By Laurie Speranzo
IFL Math Fellow, inspired by Sally Fisher, math teacher, Syracuse City School District
As teachers, especially those of us who specialize in a content area, we are considered content experts. So where is the line between asking students to acquire our content knowledge and orchestrating opportunities for them build their own? Take the quiz and then read about the practices to distinguish between those that wish content knowledge into students from those that support their construction of it!
|1. Do you ask students “What is the first step you need to take to solve this problem?”|
|2. Do you look for a student-created diagram that matches the equation a student writes?|
|3. Have you considered if the rule you are teaching expires?|
|4. Do you frontload vocabulary for a math unit?|
|5. Are there students you continually go to because they will have the right answer?|
|6. Do you call on students until you get the answer you had in mind?|
1. Do you ask students “What is the first step you need to take to solve this problem?”
If you ask this question and have one correct first step in mind, you may be wishing your content knowledge into students. By making the task about a series of steps, not only does it proceduralize the math, but it also implies that there is only one way to start thinking about the problem.
What is “the correct” first step to solve this equation?
Is it to subtract 10 from each side?
Is it to subtract 1/4x from each side?
Is it to multiply every term by 4 to eliminate the fractions?
Yes, each of these options is viable! To say one is “correct” means that we are not allowing students to make choices and justify them.
2. Do you look for a student-created diagram that matches the equation a student writes?
If you answered “yes”, congratulations! You are looking to center student thinking and reasoning at the heart of the mathematics, rather than what the book or your own experiences suggests is the diagram to use. Focusing on the five representations of mathematical thinking and how they connect is a way to surface true mathematical reasoning and highlight mathematical structures.
Mathematical structure is not just how you organize your thinking, but it is seeing relationships and patterns! It leads to understanding of mathematics in the greater picture.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. (NGA Center and CCSSO, 2014)
Research shows that stronger mathematicians have access to various mathematical representations and multiple arrows that connect the different representations (NCTM, 2014). By encouraging students to work from a diagram that they or a peer has created and make connections to other representations, there is a greater chance that they will be able to identify the mathematical structure they are studying. For more information on the use and connection of representations, check out Mathematical Representations: A Window into Student Thinking.
3. Have you considered if the rule you are teaching expires?
Some rules expire, and if you have considered this, you probably are not wishing content knowledge into students. Rules that expire can be reinforced—sometime inadvertently or unintentionally—which leads students to false and unreliable conclusions.
The following “rules” are often reinforced by words and examples, but they expire.
“When you multiply, the answer always gets bigger…”
– until multiplying by values between 0 and 1 or multiplying with an odd number of negative factors.
“The answer to a subtraction problem always gets smaller…”
– unless it involves subtracting a negative number.
“We never subtract the larger number from the smaller number…”
– except for when the situations indicate that that this is what has to happen.
The key here is to challenge absolutes. When framed as “always” or “never,” look for contradictory examples. Often when we change the type of number with which we are working, the rule no longer applies, making it a rule that expires.
4. Do you frontload academic math vocabulary for a unit?
A “yes” to this question may suggest a wishing of definitions and the mathematical knowledge embedded with them into students. Without context, giving definitions and frontloading academic math vocabulary is unlikely to help a mathematician develop their thinking. A recent NCTM publication reads:
“…we do not learn language through “osmosis” or by solely interacting with fluent or native-like speakers; teachers must explicitly teach it (de Jong & Harper, 2005; de Jong, Harper, & Coady, 2013). However, this does not mean that teachers should focus solely on teaching particular words or phrases in isolation, as this is unproductive for student learning and an inefficient use of instructional time (Gibbons, 2015).
In other words, isolated and frontloaded vocabulary practice may lead to rote recall of definitions, but it does not offer a rich, conceptually driven understanding of the words in real-world or even mathematical contexts. This applies to frontloading of vocabulary for all learners regardless of the languages they speak because all learners are acquiring academic language.
5. Are there students you continually go to because they will have the right answer?
If you have a few students that you call on because they always have an accurate answer, you may be having them wish their answer into the other students.
Try calling on students who have divergent strategies or models to share, creating space for more than one “approved” solution path. To help with this, consider planning lessons by thinking about several possible solution paths. The Thinking Through a Lesson Protocol (Smith, et al, 2008) prompts us to consider all the ways the task can be solved: methods students will use, misconceptions, and errors.
By raising multiple paths, not just the one that is expected (by you or the book), a range of student thinking is validated while also creating opportunities to address misconceptions and errors.
6. Do you call on students until you get the answer you were looking for?
If you answered “yes”, you are probably wishing your way of thinking into all students. When planning lessons, we absolutely should be planning questions that elicit student thinking and reasoning. But if we only have one answer in our heads, the entirety of the math is that one pathway, our pathway, rather than the multiple pathways that offer an opportunity for reasoning!
Every student should have access to the mathematics in our math classrooms, but access does not mean trying to get every student to think the way you do or do it the way the book does it. Nor does it mean that every student will solve the problem the same way. Forcing conventions and procedures before reasoning results in missed opportunity for students to construct their own knowledge. This also limits the opportunities for student to see themselves and their unique assets as vital contributions to the mathematical community in their classroom.
Chval, K.B., E. Smith, L Trigos-Carrillo, R.J. Pinnow (2021). Teaching Math to Multilingual Students: Positioning English Learners for Success. Thousand Oaks, CA: Corwin Publishing.
National Council of Teachers of Mathematics (NCTM). (2014). Principles to Actions: Ensuring mathematical success for all. Reston, VA: NCTM.
National Governors Association Center for Best Practices (NGA Center) and Council of Chief State School Officers (CCSSO). (2014). Mathematics. Common core state standards for mathematics. Retrieved from http://www.corestandards.org/Math/Practice
Smith, M.S., Bill, V., & Hughes, E.K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle School, 14(2), 132-138.