As we continue to discuss the science of reading, let’s not neglect the role of background knowledge.

By Sara DeMartino

The reading rope (Scarborough, 2001) has been making the rounds in both district and policy conversations about the science of reading. The rope is a useful tool for visualizing the many threads that make up becoming a skilled reader, although some will argue that it does not capture the full picture (see Duke & Cartwright, 2021 for a discussion of the different models of reading development). On its most basic level, the rope is a presentation of the processes and knowledge that fluent readers bring with them to reading tasks.

While the rope is a representation of reading development that’s pretty familiar to reading specialists and elementary school teachers, it might be new to secondary teachers who may not have had any or a limited amount of teacher preparation work on learning to read (and who may be nervous about the prospect of having to teach phonics).

Reading Rope

From “Connecting Early Language and Literacy to Later Reading (Dis)abilities: Evidence, Theory, and Practice,”
by H.S. Scarborough, 2001, in S.B. Neuman and D.K. Dickinson (Eds.), Handbook of Early Literacy Research
(Vol. 1, p. 98), New York, NY: Guilford. Copyright 2001 by The Guilford

The general expectation is that by the time students enter middle school, word recognition becomes more automatic and students are working to weave the processes of word recognition and language comprehension together to form a solid (and complex) rope. Because curriculum and policy traditionally assumes (and science tells us) the foundation for the development of these processes has been built in elementary school, many secondary teacher preparation programs barely touch on or completely neglect classwork on learning to read and phonics instruction. However, the reality is that students are entering secondary classrooms with unfinished learning for multiple reasons (Covid, lack of high-quality resources, balanced literacy that wasn’t quite balanced, etc.) and phonics is being asked of many unprepared middle school teachers as a way to try and fill the gap in reading achievement. While phonics and the work in the word recognition strand are critical to becoming a skilled reader, what’s often neglected in conversations around reading policy is the work encompassed by the language comprehension strand (the top strand) of Scarborough’s rope, particularly the role that background knowledge plays in reading comprehension. This is a piece of work that secondary teachers can immediately add into their instruction without specialized knowledge about teaching phonics.

Knowledge plays a key role in building language comprehension. Looking across a school year, this means that instructional units should cohere and help students build and access knowledge across texts within a unit as well as across instructional units. This includes both contextual or conceptual knowledge and literary knowledge. Research has shown that background knowledge is a strong predictor of comprehension, even when texts are beyond what has been deemed a student’s independent reading level (Hirsch, 2003; Cabell & Hwang, 2020, Shanahan, 2023). Knowledge that contributes to students’ ability to read and comprehend complex texts includes:

      • tier 2 & tier 3 vocabulary
      • genre specific knowledge
      • context for writing
      • content specific knowledge (i.e., scientific or historical knowledge relevant to the text)

Building and accessing knowledge doesn’t need to add days to instruction, but it needs to provide enough of a foundation that students aren’t left wondering, for example, why Swift (Jonathan, not Taylor) was allowed to suggest eating babies in “A modest proposal” (a quick overview of satire and a little context for Swift’s writing is useful in reducing that confusion). Although there are a variety of ways to address background knowledge, below is one example of how you can help students build and access background knowledge that will aid in comprehension of texts.

Example: Accessing & building contextual knowledge

Context of the background task: Eighth grade students are beginning to engage in a unit comprised of speeches and narratives about equity and freedom written before and after the Civil War. Students have had some work on the Civil War in their history class, but not with the texts in the unit. During the unit, students will be asked to read Lincoln’s “Gettysburg Address,” excerpts from Douglass’s Narrative of the Life of Frederick Douglass, and Truth’s “Ain’t I a woman?”  Students will begin the unit with comprehension work on “Ain’t I a woman?” and the teacher has recognized that while her students have had work on pre-Civil War America and Reconstruction, it would be helpful to spend time situating Truth’s speech.

