Engaging Math Thinking Tasks with a Low Floor and High Ceiling

            In his book, Building Thinking Classrooms in Mathematics, Peter Liljedahl advocates having students work on challenging, highly engaging tasks in mixed-achievement groups of two (at the primary level) or three (in grades 3 to 12). Each group works collaboratively on a vertical whiteboard, standing up, sharing one marker, with the teacher observing their work and, as described in the summary just above, answering only “keep-thinking” questions. Liljedahl suggests starting with fun problems to establish a collaborative culture, then segueing into the regular curriculum, always posing open-ended tasks that challenge students to put their heads together and think at higher levels. The teacher encourages students to observe and learn from other groups’ work and brings closure at the end of each class. From his book, here are some examples of the types of tasks Liljedahl suggests: 

Lower elementary: 

How many squares are in this image? 

             (Image: 4x4 table of squares)

You have 16 jellybeans and four jars: 

    - Place the jellybeans in the jars so that each jar has either 3 or 6 jellybeans. Are there some things that are not possible? 

    - Place the jellybeans such that each jar has one more than the jar before it. How many ways can you do this? 

    - Place the jellybeans so that each jar has twice as many as the jar before it. Three times as many. The 

Ice Dream ice cream shop has 10 flavors of ice cream. How many different two-scoop ice cream cones can you make? What if there were 11 flavors? What if there were 12 flavors? How about with 20 flavors? What if each cone had at most three scoops? 

A farm has some chickens and some pigs. One day the owner notices that the animals have a total of 22 legs. How many chickens and how many pigs might there be? Can you come up with another solution? And another? Can you come up with all the solutions? How do you know that you have all the solutions? 

Upper elementary and middle school: 

If 6 cats can kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes? 

If I were to write the numbers from 1 to 100, how many times would I use the digit 7? What if I wrote 1 to 1000? How many zeroes? 

Select four numbers from 1 to 9 at random. Using these four numbers and any operations, make the values from 1 to 30. 

How many ways are there to make a dollar using only nickels, dimes, and quarters? 

I have a four-minute egg timer and a seven-minute egg timer – the kind you turn over and let the sand run through. Can I use these to cook a nine-minute egg? If so, how long will someone have to wait for their egg? 

High school: 

Decompose 25 using addition, for example: 25 = 10+15, 25 = 10 + 10 + 5… What is the biggest product you can make if you multiply the addends together? 

You want to arrange four candles on a birthday cake. How many ways can you place the candles such that there are no more than two different distances between two candles? 

 From Building Thinking Classrooms in Mathematics by Peter Liljedahl (Corwin, 2021)

Recommended Books that Feature Unreliable Narrators

             In this School Library Journal feature, retired librarian Steven Engelfried touts books with the unusual characteristic of deliberately misleading storytelling: 

- The Assassination of Brangwain Spurge by M.T. Anderson, illustrated by Eugene Yelchin, grade 4-7 

- The Ogress and the Orphans by Kelly Barnhill, grade 3-7 

- Simon Sort of Says by Erin Bow, grade 5 and up 

- The Probability of Everything by Sarah Everett, grade 3-7 

- Invisible by Christina Diaz Gonzalez, grade 4-7 - Ground Zero: A Novel of 9/11 by Alan Gratz, grade 4-7 

- The Worlds We Leave Behind by A.F. Harrold, illustrated by Levi Pinfold, grade 5 and up 

- Scary Stories for Young Foxes by Christian McKay Heidicker, illustrated by Junyi Wu, grade 4 and up 

- Linked by Gordon Korman, grade 4-8 

- The Windeby Puzzle by Lois Lowry, grade 5 and up 

- When Sea Becomes Sky by Gillian McDunn, grade 3-7 

- The Many Assassinations of Samir, the Seller of Dreams by Daniel Nayeri, illustrated by Daniel Miyares, grade 4-8 

- The List of Things That Will Not Change by Rebecca Stead, grade 5-8 

 “Telling It Like It Isn’t: Unreliable Narrators Keep Readers on Their Toes” by Steven Engelfried in School Library Journal, November 2023 (Vol. 69, #11, pp. 38-40)

Please Note: This summary is reprinted with permission from issue #1013 of The Marshall Memo, an excellent resource for educators.

Which Student Questions Should Teachers Not Answer?

            In this chapter of his book, Building Thinking Classrooms in Mathematics, Peter Liljedahl (Simon Fraser University) says research and his own classroom observations show that teachers typically answer between 200 and 400 student questions a day – some teachers as many as 600. The problem is that the hard work of answering all those questions often lets students off the hook from doing their own thinking. 

