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.

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.

Consider this 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 [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 [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