Numberless Word Problems

“They just add all the numbers. It doesn’t matter what the problem says.”

This is what a third grade teacher told my co-worker while she was visiting her classroom as an instructional coach. She didn’t really believe that the kids would do that, so she had the class come sit on the carpet and gave them a word problem. Sure enough, kids immediately pulled numbers out of the problem and started adding.

She thought to herself, “Oh no. I have to do something to get these kids to think about the situation.”

She brainstormed for a few moments, opened up Powerpoint, and typed the following:

Some girls entered a school art competition. Fewer boys than girls entered the competition.

She projected her screen and asked, “What math do you see in this problem?”

Pregnant pause.

“There isn’t any math. There aren’t any numbers.”

She smiles. “Sure there’s math here. Read it again and think about it.”

Finally a kid exclaims, “Oh! There are some girls. That means it’s an amount!”

“And there were some boys, too. Fewer boys than girls,” another child adds.

“What do you think fewer boys than girls means?” she asks.

“There were less boys than girls,” one of the students responds.

“Ok, so what do we know already?”

“There were some girls and boys, and the number of boys is less than the number of girls.”

“Look at that,” she points out, “All that math reasoning and there aren’t even any numbers in the problem. How many boys and girls could have entered into the competition?”

At this point the students start tossing out estimates, but the best part is that their estimates are based on the mathematical relationship in the problem. If a student suggested 50 girls, then the class knew the number of boys had to be an amount less than 50. If a student suggested 25 girls, then the number of boys drops to an amount less than 25.

When it seems like the students are ready, she makes a new slide that says:

135 girls entered a school art competition. Fewer boys than girls entered the competition.

Acting very curious, she asks, “Hmm, does this change what we know at all?”

A student points out, “We know how many girls there are now. 135 girls were in the competition.”

“So what does that tell us?”

Another student responds, “Now that we know how many girls there are, we know that the number of boys is less than 135.”

This is where the class starts a lively debate about how many boys there could be. At first the class thinks it could be any number from 0 up to 134. But then some students start saying that it can’t be 0 because that would mean no boys entered the competition. Since it says fewer boys than girls, they take that to mean that at least 1 boy entered the competition. This is when another student points out that actually the number needs to be at least 2 because it says boys and that is a plural noun.

Stop for a moment. Look at all this great conversation and math reasoning from a class that moments before was mindlessly adding all the numbers they could find in a word problem?

Once the class finishes their debate about the possible range for the number of boys, my co-worker shows them a slide that says:

135 girls entered a school art competition. Fifteen fewer boys than girls entered the competition.

“What new information do you see? How does it change your understanding of the situation?”

“Now we know something about the boys,” one of the students replies.

“Yeah, we know there are 15 boys,” says another.

“No, there are 15 fewer, not 15.”

Another debate begins. Some students see 15 and immediately go blind regarding the word fewer. It takes some back and forth for the students to convince each other that 15 fewer means that the number of boys is not actually 15 but a number that is 15 less than the number of girls, 135.

To throw a final wrench in to the discussion, she asks, “So what question could I ask you about this situation?”

To give you a heads up, after presenting to this one class she ended up repeating this experience in numerous classrooms across our district. After sharing it with hundreds of students, only one student out of all of them ever guessed the question she actually asked.

Do you think you know what it is? Can you guess what the students thought it would be?

I’ll give you a moment, just in case.

So all but one student across the district guessed, “How many boys entered the art competition?”

That of course is the obvious question, so instead she asked, “How many children entered the art competition?”

Young minds, completely blown.

At first there were cries of her being unfair, but then they quickly got back on track figuring out the answer using their thorough understanding of the situation.

And that is how my co-worker got our district to start using what she dubbed Numberless Word Problems – a scaffolded approach to presenting word problems that gets kids thinking before they ever have numbers or a question to act on.

Recently we shared this strategy with our district interventionists and several of them went off and tried it that week. They wrote back sharing stories of how excited and engaged their students were in solving problems that would have seemed too difficult otherwise. This seems like a great activity structure for struggling students because it starts off in a nonthreatening way – no numbers, how ’bout that? – and lets them build confidence before they ever have to solve anything.

Do I think that every word problem should be presented this way? No. But I do think this is a great way to prompt rich discussion and get students to notice and grapple with the relationships in problem situations and to observe how the language helps us understand those relationships. To me this is a scaffold that can help get students to attend to information and language. As many teachers like to say, standardized tests are as much reading tests as they are math tests.

Perhaps you can use this activity structure when students are seeing a new problem type for the first time and then fade away from using it over time. Or maybe you have students who have been doing great understanding word problems, but lately they’re rushing through them and making careless errors. This might be an opportunity to use this structure to slow them down and get them thinking again.

Either way, if you do try this out, I’d love to hear how it went.