            So what should teachers do? “The answer lies not in whether or not we answer students’ question,” says Liljedahl, “but which questions we answer.” He’s found there are three types of questions in classrooms: 

• Proximity questions – Students ask these questions when the teacher happens to be close by, and most often the questions are about something kids don’t really need help on – for example, For Question 3, were we supposed to find all the answers? or Are we doing this right? Why do students ask proximity questions? Because of highly socialized roles in classrooms, says Liljedahl. Asking a question is one of the most “studently” things a kid can do, and answering the question is one of the most “teacherly” things a teacher can do.

            Sometimes a shy student will take advantage of the teacher’s proximity to ask a question they wouldn’t raise their hand or stand in line to ask, but most of the time, proximity questions are socialized filler and add little value. Sometimes these questions are a way for a student to distract the teacher from catching them on their cell phone or some other off-task behavior. 

• Stop-thinking questions – Examples of this type: Is this right? Is this going to be on the test? Do we have to learn this? “These questions,” says Liljedahl, “are motivated by the reality that, for students, thinking is difficult, and it’s hard to decide for themselves that what they are doing is correct. If they can just get you to do that for them, their life would be so much easier.” 
• Keep-thinking questions – These clarifying or extension questions are asked when students are motivated and engaged with the task at hand – for example, We’re having trouble here. Were we supposed to do this for all the possible sizes? or Are we supposed to now look at the general case? 

            Liljedahl has found that about 90 percent of the questions students ask are in the first two categories. The result: teachers are knocking themselves out doing something that contributes very little to students building their thinking muscles. In many cases it completely shuts down thinking. 

            The solution, says Liljedahl, is for teachers to answer only the third type – the keep-thinking questions. This requires knowing how to quickly spot proximity and stop-thinking questions and developing a skillset for deflecting them. He’s found that most teachers have no trouble discerning the type of question being asked. Most of the questions students ask immediately after being given an assignment are for clarification or to avoid having to do the work of seeing what’s being asked and deciding how to solve the problem. The moments just after an assignment has been given are when the most questions are asked, and it’s wise for the teacher to not circulate at this point. 

            Once students have their heads into the assignment, circulating is important and the teacher needs to decide on the effect answering a question will have: Are they asking for more activity or less, more work or less, more thinking or less? Students may be inventive, making a statement and quizzically raising their eyebrows at the end: We are thinking this is correct. [What do you think?] This is correct [Right?] I think we are going the right way. [Right?] “Don’t be fooled by these pseudo-statements,” advises Liljedahl. “If their tone is inviting a response from you, they are really asking a question – and by the nature of the disguise, it is almost always a stop-thinking question.” 

            But how can teachers get away with not answering stop-thinking and low-value proximity questions? “Students can be very persistent in their efforts to get you to help them reduce their workload,” says Liljedahl, “and how you respond to this is important.” One approach is answering the question with a question – for example: 

    - Isn’t that interesting?

    - Can you find something else? 

    - Can you show me how you did that? 

    - Is that always true? 

    - Why do you think that is? 

    - Are you sure? 

    - Does that make sense? 

    - Why don’t you try something else? 

    - Why don’t you try another one? 

    - Are you asking me or telling me? 

            Some of the teachers Liljedahl was working with said this strategy was successful, but others found it was a slippery slope. Here’s how one dialogue with a student went: 

    - Why don’t you try something else? 

    - Like what? 

    - Maybe you need to consider the cases where x is negative. 

    - You mean like this? 

    - Right! 

            Answering a question with a question was only effective, Liljedahl found, when it was immediately followed by the teacher walking away without saying more. This annoyed students, but teachers found that within a few days using this approach, there were far fewer proximity and stop-thinking questions… 

            Except in primary-grade classrooms. “If a six-year-old asks a question and it is not answered,” says Liljedahl, “they ask it again. If it is still not answered, they ask it again. And if it is still not answered they do something that a 16-year-old does not. They reach out and touch the teacher – tap them on the arm or pull on their clothing. And if the teacher walks away, they follow. I have multiple videos of kindergarten and grade 1 teachers walking around the room with a row of little ducklings following them.” 

            With young students, when the teacher didn’t respond a question, they assumed it hadn’t been heard and felt ignored. So Liljedahl came up with a modification for primary grades: look at the student asking the question, maybe ask a question, smile, and then walk away. This turned out to be effective for all grade levels. Students knew their question had been heard and that the teacher’s decision not to answer was deliberate. 

            “Many students took this to mean that they needed to do more work,” says Liljedahl. “Over time, the students began to see the smile and walking away as a sign that the teacher had confidence in their ability to resolve the question on their own. There were still a few students who were frustrated by these encounters. But they were thinking more – or no longer having the teacher do their thinking for them.” 

