Games: In my opinion, there’s no better way to build fact fluency, develop strategic thinking, and increase enjoyment of math.
I created this overview for my 4th About 2/3 of the way through the school year, I showed them the one-page blank version of the overview (the one without problems) and told them, “These are all of 4th grade math standards, packed into 20 boxes! You can do this!” Seeing this limited amount of math made the Common Core standards feel manageable to them. We made a plan to spend the rest of the year mastering each of these concepts.
I handed them a copy of the one-page blank overview and asked them to write down a 1, 2, 3, or 4 in the top corner of each box (see scale below) as we talked about the types of problems that fit each learning target. I told them to be honest with themselves and me. I also pointed out that we had not yet covered all of the concepts, and that we would continue to strengthen our skills in all of these areas. I wanted them to feel OK about where they were. After all, this was about getting each kid to a “3” or “4” – it was OK if they weren’t there yet.
1 = I do not know how to do this!
2 = I think I can get started, but I’m not confident I can get an accurate answer.
3 = I can answer this type of problem accurately.
4 = I know two or more strategies for answering this type of problem.
In my classroom, I had kids rotate through math stations throughout the week. For a few weeks, one of the station activities was working through the large version of the overview with sample problems. I would circulate with a pen and put a star by problems that had been solved accurately, and where I could see work shown. For the most part, I did not focus on how the problem was solved. It was OK for the kids to use inefficient strategies. The focus was on understanding the concept and solving the problems accurately.
After kids had a chance to work through the overview, I handed back the one-page version of the overview, where they had pre-assessed themselves. I asked them to write down a new self-assessment score for each concept. Afterwards, we had a class conversation about the growth the kids had made, and which concepts still felt shaky. I took these and the worked-on packets to help me plan and differentiate lessons.
After a few weeks of math activities and support informed by the packets and self-assessment, I handed fresh copies of the overview to students, with new problems to match each learning target. These were to be treated as assessments. Students worked independently on them for a few math periods. As students handed them in, I checked them over and asked them to try again on problems they hadn’t solved accurately. I marked problems with a star if they were completed accurately, or an arrow if they needed to be tried again. This arrow helped me remember later which students had trouble with which problems. Once a student had completed the packet with 100% accuracy, I asked him or her to make example posters for each learning target that showed 2-3 neat strategies for solving each problem. These students also began to work as student leaders, helping students who had done as much as they could independently.
Along with several whole group and 1-on-1 conversations about this process, students walked their parents through the process and showed their work during student-led conferences. They showed their parents their self-assessment scores, their practice work, and their assessments.
In the days leading up to the end of grade standardized tests, I used the overview as a reminder to my nervous students that there was not an endless amount of math on the test. They had seen it all and practiced it all, and they could do it!
When I first started teaching in an inclusive classroom I was nervous. I didn’t know how I would be able to differentiate for every student, meet their diverse needs, and remain sane and happy.
I grew to LOVE inclusion.
One lesson I learned is that a learning disability is just one strand of who a student is.
I am constantly impressed with the wide variety of strengths and challenges my students present.
The rich insights from dyslexic students into the plot of read-aloud books have left me in awe.
The spot-on recommendations I’ve heard from a student for how we can solve class-wide social problems have opened my eyes to his future. I see him in a successful career, respectfully solving problems and leading a team of employees. It now breaks my heart to think that in a different setting that student would be left out of this social experience because he needs to dictate written work.
Learning about all these STRENGTHS has tuned me into how inaccurate the message about learning disabilities often is.
Even a kid who can’t read, write, or solve math problems on grade level can be incredibly intelligent in so many other areas that are just as or MORE important in real life.
I have seen mind-blowing artwork and ingenious math strategies come from students who have pushed themselves to be great in spite of tremendous challenges.
Those students need to be allowed the space and time to do this.
As teachers in inclusive classrooms it’s not our job to say, “Well, you have strengths in other areas.” It’s our job to provide activities with multiple entry points, provide blocks and visuals and oral instruction, and perhaps most importantly, reflective debrief at the end of these activities.
Sometimes we have to be really creative about how reach all students.
I plan in best practices by always having a station with a hands-on activity, a station that requires logic and reasoning, and a station where students are pushed to persevere in solving a complex problem they haven’t seen the likes of before.
The kids are spread out in 3 areas of the classroom, and I’m free to walk around, listen in, support those who need it, and sip my coffee!
