Yesterday the students sat their Basics Facts test. This is a school-wide test carried out in Term 2, which assesses for accuracy. The students are given a total of 50 problems, 2 mins/column, a total of 4 mins.
The results are as follows:
A quick look at the basic facts test shows that all students are solid on their addition and subtraction facts, except for student C who needs to work on his subtraction. All students need to know their multiplication and division facts, while at least two need to work on their mixed facts.
A further analysis of their answers shows that I need to cover the concept and definition of zero in multiplication as three students (A, C and D) did not understand the purpose of zero - nothing - in multiplication. Student D consistently wrote his answer as 4 x 0 = 4, 5 x 0 = 5, 9 x 0 = 9, etc
Secondly, student D needs guidance in the concept of division and the division sign, as 7 of his divisions were treated as multiplication. Hence 3 / 3 = 9, 6 / 2 = 12, 8 / 2 = 16, etc.
Thirdly in another instance student C and E followed a pattern of operation rather than looking at the detail of the problem. They failed to observe that the operation had changed so they need to learn to notice, recognise and respond to the operation sign whether it is addition, subtraction, multiplication or division.
Fourthly, student C gets the right answer when dealing with zero in a singular list of subtraction basic facts and a second singular list of addition basic facts. But when the basic facts get mixed up with multiplication, division, subtraction and addition in the same list, he consistently gets the wrong answer in subtraction eg. 2 - 0 = 0. Addition was not a problem: when he adds 4 + 0, he gets the right answer.
Fifthly, students A, B, C, and F need to understand that division is repeated subtraction, division and multiplication are the opposite of each other and that if they use their knowledge of multiplication facts they can easily solve division problems. A corollary of this could be the students don't know that subtraction is the opposite of addition and vice versa.
Numeral formation needs to improve for students B, E and in particular C who formed her digits poorly. It could be that they wrote hurriedly, or it could be that they need practices on how to write numbers. Regardless, it needs to be covered to reduce their error rate.
Lastly, while observing the students during their basic facts test, not one child checked or proof-read their work. In fact student C turned his test paper over, faced down after each test. While student A had given up during his division test and decided he didn't want to do anymore, hence his low score of 30%.
All the above issues need to be addressed and talked about with the students so they become aware of how they can improve on understanding their skills and techniques.
As a teacher of students who have learning needs I am not at all opposed to students knowing their basic facts. To me the issue of learning basic facts is less about what students should know, and more about how basic facts should be taught.
The traditionalist would argue techniques such as memorisation, rote learning, speed tests etc should be used because they have been proven by the test of time. The modernist points to the importance of number strategies because it teaches students how to manipulate numbers rather than just memorise them.
Professor of Mathematics Education Jo Boeler, comes down firmly on the side of the modernist in her research, 'Fluency Without Fear: Research Evidence on the Best Ways to Learn Maths Facts'. She argues that "mathematics facts are important but the memorisation of maths facts through times table repetition, practice and time testing is unnecessary" and damages student's attitudes towards maths and their long-term understanding of it.
She argues that an adult who was taught their maths facts by traditionalists might say 7 x 8 = 54 when in fact it's 56. If they're unable to correct themselves, it's likely because they lack an understanding of how numbers fit together.
The student taught by a modernist may know that 7 x 7 = 49 and they have to add 7. Or they might know 7 x 10 = 70 and subtract two lots of 7 to arrive at 56. If the student makes an error, they can correct themselves using their knowledge of numbers rather than memorised facts.
The essence of the argument is that students learn their basic facts anyway without memorisation through the process of learning various mathematical strategies. Moreover, because learning number manipulation offers students a deeper understanding of how numbers fit together, it gives them a logical pathway towards higher level thinking.
However, on occasions we have to look inside ourselves because what works as a teaching tool for one student may not work so well for another. So it follows that memorisation and rote learning techniques for learning basic facts should not be completely abandoned. Instant recall of basic facts is a necessary skill to solve number problems especially in PAT tests where speed is required and where time is limited. Moreover as I have already mentioned in my earlier blog, basic facts is the core to the the various conceptual understanding and knowledge in mathematics (27 April) and so basic facts is a requirement.
Admittedly, as a teaching technique for learning basic facts, memorisation focuses primarily on two senses: the ear (as in chanting) and the eyes (as in flash cards and the like). It can be made more successful by adding in the kinesthetic sense (as in writing out the facts by hand in a frequent maths activity).
