I thought that title would get your attention! Now just hear me out. I want you to consider that there is really very little in math that students must learn through direct instruction, that is explicit instruction from a teacher. Let me give two examples, and then I’m sure you can think of others.

Take, for example, expanded notation [eg., 234 = (2 x 100) + (3 x 10) + (4 x 1)]. Yes, students need to be directly taught the conventions for writing a number in expanded notation. They would have no way of knowing that we put parenthesis around each multiplication expression, nor could they discover it through exploration. But they can construct their own learning about what expanded notation is (describing the value of each digit using a multiplication expression) through exploration with base-ten blocks and careful questioning by the teacher. Be sure to check out this blog post for more on that.

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Another great example is decimal equivalencies. Say, for example, you are trying to teach the concept that 0.4 and 0.40 are equivalent. That is a very abstract concept for students to grasp through direct teaching. Instead, have students use base-ten blocks to build 0.4 and 0.40. Note that when using base-ten blocks for decimal values, we typically use the flat to represent ones, the rod for tenths, and the unit for hundredths. If they build those two numbers, how could they *not *see the equivalence? Have them build a few more similar pairs (0.7 and 0.70, etc.) and then have them explain why the two numbers in each pair are equivalent.

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In her book What’s Math Got to Do with It?, Jo Boaler describes traditional teaching methods as passive learning. She cites years of research that underscore how ineffective the lecture-demonstrate-practice cycle of teaching is for student learning. I love this quote: “Students taught through passive approaches follow and memorize methods instead of learning to inquire, ask questions, and solve problems.” If we want deeper understanding (and don’t we?), we must move past stand and deliver teaching.

The idea of direct teaching vs inquiry is not really a debate about small group instruction vs whole group instruction. It’s more a teaching style and philosophy. Unfortunately, direct instruction happens as often in a small group setting as in whole group instruction. The shift we need is more toward facilitating learning through thoughtful questioning and away from telling and showing students what they need to learn.

Do you notice the other common thread that is a must for learning through discovery? Students must be allowed to explore concepts with concrete materials for discovery to occur!

Can’t wait to hear your thoughts on direct instruction vs discovery!

I agree that math needs to be more inquiry based. Give students the tools and be a guide. They don’t get it until they try it on their own.

I think we need resources for this type of inquiry-leads-to-understanding teaching. I have plenty of station games and I can monitor while they play with manipulatives. But, a specific task or activity that you can tell students to do which directly leads to the “Did you see how that means this…” type of understanding, I think those activities/task/questions are rarely represented on something like pinterest or tpt. Do you have any suggestions for resources?

I agree and would love some resources for this.

Check out Marian Small’s Good Questions

I would highly recommend Denise Gaskins book Let’s Play Math. Her ideas are amazing, but with every idea she gives many other resources. She honestly could have just put together all the resources and sold them as a book individually (IMHO) 🙂

“Let’s Play Math offers a wealth of practical, hands-on ideas for exploring concepts from preschool to high school.”

That sounds like a great book, Niamh! Thanks for sharing!

NCTM has some great lessons with videos on this site: http://www.nctm.org/ptatoolkit/

If you are a member of NCTM you have access to more lessons than if not. There are also task on youcubed and on Rober Kaplinsy’s site http://robertkaplinsky.com/lessons/ I wish there were many more resources for these organized by content strand and grade level! Great article Donna!

Thank you so much for sharing those resources, Melynee!

Agree!

These tasks can be found on sites lined illustrate.math, nrich and tools4ncteachers.

Yes, there are lots of good sources. But often you can take a regular lesson, and just flip the I Do, We Do, You Do script.

For accountability purposes, have students write their names on a whiteboard and place it next to their manipulatives. Students can take a picture of their work for the teacher to check.

I agree with the fact that students learn best with hands on discovery experiences. How do you suggest getting administration that mandates Saxon math that uses daily direct instruction filling in worksheets (& then do it again at home with the nightly homework sheet.)

Basically “copy me math” in my opinion.

I was told if the Ks aren’t instructed THIS way they won’t be ready for the worksheets in 1st grade.

My heart hurts because I know this is not the best way for them to REALLY learn!!

Any suggestions would be appreciated!

Thanks

Wow, tough situation. One suggestion would be to introduce research, books and journal articles, that support the fact that students need mathematical experiences and conversations, not more worksheets. Teaching Children Mathematics, the journal from NCTM is a good start, and there are so many great books on the subject. I have a friend that was in a similar situation at her school, and she asked her principal if she could try teaching a different way. Her principal let her, and the results were so good that she ended up being put in a coach role. It’s a tough call, though. I wish you all the best!!

I love your articles! I teach kinder and I really want to start Guided Math. What would be your suggestions on how to begin? We are now working on decomposing numbers. I’d love to have suggestions

Thank you!