Talking Up Talking Points

I have been talking up Talking Points ever since I got home from Twitter Math Camp in July. Don’t know what Talking Points are? No problem. You can learn more about them on the Twitter Math Camp wiki. We had a group led by @cheesemonkeysf who dove deeply into this topic and shared their work with the rest of us. When you have a chance, I suggest reading the document titled About Talking Points. In the meantime, here’s a brief summary:

Talking Points are simply a list of thoughts. They are statements which can be factually accurate, contentious, or downright wrong. They can be thought-provoking, interesting, irritating, amusing, smart, simple, brief or wordy.

That definition aside, it’s not so important to know what Talking Points are as it is to know what to do with them and why.

Often children (and even adults) are asked to discuss their ideas or work together in groups, but it quickly becomes apparent that those involved don’t really know what is being asked of them.

You have those participants who love to talk, but they don’t really know how to consider other people’s views. On the other hand, you have participants who are quiet. They find it hard to join in the conversation.

The role of the teacher is to make explicit the kind of talk that is useful, and that is where the Talking Points activity comes in.

The activity stimulates speaking, listening, thinking, and learning. It offers ways in to thinking more deeply about the subject under discussion. It gives everyone a chance to say what is on their mind, so that others can decide whether they agree or disagree.

If you want to try out the activity, you need to create small groups of about 4 people per group. Give each group a list of Talking Points. (You can write your own, or you can start by using some of the ones on the Twitter Math Camp wiki.) The group will engage in 3 rounds per Talking Point.

Round 1

  • One person reads a Talking Point
  • Go around the group. Each person says whether they AGREE, DISAGREE, or are UNSURE about the statement AND WHY.
  • The most important part of this is that there is NO COMMENT by anyone else in the group. Their job is to listen.
  • After everyone has had their turn, proceed to round 2.

Round 2

  • Go around the group again. Each person says whether they AGREE, DISAGREE, or are UNSURE about their own original statement OR about someone else’s statement they just heard AND SAY WHY.
  • Again, there should be NO COMMENT from anyone else in the group while someone is speaking.
  • After everyone has had their turn, proceed to round 3.

Round 3

  • Go around the group one final time. Each person simply states whether they AGREE, DISAGREE, or are UNSURE about the original statement. The group takes a tally and moves on to the next Talking Point.

When time is up, the facilitator can choose whether to have the participants complete a group self-assessment. (Check out the wiki for an example.) This gives them a chance to reflect on the discussion and how their group worked together.

The facilitator can also do a whole group debrief. I highly recommend doing this because it helps the participants reflect together, and it also gives the facilitator a chance to point out and emphasize to everyone behaviors and ways of talking that were effective. Here are sample questions that could be discussed:

  • Who in your group asked a helpful questions and what was it?
  • Who in your group changed their mind about a Talking Point? How did that occur?
  • Who in your group encouraged someone else? How did that benefit the conversation?
  • Who in your group provided an interesting additional idea and what was it?
  • What did your group disagree about and why?

You might be thinking to yourself, “Oh this activity is just like ___.” I’ve had several people tell me that, but once they have participated in their first Talking Points activity, they see how much deeper the conversation is than in the other activities they were thinking of.

The first time I tried this activity was with our district interventionists. This school year we are leading PD sessions with them about 10 times spread across the year, and one of the recurring themes is knowing their learners in general and using strategies to foster a growth mindset in particular. Before we dove into this topic, we created some Talking Points to get them talking with one another about their beliefs about intelligence and learning. Here is the set we used:

Talking Points About Intelligence and Personal Qualities

  1. Your intelligence is something very basic about you that you can’t change very much.
  2. You can learn new things, but you can’t really change how intelligent you are.
  3. No matter how much intelligence you have, you can always change it quite a bit.
  4. You can substantially change how intelligent you are.
  5. You are a certain kind of person, and there is not much that can be done to really change it.
  6. No matter what kind of person you are, you can always change it substantially.
  7. You can do things differently, but the important parts of who you are can’t really be changed.
  8. You can always change basic things about the kind of person you are.

The Talking Points were successful in getting the interventionists to start talking about ideas of growth mindset without us have to tell them anything. Many of them already felt strongly that intelligence can change (which made me happy considering the population they serve!) but I appreciated the discussion in a few of the groups where one person was able to point out some nuance in the language that made the others in the group consider the statement more carefully.

For example, in one group someone raised the point that she has an uncle with Down syndrome, and over the course of his life she doesn’t feel that his intelligence has really changed all that much. This personal experience made it difficult for her to fully agree with the Talking Points statements.

When the groups were finished discussing their Talking Points, we debriefed as a whole group. One of the recurring comments they made was how nice it was to have a chance to give their opinion and feel like it was heard by the rest of the group. Because of the “No Comment” rule, they knew that no one was going to interrupt them, and if someone tried, the other group members quickly reminded them of the rule.