When there are a few minutes left, we gather around the circle and discuss math strategies we used and character traits that helped us, like perseverance and flexibility.
My students know where math tools are, and they recognize that my expectation that they be fairly independent is part of the respect I have for them.
I find that very few students can’t grasp grade-level math concepts, even if they’re challenged by accurately solving the corresponding problems. For example, most students who struggle with dividing large numbers can tell you exactly what division is.
It is most often not going to help them to dumb-down a math problem. Instead, we work together to figure out what strategies and tools will work for matching accuracy to conceptual understanding.
When my class shares at the end of math, we hear about the struggles of perfectionists, kids with learning disabilities, and kids who have trouble working collaboratively. We also hear about successes, and the work that went into them.
An inclusive classroom is well-rounded, and that means it’s a classroom that is balanced by kids with strengths and challenges in all different areas. Some students (with or without IEPs) have trouble with social skills, while others shine with exceptional interpersonal skills.
I try my best to help my students see that we’re a community there to support each other. We can all grow if we share our strengths and empathize with each other’s challenges.
They key is the way we, as teachers, support all students in embracing the differences. Inclusion is a way to build empathy and tolerance in children. Not surprisingly, it’s open communication and respect that fosters understanding instead of stereotyping and teasing.
If you have read Why I Love Inclusion, well, then you know that I love inclusion.
Unfortunately, I find that by the time some students get to me in 4th grade, they feel they can’t do things that I believe they can do. They have generalized their own learning disability too broadly, and given up on being “good” at skills for which I feel they have potential.
The Yale Center for Dyslexia and Creativity says, “One thing we know for certain about dyslexia is that this is one small area of difficulty in a sea of strengths.”
The following are some quotes from an article by Sally E. Shaywitz, M.D. that appeared in Scientific American. The full article can be found here.
“Testing the youngsters yearly, we found that dyslexia affects a full 20 percent of schoolchildren…”
“This basic deficit in what is essentially a lower-order linguistic function blocks access to higher order linguistic processes and to gaining meaning from text. Thus, although the language processes involved in comprehension and meaning are intact, they cannot be called into play, because they can be accessed only after a word has been identified.”
“Linguistic processes involved in word meaning, grammar and discourse—what, collectively, underlies comprehension—seem to be fully operational, but their activity is blocked by the deficit in the lower-order function of phonological processing.”
“The phonological model crystallizes exactly what we mean by dyslexia: an encapsulated deficit often surrounded by significant strengths in reasoning, problem solving, concept formation, critical thinking and vocabulary.“
“It is true that when details are not unified by associated ideas or theoretical frameworks—when, for example, Gregory must commit to memory long lists of unfamiliar names—dyslexics can be at a real disadvantage. Even if Gregory succeeds in memorizing such lists, he has trouble producing the names on demand, as he must when he is questioned on rounds by an attending physician. The phonological model predicts, and experimentation has shown, that rote memorization and rapid word retrieval are particularly difficult for dyslexics.“
That is my theory.
Most of us were taught to memorize times tables, memorize the standard algorithms, memorize the steps of long division.
Most of us were not given opportunities to problem solve, figure out strategies on our own, or put math into context.
Many teachers are still teaching this way.
Students with IEPs are often provided remedial help. If she doesn’t have facts memorized, she’ll have help memorizing them. If she can’t keep a procedure straight, she’ll get extra practice doing it over and over again.
This all seems like common sense. But teachers are missing so many opportunities to allow her to internalize higher level math concepts through her strengths: visual spatial strengths, reasoning strengths, conversational strengths, and, critically, the ability to form conceptual frameworks.
She is being pulled aside and being held back.
Teachers can help students with dyslexia use their strengths to overcome their challenges. Math is actually the perfect subject for this.
Through hands-on experiences and rich math discussion, students with dyslexia can reach and exceed grade level expectations of math concepts.
Teachers can encourage students to figure out how to solve math problems on their own, suggesting that they use math strategies they are already comfortable with. For example, holding back on showing a more efficient strategy, and allowing kids to use repeated addition, blocks, or some other well understood strategy to solve newly introduced large multiplication problems. Is this time consuming? Yes? Does this allow for seamless understanding? Yes.
Teachers can offer activities that require students to practice math in context. For example, ask students to work together to solve a complex word problem that is realistic. The word problem should be written and read aloud several times – as many times as needed – throughout the activity.
Teachers can ensure that conceptual understanding becomes solid by providing time EVERYDAY for students to share strategies they used, and discuss similarities, differences, and patterns among their classmates’ work.