Sojourner Truth

To access and build background knowledge, the teacher does the following:

Whole group work (5 minutes): The teacher showed students an image of Sojourner Truth and explained that Truth was born into slavery in 1797 in New York. She was bought and sold four times before running away in 1827 with her infant daughter. An abolitionist family bought her freedom for $20. Truth became an abolitionist herself and fought both for the rights of slaves and the rights of women. Truth died in 1883.

Pair work (15 minutes): After a brief discussion about Truth, the teacher provided students with a list of events and asked them to work with a partner to put these events on a timeline. The events students are asked to work with are:

      • Start of Civil War
      • End of Civil War
      • Emancipation Proclamation
      • Women’s Rights Convention (where Truth gave her speech)
      • Women get the right to vote (19th amendment)
      • Black men get the right to vote (15th amendment)

Because students have had instructional work in history about this time period, there doesn’t need to be a whole lot of time spent on research of events. But it is important that students understand that Truth’s speech was given prior to the start of the Civil War.

Whole group presentation (5 minutes): After seeing that students had pretty much ordered the events correctly, rather than a gallery walk, the teacher asked one group to present their timeline and then asked students if they agreed with what had been placed on the timeline or if they had something different.

Individual quick write (5 minutes): Once the class was on the same page about the order of events, the teacher asked students to jot some quick noticings and wonderings about Truth and the speech she gave based on the timeline of events.

 Pair/trio share (6 minutes): Students had a quick opportunity to talk about what they wrote with a partner and hear their partner’s thinking before being asked to share some ideas with the whole group.

Whole group discussion (10 minutes): Finally, students shared their ideas with the group, getting opportunities to agree, disagree, and add on to the ideas of other students. The teacher charted students’ thinking to keep track of their ideas. She also made sure to have students keep their writing in a place where they could access it after reading Truth’s speech.

The background knowledge piece took about half of the teacher’s instructional block to complete prior to asking students to engage in the comprehension task for Truth’s speech. The comprehension task asked students to read the speech and then respond to the question, “What’s Truth arguing in this speech?”

The background work became important artifacts that students revisited throughout the unit. After reading Truth’s speech, students revisited their noticings and wonderings about Truth to reflect on or revise what they originally thought. The teacher also asked students to reflect on the significance of Truth’s ideas considering when the speech was given in relationship to the other events on the timeline. As the class engaged with the other texts in the unit, the teacher asked students to go back to the timeline and add events about Frederick Douglass and Lincoln’s “Gettysburg address” to the information they had already noted. Students were able to use the timeline and the quick writes that they composed across the unit as tools to help them write an essay about the relationship among the ideas of the three authors in the unit.

In her reflections about the work, the teacher said that adding in this background piece deepened students’ understanding of the context in which all three authors were speaking/writing and helped students make connections between the reading they were being asked to do in English and the work they had already done in history. She noticed in past years that students had misconceptions, especially about Truth’s speech – thinking that she was speaking after the Civil War and after slavery had been completely abolished. By addressing the context in which Truth was speaking first, students were able to pull in more of what they understood about the time period to their writing and discussions about the speech.

The Importance of students having a coherent knowledge base should not be neglected as we support students to become fluent readers and as we continue discussions about better integrating the science of reading into instruction. It aids in comprehension, analysis, and interpretation of texts. We’d love to hear about how you’ve worked to build and access students’ knowledge base before you’ve engaged with a text.

Citations:

Cabell, S. Q., & Hwang, H. (2020). Building content knowledge to boost comprehension in the primary grades. Reading Research Quarterly55, S99-S107.

Duke, N. K., & Cartwright, K. B. (2021). The science of reading progresses: Communicating advances beyond the simple view of reading. Reading Research Quarterly56, S25-S44.

Hirsch, E. D. (2003). Reading comprehension requires knowledge of words and the world. American educator27(1), 10-13.

Scarborough, H.S. (2001). Connecting early language and literacy to later reading (dis)abilities: Evidence, theory, and practice. In S.B. Neuman & D.K. Dickinson (Eds.), Handbook of early literacy research (Vol. 1, pp. 97–110). New York, NY: Guilford.