A few weeks later, I was asked to share Talking Points with our district leaders at a Vertical Leadership Team meeting. This meeting consisted of all of the principals from all of our elementary, middle, and high schools as well as many other district leaders, about 120 folks in total. No pressure! I was nervous about what they’d think, but happy to have the forum to share the activity.

As with the interventionists, I was using the activity to lead in to a discussion of growth mindset research, but this time I revised the statements because I wanted them to each feel unique. In the first set I used, I felt like several statements said the same thing but in different ways. Here are the statements I finally settled on:

Talking Points About Learning and Intelligence

  1. You can learn new things but you can’t really change how intelligent you are.
  2. You are a certain kind of person, and there is not much that can be done to really change it.
  3. When you are learning something new, you should avoid making mistakes at all costs.
  4. You are smart when you can complete tasks quickly and accurately.
  5. The people who are the best in their field tend to be naturally good at what they do.

I only had 20 minutes to introduce Talking Points, share how to do them, have everyone in the room try them, and then debrief. It was a rush to get it all done, but it went smoothly enough. I definitely think 30-35 minutes is probably better for a first introduction.

I love walking around and listening in on different discussions while the groups are going through the Talking Points. While listening to the district leaders, I loved hearing one woman say, “Well, I did agree with the statement, but now I disagree because of what you two just told me. I just didn’t know that before.” It’s not required that anyone change anyone else’s mind during Talking Points, but it sure is powerful that it has that ability to happen based on just three short rounds of sharing opinions.

Someone in my department used Talking Points just over a week ago with a team of instructional coaches. While each grade level has some common ELA vocabulary that is used, the way it is used has not always been consistent across grades. Before the coaches started preparing for upcoming PD sessions, she had them go through some Talking Points about the vocabulary. She said it was such a quick and powerful way to gain clarity as a group before they started their planning.

Ever since I’ve shared Talking Points, it has started to slowly spread in my district, and I’m excited to see the creative ways it is used to deepen conversations among both students and staff.

 

Kickoff! #ElemMathChat

Tonight we kicked off a new weekly Twitter chat, #ElemMathChat. Hooray! As the name implies, the chat is designed for elementary school folks to talk about math.

ColorfulFireworks

I’ve been so excited to get this chat started! For the past two years, I’ve been a member of the MathTwitterBlogoSphere whose membership is primarily composed of middle and high school teachers. There are a few of us elementary-minded folks. We have appreciated all of the interactions we’ve had with the MTBoS. However, after meeting up at this year’s Twitter Math Camp, we decided our mission this year is to grow the elementary-side of the MTBoS.

And so it begins.

Tonight’s chat was a huge success! We had a great turnout with educators from around the US and Canada. (Thanks for catching my mistake @ChrisHunter36!) Everyone seemed excited about having a forum to discuss elementary math specifically. One person even commented that she was happy to have a place where she could be taken seriously. She said she’s tired of being considered “cute” for teaching first grade.

Our topic for the first chat was balancing problem solving with teaching/covering math skills. If you want to catch up on the conversation, you can check out the Storify put together by @davidwees. While the overall conversation was energetic and interesting, I was left a tiny bit disappointed.

I think it’s because I was the one who suggested this topic. Balancing problem solving and covering math skills is something I have struggled with myself as a teacher, and now as a district curriculum specialist, I am hearing from numerous teachers who are struggling to find the same balance themselves. So going in, I had some clear ideas of what I wanted to talk about and get out of the discussion.

The first question was “How do you define problem solving in the elementary math class?” This generated some interesting discussion. Some key points that rose to the surface for me were that problem solving involves thinking critically, collaborating, and using math as a tool. I especially like the “math as a tool” metaphor because it gives meaning to why we’re learning it in the first place. I think it’s often an implied message, but one educators need to try to make more explicit. I also liked how people described problem solving as a time to make kids get out of their comfort zones and make their brains sweat. I love the image that conjures in my mind.

The interesting thing that came out of this first question is that everyone seems to have different ideas about what problem solving is. Some people talked about it in a way that sounded like solving word problems, whereas others referred to rich and engaging tasks that focus more on the process than the endpoint. This is one area where Twitter chats can frustrate me. The conversation is happening so fast with so many people talking simultaneously that it can be challenging to pull the threads together into a coherent whole.

Maybe that’s what I need to learn how to do as a moderator. Instead of following my script of questions, I could have stopped and made question 2 be “So I’ve heard problem solving described as ___, ___, ___, and ___. What is one definition we can all agree on?” The conversation over the rest of the hour felt weaker because we didn’t necessarily have an agreed-upon definition to base our discussion on.