            What if students insist that the teacher answer a stop-thinking question? If it’s an Is this right? question, Liljedahl suggests being explicit: “I’m not going to answer that question. Me telling you that it is right is worth almost nothing. If you can tell me that it is right, however, that is worth everything.”

            Should teachers explain the three types of questions to students, telling them which will be answered and which won’t? Liljedahl has found this is a good idea, but only after a couple of weeks of responding only to keep-thinking questions, and with the others, answering a question with a question, smiling, and walking away. When it’s handled this way, students say, “So that’s what’s going on!” and “It’s cool that he told us that.” When working in groups, students also begin monitoring themselves: “Dude! She’s not going to answer that. That’s a stop-thinking question.” 

            Does answering a question with a question and smiling and walking away work for all students? There are some “who can’t get past the fact that you have not answered their question,” says Liljedahl. “This may be because they are insecure about their own abilities, have learned helplessness, or have a spectrum disorder – such as obsessive compulsive disorder – that does not allow them to move forward without resolution.” And students have years of experience in classrooms where their questions get answered. Teachers need to read the situation and know when “a nod, a wink, or an encouraging remark – ‘I have complete confidence that you can figure this out’ – is needed.” 

            Should parents be told about the question-answering policy? Absolutely, says Liljedahl – and parents should hear about it first from the teacher, with an explanation that the goal is to encourage students to do their own thinking and that answering only certain types of questions is part of a deliberate strategy. 

“How We Answer Questions in a Thinking Classroom” by Peter Liljedahl, Chapter 5 in his book, Building Thinking Classrooms in Mathematics (Corwin, 2021); see Memos 976 and 992 for two other summaries of Liljedahl’s work. Jenn David-Lang recently did a thorough summary of the book in The Main Idea and is making it available to Marshall Memo readers here. Liljedahl can be reached at liljedahl@sfu.ca.

Please Note: This summary is reprinted with permission from issue #1013 of The Marshall Memo, an excellent resource for educators.

Discussion-Worthy Tasks for Upper Elementary Math Groups

            In this Mathematics Teacher article, Nicola Hodkowski (Digital Promise) and Carolyn Carhart-Quezada (Cignition) say students learn more about mathematics when they discuss with classmates, connecting what they know with formal math content. To spark meaningful discussions, say the authors, teachers need to pose open tasks – that is, questions with multiple entry points, varied solution strategies, and more than one right answer. Well-chosen tasks allow students to “discuss, argue, represent, hear, and compare one another’s viewpoints.” 

            Hodkowski and Carhart-Quezada came up with a series of open tasks to help below-level fourth and fifth graders better understand fractional reasoning. Working on the tasks online in groups of four, facilitated by a tutor, students made significant progress. “By prioritizing conceptual understanding and promoting mathematical discourse,” say the authors, “we were able to both change traditional tutoring and empower our students to see themselves as doers and sense-makers of mathematics.” 

            Hodkowski and Carhart-Quezada developed five types of open tasks for their program and thought about where each type was most helpful: 

• Multiple strategies – More than one strategy can be used, but the solution or answer is the same – for example: The answer is ½. What is the question? 
• Multiple outcomes – Solution strategies may be similar, but solutions are different – for example: Add something to ¾ that sums to a number close to 1 but not exactly 1. Who is closest? How do we know? 
• Sorting and ordering – Students are asked to invent a sorting criterion, with different criteria leading to different sorting – for example: With the following fractions, quickly decide if they are bigger or smaller than ½. How did you decide? 3/11, ¼, 5/6, 7/8, 7/10, 6/12 
• Justification – Students need to explain and justify their answers to groupmates – for example: Ana says that 1/8 is bigger than 1/3 because 8 is bigger than 3. Margo thinks that Ana is not right. Who do you agree with? Explain your reasoning. 
• Group challenge tasks – Students work in pairs and discuss outcomes with another pair – for example: 6 people are going to share these 5 candy bars equally [show 5 rectangles]. Write a fraction that shows how much one person gets. 

            Students’ prior knowledge and readiness determine which type of problem will spark the best discussions. With students who are new to a concept, multiple-strategy tasks seem more engaging, allowing for more exploration as the group works toward a single solution. With students who had already been introduced to a concept and are working to deepen their understanding, a multiple-outcome task is more appropriate, with students discussing how they came up with different answers.                Hodkowski and Carhart-Quezada noticed that at first, students talked less and the tutors talked more – the opposite of what was intended. Why? Students weren’t used to having discussion in math classes and lacked confidence in their insights. “We quickly realized,” say the authors, “that rituals, routines, and collaborative norms were needed when using open tasks.” The tutors were prompted to establish these norms: 

• Explain why, not just the right answer. 
• Show active listening. 
• Compare and critique with groupmates. 
• Summarize what you’ve learned. 