Shanahan, T. (2023, February 4). More on Hanford: Phonics reform and literacy levels. Shanahan on Literacy. https://www.shanahanonliteracy.com/blog/more-on-hanford-phonics-reform-and-literacy-levels

IFL Recommends for January 2024

Welcome to 2024! We start the year off with recommendations that come from an educational consultant and language researcher, a political podcast, and two stories from the Library of Congress’s Bibliomania blog.

Brenda Robles

Brenda Robles

Mathematics Fellow

Brenda says, “I follow Dr. José Medina on TikTok. Listening to his Monday message is such an inspirational way to start every week. His messaging uplifts the use of Spanglish and gives me hope that we are moving towards a society where we honor each other’s linguistic repertoires. The following is one of my favorite videos of his.”

Medina Monday Message
Dr. José Medina

“On my goodness, look at you moving the parts of language that you needed.”
– Dr. José Medina, language researcher and author

Get motivated here

Not on TikTok? You can still play the one that is linked above. You can read more about Dr. Medina’s work on CNN.

Jose Medina

Aaron Anthony smiling for the camera

Aaron Anthony

Director of Operations

Aaron says, “I’m recommending a new-to-me political podcast from The Ringer called “Somebody’s Gotta Win.” I like it because it’s a great balance of smart, funny, unbiased, and informative thanks to its host, Tara Palmeri. I know lots of folks are just sick of political news, but not me. I like following politics closely because I think it’s really important and directly tethered to our mission at the IFL of improving education. The podcast is available wherever you get your podcasts—give it a listen and see what you think.”

 

Somebody’s Gotta Win

Tara Palmeri, host

“I’m trying to inform. I’m trying to let people hear different perspectives and make their own decisions. I’m trying to be honest and have like the conversations that maybe aren’t necessarily polite and kind of let people talk their book but also push back on it because everybody has an agenda in this town.”
– Tara Palmeri, senior political correspondent for subscription news platform Puck

Listen here

Somebody's Gotta Win Podcast

Tracey Tomei headshot

Tracy Tomei

Manager of Instructional Design and Product Development

Tracy says, “The Library of Congress’s Bibliomania Blog features stories and facts about their rare collections usually accompanied by some beautiful imagery. These two recent posts stuck out to me to share with the IFL. The first is about medieval techniques to retain and learn information with memory and images. And the second discusses the use of gamification and student engagement with a look at some flashcards from the early 1500s.”

 

In Your Mind’s Eye: Strange Mental Architecture to Help You Remember

Marianna Stell

“Classical mnemonic theories emphasized that students were more likely to remember strange and surprising images. Like marketers today, theorists of the past understood that memory is more easily stimulated when information is contained within an image that provokes an emotional response such as surprise, delight, or amusement.”
– Marianna Stell, reference librarian

Read more here

memorization

flashcards

Gamification: Thomas Murner Makes Learning Fun

Marianna Stell

“What is an effective means of engaging students in the classroom? Turn learning into a game.”
– Marianna Stell, reference librarian

Read and see more here

How Early Childhood Trauma Changes the Brain: A Conversation with Dr. Jamie Hanson Bridging to Research

Lindsay Clare Matsumura, Senior Scientist at the Learning Research and Development Center with Angela Allie, Executive Director of the IFL

Dr. Jamie Hanson is an assistant professor of psychology and a research scientist at the University of Pittsburgh’s Learning Research and Development Center. His research focuses on changes in the brain’s neural circuitry as a result of early life stress and how these changes impact outcomes for children in adolescence.
How did you get interested in studying brain development in young children?