Teachers can create activities in which students must use several models to represent the same math problem. This helps all students see the connections between visual models, written equations, and physical models.
What if I asked you to copy down by hand pages of text in a language you didn’t know? It would be painfully boring, and you’d soon be frustrated or even angry.
But what if I taught you the language first? Then you’d be engaged with the content. You’d know if you were transcribing a fairy tale or top secret documents. You would be in on the story.
Some students feel that practicing math problems with pencil and paper is like the first scenario. They see practice problems as writing and rewriting symbols following a memorized procedure. They’re not in on the story – they’re simply writing in a language foreign to them.
While we should limit the amount of paper-pencil practice students do, it is also important for them to do some.
By first providing concrete (hands-on) experiences that help students understand the math behind symbols and equations, you’ll see that more students can stay engaged with practice problems when they are required to do paper-pencil practice.
Hands-on activities not only lead to stronger student engagement, they help students reset their math mindsets.
One of my FAVORITE math activities that is fun, challenging, and super engaging is The Stolen Pendant.
In this activity, kids learn that a beautiful necklace pendant has been stolen by 3 thieves, Bandit, Rascal, and Clyde. The thieves have split up the jewels and sold them off. The kids need to help Detective Sherloff Homie do a lot of math to ensure that the thieves (who have been caught) properly pay back the necklace’s owner.
Students use what they know about the whole and fractional parts, proportional reasoning, multiplication and division, and mathematical discourse (math talk) to work together or independently to solve the case!
As kids engage in this fun and complex activity, they use many problem solving strategies: they build and deconstruct pattern block designs, explain their reasoning, connect their concrete understanding to symbols and equations, and write about their process.
Using the provided discussion questions, teachers can take advantage of the positive vibe in the classroom to help kids realize the power they have to solve really challenging math problems.
When kids get to have an experience like this tile design activity, sometimes they forget they’re doing math. The math is so in context, so authentic, that it just feels like a seemless part of the design process.
They’re moving, talking, and having fun. And at the end, it’s the teachers job to host a reflective discussion that helps students realize that they just tackled a complex math challenge. That last reflection component is the key to resetting kids’ math mindset.
Hands-on learning helps students make their own sense of math. Activities with blocks or fraction tiles boost proportional reasoning. Activities with counters can support number sense and organization. Dice activities foster subitizing, and games motivate efficiency and fluency.
Set up math stations so even when students are sitting and working, they’re not at their usual work space. Even that small novelty changes the dynamic of the classroom.
Beginning class in a circle on the floor to introduce activities and ending class with a reflection circle of questions and shares means students are moving to at least 3 spots during their math period.
Help students make strong math connections through talk. As they work, ask questions that connect physical reality to abstract math. Explaining concepts helps us process information at a deeper level – so take a break from teaching and leave the explaining to the kids!
Here are some questions that get kids talking:
What symbol can I use to represent pushing two sets of blocks together? (addition +)
What are my choices if the two sets have the same number of blocks? (x or +)
What if I break a set of blocks into equal groups? (division ÷)
How would this be different if you had 10 more counters?
How many more counters would you need to get to the next 100?
How is your strategy different from your partners?
How is multiplying with 2 digit numbers different from 1 digit numbers?
Take advantage of the fun of hands-on activities and games to steer students towards a growth mindset. When kids are having fun, they don’t always realize the perseverance and problem solving skills they’ve used. Highlight those skills, and help students see that they have what it takes to work through challenging problems, whether they’re part of a game or an assessment.
At the end of class, ask direct questions about perseverance and problem solving:
Who got stuck and had to change gears at some point?
Who figured out strategies during the game that helped them do better and better?
How did you use math you already knew to solve this more complex problem?
What did it feel like to get stuck, and what did you do?
How can you use the perseverance strategies from this game to help you do math in the future?
Click here to check out one such activity:
The Great Gingerbread House Project is a math project that I developed 5 years ago, and I’ve done it with 4th graders every year since. I’ve tweaked and improved it every year, and now I’m so excited to share it.
The Great Gingerbread House Project is a lot of work for students… they make drafts and final copies of everything: floor plans, calculations of area and perimeter, and front and side views of their gingerbread houses. They do lots of math and have tons of fun!
Together with a partner, students collaborate to create a floor plan for a gingerbread house that cannot exceed a certain area.
They envision their gingerbread house and create to-scale drawings of front and side silhouettes.