Question 2 also had some problems: “How do you define math skills?” This is where I had a clear idea of what I meant, but the majority of the group was on a different wavelength. Since we had just talked about problem solving, everyone seemed to think that I meant the Standards for Mathematical Practice or general thinking skills that are needed to solve problems. What I really meant, and I did try to clarify, are the nuts and bolts skills that teachers need to teach their kids: adding and subtracting whole numbers within 1,000, multiplying fractions with whole numbers, interpreting dot plots, and measuring angles, to name a few.

Here’s an example to illustrate the tension I was thinking about when suggesting this week’s topic. Learning a skill like long division takes time and effort. It is a very structured thing to do, but until students understand it, they are prone to making many errors. Can I do a few problem solving activities and have my kids somehow come away from the experiences as masters of long division?

You may be thinking right now, “But kids don’t actually have to know long division in order to solve problems. They just need a strategy that makes sense to them.”

I agree with you. However, in Texas and in Common Core, the standards do explicitly state that students learn to divide using the standard algorithm. So like it or not, it’s a skill that students are expected to learn.

Here’s where the tension comes in. Long division is just one skill. There are numerous other skills students are also expected to master in any given grade. How do you ensure the nuts and bolts mastery while at the same time providing ample opportunity for the types of activities that require critical thinking, collaboration, and brain sweating?

And please don’t take any of this the wrong way; I don’t fault anyone in the chat for not providing me a satisfying answer. To be honest, I don’t think a one hour Twitter chat is going to be the place to find concrete answers to big questions like this. It doesn’t mean I don’t want answers (hence the tiny bit of disappointment I felt), but I have learned over the past two years what Twitter can and can’t do.

What it can do is bring together like-minded people to fuel conversations and build relationships. The more I connect with people on Twitter, the more I get to know them. I can start chatting with them outside of our weekly chats. Perhaps I ask for help with a problem I’m having, or perhaps we set up a Google Hangout to have an actual conversation about a particular issue (good-bye 140 character limit!), or maybe we even collaborate on a proposal for a national conference.

Valuable professional relationships can grow from short, weekly conversations. It’s why I’m still here two years later, and it’s why I’m excited to get this specific chat launched. I’m eager to meet like-minded elementary folks and start forging some new professional relationships.

Adventure Time

One thing I’ve learned in my first month and a half at my new job is to be prepared for anything. A few weeks ago, my boss told me and the elementary ELA lead that we were going to be providing professional development to our district’s 107(!) interventionists. Not only that, but we would be providing them a total of 14(!) all-day PD sessions over the course of the school year.

On one hand, how cool is it that we get to work with every single one of our campus interventionists to create some shared vision around Response to Intervention and to help build their knowledge and skills of math and reading intervention?

On the other hand, holey moley! That is quite an undertaking on top of our other job responsibilities.

Just over a week ago, we held our first session. Thankfully over the course of planning for it (and getting feedback from interventionists who couldn’t believe they were going to be off campus 14 full days during the year) this adventure did come into better focus. For example, instead of having to plan 4 hour sessions each time, my partner and I only have to prepare 2-2.5 hour sessions on math intervention. In addition, instead of meeting 14 times during the school year, the higher ups knocked that number down to 9.

It’s still a big undertaking, no doubt about it, but it is feeling more and more manageable. On top of that, our first session went beautifully. Our primary goal, which I’m happy to report we achieved, was getting buy-in from the interventionists, many of whom had no idea this was even going to happen until a few days before the first meeting.  Now that the first session is over, I’m excited for the work ahead.

Since this is such a big project, I’m going to try to blog and reflect about it this year to see what I learn from it to apply to future endeavors. I also want to share my experiences in case anyone else out there can benefit from them.

The day started with all of the interventionists in a large group to hear from the higher ups about the purposes of these meetings and why they felt they are important enough to warrant so much time away from school. The district leads for RtI and dyslexia also went over some important changes that the interventionists needed to know about. When all of that was over, the interventionists broke up into groups to attend either a math session or ELA session.

Show time!

I opened the math session with everyone sitting in a community circle. My background is in a program called Tribes, and while I mostly used it with my students, the community building ideas apply to adults as well. The first thing I did was have everyone go around and introduce themselves, describe the types of intervention they provide on their campus, and then tell everyone a movie, book, or TV character that best represented how they felt at that moment.

I told them I felt like Lucy from The Lion, the Witch, and the Wardrobe, specifically the scene where she parts the coats in the wardrobe to reveal a newly discovered snowy landscape before her. I told them I felt like I knew what I was getting into with this job, but being asked to do these sessions this year opened me up to exciting new possibilities I hadn’t imagined. I added that I hoped nobody would be turned to stone or sacrificed on a stone table during the year.