Laney Sammons, author of Guided Math, and Debbie Diller both have great books for getting started with guided math. Here’s a great post from my friend Meg at the Teacher Studio. Jump in and good luck!!

Yes we need direct instruction! There are many students with disabilities that cannot learn without direct instruction. Direct instruction is the most effective way to teach children to learn to read. Students with math disabilities need direct instruction with concepts taught using CRA and limited strategies. These strategies need to be explicitly taught, explained, modeled, and reviewed with relentless consistency. Along with this direct instruction, good teachers will use questioning to help students develop and understand math concepts. The questioning is essential to students developing understanding of concepts. But some students cannot make the leap to developing their own understanding. They need a lot of concrete support.

This was my first reaction as well. Another point is that children need to have a learning ethic. They need to be taught that in playing with the manipulatives, they are expected to make and report on discoveries…to LEARN something and prove that they did so. When kids are playing, they are not necessarily making the connection that they are actually supposed to be learning from these experiences. That language needs to start even before Kindergarten. Academic self discovery only happens with exploration if they know what it means to learn and that our activities are done for a reason.

Good point. Many children do not grow up in a culture that provides experiences and develops the language base needed for making connections and learning independently. By expecting these students to learn in the same way as their peers will only further the distance between the haves and the have nots.

Amen! I am currently teaching high school math. I LOVED my 9th grade Algebra 1 teacher and that’s why I got into teaching after a successful business career. What did I love about his class? Structure. Walk in, sit down, take notes, copy some examples, out the door, home to do 20 problems, success! Most of the first semester, if not all, was review of pre-Algebra. And I NEEDED that! It strengthened my foundation and gave me confidence. When we got into the second semester and new material, again, I had the confidence. Now, we do very little review: “kids should have had this in a previous course” (CPM curriculum…ugh!!). If I was in a math class today as a high school student, I would have hated it!

When you start a new job, your employer doesn’t say “Sit down and explore! See if you can figure out how to do this job. It will deepen your understanding!” NO!! You are “directly taught” because it is the most effective way to train and to instill in your employees what and how they need to do the job.

Our school has been using Envisions 2.0, which many of our teachers are struggling with implementation. Many of our teachers feel we spend all of our time in direct whole group instruction, and hardly any time with small guided groups. Inquiry based and games are parts that what we would like to do but feel trapped by all the items we are required to do as part of each lesson. We have 2 pages (front and back) plus a homework piece that goes home at night. The strategies we are using are varied, and there is not time in our scope and sequence for slowing down and getting comfortable with them. We introduce it one day, and move on the next day. It is a spiraling curriculum, but it is wearing our kids and teachers out (we’re in first grade) trying to keep up the pace. I feel like many of my kids are not getting the topics, but we’ve been told to keep pushing on…Just wondering how to fit in some of your great ideas on this site. I’ve also had a chance to have professional development with Math Solutions while working in another district, and really am excited to implement some of their ideas, but again feel my hands are tied. Any suggestions would be greatly appreciated!

Monique, I don’t have any suggestions unfortunately but wanted you to know that I’m in the same boat with 4th grade Envisions. I feel like I’m killing the joy of learning based on my districts insistence on following the textbook. 🙁

Melissa,

Our district signed up for a new version of Envisions, but the new version is so different from the old. We, the teachers, fill that we are leaving huge gaps in student learning with this curriculum. We teach a concept one day and are told to move on, regardless if the kids get it…we just need to make it through all the topics with no time for differentiation. It is frustrating and everyone is feeling it. I don’t worry about my high kids anymore as it feels like that is who I am teaching to all the time. My low babies, and some middle of the road kiddos, are getting left in the dust. There’s no time to get them up to speed. It’s sad.

I hear both of you loud and clear- what a piece of junk! We unfortunately had Envision on our last adoption– after we had a correlation with CCSS done, the district encouraged teachers to NOT use the book. We adopted again last year, this time opting for a resource that is significantly better- Curriculum Associates. Envision didn’t do the standards justice. Total FAIL.

This is so true! That’s why the math workshop approach works so well! Give a mini lesson and then let them explore with manipulatives! We have to teach them how to have genuine math conversations!

This is a provocative post! Thank you.

I ask myself, “How can I deploy the human resources within my classroom to see everyone move forward?” This is a the-answer-is-in-the-room attitude that can be communicated to students via appeals to their citizenship and collaborative instincts that “Our ship doesn’t go forward unless everyone’s aboard.”

So-called “direct instruction” is an important part of a collaborative classroom, and is best viewed as a form of whole-class modelling. It can never be purely didactic. On a good teaching day, direct instruction is nothing less than assessment for learning, an on-the-fly cycling of questions, strategy sharing, confirmations, and feedback that most everyone in the group — including teacher — needs to confidently proceed.