I got interested in this area after reading a series of papers that came out of a research unit at the National Institute of Mental Health that described the basics of how the brain grows and develops from childhood into adolescence into young adulthood. After learning about basic brain development, I started working with families that lived in very stressful conditions. For example, I worked with kids who had lived in Russian and Romanian orphanages after Ceausescu was deposed and the Soviet Union fell. Even though these kids had experienced pretty aberrant and abhorrent conditions, there was a lot of variability in the challenges they were having. For example, I met some kids who had lived in an institution for seven years, and they were fine, and you wouldn’t know that they had suffered that kind of intense trauma. Then, I worked with kids who had been in an institution for shorter periods of time, and they had a lot of challenges. Understanding why some kids were able to move through things and others have lifelong scars motivated me to understand how brains may change in response to stress and how these changes can impact the trajectory of a child’s life.

What are some of the main “takeaways” from your research looking at how the brain changes as a result of early life stress and trauma?

The biggest takeaways are first, that there are a lot of changes in brain circuitry involved with executive functioning. These are the control-based processes that help us think about thinking, plan for the future, inhibit distraction, and regulate behavior in the face of important or emotionally salient events. My colleagues and I started to see lots of changes in portions of the brain involved with those processes, namely in the prefrontal cortex. Basically, this front third of the brain is often smaller and has smaller rates of growth in those who have experienced trauma. There is less physical tissue there and often lower levels of activity.

The other kind of big set of changes happens in some brain regions involved with learning and emotion: the amygdala and the hippocampus. The amygdala is central to processing vigilance— telling us what to pay attention to in our environment. The hippocampus is central to learning about relationships—relationships within our environment and in different contexts. Differing kinds of early experiences can change the developmental trajectories of these parts of the brain. Particularly, we can see smaller regions over time, we think, because that region might be just working so hard it actually “burns itself out.”

With the hippocampus, this region also tends to be a little bit smaller. So again, you have tissue loss and often lower levels of activity. We think this relates to challenges with remembering information and not updating knowledge used in different environmental contexts (or settings). This could lead to perhaps overgeneralizing a learning or behavioral strategy too much.

Can you say more about what you mean by less material or tissue in the different brain regions?

Yeah, so for the hippocampus and the prefrontal cortex, we see smaller amounts of tissue. We don’t have a great idea, though, if this is because there are less cells in these areas or if it is because the cells are smaller and less complex. Based on studies of non-human animals, we also know that there can be changes to the brain’s neurons. These are the cells in the brain that send and receive chemical and electrical signals and allow you to think, move, and feel. Neurons have these branchy bits called dendrites, which can retract with stress. In other words, if a rat has been exposed to a good deal of stress, their dendrites actually have less fibers—almost like a bush that has been trimmed.

Why do you think incredible stress causes these changes?

I think there are a couple of different possibilities, and my research lab is trying to arbitrate between them. So first, stress, very extreme levels of challenging things, activates the hypothalamic pituitary adrenal (HPA) axis which increases cortisol. Cortisol in consistently high dosages causes brain cells to shrink and eventually in high enough doses to die. You have this chronic HPA stress system activation, and over time, that can alter the brain significantly.

The other thing is that the brain is built by experience. Different kinds of experience will cause things to strengthen or weaken like a muscle, so you can see enrichment and deprivation effects depending on the resources you have access to in your environment.

Finally, there is consistent reciprocity between the brain and behavior. An experience can exert an impact on the brain itself, often through a physiological cortisol-based means. But brain changes can lead to psychosocial challenges that also then affect the brain. For example, if you’re physically abused, that can cause an increase in cortisol. But, in the case of physical abuse, you have a lot of attachment disturbances, and so this might influence how you’re interfacing with your caregiver; so that’s going to cascade back to the brain where the specific adversity causes additional brain changes.

How do these early childhood brain changes influence outcomes for adolescents?

I think it’s very important to think about the lived experiences of people and the phenomenology of adversity, because any experience might be quite different and might exert quite different effects depending on what’s happening in an environment. We’re still unpacking this.

For example, if you experience physical or sexual abuse in your home, this is going to cause you to be pretty vigilant and keyed up to threats in that environment. You’re going to really watch yourself and make sure that after this really difficult thing happened, you are safe. This makes a lot of sense in the environment where the abuse happened. A lot of times, though, that kind of “teed-up-ness” comes out in other kinds of spaces that kids spend time in.