As the interventionists went around the circle, I was impressed with how thoughtful, creative, and telling their responses were. Listening to them talk, I picked up on themes about wanting to be in control but feeling overwhelmed, about wanting to do the best job possible for their students, and about recognizing this opportunity to grow as a leader on their campus. While at first I had felt silly asking them to name a movie, book, or TV character, it ended up being a great way for them to share their feelings and realize that many people in the room felt the same way.

After introductions we moved into an activity called Talking Points. I learned about this activity in July from @cheesemonkeysf at Twitter Math Camp. I love Talking Points! They provide a way for people to improve exploratory talk, to dive deeper and have more meaningful conversation. You can download instructions and see examples of Talking Points on the Twitter Math Camp wiki.

The Talking Points that I had the interventionists do were all statements related to growth and fixed mindset. The statements didn’t use those exact words, but rather they got the groups talking about things like whether intelligence is something that can or cannot be changed. I did this on purpose because I wanted to know their current thinking on the matter. As interventionists, these people work with students who are having a difficult time in school. They not only need academic support, but they need someone who can help motivate them and encourage them to believe that they can learn. Like I told the interventionists, “If you didn’t believe that these kids can learn, then why would you bother showing up to work every day?”

Thankfully the interventionists seemed to believe in growth mindset by and large, so the groups tended to agree with each other, which I’m ultimately okay with. I was happy to see some dissenting opinions here and there though. Those groups were able to tease out some interesting ideas that the other groups missed out on.

After debriefing the Talking Points, I gave the interventionists a copy of our district’s math goals. I asked them to read the goals and do a quick write about how these goals are currently being met (or not) in their campus’ intervention program. Then they talked about their notes with the other folks at their table. This discussion was interesting because some of the interventionists didn’t even know we had district math goals. It also got some of them questioning whether their current intervention program was meeting the goals.

I used this discussion to segue into the foundation for our work this year, the What Works Clearinghouse guide on RtI: Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. The guide was put together by a panel including a research mathematician who is active in K-8, two professors of math education, several special educators, and a math coach. It provides 8 specific recommendations to help schools implement math intervention. The recommendations are based on the best available research evidence and the panel’s expertise in mathematics, special education, research, and practice.

Here are the 8 recommendations:

  1. Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk.
  2. Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8. These materials should be selected by committee.
  3. Instruction during intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.
  4. Interventions should include instruction on solving word problems that is based on common underlying structures.
  5. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.
  6. Interventionists at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.
  7. Monitor the progress of students receiving supplemental instruction and other students who are at risk.
  8. Include motivational strategies in tier 2 and tier 3 interventions.

All of the work we do this year is going to revolve around these 8 recommendations. To help kick it off, I had the interventionists look more closely at recommendations 2, 3, 4, and 5. I copied the sections about each of those four recommendations from the guide, and I had each interventionist read one of the sections. When they were done, they got together with the other people who read about the same recommendation. They collaborated to create two slides. The first slide contained up to 4 key points about their recommendation. The other slide contained noticings and wonderings their group had based on what they read. When the session was over, I put all of the slides together into one presentation that the interventionists could keep as a reference or that they could share with others on their campuses.

The interventionists were very receptive to what they read in the 4 recommendations we focused on in this session. They liked that the emphasis is supposed to be on number concepts rather than trying to keep up with what the teacher is doing in the classroom. Some of them said that by trying to fill gaps instead of focusing on key concepts they often feel like content mastery teachers rather than interventionists. They are hoping they can better define their role through our work this year.

We ended the session by revisiting our district math goals to see how they related to the 8 recommendations. It was very easy for the interventionists to see that these recommendations align very well with our district’s math goals. That’s not to say that their work with their students in meeting these goals will be any easier per se, but it is reassuring to know that the work they are doing with their students is going to be meaningful and supported by research.

All in all, the interventionists left excited about the adventure we’re embarking upon. They’re especially happy that they’ll have the opportunity to get to know each other better so they can utilize each other as resources. I couldn’t ask for a better outcome from our first time together. Part of me is even a little sad that I only get to plan 8 more sessions instead of 13.

Just a little. I still have plenty of other work to do.

Of

Yesterday on Twitter, I took part in an impromptu discussion of fraction multiplication. I’ll be honest. I often get frustrated diving deeply into meaty topics on Twitter because I’m limited to 140 characters per tweet (less when you take into account the fact that the handle of everyone tagged in the tweet is deducted from the total). However, this ended up being a very enjoyable conversation and reminded me of the power of connecting with folks from all over.

Tracey Zager was nice enough to Storify the conversation. If you’re interested to hear how a small group of elementary educators unpacked this topic, take a look. If you’re an elementary teacher yourself, especially an upper elementary teacher, you may appreciate it because you’re likely having similar conversations within your own district or campus.

