Approaching Mental Math Lessons

March 28, 2020

Note to the Reader: This essay references five different mental math problems.  For reading ease, I recommend selecting one topic and then following it exclusively throughout the article (see Table 1.)

The term Mental Math is often blended with rote learning, memorization, and automatization, but these surface designations are traps for unenlightened and/or disinterested educators. 

Succeeding in mental math lessons not only requires strong arithmetic skills, but also proficient place value understanding.  To teach these topics well, educators must engage their class in deep mathematical thinking.  Accurately performing mental arithmetic is always the lesson’s end goal, but doing so is simply a byproduct of deconstructing foundational knowledge and piecing together a variety of skills.  When left to their own devices, children often solve problems using inefficient strategies.  Teachers, therefore, play a vital role in keeping their students mentally adroit.

There are seven prominent stages to delivering mental math lessons.

I. Planning the lesson.

During the lesson planning process, the teacher considers their class’ collective skill and place value understanding.

Invariably, they will determine that at least some of their students aren’t comfortable enough with prerequisite skills to comfortably engage with the lesson tasks. When they conclude that much of their class won’t gain meaning from the lesson, they either modify its content or skip the lesson entirely (see Table 2).

If they decide on the latter, they use it as extension work during future lessons. If they choose to use an instructional day on the topic, they select sample problems that they’re confident ten percent or more of the class will answer correctly.  They then determine the solving method(s) that they’ll want their students to practice (see Table 3).

The written explanation(s) becomes an articulation of the mental solving process.

II. Beginning the lesson.

The teacher starts the lesson by projecting the selected problem and directing their students to try and solve it without a writing tool.  To add a piece of theater, they might tell the children to literally or figuratively sit on their hands.

They then provide the class with an opportunity to engage in productive struggle. During this time, the teacher carefully observes individual students, trying to gauge the mental effort that they’re exerting.  This can last anywhere from thirty seconds to a few minutes.

Regardless, the teacher always sides on providing extra time rather than cutting it short.  When faced with a challenging problem, many children have a tendency to give up easily, especially if they don’t think they can mentally solve it.  By prolonging the silent, thinking time the teacher increases the likelihood of students persevering.

III. Students record their answers.

Once the teacher feels that all children either have an answer or have given up, they tell their class to take out a writing tool.  Those who are confident that they have a correct answer record it along with an explanation as to how they figured it out.  Students who are unsure or stumped try to solve the problem using any written method they want (see Table 4).

IV. Signal for a communal response.

After every child has solved the problem, the teacher signals for students to share their answers using a whiteboard or choral response.  During this elicitation, they carefully look and/or listen for differing replies.  Contradictory answers, they understand, provide opportunities for rich mathematical discussions.  In the end, the instructor provides the correct answer.

V. Students share their thinking.

With the answer revealed, the teacher prompts their students with the question How do you know? before directing them to Turn and talk to their partner.  As children discuss, the educator probes the room, listening to student explanations and cataloguing responses they’re expecting to hear, as well as outlying strategies (see Table 5).

VI. Students share their thinking with the class.

Following partner discussions, the teacher re-centers the class and asks for two or three volunteers to share their thinking.  Although they don’t reveal it, they call on students so that strategies are sequenced from least to most complex.  This intentionality validates all methods, while illuminating the efficiency of individual strategies.

Knowing that even their most advanced students might struggle to articulate their thinking, they are prepared to format anticipated responses on the board.  This provides children with a unifying mental image, making the strategies comprehensible to everyone (see Table 6).

If no student shares the predetermined method(s), the teacher provides them, often creating a fictitious conversation, e.g. “I heard someone say…” or “I was talking about this problem with another teacher and they said that you could solve it by...”

VII. Students practice analogous problems.

After all strategies have been expressed, the teacher provides analogous problems and directs students to solve them using the shared method(s) (see Table 7).

Once students have demonstrated that they can successfully perform each strategy, they practice more problems using their preferred method, which includes solving mentally.

By requiring students to express computational and/or place value strategies, the teacher ensures that every child makes mathematical progress even if they fall short of the lesson’s objective.