Many kids who have been physically abused have what’s called a “hostile attribution bias.” They believe that neutral things are more hostile. For example, a teacher might say something with benign intentions, but the kid actually sees hostility in what the teacher said, and this can lead the kid to be reactive, aggressive, or respond in a way that is unexpected or different. But it actually makes good sense when you think about the full life of a child. If they live in dangerous environments, it’s adaptive for them to be keyed up to any discrete displays of aggression or potential threat in those places. However, it may not be good to be as keyed up at school because a kid might really misread innocuous things that happen. They might think their teacher is out to get them or somebody who bumped into them at the water fountain did it on purpose, and that’s not necessarily true. In a way, we’re asking kids to “emotionally code switch” – to change between one belief and set of approaches in the world and then completely change to another. This can be hard for some kids, especially if a caregiver or school practitioners are not cued into what’s happening with the kid or doesn’t have training in trauma-related pedagogy.

Are there ways to compensate for the changes that happen in the brain as a result of early life stress? Are negative changes reversible?

First, when we talk about stress and adversity, it’s important to also think about resilience. For every kid who has had a really difficult experience and some challenges as a result, there are lots of other kids who don’t have those challenges. Even in really extreme cases where you see incredible amounts of stress and a two- to three-fold rise in behavior problems or depression, that still means that 30 percent of kids who faced these same stressors didn’t have these kinds of challenges.

One of the biggest sources of resiliency that I think is relevant for your audience are supportive adults outside the home. That’s one of the most well-replicated resilience findings. Having a supportive human and adult outside of your house is something that is massively buffering against the effects of stress, because I think you see a different vantage point in the world working in a different way.

Second, I do think that some of the negative effects of stress on the brain can be alterable. In fact, this is the predicate of therapy in adults. We often reverse some negative information patterns that we have if we’re depressed or anxious. Sometimes, it’s compensation, but a lot of times it’s reversal. We’re able to stop these patterns. In fact, researchers have studied people going through (talk) therapy, and you see changes in the brain as a result.

The other thing connected to reversal, or changing and disrupting some negative patterns, is compensation. One thing that we have to be careful about though is that some brain changes or patterns likely arose out of usefulness. I mentioned earlier that a kid may be hyper-vigilant to things in their environment for good reasons, though that vigilancy could cause trouble in a school setting. However, you don’t want to turn that vigilancy off. For instance, if you live in a rough neighborhood, this vigilancy might be helpful and adaptive, saving you from some harm or violence. Put another way—it might be good to look tough and “puff out your chest” so that you don’t get jumped going home. If someone just turned that behavior off or reversed this response, you might create other challenges for kids.

I think instead, we can maybe think about compensatory experiences that could alleviate or allow for some of the consequences of adversity to be lessened, or at least to be felt less by youth. If we were able to get kids to understand safe contexts and spaces where they’re okay and then say, “I know what it is like in this neighborhood. I know what it’s like on the walk home. And I get that. But hopefully, you can turn a little bit of that off or at least turn it down a little bit when you’re in school.” This is the emotional code-switching idea I talked about earlier. Creating safe, warm, and supportive spaces for kids could be a form of contextual learning. Kids could come to understand that the world may not be a warm and fuzzy place, but they can still feel warm and comfortable in their school. Schools themselves could be an intervention.

IFL Recommends December 2023

This month’s recommendation contains an article and a press release about the benefits of being multilingual and a podcast about the importance of listening to teachers’ voices.

Laurie Speranzo

Laurie Speranzo

Mathematics Fellow

Laurie says, “I am fortunate to be working this year in several districts where the majority of the student population does not use English as their language at home. In speaking with the teachers though, it came to light that students do not see their multiple languages as a true asset. Being bi- or multi-lingual is a strength as these two articles show. In addition, the second article (press release) includes a comprehensive toolbox of resources.”