Then, they find the area and perimeter of the floor plan, and determine how many graham crackers they’ll need for the floor, walls, and roof.
This inclusive project works for ALL students. It is visual, hands-on, and has many entry points for solving each math problem.
At each step of the project, partners work through multiple drafts to improve accuracy, clarity, and neatness.
In the end, the kids get to build the house they’ve worked hard to design. They compare their predictions to the number of graham crackers and chocolate chips they actually use.
The bundle comes with activities to introduce area and perimeter, everything needed (besides the sweets!) for the project, and reflection questions to wrap it all up.
This Monday and Tuesday, November 28 and 29, all of my teaching materials on Teachers Pay Teachers are 20% off!
All you need to do to enter to win is sign up for my monthly newsletter by clicking the link below:
I’ll choose a winner at random on Wednesday, November 30, 2016. GOOD LUCK!
PS: I suggest you use this sale to purchase this DIVISION STRATEGIES video. The video shows parents and teachers 6 methods for dividing large numbers by one-digit divisors. These strategies are perfect for 3rd, 4th, and 5th graders.
See you at the sale!
In 2014 I did some action research for the Governor’s Teacher Network. I was researching perseverance in elementary students, and I created a plan to gather data every few weeks.
I developed a rubric for perseverance, and I was clear with my 4th grade students about how I would be scoring their math assessments. I scored every problem separately, using a 4, 3, 2, 1 system:
Why? Because on the very first assessment, before I could implement the other aspects of my plan, the students out performed any normal group of beginning-of-year 4th graders. Almost all of them tried to earn a “4” on every problem, attempting to solve each problem with two strategies.
Combine that mystery with a fixed mindset of “I’m not great at math,” and you have students who look like they’re not willing to try for that “A,” but they actually just don’t know how. They feel that the “A” is not for them. Clearly explaining how students earn points invites more of them to try.
It’s not just students who feel that grading is a mystery. Many teachers are overwhelmed by creating their own grading system. And parents LOVE the clarity of this rubric.
I gave students a “4” for accurately using ANY two strategies; abstract, representative, and concrete strategies were OK. Even on assessments, students could use hands-on tools, like base ten blocks, counters, or other manipulatives.
This practice does not only improve the number score on their assessments, it makes use of every moment of math class for students to practice problem-solving and self-differentiating. This helps students grow faster as mathematicians. By allowing a student who only understands a concept in a concrete way to use a concrete solution, he is not being left out.
Caveats: as the year progressed, I was clear about changing expectations. For example, by the end of 4th grade, the Common Core Standards require 4th graders to use the standard algorithms for addition and subtraction. Therefore, by the end of the school year, I required students to use the standard algorithm as one of their two strategies in order to earn a 4. Also, throughout the year I continually showed students when strategies were too similar to count as 2 different strategies. Dividing 84 by 4 by separating 84 ones blocks into 4 groups and then drawing the same process did not count as 2 strategies (even though that step of moving from concrete to representative is important and valuable during math practice!).
Have you seen some of the end of grade test questions elementary students need to solve these days? They can be REALLY complex, and more often than not they are multi-step problems.
Checking solutions with inverse operations works well sometimes, but kids get mixed up working backwards from their solution to a multi-step problem. Also, sometimes kids miscopy the numbers from their word problems, and checking with the inverse operation won’t help them find that mistake.
Students catch many of their mistakes when they solve a problem once with one strategy, and then start all over again with a different strategy.
This is especially helpful for advanced students who sometimes rush through solving problems or those who are overly confident and don’t check their work. When they know that the only way they can get the highest score is to solve the problem twice, they are more likely to catch mistakes they made the first time. Also, pushing these students to come up with a second strategy is a great way to engage those students who sometimes feel bored with problems they think are “too easy.”
Students who are frequently asked to use 2 strategies begin to see patterns and structures in math that make sense to them. They begin to understand how numbers work.
Over and over again, I see students using place value-based methods throughout the operations. Once students realize they can break down an addition problem by place value, they try a similar method for subtraction. They break up multiplication problems by place value, and when division is introduced they break apart place value again.
I notice similar trends for students who like using math tools, like base 10 blocks. They realize that using manipulatives turns the abstract into concrete… the numbers in equations become blocks, and the operation symbols become actions (separating blocks, bringing them together, etc.).
I hope that using this simple grading rubric helps you as much as it’s helped me. It’s all about empowering students, right? If they know how to reach their goals, they’re much more likely to try.