Tutors supported these norms by saying, “Yes, and…” rather than immediately telling the answer and nudging students for “deeper noticings and wonderings about others’ work.” 

            But in some groups, setting norms was not enough. Over time, Hodkowski and Carhart-Quezada worked with tutors to develop several other ways to jumpstart discussions: 

• Individual think time before sharing in the group 
• Giving students 20-30 seconds to ponder the problem before beginning the group discussion.
• After presenting the problem, the tutor divides a slide into four sections, one for each student; students solve the problem on their section and then look at their groupmates’ solutions and discuss. 
• Students think about the task alone, then work together on solving it on one slide, then the teacher chooses one student to explain their solution, discuss, and decide on the correct solution. 
• Taco talk – Students are assigned to be tomato, lettuce, cheese, and taco shell, the first three present their ideas in that order, then the fourth student (taco shell) wraps up the discussion by explaining what their groupmates said. 

 “Let’s Give Them Something to Talk About” by Nicola Hodkowski and Carolyn Carhart-Quezada in Mathematics Teacher: Learning & Teaching PK-12, November 2023 (Vol. 116, #11, pp. 837-844); the authors can be reached at nhodkowski@digitalpromise.org and carhart-quezada@cignition.com.

Please Note: This summary is reprinted with permission from issue #1012 of The Marshall Memo, an excellent resource for educators.

Literacy As a Springboard for Student Empowerment

            In this article in Language Arts, Chris Hass (James Madison University) says that as a second- and third-grade teacher, he used to ask his students, What is reading? At first they said reading was figuring out unfamiliar words and learning new vocabulary, but as he continued to ask the question, their responses evolved: 

• Reading is imagining you’re the character in a book. 
• It’s trying to figure out what might happen next. 
• It’s forgetting what’s going on around you because you’re so into the story. 

“The power of returning to this question over time,” says Hass, “was that it scaffolded my students to continue thinking more deeply about their practice as well as their relationship to reading.” 

            Then he began asking a follow-up question: Why is it important that we’re growing as readers, writers, and speakers? At first, students’ responses were pedestrian: to get ready for fourth grade, do well on “those tests,” go to a good college and get a well-paying job. 

            But one day a quiet student raised her hand and suggested, “So we’ll be able to stand up for ourselves.” This comment got the class thinking about literacy at a different level. Students recalled reading Malala Yousafzai’s story, doing a project on saving sea animals, lobbying a state legislator, and writing letters to a city council member. 

            “Once they had the eyes to see how literacy and democratic practices go hand in hand,” said Hass, “they could not help but continue to share more expansive visions of literacy…” – talking out disagreements, reading a news article about unfair dress codes, exploring reports of problems around the world. Hass built on this pivotal discussion for the remainder of the school year, continuing to see literacy as a tool for students standing up for themselves and others. There were three main vehicles for the theme: 

• Keeping classroom journals – As part of morning meetings, students added to journals (made of stapled sheets of art paper) that explored a wide range of questions: Why do worms crawl on the driveway when it rains? Why do things look darker when they’re wet? Why do people want to ban books at school? Why aren’t there any female presidents? Why is there racism? “The discussions that grew from these questions,” says Hass, “created a culture of inquiry in the classroom – one that positioned each of us to think more critically about the workings of the world around us.”
• Inquiring about activism – Hass purchased a set of books with stories about different types of community advocacy – the 1909 Shirtwaist Factory strike, protecting Egypt’s treasured books, African-American freedom fighters, how Selma’s teachers changed history – and led discussions about the problems confronted in each story. Students took the books home and reported back on insights from discussions with family members. Realizing that they all had the power to create change, students generated a list of what taking action looks like in practice. 
• Acting on their convictions – Hass asked each student to choose one issue from the list that was especially important to them. Kids read a variety of texts on their chosen issue, conducted surveys and interviews, and created a culminating project. There were petitions, schoolwide signature drives, and letters sent to city officials and state legislators, school board members, food service providers, and the school’s principal. 

            “By the end of the school year,” Hass says, “my students learned there is much more to reading than simply decoding words or gaining skills to do well on a test.” Students were informed about issues in their communities and around the world, built understanding and empathy for others, became more-critical consumers of information, and took action on issues that were important to them. 

“Learning to Stand Up for Themselves: Using Literacy As a Vehicle for Change” by Chris Hass in Language Arts, September 2023 (Vol. 101, #1, pp. 65-68); Hass can be reached at hasscl@jmu.edu.

Please Note: This summary is reprinted with permission from issue #1009 of The Marshall Memo, an excellent resource for educators.