6 Reasons Being Bilingual Makes You an Asset to Your Company
Ava Roman
“Diverse teams consider multiple perspectives to solve problems. They bring greater creativity, innovation and flexibility to the floor, developing more complex approaches that consider how the same processes impact different individuals.” – Ava Roman, managing editor of Revivalist, a women’s lifestyle magazine that empowers women to live their most authentic life Read the article here

Bilingual

Biden-Harris Administration Launches “Being Bilingual is a Superpower” to Promote Multilingual Education for a Diverse Workforce
U.S. Department of Education “The younger generations aren’t getting ‘worse.’ The kids aren’t getting ‘worse.’ We’re just failing them. Kids are so inherently good and pure and wholesome.” – Ali Levasseur, third-grade teacher Read it here

 

Shamira Underwood

Shamira Underwood

Mathematics Fellow

Shamira says, “With support from the Chan Zuckerburg Initiative, the Voices of Change Writing Fellowship gave space to a group of school leaders, support professionals, and teachers to use their voices for powerful storytelling on a wide range of topics impacting education today. Their dynamic stories were published in 2021-2022 and can be found and read here. Towards the end of 2022, one of the Writing Fellows, Jennifer Yoo-Brannon, participated in an EdSurge podcast in which she interviewed a veteran teacher who ultimately decided to leave the profession. She is not alone in the mass exodus of teachers leaving classrooms across our nation. Listening to and reading the stories of those doing the work of educating young minds in a changing educational landscape have implications for policy, research, and practice. We should all be listening.”

Exit Interview: Why This Veteran Teacher is Leaving the Profession

EdSurge Podcast, interview by Jennifer Yoo-Brannon

“It’s like leaving a relationship, that you love this person, but you realize life is too short to feel this beat up. So you have to make a choice.”
– Diana Bell, veteran teacher of more than 18 years who recently decided to leave teaching

Read more and listen here

 

 

Voices of Change Writing Fellowship In

 

Supercharging Three Common Practices in Math Classrooms 

By Kristin Klingensmith, Beatriz Font Strawhun, Brenda Robles

We have been in numerous math classrooms across the country and have seen a variety of practices and routines being used to support student learning. On any given day prompting students to turn and talk, asking students to use and connect representations, and facilitating whole group discussions offer the chance to increase student engagement with each other and the content. In this article, we juxtapose two scenarios for each of these practices and consider how small refinements can supercharge a lesson and unlock the potential for greater student-to-student engagement and deeper engagement with the content.

Math students working at the whiteboard

Prompting Students to Turn and Talk

Scenario A – Sharing with a Friend

During a Which One Doesn’t Belong (WODB) warm up activity, the teacher has students seated in rows of two. Students are asked to share their answer with their elbow partner. After they share which one they think does not belong, the teacher prompts the students to explain to their partner why they think their selection should be eliminated. The teacher asks three students at random to explain their thinking to the whole group. The teacher asks students if they agree or disagree with what was shared. Then the warmup ends.

Benefits of Turn and Talk in Scenario A:

  • The teacher positions students as holders of knowledge by creating space for them to share with a peer.
  • Students listen to others and can share their own thinking giving them a chance to rehearse and maybe solidify their own reasoning.
  • Students are exposed to different ways of thinking.
  • Students may acquire more academically precise language from their partner.

 

 

A Bit About WODB Activities

Which One Doesn’t Belong activities are designed to spur critical thinking. In a WODB, participants are presented with a set of items. They are asked to identify which one of the items doesn’t belong and explain their reasoning. The goal is to encourage participants to think critically, make connections, and justify their choices based on specific criteria. The key aspect of a WODB activity is that there can be multiple valid reasons for selecting any of the items as the one that doesn’t belong, making it a flexible and open-ended learning tool.

Now give it a go! Which would you say doesn’t belong and why?

