to support your students?

Do your questions focus on what you hear from students or do you funnel students toward a particular strategy you feel is best for the students? Does it matter?

One way of thinking about teacher questioning is considering the difference between “focusing” and “funneling” (Herbal-Eisenmann & Breyfogle, 2005; Wood, 1998).

When student sense making guides teacher questioning provided in response to student contributions, the teacher is focusing on student thinking. This type of questioning allows for connections between what students provide in their answers and the goal the teacher has for the lesson. Here, both the teacher and the students are engaging in the cognitive activity of making sense of the problem and solution.

When teachers use funneling questions, they guide students toward a particular outcome within the lesson regardless of what the students provide. This often occurs when teachers are looking for a specific strategy that they want students to use and they accomplish this by asking questions and collecting responses from students until they hear the student contribution they seek. Sometimes teachers will start from a student’s incorrect response and ask a series of questions leading the student to the strategy they desire. Teachers may also use the funneling question structure to lead students through the steps of a procedure or process. In funneling, the teacher is the one engaging in the cognitive activity (Wood, 1998).

Let’s look at a hypothetical example to help understand the difference between the two: Imagine a lesson where the goal is to identify the slope of line segments from a graph. One way to begin this lesson might be to allow students to explore the slope of segments as sides of triangles on a coordinate grid (see below).

Exploring slope using triangles on a coordinate grid

Consider the following two classroom vignettes:

Vignette 1

Teacher: What do you notice about segments BC and DF?

Calvin: I see that the change for segment BC goes down 4 and over 2 and the change for DF goes down 2 and over 4, hey, that means they are the same…

Teacher: They cannot be the same, as one goes down 4 and the other goes down 2.

Calvin: No, I mean that the triangles are the same.

Teacher: But that is not what we are looking for, what about the vertical change compared to the horizontal change? How are they different?

Calvin: What do you mean?

Teacher: How do you explain the ratio between the vertical change and the horizontal change?

Vignette 2

Teacher: What do you notice about segments BC and DF?

Calvin: I see that the change for segment BC goes down 4 and over 2 and the change for DF goes down 2 and over 4, hey, that means they are the same…

Teacher: Explain more about what you mean.

Calvin: The triangles are the same.

Teacher: How do you know they are the same?

Calvin: I can see that they are both right triangles and ABC has sides that are 2 and 4 units long and so does DEF.

Teacher: That is interesting. And how do the relationships of the sides BC and DF compare to each other?

Calvin: They are the same length.

Teacher: And how do those lengths connect to how we can describe the ratio of the vertical change to the horizontal change?

In the first vignette, the teacher is working to guide the student to see the rate of change of the segments and is using “funneling” to direct students toward the desired outcomes of the lesson. Funneling allows the teacher to take control of the thinking of the lesson and may occur in exploration when time is pressing or when the teacher wants to direct students to think in the same way as the teacher. What is lost is the opportunity to value the students’ thinking and to connect the learning goal to what students provide during instruction.

In the second vignette, the teacher uses the student’s response and guides the thinking toward the learning goal. Notice how the teacher builds off of what the student has offered and guides the thinking of the student toward the learning goal. This can take more instructional time but it helps students make connections between their current reasoning and the learning goal.

Here you get a sense of some of the differences between the focusing and funneling question structures. Consider your own questioning:

What role do students’ responses play in your questioning?

How do you use questioning to learn more about students’ thinking, and how do you use questioning to guide your lesson?

How does your questioning in response to student contributions differ when your students are exploring a concept, linking the concept to a procedure, or practicing a procedure?

In examining your questioning, you are demonstrating how you value student thinking. It is a difficult task and requires you to make sense of mathematics for teaching, to adjust to student-provided strategies as a guide for the development of understanding. Listening carefully to students and building off of their ideas takes a great deal of energy – but the rewards are clear. Students are able to connect their current ideas to the new concepts and procedures we need them to understand, and they will remember them better. And you will find that in working to make sense of student thinking and developing questions that link student thinking to your instructional goal, you deepen your own understanding of mathematics!

I am interested in your reactions to this blog, as well as your own examples of funneling and focusing and how you use them in the classroom. Feel free to respond on Twitter and tag me (@ed_nolan) – include the hashtag #DNAmath or any of the grade level hashtags of #MSMTK2, #MSMT35, #MSMT68, or #MSMTHS to bring more mathematics educators into the conversation.

Herbal-Eisenmann, B. A., & Breyfogle, M. L. (2005). Questioning our “patterns” of questioning.  Mathematics Teaching in the Middle School, 19(9), 484-489.

Wood, T. (1998).  Alternate patterns of communication in mathematics classes: Funneling or focusing?  In H. Steinbring, M. G. Bartolini Bussi, & A. Sierpinska (Eds.), Language and communication in the mathematics classroom (pp. 166-178).  Reston, VA: National Council of Teachers of Mathematics.