One question that came up several times during the conversation was how and why the word “of” means multiplication:

2014_09_Tweet_Of-Is-Mult-02

2014_09_Tweet_Of-Is-Mult-01

As Math Minds’ tweet alludes to, there is a whole issue of teaching keywords and the damage they cause students, but that’s not what I’m focusing on here. Today I want to think through the idea of how the simple two-letter word “of” is related to the operation of multiplication in the first place.

This question made me think of the chapter I’m currently reading in Kathy Richardson’s book How Children Learn Number Concepts: A Guide to the Critical Learning Phases. It’s very timely that I’m reading a chapter titled “Understanding Multiplication and Division”. Richardson begins the chapter with the following quote from Keith Devlin (I like that I’m quoting a quote from a book.):

 “…in today’s world we are faced with a great many decisions that depend upon an understanding of quantity. Some of them are inherently additive, some multiplicative, and some exponential. The behavior of those three different kinds of arithmetical operations differs dramatically…”

Richardson goes on from there to discuss the need for elementary teachers to differentiate additive and multiplicative thinking.

“Central to understanding multiplying is the idea that the two numbers (factors) in a multiplication equation have two different meanings: one number describes how many equal groups there are and the other describes the size of each of the groups.”

And when we describe the relationship between the two numbers verbally, the word “of” can become an essential part of our description. Here’s an example from Ask Dr. Math:

Suppose items come 8 to a box.

If I have 2 of these 8’s, I multiply to find the total, 2 × 8 = 16.

(There are 2 equal groups and the size of each group is 8.)

If I have ½ of an 8, I multiply: ½ × 8 = 4.

(There is ½ of a group and the size of the group is 8.)

Going back to Richardson’s book, she goes on to describe the types of multiplication situations students should encounter in elementary school:

  • Equal groups (equivalent sets)
  • Rate / Price / Length
  • Rectangular arrays
  • Multiplicative comparison (scale)
  • Combination problem (Cartesian product)

It’s the multiplicative comparison (scaling) situations that lend themselves best to understanding fraction multiplication. I found it very telling that in the CCSS grade 4 Operations & Algebraic Thinking domain, there is a standard that says students should interpret multiplication as a comparison. The standard uses whole numbers in its example, but the pump has been primed. Then in grade 5, this idea is embedded in the Number & Operations – Fractions domain in a standard that says students should interpret multiplication as scaling (resizing).

The trouble seems to be that up until fraction multiplication, the act of multiplying two whole numbers has always resulted in a larger number, and it has been easy for teachers and students to view it as repeatedly adding that quantity over and over. However, this idea of a quantity growing larger through repetition is only half of what’s going on. If quantities can grow larger, then they can also grow smaller, and our language to describe this needs to adjust accordingly. Instead of having 3 times as much or double the amount, we can now consider 2/3 of a quantity or half as much.

I don’t think the problem is that the word “of” doesn’t mean multiplication per se, but that as elementary educators, we haven’t opened ourselves up to needing different language to describe something new students are learning to do, namely using the operation of multiplication to decrease the size of a quantity.

Although, after writing all that, I want to revise my thinking. Richardson goes on to describe how children have difficulty grasping the word “times” when they first learn about multiplication. She recommends teachers use phrases such as “groups of”, “rows of”, “piles of”, “stacks of”, etc. There’s that word “of” again, and Richardson is advocating using it with children well before they learn about fraction multiplication.

If students can visualize and make sense of:

  • 2 groups of 5
  • 3 rows of 6
  • 7 piles of 10
  • 3 stacks of 9

Then we should be able to extend to this later on:

  • 1/2 group of 5
  • 2/3 row of 6
  • 1/10 pile of 10
  • 1/3 stack of 9

We come back to the idea that the two numbers in a multiplication situation have two different meanings. The first number in each example is the number of groups, whether it’s 2 groups or 1/3 of a group. The second number is the size of one whole group, whether the whole group is 4 pans of brownies or 1/3 pan of brownies.

So ultimately it seems that the word “of”, and phrases built around it, are mostly there to help students to visualize and make sense of a new kind of thinking. Up until grade 3, most students have been focused on additive thinking, so this is quite the paradigm shift for them, and they will grapple with it for several years. As a result teachers need to use familiar language and phrases, which include the word “of”, to help students expand their understanding of how we operate on quantities.

Interpretation Frustration [UPDATED]

[UPDATE] I wrote an email to TEA and heard back from them within three days. I’m very impressed! The person who wrote me went over each of my concerns one by one:

1. The example for 3(3)(E) in the side-by-side document discusses the partitioning of objects and the fraction as a concept of division, not a numerical representation. The goal would be that a student realizes that each student would receive five half-pieces of cookie. With this basis, students can the develop improper fractions and mixed numbers in grade 4.

