Last Friday, I attended a virtual conference held by the British Columbia Association of Math Teachers.
The keynote speaker, Michael Pruner, challenged us as math teachers to slow down. To do so, he admitted, would require us to focus less on testing and more on how children actually learn. Although admittedly disruptive to the system, he said, it is our duty as educators to bring about this change.
This was such a welcome message. Corrinne and I have been saying this for years – lonely voices in the wilderness! Many teachers have nodded at us in understanding, but added regretfully, “My district/admin/parents/curriculum/testing requirements won’t allow it.”
And yet, WE are the professionals. Let us claim our own expertise. We are the ones who must take this to the streets, to the board meetings, to the ballot box and to the classrooms.
Here are our suggestions for slowing down:
1. Use Manipulatives More.
In our own anecdotal experience (still… >60 years combined!), about one third of all students are visual learners, and therefore do poorly with the traditional ‘memorize and move on” approach to math. These children need three times – four times – eight times as much time at the concrete level as we’re currently giving them. When not given enough time, these students give up, try to memorize long enough to pass a test, and label themselves as lifelong failures at math. Yet – given the time they need – they can learn math, all the way through high school math and beyond. We’ve seen it over and over.
Because the concrete level is where humans actually learn to make sense of mathematics. Without it, we limp on to higher mathematics blindfolded, befuddled, reciting mysterious, magical formulas.
The pyramid at right shows the relative importance of the 3 levels of math learning.
There are many research studies that back up the effectiveness of this approach (C>P>A) to teaching math.
*HERE is a link to a long list of studies on the value of C>P>A.
*This pdf is from my BCAMT presentation last Friday on using manipulatives.
*This video is of my upcoming CMC presentation on manipulatives.
2. Teach in Longer Units.
Figuring this out was one of our greatest challenges over the last 20+ years. As teachers, don’t we all set goals that feel impossible? We want to both:
(a) meet the needs of the visual learners, the step-by-step learners, the ponderers, the discouraged, and
(b) meet the needs of the racers, the abstract thinkers, the memorizers, the gifted.
For Corrinne and me, the solution was to teach in longer units, including one year-long topic per grade level.
For 2nd grade, for example, it’s 3-digit subtraction. For 5th grade it’s fraction operations. We chose skills whose mastery is indispensable in the development of a child’s math understanding.
We stick with that topic ALL YEAR, keeping most of it at the concrete/pictorial level. We introduce it in September with investigations, games, etc. and then we then go on to other units as usual, but do at least one problem a day (we call it our daily vitamin!) that reinforces the concrete/pictorial understanding of that original topic, and we withhold the algorithm until everyone has had a chance to build confidence at the first two levels. Because of the distance between introduction of the unit in September and its comeback in April/May, we call this a “Sandwich Unit”.
Second grade teachers – wouldn’t it be great to know all your incoming students mastered at least addition and subtraction to 20 (or 30?) before entering your class? Sixth grade teachers, would you rejoice if every student entering your grade in September had a firm grasp (at least at the pictorial level) of addition and subtraction of fractions?
Amen. Yes, this level of mastery is worth the time invested.
3. Ask, Don’t Tell.
This one was and continues to be difficult for us. We teachers so badly want to help, to explain, to facilitate, to lay bricks in the Yellow Brick Road of Understanding. And yet, the truth is that the student must build that road. It is our job as teachers to find problems “just hard enough” to cause students to need manipulatives. Then it is our job to be quiet. I pretend not to know the answers, to look at my students’ blocks in focused puzzlement, to talk about what I see, and ask myself what I might… maybe … do next?
This takes time – if your students are not used to it, don’t expect miracles the first week. But with time, students learn to depend on their own thinking, their own ability to “ask the blocks”, their own innate intelligence, regardless of speed. Isn’t this what we want in the long run?
4. Differentiate by Speed.
Every one of our classwork or homework sheets has 3 levels: (1) A concrete/pictorial level, (2) A second level that transfers to word problems and algorithms, and (3) An optional challenge level.
In our experience, we’ve found that most (80%-ish) students often finish Level 1 and Level 2 in the time given. Half of those (40%-ish) have time to go on to the challenge level, and do at least part of it. The other 20% finishes at least Level 1, meaning that that’s where they are, developmentally. They still need practice at the concrete level before they can build a foundation and bridge to the abstract level.
5. Spend Less Time on Assessment.
One of the most striking things I ever heard Jo Boaler say was, “You can teach like a superhero, building conceptual understanding, doing everything right, meeting everyone’s needs, and then ruin it all with one assessment. We’ve all seen this happen. A child gets a score that is lower than that of the classmates around them, a quiz pockmarked with x’s. The child’s immediate translation is “I stink at math.”
Instead, let’s learn to tell that child, “It’s fine to just work on Level 1. That’s where we all show true understanding. Use the blocks (we call them the “Math Whisperers”) and you’ll figure it out.” Gradually, that child can conquer Level 2 as well.
Additionally, let’s keep assessments short (3-4 problems is enough), and use them for diagnostic purposes only. Feedback to parents becomes “Your child has mastered 3-digit subtraction with blocks, and is showing a growing ability to represent that understanding with drawings and mental math.” We include photos of that process. Children own their own progress, and feel proud of what they can do.