WODB

Scenario B – Sowing Seeds for a Discussion

During a WODB warm up activity, the teacher has the students seated in groups. Students have a variety of tools, writing utensils, and their own copy of the WODB. Students are asked to circle which one does not belong and stop and jot why they would eliminate it. After having some individual think time, the teacher asks students to turn and talk about which one they eliminated and why they kept the other ones. Meanwhile, the teacher circulates the room listening to the turn and talk between shoulder partners. The teacher purposefully selects a shoulder-partner pair of students to share what they heard from each other. After they share, the teacher asks the class about the connections they can make to their own thinking. The teacher continues to ask a series of questions that invite other students into the conversation by linking their thinking to that of their peers and asking students to revoice others’ thinking. The teacher uses questions like

  • Thank you, Sam. Justin, you also said you thought B did not belong? Was your reasoning similar to or different from Sam’s?
  • How is your thinking related to Jing’s?
  • Who can say back what Jing’s reasoning was?
  • Marco, how was your argument different?
  • Who agreed with Marco? Elena, say it for us one more time. What was the reason both you and Marco said C did not belong?

The teacher also encourages students to pose questions to each other as clarification is needed. They also mark the key observations and critical ideas about mathematics that students share by charting the contributions publicly to be used later. The warmup ends with the teacher saying that they will return to the claims and ideas that have been charted during the lesson.

Benefits of Turn and Talk in Scenario B:

Students receive all the same benefits expressed in Scenario A plus the following.

  • The teacher has a better understanding of the students’ understanding of the concept by listening to the turn and talk.
  • The teacher sets an entry point into the discussion by intentionally selecting which student pair will share first. This decision may be based on the most common thinking to surface in the group, the earliest access point into the conversation, a misconception that surfaced, or a generalized claim, to name a few.
  • Shifting the focus of the turn and talk from just what did you eliminate to how are the remaining items connected increases the likelihood that students will engage with the mathematics.

Some tips for maximizing the use of a Turn and Talk:

  • Layer learner-centered routines like asking students to engage in a stop and jot before having them do a turn and talk. This gives students a bit more time to get their own thinking in place before sharing with a partner. Including a stop and jot tethers oral communication with written communication which can help to deepen a student’s understanding.
  • Vary the questions posed to include reasoning for both examples and non-examples, like asking students to reason about why a particular item was eliminated and why were the other kept.

Using and Connecting Mathematical Representations

Scenario C – Expecting a Representation

A class is part-way through a unit on operating with negative integers.
The teacher asks students to solve a problem using some sort of visual representation. Some students make a vertical number line, some create
a horizontal number line, others use a chip board.

As the students work through the problem, the teacher asks students to find someone else in the classroom who created the same representation to solve the problem and share their solution. The teacher then asks students to find a partner that used a different strategy than their own and explain how they solved the problem to each other.

Benefits of creating and discussing Mathematical Representations in Scenario C:

  • Students connect their own understanding to that of others.
  • Students talk through their own solution strategy that include visual and physical representations.
  • Students see other pathways to the solution that include different visual and physical representations.

Mathematical Representations and the Connections Among Them

Use and Connect Representation

National Council of Teachers of Mathematics (NCTM). (2014). Principles to Actions: Ensuring mathematical success for all. Reston, VA: NCTM.

Adapted from [Lesh, Post, Behr (1987) Representation and Translations Among Representations in Mathematics Learning and Problem Solving] and modified from [Huinker, D. (2015) Teaching Matters: Actions for Attaining High-Leverage Teaching in Every Mathematics Classroom]

Scenario D – Leveraging and Connecting Multiple Representations

A class is part-way through a unit on operating with negative integers. The teacher asks students to solve the negative integers problem using whatever representation and strategy they like. The teacher walks around to see how the students are working. The teacher then groups the students into triads in which each student has a different representation and/or strategy for arriving at the solution. For some triads, this includes student work that contains an error or misconception. The teacher asks the students in the triads to look at each other’s work to identify how each of the parts of the problem are represented even though the representations and approaches are different. Additionally, students are expected to construct an explanation in writing and be ready to share their collective thinking about how all three pieces of work are related to each other and the mathematics of the task.

