# We Need More Drawing in Math Class

I worked with a 6th grade student in our In School Intervention classroom today.  I do this every day.  A boy had been assigned a set of pages from a workbook and had begun the work on his own. From what he had already completed, I could tell that this student had a strength in math. He was unsure of what he had done, so he was looking for validation from me.  We looked at this next problem together:

He said, “so you gotta subtract right?” I said, “yes you do, but that’s if you’re looking for the answer. This is asking you to write an equation.”

He began to write y = x – 14.

I talked to him about variables, and how many unknowns were in this problem and he could tell that there was only one.

I asked him to draw a picture of the situation.  Here is what he drew.

HMMMM….(maybe they’ve recently worked on area and perimeter???)

I then asked him to reread the problem and prompted him to draw the house and the airport.

He drew the line between the house and the airport, and I had to do some questioning to get him to realize it represented the distance of 29 miles.  We reread the problem and I asked him about the 14 and how he would represent that on the drawing.

He thought to himself for a moment and then began drawing the tick marks.  I could tell that he was counting them.  He labeled the 14th tick mark and the 29th, and then circled the 14.

I drew the bracket and asked him what that would represent.  He said with a questioning tone, “the remaining distance?” Thinking for a minute, he then said, “oh I know…would it be 14 + x = 29?”

We did one more problem together that we represented with a drawing, and then I left him to be independent in the rest of his work.

He came over to show me one of his drawings and the equation he had written:

(In a previous part of the problem they were told there are 150 seats in coach seating.)

We high-fived!

Teaching students how to create math drawings is a valuable sense-making tool that can be over looked in math class.  I know numerous reading language arts teachers that ask students to draw pictures to portray what’s going on in a story.  We need more of that in math class.

# Noticing Subtleties…Reflection

So I wrote this post yesterday, and today we tried it out.

The purpose was to give students an opportunity to notice the subtleties in the language associated with three different mathematical scenarios they may need to represent.

We read each scenario out loud and then asked the students to compare and contrast each problem type with their table partner.

Some students began by discussing the similarities and differences of the contexts…

Others began by making a list of what they noticed on the back…You can see that this student paid attention to more of the mathy parts~understanding what was meant by a one-variable versus a two variable equation.

And then you’ll see below where students were able to make sense of each scenario and the math required.  However, the first student used an equation in two variables for the first scenario and created a table to find the solution.

We didn’t get to the whole class conversation part of this lesson…I want to talk about each problem type and how to recognize the differences.  Notice that two of the students above wrote the equation for the two variable scenario, but the third student created a table.  I think we need to talk about why that is.  Also, I think I may want to do three more scenarios that would produce equations in standard form, to see if they would recognize the differences then.

I definitely think that this was a useful exercise and would do it again.

One little shout out…one student pulled out their phone because they wanted to check out the equations for the third scenario on their Desmos app!!!

# Noticing Subtleties

This just happened…

I walked into an 8th grade classroom finishing up writing and solving systems of equations given word problems.  They began this learning last week.

I overheard the teacher and a student having a discussion about drawing pictures to represent the problem versus trying to write the equations.  This told me that the weekend was too big of a gap from the examples that they had done last Thursday (they had a sub on Friday).

Here is a problem from the set that they were working on:

I briefly walked over to a different student and noticed that she only had one equation to represent this cars and motorcycles problem. She defined the variables correctly, but had written a single equation that mixed the information about the number of wheels and the total number of vehicles.  She hadn’t made sense of what her equation actually meant.

But, I thought that maybe it goes further than that.

I think the students need to see the difference between problems that require a one-variable equation, a two-variable equation, and a system of equations side by side. Here is what I came up with:

It’s nothing fancy or mind-blowing…I just want to see if this gives them a structure to look for when they are deciding how best to represent a given scenario.  I want them to see that there are two unknowns in the third case, and that they need to create an algebraic representation for Eli’s savings and for Lucas’ savings.

I’m wondering if we should ask them to notice and wonder???  Or simply ask…what is the same and what is different…

I’m also wondering if I should remove the headings from the table before we ask them to notice and wonder???

We’re going to try this tomorrow…so I’ll let you know how it goes.

# Vacas y Pollos ~ Best Day Ever

Yesterday I was talking to our Algebra teacher.  She had asked the Spanish teacher in the building to translate the  problem below so that she could pose it to her students that take both Algebra and Spanish.

Our population of students that are emerging bilingual has grown significantly over the past several years.  The grade 8 students are also working on systems of equations and I had the perfect teacher in mind to pose this problem (written in Spanish) to his students as well.

His last class has the greatest number of Spanish speaking students and I wanted to see the look on their faces when he posted this problem on the board…

Priceless. I watched the biggest smile appear on their faces! The look of joy of what they knew. The boy that participates the least read the problem out loud so quickly and proudly…it was AWESOME! The rest of the students were in awe. They tried to figure out pieces of the problem. But, the students that usually have to spend time using Google translate to figure out what was going on, were the first ones to get to dig in this time.

One girl then said…”wow…they have to do this all the time.”

Yep.

