When I was a teacher, my colleagues and I had a nickname for a problem we saw among some of our students: “cookbooking.” Cookbooking was when a student would solve math problems like someone who can only cook by using a recipe. They would follow the steps but didn’t understand why they led to the right answer. Because they only had a surface-level, procedural understanding of what they were doing, they couldn’t extend their knowledge to new situations, explain their thinking, or provide meaningful definitions. They could make biscuits, but not pie crust, even though the recipe for both is the same. We wanted our students to be master chefs who fully understood why they were adding each ingredient, able to critique and correct recipes, and improvise their own creations, but cookbooking students weren’t there yet.
My fellow teachers and I realized that if we wanted to get our students to a new level of mastery, we had to try something different—we had to, as the old Reese’s Peanut Butter Cups commercial said, “get some chocolate in our peanut butter.”
Our change to instruction was simple, and the results were fantastic. We increased language supports and demands in our math classes, which led to students achieving more and classes becoming more enjoyable. The funny part was that we didn’t even have to come up with new strategies; we could liberally steal them from the supports that helped our English Learners (ELs) thrive.
Here are some of the best practices we adopted and some suggestions for how you can implement them:
Scaffold language at the word, sentence, and discourse levels.
We were experts at scaffolding mathematical processes, but our students needed language scaffolds, too.
Word Level: Assess students’ familiarity with the vocabulary for the upcoming unit and use this information to guide your word-level scaffolding. Word walls, word banks, flash cards, glossaries, and vocabulary lists are all great word-level scaffolds. While ELs are often directly taught how to use these tools to support their academic vocabulary growth, don’t forget to teach your native English speakers these skills, too. Many students, as children of the internet age, don’t have experience with printed reference materials and may not be able to use them efficiently without help.
Sentence Level: Provide sentence starters when you ask students to work with partners or write about their thinking. Consider creating a variety of sentence starters and posting them somewhere prominently in your physical or (if you’re teaching remotely) virtual classroom to encourage students to use the starters regularly.
Discourse Level: Model the math conversations you want students to have. A fifth grade teacher and special education team did this beautifully at my school. At the beginning of the school year, they’d sit together at the front of the classroom and pretend to be students engaged in partner work. They had students identify what was or wasn’t successful math communication. They repeated this routine, as needed, throughout the school year to great success.
In your own classrooms, be sure to praise thoughtful math conversations when you hear them, ask students to relate something meaningful that their peers said, and listen authentically to students’ responses to reinforce their importance.
Provide regular and varying opportunities to use language for genuine communicative tasks.
Like any other skill, students need to practice their math communication skills to see improvement. Here are some examples of discourse opportunities you can use to get students talking and writing:
- Ask students to rephrase what you said, they said, or their table partner said.
- Have students write how they know their answer is reasonable.
- Share a student’s work with the class (with their permission, of course), and then call on other students to explain what they see.
- Provide opportunities for students to reflect and write about their progress.
- After students work in partners, ask each of them to explain what was similar or different between their problem-solving strategies.
- Give students an example of a flawed mathematical argument and have them correct or improve it.
- Use Discourse Cards to start conversations and add a game-like element to class.
Focusing on improving the frequency and quality of students’ language output in math class had an added benefit of shifting the student to teacher cognitive ratio in favor of the students. When we said less, students had to think more. If you struggle with this, try limiting yourself to three declarative statements per math session.
Use students’ knowledge and language assets for learning.
Sending home information about the skills and frameworks you’re planning on using can help you get better family buy-in and extend the conversations you’re having in class to students’ dinner tables. Also, picking topics for problems and projects that connect with students’ interests and culture makes talking about them easier. For example, my students were obsessed with basketball, so systems of equations word problems about games like this one resonated with them:
In a free-throw competition, each basket made is worth 5 points and each basket missed is worth -3 points. If Mr. Shiggs took 16 shots and had a final score of zero points, how many baskets did he make? (Answer: Six.)
Provide opportunities for students to work with multiple modes and representations.
When students struggle to engage deeply with a topic (like writing systems of equations), try to incorporate more hands-on problems, visual models, and manipulatives.
The free-throw game question was an opportunity to play with the problem in real life and model it as a group. I asked the follow-up question, “If Mr. Shiggs took 12 shots, would it be possible for him to get a final score of zero? Why or why not?” Students talked about the question with their table partners, and then we played a small version of free-throw basketball to elicit patterns and strategies. Having seen the parameters of the problem play out, students could more easily talk about what kind of equations they might write, what their variables would mean, and how the score could or couldn’t change. Every student had the foundation they needed to discuss the problem.
Universal design is the idea that accommodations necessary for some people make things better for all people. The ideas we integrated into our teaching that ELs needed made rich learning more accessible to all our students. Everyone was more likely to engage deeply with math lessons, and we were more likely to catch and address students who had only surface-level understanding of concepts right away.
To extend my metaphor from earlier, when our junior chefs learned the key to the perfect pie crust (work the dough as little as possible), we were there to ask them what other dishes it could apply to (e.g., biscuits, rough puff pastry, and scones) and make sure they knew why it worked (the chunks of butter in the dough make it flaky and overworking destroys butter chunks). By the end of the year, not only were they solid, flexible math students, but they started to see when they were cookbooking and to take steps to correct it on their own.
If you’d like to see some examples of how Ready Classroom Mathematics implements these and other best practices, download a sample here.
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