An even more successful approach uses all senses at once on a device. Susan Koscinski and David Gast claimed in their research, 'Computer Assisted Instruction With Constant Time Delay to Teach Multiplication Facts to Students With Learning Disabilities', that, "results indicate that the computer-assisted instructional program was effective in teaching multiplication facts to students", who have more difficulty in learning basic maths facts than "their other non-handicapped peers". Thus repetitive learning of basic facts has its place, and by extension, it can also benefit those of normal ability who gain more traction using memorisation and rote learning techniques as well.
Hence, one of the teaching tools I will be using to help my learners to increase their basic facts knowledge and their speed of recall will be the use of online interactive programmes.
The traditionalist would argue techniques such as memorisation, rote learning, speed tests etc should be used because they have been proven by the test of time. The modernist points to the importance of number strategies because it teaches students how to manipulate numbers rather than just memorise them.
Professor of Mathematics Education Jo Boeler, comes down firmly on the side of the modernist in her research, 'Fluency Without Fear: Research Evidence on the Best Ways to Learn Maths Facts'. She argues that "mathematics facts are important but the memorisation of maths facts through times table repetition, practice and time testing is unnecessary" and damages student's attitudes towards maths and their long-term understanding of it.
She argues that an adult who was taught their maths facts by traditionalists might say 7 x 8 = 54 when in fact it's 56. If they're unable to correct themselves, it's likely because they lack an understanding of how numbers fit together.
The student taught by a modernist may know that 7 x 7 = 49 and they have to add 7. Or they might know 7 x 10 = 70 and subtract two lots of 7 to arrive at 56. If the student makes an error, they can correct themselves using their knowledge of numbers rather than memorised facts.
The essence of the argument is that students learn their basic facts anyway without memorisation through the process of learning various mathematical strategies. Moreover, because learning number manipulation offers students a deeper understanding of how numbers fit together, it gives them a logical pathway towards higher level thinking.
However, on occasions we have to look inside ourselves because what works as a teaching tool for one student may not work so well for another. So it follows that memorisation and rote learning techniques for learning basic facts should not be completely abandoned. Instant recall of basic facts is a necessary skill to solve number problems especially in PAT tests where speed is required and where time is limited. Moreover as I have already mentioned in my earlier blog, basic facts is the core to the the various conceptual understanding and knowledge in mathematics (27 April) and so basic facts is a requirement.
Admittedly, as a teaching technique for learning basic facts, memorisation focuses primarily on two senses: the ear (as in chanting) and the eyes (as in flash cards and the like). It can be made more successful by adding in the kinesthetic sense (as in writing out the facts by hand in a frequent maths activity).
An even more successful approach uses all senses at once on a device. Susan Koscinski and David Gast claimed in their research, 'Computer Assisted Instruction With Constant Time Delay to Teach Multiplication Facts to Students With Learning Disabilities', that, "results indicate that the computer-assisted instructional program was effective in teaching multiplication facts to students", who have more difficulty in learning basic maths facts than "their other non-handicapped peers". Thus repetitive learning of basic facts has its place, and by extension, it can also benefit those of normal ability who gain more traction using memorisation and rote learning techniques as well.
Hence, one of the teaching tools I will be using to help my learners to increase their basic facts knowledge and their speed of recall will be the use of online interactive programmes.
Very interesting point of view of what is traditional learning of multiplication. Interesting about using interactive programmes on devices. Coming from a traditionalist background of learning, we also had interactive games to help us learn the tables. We very rarely did drill or recital learning. Do your students actually understand that times tables are repeated addition? Do they understand what 7x8 actually mean and the different ways you can solve that questions. Your inquiry is sounding exciting.
ReplyDeleteHi Teresa, thank you for the sharing these comparative studies. I have noticed, and had learners feedback, that computer assisted instruction is less pressure i.e. if I am wrong I'm not going to be judged and can try again at my own pace. At the same time there is that element of competition, in this case competing against myself, that some learners enjoy.
ReplyDeleteI am wondering if memorisation of maths facts through times table repetition in conjunction with applying this in real scenarios that learners would enjoy and relate to would make a difference? There was some conversation about this at our Sensemaking session with Woolf Fisher recently specifically looking at what could be happening at Y10 where their data is showing improvement. Some of the suggestions all involved strategies for increased engagement including competitions and a shared responsibility approach with whānau. Tamaki College are using computer assisted instruction and results are shared with whānau, which is helping to engage learners. Might be interesting to find out more about this from Russel Dunn or Christine David.
Look forward to hearing how you progress.
Fiona
Fiona, thanks for your feedback. I am in favour of computer assisted instructions, as long as the learners can understand the steps and why the steps have been taken, otherwise the teacher needs to be beside the child in case a misunderstanding occurs in the viewing.
ReplyDeleteTereas