Best day ever.

# Un-quiz

My first year out of the classroom was 2010.  I was talking to a colleague today about a type of assessment I used to do in my classroom.  I called it an un-quiz.

This is a variation of a typical multiple choice quiz, except that I asked for a WRONG answer. Along with the wrong answer choice, the students had to give a reason why it was a wrong answer.  (Be sure that your wrong answer choices provide valuable information!) This allowed me to see how they reasoned about wrong solutions, language they used, AND the students had multiple means of showing their knowledge.

I haven’t used this strategy in a while…if you do something similar or try it for the first time, I’d be interested in hearing about what you think.

# Equal and Opposite

I don’t know if this is the right title for this post…it feels like it is.  I get to its point at the bottom…

I was supporting an 8th grade teacher that was implementing the Classifying Solutions to Systems of Equations formative assessment lesson from Mathematics Assessment Project.

The students had not yet worked with linear equations in any form other than slope-intercept form.  The students already understood the different types of solution a system might have.

The students started out by completing the assessment task.  I wanted to focus on this part below:

I’ve encouraged this teacher to build a strong foundation for students being able to create and complete a table of values.  Creating a table should be a go-to strategy for students if all else fails.

I stood in the back of the room and I looked at the second equation…

I wondered if I could make a connection for kids…if it would be too much, or would they follow…Here is the gist of what I did:

I interrupted the class to pose a question.

Me: What do we know about an equation that is solved for y?

Students: We know the y-intercept and we know the change in y over the change in x.

Students: It’s solved for x.

Me: What do you think that might tell you?

Students: The x-intercept???? (Imagine a questioning tone here…) Is that a thing?

Me: Yes it is a thing. What about the coefficient of the y? What do you think that might tell you?

Students: The change in x over the change in y??? (Imagine a questioning tone again)

Me: Yep. Let’s pull up Desmos and check it out.

I remember being a math student that naturally made these sorts of connections~that if something happened mathematically one way, I could predict what would happen in an opposite direction.

I want the students to know and wonder about these things for themselves.  But, I think we need to make it obvious to them sometimes that they can do this thinking on their own.

# Teach Math as a Story

I had the opportunity recently to complete peer observations outside of my content area. I scheduled a half day of observations with a sixth grade social studies teacher new to our building.  We observed in two eighth grade social studies classrooms and one sixth grade classroom.

If you’ve never observed outside of your content area I highly recommend it.  I think it forced me to pay closer attention to the general instructional practices, rather than focusing in on the content.

In the sixth grade classroom the teacher began by reviewing what the students discussed in the previous class.  They were in the middle of learning about Julius Caesar and the fall of the Roman Republic. The students eagerly answered her questions and were incredibly engaged with the “story” she was telling.

I was engaged with the story, intrigued by the cast of characters and happenings that she described.  The students were making predictions about what would happen next and the teacher responded, “just wait…maybe we’ll see today…”

The students knew the characters in this story, they understood how they related, they recalled the parts of the story that were told to them in the previous class, they made predictions about what would happen next. The teacher also knew this story, oh so well, that she could add on interesting and important details and maintain the curiosity that she had sparked.

I wondered…can we teach math as a story?

It is an excerpt from a book and it’s mostly about using stories in the math classroom, but it has useful think-a-bouts like this…

“In our description of how to teach mathematics, we are not concerned with fictional stories about the topic, but rather we are concerned with how we can shape the topic to enhance its attraction to students. In doing this, we will not be falsifying anything, or giving precedence to entertaining students over educating them. Instead, we will be engaging them. We see engaging students with mathematical activity as a crucial aspect of successful education as, and it is the real vividness and importance of this subject in which we want to engage students.

In summary, the great power of stories, according to Kieran Egan (1986, 2004, 2008), is in their dual mission: they communicate information in a memorable form and they shape the hearer’s feelings about the information being communicated.”

I did some additional searching through Peter Liljedahl’s work and found this interesting article that seems related to what teacher planning might look like in order to teach math as a story.

In this article, Zazkis and Liljedahl contrast a typical lesson plan to what they’ve termed a lesson play.

“In terms of the pedagogical features of the lesson play, we wish to draw attention to some aspects of its format. The structure of the lesson play – as a dialogue occurring overtime with possibilities for different points of view – allows for the portrayal of the messy, sometimes repetitive interactions of a classroom. This structure stands in stark contrast to a necessarily ordered and simplified list of actions such as: take up homework, state definition, provide examples,give problems, and evaluate solutions.”

Crafting a lesson play provides for the improvised interactions that may occur with teaching math as a story-being able to respond and shift according to responses from students.

I don’t think any of this is dissimilar from the ideas in books such as 5 Practices,  but I now have a different analogy that I’m considering. As I continue the thinking that I’ve started here, I want to keep in mind these things in terms of how I work with the math teachers in my building:

1. On the macro level-How can I help teachers to tell the math story as a set of interconnected ideas and concepts?
2. On the micro level-How can I help teachers to consider a lesson play, so that the day to day story is just as interesting as the year long story they are telling? How do we get students to want more?