I blogged about this last year with some posts about using guided math groups where the students were learning key math ideas that were helping them to develop basic fact fluency. With so many articles in the news lately about “back to basics” etc. because of the newly released EQAO scores for grade 6 in our province I inspired me to make this my first post of the new year. Most of the articles from the media want to create drama that gets people to click on their article of buy the paper by using terms like “discovery math” and not enough “basics” in the article title to try and highlight that as the problem with our math classrooms. That is another post completely but I want to quickly point out that there has never been a label called “discovery math” in any document that I can find released for math education in the province of Onatrio. They all promote a balanced math program.
Here are two pics are from our Effective Guides to Instruction; Volume #1-Foundations of Math Instruction. I wanted to share these two pics because it again shows our Ministry documents promoting a balanced math program with Guided( which includes some direct instruction), Shared and Independent math.
I wanted the focus of this post to highlight that teachers are still teaching basic facts or fact fluency in their classrooms. The difference is many are now doing it in a way that is foreign to most of the classrooms our students parents and myself would have attended. I want to share personal story before I detail some of the differences we now see. I hated math in school, despised it with all my heart, now it is passion and I see it in such a different light. One of many areas I struggled in was memorizing my times tables and simple addition and subtraction facts. I couldn’t learn from flash cards or endless games of “travel around the classroom” that would have seen me shaking and sweating with nervousness as the person who was flying around the class beating everyone moved closer to my desk. First of all my mind doesn’t think fast like that, sometimes I need process time and even if I did know it I would cease up with anxiety. This was majority of my math experiences, memorize this fact, this formula and then watch the teacher do it and then try and repeat it many times even though I knew I didn’t understand it in the first 10 questions! I was never shown other ways to do problems or math concepts and I was certainly never shown that there are strategies, patterns and games that can help you develop auto-recall of your facts. Surprise! Writing math facts out hundreds of times didn’t work for me. Maybe it did work for you and that is great but for many students it didn’t work and even for some that it did work for they maybe didn’t understand what it meant. Here lies the focus of this post, and that is to highlight how we are getting students to develop auto-recall of their basic facts and have a strong number sense at the same time. Let’s make it clear nowhere in our curriculum or in our ministry documents has it said students don’t need to have automaticity of their facts. If there is I have never seen it.
Here are some pics of some of the Ontario Ministry documents that shine a light on the focus of learning basic math facts and computational fluency :
Below is a pic of one of the Effective Guides to Instruction: Volume 5- Teaching Basic Facts and Computations. This whole guide is dedicated to teaching students how to become proficient at basic facts.
This next picture is from our curriculum where it clearly states multiply to 7×7 and divide up to 49 by 7 by the end of grade 3. Some would argue it doesn’t clearly state memorize but after my post I hope it will be clearer that it means the same. We want students to be able to do this by the end of grade 3. Some teachers think that the second part, up to 9×9 which is for the end of grade 4 probably should be included with the grade 3 expectation and I would agree with that. Why stop at 7×7 when kids are moving to proficiency!
What is the difference between how we learned facts (in most cases, I have heard of some teachers in the past teaching strategies) in the past compared to now?
The simple answer is students are allowed to develop strategies that help them access facts and answer problems until, with enough practise they become automatic. Alex Lawson’s book “What to Look For” has been a valuable asset to help guide teachers to give students the best learning opportunities to move from direct modelling to proficiency. Lawson has an article on edugains where she explains this very clearly and compares it to how most of us were taught. Here is pic of her developmental continuum for moving from direct modelling to proficiency:
Dr. Lawson explains it like this, imagine if all the strategies on the above continuum between counting three times (direct modelling) and proficiency were covered up. That is what most of our past teaching and learning of basic facts would have looked like. We would start in Kindergarten/Grade 1 by answering simple addition and subtraction problems by direct modelling the problem on counters or our fingers and then counting to get the answer. Then we would have quickly moved to trying to memorize the facts through activities like flash cards, worksheets or writing them out multiple times until we hopefully became proficient. We would therefore not learn or be introduced to all these strategies that research now shows kids actual develop through on their way to proficiency. You often talk to many adults who developed many of these strategies on their own because it makes the math easier to do! The same principle applies to multiplication and division. I chose Lawson’s continuum because our board is using it a lot currently. There are other resources that promote this same instructional approach. (Clements, Richardson, Small, Van De Wall are some other good examples). Here is a pic of Lawson’s continuum for multiplication and division:
How do we develop students through these strategies until it becomes automatic for them?
Teachers are using many different instructional strategies for this but two that stand out are number talks and math games. Number talks allow students to share their thinking and strategies with their fellow classmates while seeing different strategies modelled on the board with an appropriate model to support conceptual understanding. Math games which is just as important as the number talks allows the students to practice their strategies and computations in fun way. There are others but I wanted to highlight these two especially games because in essence it is drill (practice) but in much more effective setting then say flashcards or drill worksheets.
The last piece of this post is I wanted to share three videos that show one student’s journey along the continuum from counting on to proficiency. The journey goes from beginning of grade 1 to near the end of grade 1. He now knows all his addition/subtraction facts to 20 automatically and is now applying the strategies to larger number combinations while also beginning to work on the multiplication continuum. Mrs. Kennedy had her students playing games, doing number talks and also working in small guided groups with her on key ideas that help students develop these strategies. Enjoy his journey!
Video #1- October 2016 Beginning of Grade 1
Video #2- January 2017 Middle of Grade 1
Video #3- April 2017 End of Grade 1
Here is pic of a continuum that highlights his journey with each strategy he used along the way.
Hope this helps show that students are most certainly still learning their facts and becoming proficient to the point of auto-retrieval. Maybe this isn’t happening in all classrooms but it is in the schools I work at and I certainly see a lot of like minded educators on twitter teaching this way. It matches what our Ministry of Education in Ontario and our board promotes. It also works! The journey is different now which opens it up for all kids to learn their facts!
I am not afraid to admit that I did not know all my facts when I became a teacher, I do now! It comes from being introduced to these strategies and just playing with the math! I still have to think about some, for instance 8×7 and 9×7 confuse me sometimes but I now can quickly do 8×5=40 and 8×2=16, put them together and you get 56. I just used the partial product strategy (distributive property) in under 5 seconds to access my fact I was stalled on. Think of the power this gives a student who use to think if they didn’t know the answer to the flash card/worksheet they just moved on and didn’t get it! That was me! Not anymore and I work now to give all students this opportunity. If you were a memorizer and you got it, congratulations but in my experience that just isn’t the case with most students. Teaching facts may have prevented a lot of math phobia that many parents show today. My 75 year old mom shared a story last week with me that highlighted the anxiety and sick feeling she would get when her teacher use to draw a clock on the board, then point to two numbers and pick someone randomly in their seat to answer. If you didn’t get it then you got centered out in front of the whole class. She still remembers it to this day 60 plus years later! That is just another reason why the process we use now makes me so happy. Thanks for reading.