Saying that the answer should not be a numerical representation sounds like splitting hairs. The text of the standard says: solve problems involving partitioning an object or a set of objects among two or more recipients using pictorial representations of fractions with denominators of 2, 3, 4, 6, and 8;

If I look at the phrase “solve problems” I’m led to believe I’ll get a numerical answer like I get when I solve other math problems. Yet I think he’s focusing on the phrase “pictorial representations” to say that their answer is meant to be less formal. All in all, I feel students can solve problems like sharing five cookies among 2 people, but the avoidance of 5/2 as an answer leaves me scratching my head. What purpose does it serve?

2. Much like with 3(3)(E), the focus of [3(7)(A)] is the division of the line segment. In the given example, the mark is ¼ of the distance between the numbers 16 and 17 on the number line.

This is some shady logic. The standards don’t ever mention mixed numbers in any elementary grades, but apparently they are implying that because a mixed number is composed of a whole number (3rd graders should be comfortable with those by now) and a fraction less than one (introduced in grade 3), they are fair game. I’m not against this interpretation. What I don’t like is the vague language that leaves it open to interpretation in the first place. I feel like I need to hire a lawyer to help me make sure I’m interpreting the language of these standards accurately!

3. You are correct; the last stand-alone measurement standard is 2(9)(D). However, students can be asked to measure the side lengths of a polygon in 3(7)(B).With the process code of 3(1)(A), a ruler could stand in for a number line in 3(7)(A).

And yet another example of relying on implication rather than writing standards that were clear and easy to follow in the first place. And he ends with my favorite line that I’ve heard over the years:

4. Please remember that the Texas Essential Knowledge and Skills are minimum standards and are not intended to limit what is taught.

It’s the “Get Out of Jail Free” card. I don’t think the issue is that teachers are scared of teaching beyond the standards. The problem is teachers trying to get a good grasp of what the bare minimum is in the first place. After reading through the TEKS, which are technically the standards, teachers can walk away thinking they know where the bar is. However, based on supplemental documents and email clarifications, the bar seems to be in a state of flux, leaving teachers unsure of how high their students need to jump. This doesn’t seem like a fair position to put teachers (or their students!) in.

Original post follows.

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I blame the TEKS formy headache today. Specifically the grade 3 TEKS. They are not on my good side right now.

To give you some background, for the past few years I designed curriculum based on the Common Core Standards. I’ve also designed materials for Texas, but lately it was kind of secondary to the Common Core stuff. I’ve grown to love the Common Core standards. There is a lot of thought and care into the progression of topics from grade to grade. They aren’t perfect, but I value how much they do make sense, especially if you read the accompanying progressions documents.

Several years ago, Texas decided to write some new math standards. They didn’t want to adopt Common Core…because Texas…but it was clear the writing team appreciated those standards, too. The first draft of the new Texas standards had so much Common Core language in them, they may as well have been the Common Core. But then the Texas standards went through a round of revisions and what came back looked like someone had hacked off pieces of the Common Core standards, shuffled them around a bit, and called the final product new Texas standards. Needless to say, I’ve been unimpressed.

However, in my new job, I am working squarely in a Texas district in the state of Texas so the Texas standards (TEKS for short) are my focus from here on out. Lord help me.

Today, while putting together assessment materials for a grade 3 unit on fractions, I started to come across some inconsistencies in the language of the TEKS. It started with 3.3A and 3.3B:

3.3A represent fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines;

3.3B determine the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line;

Remember, I come from a Common Core background. Their standards say this:

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

And this:

Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Can you spot the main difference? In Common Core there is no specification that the numerator a has to create a fraction that is less than or equal to 1. You could just as easily make 5/4 as you could 3/4. In the new TEKS, however, there is a clear specification that third graders are working with fractions greater than 0 but less than or equal to 1. (By the way, what’s with the fractions having to be greater than 0? Anything wrong with discussing 0/4?)

Ok. I can handle that. But what’s this grade 2 standard over here say?

2.3C use concrete models to count fractional parts beyond one whole using words and recognize how many parts it takes to equal one whole;

Oh, so in second grade it’s okay to count fractional parts above one whole, but we need to stop in grade 3? Apparently that’s the case because improper fractions aren’t brought up again until this grade 4 standard:

4.3A represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b;

Weird. Let’s introduce an idea in grade 2, completely skip it for a year in grade 3, and come back to it in grade 4. Well, at least that’s settled…I think.

Let’s look at another grade 3 standard:

3.3E solve problems involving partitioning an object or a set of objects among two or more recipients using pictorial representations of fractions with denominators of 2, 3, 4, 6, and 8;

In the Side-by-Side comparison documents provided by the Texas Education Agency, we see the following example provided to help clarify 3.3E:

Examples of problems include situations such as 2 children sharing 5 cookies.