Benefits of creating and discussing Mathematical Representations in Scenario D:

Students receive all the same benefits expressed in Scenario C plus the following.

  • Students must make connections between and among various mathematical representations.
  • Students reason about the mathematics more deeply as they translate it across different representations and/or strategic approaches.
  • Students critique the reasoning of others; in some cases, this may include revision of an error or misconception.
  • Students work together to construction an explanation in writing (which serves as another representation) and requires synthesis across the ideas shared by the three students.

Some tips for maximizing students use and connection of representations:

  • Expect students to create and explain various mathematical representations and orchestrate opportunities that require students to make connections among the representations to reason about how the math is translated in each of them.
  • Select or create problems that allow for a variety of representations and strategic approaches AND allow students to choose what works for them so that there will be a range of solution paths that can be shared and discussed.

Facilitating Whole Group Math Discussions

Scenario E – Having a Show and Tell

Students have been working on a high-level task. They have had a chance to work on their own and in a small group. While they worked, the teacher walked around to learn about their current thinking and pose questions to advance their thinking.

Following the small group time, the teacher pulls the students together for a whole group discussion. The groups of students take turns sharing their thinking by holding up their papers from their desks as their peers listen.  When one group finishes the next group starts. Each group gets to share, and by the end of the discussion, all six groups have shared their work.

Benefits of facilitating whole group discussion in Scenario E:

  • The teacher positions students as holders of knowledge by creating space for them to share with a peer.
  • Students have to be able to explain the work that they did with peers.
  • Students get to listen to how other groups thought about and solved the task which may expose them to different ways of thinking.
  • Students may acquire more academically precise language from hearing other groups discuss their work.
Scenario F – Sharing, Connecting, and Growing Collective Reasoning

Students have been working on a high-level task. They have had a chance to work on their own and in a small group. While they worked, the teacher walked around to learn about their current thinking and pose questions to advance their thinking. As the teacher visits and revisits each group, the teacher identifies the portion of the group’s work that the students will be invited to share during the whole group discussion. The teacher shares this with each group before moving onto another group.

Following the small group time, the teacher pulls the students together for a whole group discussion. The teacher strategically calls on group members to show and share the starred portions of their work, making a communal poster of work associated with the task. As the groups take turns sharing their thinking and building onto the communal work, their peers listen and are asked to make connections between the existing information and the new information being shared. Many of the groups share their thinking and by the end of the discussion, members from all six groups have actively engaged in the discussion by sharing and/or making connections between and among their work and what has been shared by their peers.

Benefits of facilitating whole group discussion in Scenario F:

Students receive all the same benefits expressed in Scenario E plus the following.

  • Students are positioned as the authors of the work from which the collective group develops a common and shared understanding of the mathematics.
  • The teacher sets the tone of the discussion and can work to ensure that the thinking of all groups surfaces in the communal work.
  • Student thinking from multiple entry points can be honored since the teacher is selecting the portions of the work each group will share.
  • Students have access to a shared visual record of their collective thinking which can serve as anchor for this (and future) discussions.

Some tips for maximizing the facilitation of a whole group mathematics discussion:

  • Monitor and identify specific elements of work that can be shared by each of several groups to ensure that a variety of thinking is reflected, being sure to look for a range of entry points and opportunities to discuss common misconceptions or errors to get at the “why” and “how” of mathematical concepts, rather than just the “what.”
  • Encourage students throughout the discussion to respond to and build upon each other’s ideas with the goal of prioritizing deep and shared conceptual understanding over mere reporting out of procedures used to arrive at a calculation.

In this article, we hoped to highlight the impact that different ways of implementing common practices have on students’ opportunities to engage with the mathematics and each other during a lesson.  Though all the illustrated scenarios offer some benefits to the learners, we can see that minor tweaks supercharge these common practices and significantly impact the opportunities afforded students.