So true! When students build their own understanding of math through interactions and explorations, the learning stays with them.

I generally start each lesson with some kind of exploration, and when frustration levels approach the point of being non-productive I jump in with a mini lesson. (Side note: by 8th grade students who have not learned math using discovery tend to reach that point pretty quickly. ?.) The exception? When it is two weeks till “the test” and you still have 2 units to cover…

Yes yes yes! Except for a few “nuts and bolts” as in the example, the only way to learn and understand math is to dive in and mess around and figure it out. I learned that as an adult in grad school. I had shut myself off from math since middle school and had to start over. Tears, struggles, and epiphanies ensued. Our students need the same opportunities. “Not yet but getting there” is not a bad place to be – help your students embrace it.

If you look at the whole body (1000’s upon 1000’s) of research studies you will find that there should be a place for both direct instruction and hands-on exploration. To throw one out without the other is another disaster in mathematical learning waiting to happen. Why is it when something new comes along, we are ready to throw out hundreds of years of research of proven strategies. The old saying is “we throw the baby out with the bathwater”. Do you remember “Whole Language”. There is a perfect example of turning away from proven research to something entirely different. And those of you who remember those years know that it failed miserably as test scores fell dramatically. After spending millions of dollars to train everyone in “whole language” we ended up abandoning it and going back to those principles that did work as backed by research. I can see the same thing happening here. I have no problem with hands-on learning. I have a problem of abandoning everything else that is tried and true according to research just to do hands-on.

Monica, inquiry learning is not “something entirely different.” It been considered best-practice for math instruction for years. And I’m not saying in the post that it’s one or the other–of course it is a balance. But currently in US classrooms, whole-group direct instruction is sadly the norm, and that is not an effective way to meet the diverse needs of students or develop the mathematical thinking required of our students to find success in a job market that values problem solving and critical thinking skills.

“Where all think alike, no one thinks very much”, Walter Lippmann.

PISA shows inquiry-based learning a failure, alas

http://intellectualmathematics.com/blog/pisa-shows-inquiry-based-learning-a-failure-alas/

Also, from https://www.acer.edu.au/files/FarkotaThesis.pdf

“From the research and literature reviewed in this study it is apparent that the domain of mathematics teaching in the western world is in a sorry state and there is no consensus on how this should be addressed. Almost all of the problems associated with student learning relate back to curriculum and teaching method and unfortunately, these problems are compounded in the transition years. After examining the literature on student-directed approaches to learning alongside that relating to Direct Instruction it was concluded that the empirical data heavily favoured the latter as being the more effective…”

Too many math education professors have been focusing their attention on the psychological rather than the mathological. It’s not too late to learn how elementary math SHOULD be taught. The ancient Chinese and Indians understood the foundations of math many centuries ago.

If a math educator cannot explain the meaning of a negative multiplier, for example which was taught 2100 years ago in China, that educator will likely benefit greatly from reading the math conference presentation and history paper below.

http://t.co/xKXUKlQs6a

http://bit.ly/LostLogicOfMath

Donna, only when elementary math is understood by teachers, will the debate about direct instruction and inquiry based learning matter. I doubt the Chinese, Japanese and Russians are worried about growth mindset rah-rah seminars. I’m sure they they just get on with teaching math, which is what highly skilled and respected teachers are paid to do.

It is said, “Those who do not learn history are doomed to repeat it.” This applies to math more than nearly any other subject. The CCSS and most likely every other set of math standards in the West would be laughed at by ancient Chinese mathematicians. China only adopted Hindu Arabic numerals a little over a century ago. The West built its math pedagogy on Greek geometry which has neither zero nor negative numbers, and East vs. West PISA results reflect this fact.

Thank you for this article – always love your thought provoking blogs 🙂

Teaching through direct instruction does not support the development of the behaviours of the mathematical proficiencies (including making connections between key ideas, reasoning, justifying, explaining etc)

A problem solving approach to teaching maths can also be highly focused, explicit and intentional – as well as supporting mathematical behaviours.

Love your work!

Thank you for the kind words!

I wish all math teachers would do this. It would be great, but they don’t. By the time I get them in 7th grade they are SO passive and have such math anxiety about getting things wrong! I battle it the best I can, but a systemic change is needed. Hopefully, the one Jo Boaler has started will spread!

I agree that kids should learn by exploration, but I see 2 caveats: 1.our entire education system is not structured to allow time to explore, and 2. believe it or not, some people need the algorithm first, then go back & see why it works.

One major hurdle to learning by exploration is time. In my experience, it takes a long time for many kids to actually figure out a concept by exploration. We have such a limited amount of time, and such pressure to have them be able to “be ready for the test,” to write paragraphs about their math thinking (my students are very slow writers and often slow to be able to explain their thinking,) and to have something filled in to show that they worked on the lesson.