I can buy students solving this problem. That’s fine, but how do you rationalize the answer? You are either going to end up with 5/2 which contradicts the rigidity of 3.3A and 3.3B, or you’re going to end up with 2 ½ which is a mixed number. By the way, did I mention the term mixed number doesn’t appear in the TEKS at all across grades K-5? At all. Can you see why this might make my head hurt a bit?

My guess is that they are cheating a bit in their interpretation of 3.3A and 3.3B. By having students use mixed numbers, they are really only writing a whole number combined with a fraction less than one. Do you get it? The number 2 ½ doesn’t break their rule because the fractional part is less than 1.

So students are likely going to be held accountable for understanding mixed numbers in grade 3 even though they aren’t mentioned in the standards and several of the grade 3 standards explicitly state students work with fraction less than or equal to 1. (Good luck third grade teachers!)

I’m pretty sure this is how they are interpreting it because of how they interpret another standard. In the old TEKS we had this standard:

Old 3.10 The student is expected to locate and name points on a number line using whole numbers and fractions, including halves and fourths.

On this year’s high stake test (STAAR), the students had to locate the mixed number 16 1/4 on a number line. Do you think they would ask the same thing based on the wording of the new TEK? I sure can!

3.7A represent fractions of halves, fourths, and eighths as distances from zero on a number line;

And that’s not all! Looking at the TEKS related to fractions on a number line got me thinking about measuring to fractions of a unit. Guess what! That’s a whole new can of worms. Here is the linear measurement standard from grade 2:

2.9D determine the length of an object to the nearest marked unit using rulers, yardsticks, meter sticks, or measuring tapes;

In which grade level do you think they specify measuring to the nearest half, fourth, or eighth of an inch? If you guessed “they never specify it”, you’re right! The standard 2.9D is the FINAL linear measurement standard in the TEKS. The only mention I could find about measuring to fractions of a unit comes from the grade 5 Side-by-Side document put out by TEA. Here’s the standard:

5.4H represent and solve problems related to perimeter and/or area and related to volume.

And here’s how the Side-By-Side “clarifies” it:

Because fluency with the addition and subtraction of positive rational numbers is expected within the Revised TEKS (2012), lengths may reflect fractional measures with perimeter.

So the wording of the standards themselves never brings up fractional measures in grades K-5. The only way you would even know this grade 5 standard uses fractional measures is if you happen to cross reference it in the Side-By-Side document which is available on a completely different website from the standards themselves. I’m not even sure they’re available on the Texas Education Agency website.

Can you see why I had a headache today?

I did email someone at TEA today to request clarification. I can’t imagine I’m the only person who finds these particular standards unclear and confusing. If I hear back, I’ll be sure to share! I don’t want anyone, teachers or students, to suffer as I did today.

 

One Month In

It’s hard to believe that in 3 days I’ll have been at my new job for a month! I can’t remember if I posted about it already or not, but I left my position at McGraw-Hill Education to become the lead curriculum specialist for elementary math at my old school district.

By the way, my job title is a mouth full and I feel pretentious every time I have to tell it to people. I’ve tried finding ways to shorten it, but it just doesn’t work. If I say I’m a curriculum specialist, then it sounds like I do it all. While technically I can do it all as an elementary teacher, I don’t do it all now. And I can’t just say I work in elementary math because that doesn’t really feel very descriptive. Oh well. As far as problems go, I can live with this one.

Time has flown by, mostly because I’ve been so busy. I’ve been enjoying meeting lots and lots of new people and learning the ropes. My first big hurdles on the job relate to our new textbook adoption. In the spring, our district adopted a curriculum called Stepping Stones. It’s by a company called Origo. It’s a digital curriculum, meaning there is no printed book for the teachers. All of the lesson plans are online.

After a week or so on the job, I was tasked with compiling a list of all the elementary teachers in the district who teach math so we could create their online accounts. Basically we don’t want to spend money buying content licenses for teachers who aren’t going to use the curriculum at all. Making a list sounds easy, right? Just contact HR and get one from them. Yes? Maybe?

No, not so much. HR was able to give me an initial list, but it took communicating with Assistant Principals at our 33 elementary schools and making lots of edits to a spreadsheet to finally nail it down.

The next big hurdle was putting on implementation training for about 1,200 teachers spread out at 6 campuses across the district. Thankfully I have a wonderful partner in crime on my team who knows the district inside and out. She was invaluable in getting everything organized and ready to go. The trainings took place last Wednesday, and all in all, they went amazingly well. Yay!

With those two big tasks under my belt, I’ve moved on to an ever growing to-do list. Currently we’re wrapping up the curriculum documents for the second nine weeks for grades K-5. Then we have to write second semester timelines and unit guides which haven’t even been started yet. We also have Curriculum Based Assessments (fancy name for benchmark exams) to prepare, a department website to update, and preparations to make for upcoming trainings we’re doing with our interventionists.

Whew! Good thing I like what I do.